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Elements of Dialectical Contextualism

Posprint in English (with additional illustrations) of  an article appeared in French in the collective book (pages 581-608) written on the occasion of the 60th birthday of Pascal Engel.

Abstract In what follows, I strive to present the elements of a philosophical doctrine, which can be defined as dialectical contextualism. I proceed first to define the elements of this doctrine: dualities and polar contraries, the principle of dialectical indifference and the one-sidedness bias. I emphasize then the special importance of this doctrine in one specific field of meta-philosophy: the methodology for solving philosophical paradoxes. Finally, I describe several applications of this methodology on the following paradoxes: Hempel’s paradox, the surprise examination paradox and the Doomsday Argument.

In what follows, I will endeavour to present the elements of a specific philosophical doctrine, which can be defined as dialectical contextualism. I will try first to clarify the elements that characterise this doctrine, especially the dualities and dual poles, the principle of dialectical indifference and the one-sidedness bias. I will proceed then to describe its interest at a meta-philosophical level, especially as a methodology to assist in the resolution of philosophical paradoxes. Finally, I will describe an application of this methodology to the analysis of the following philosophical paradoxes: Hempel’s paradox , the surprise examination paradox and the Doomday Argument.

The dialectical contextualism described here is based on a number of constitutive elements which have a specific nature. Among these are: the dualities and dual poles, the principle of dialectical indifference and the one-sidedness bias. It is worth analysing in turn each of these elements.

1. Dualities and dual poles

To begin with, we shall focus on defining the concept of dual poles (polar opposites)1. Although intuitive, this concept needs to be clarified. Examples of dual poles are static/dynamic, internal/external, qualitative/quantitative, etc.. We can define the dual poles as concepts (which we shall denote by A and Ā), which come in pairs, and are such that each of them is defined as the opposite of the other. For example, internal can be defined as the opposite of external and symmetrically, external can be defined as the contrary of internal. In a sense, there is no primitive notion here and neither A nor Ā of the dual poles can be regarded as the primitive notion. Consider first a given duality, that we can denote by A/Ā, where A and Ā are dual concepts. This duality is shown in the figure below:

The dual poles A and Ā

At this point, we can also provide a list (which proves to be necessarily partial) of dualities:

Internal/External, Quantitative/Qualitative, Visible/Invisible, Absolute/Relative Abstract/Concrete, Static/Dynamic, Diachronic/Synchronic, Single/Multiple, Extension/Restriction, Aesthetic/Practical, Precise/Vague, Finite/Infinite, Single/compound, Individual/Collective, Analytical/Synthetic, Implicit/Explicit, Voluntary/Involuntary

In order to characterize more accurately the dual poles, it is worth distinguishing them from other concepts. We shall stress then several properties of the dual poles, which allow to differentiate them from other related concepts. The dual poles are neutral concepts, as well as simple qualities; in addition, they differ from vague notions. To begin with, two dual poles A and Ā constitute neutral concepts. They can thus be denoted by A0 and Ā0. This leads to represent both concepts A0 and Ā0 as follows:

The dual neutral poles A0 and Ā0

The dual poles are neutral concepts, i.e. concepts that present no ameliorative or pejorative nuance. In this sense, external, internal, concrete, abstract, etc.., are dual poles, unlike concepts such as beautiful, ugly, brave, which present either a ameliorative or pejorative shade, and are therefore non-neutral. The fact that the dual poles are neutral has its importance because it allows to distinguish them from concepts that have a positive or negative connotation. Thus, the pair of concepts beautiful/ugly is not a duality and therefore beautiful and ugly do not constitute dual poles in the sense of the present construction. Indeed, beautiful has a positive connotation and ugly has a pejorative connotation. In this context, we can denote them by beautiful+ and ugly.

It should be emphasised, second, that the two poles of a given dual duality correspond to simple qualities, as opposed to composite qualities​​. The distinction between single and composite qualities can be made in the following manner. Let A1 and A2 be simple qualities. In this case, A1 ∧ A2, and A1 ∨ A2 are composite qualities. To take an example, static, qualitative, external are simple qualities, while static and qualitative, static and external, qualitative and external are composite qualities​​. A more general definition is as follows: let B1 and B2 be single or composite qualities, then B1 ∧ B2 and B1 ∨ B2 are composite qualities. Incidentally, this also highlights why the pairs of concepts red/non-red, blue/non-blue concepts can not be considered as dual poles. Indeed, non-red can thus be defined as follows as a composite quality: violetindigobluegreenyelloworangewhiteblack. In this context, one can assimilate non-blue to the negation-complement of blue, such complement negation being defined with the help of composite qualities​​.

Given the above definition, we are also in a position to distinguish the dual poles from vague objects. We can first note that dual poles and vague objects have certain properties in common. Indeed, vague objects come in pairs in the same way as dual poles. Moreover, vague concepts are classically considered as having an extension and an anti-extension, which are mutually exclusive. Such a feature is also shared by the dual poles. For example, qualitative and quantitative can be assimilated respectively to an extension and an anti-extension, which also have the property of being mutually exclusive, and the same goes for static and dynamic, etc.. However, it is worth noting the differences between the two types of concepts. A first difference (i) lies in the fact that the union of the extension and the anti-extension of vague concepts is not exhaustive in the sense that they admit of borderline cases (and also borderline cases of borderline cases, etc., giving rise to a hierarchy of higher-order vagueness of order n), which is a penumbra zone. Conversely, the dual poles do not necessarily have such a characteristic. Indeed, the union of the dual poles can be either exhaustive or non-exhaustive. For example, the abstract/concrete duality is then intuitively exhaustive, since there does not seem to exist any objects that are neither abstract nor concrete. The same goes for the vague/precise duality: intuitively, there does no exist indeed objects that are neither vague nor precise, and that would belong to an intermediate category. Hence, there are dual poles whose extension and anti-extension turns out to be exhaustive, unlike vague concepts, such as the two poles of the abstract/concrete duality. It is worth mentioning, second, another difference (ii) between dual poles and vague objects. In effect, dual poles are simple qualities, while vague objects may consist of simple or compound qualities. There exist indeed some vague concepts which are termed multi-dimensional vague objects, such as the notion of vehicle, of machine, etc.. A final difference between the two categories of objects (iii) lies in the fact that some dual poles have an inherently precise nature. This is particularly the case of the individual/collective duality, which is susceptible to give rise to a very accurate definition.

2. The principle of dialectical indifference

From the notions of duality and of dual poles which have been just mentioned, we are in a position to define the notion of a viewpoint related to a given duality or dual pole. Thus, we have first the notion of viewpoint corresponding to a given A/Ā duality: it consists for example in the standpoint of the extension/restriction duality, or of the qualitative/quantitative duality or of the diachronic/synchronic duality, etc.. It also follows the concept of point of view related to a given pole of an A/Ā duality: we get then, for example (at the level of the extension/restriction duality) the standpoint by extension, as well as the viewpoint by restriction. Similarly, the qualitative viewpoint or perspective results from it, as well as the quantitative point of view, etc.. (at the level of the qualitative/quantitative duality). Thus, when considering a given object o (either a concrete or an abstract object such as a proposition or a reasoning), we may consider it in relation to various dualities, and at the level of the latter, relative to each of its two dual poles.

The underlying idea inherent to the viewpoints relative to a given duality, or to a given pole of a duality, is that each of the two poles of the same duality, all things being equal, deserve an equal legitimacy. In this sense, if we consider an object o in terms of a duality A/Ā, one should not favour one of the poles with respect to the other. To obtain an objective point of view with respect to a given duality A/Ā, one should place oneself in turn from the perspective of the pole A, and then from that of the pole Ā. For an approach that would only address the viewpoint of one of the two poles would prove to be partial and truncated. The fact of considering in turn the perspective of the two poles, in the study of an object o and of its associated reference class allows to avoid a subjective approach and to meet as much as possible the needs of objectivity.

As we can see it, the idea underlying the concept of point of view can be formalized in a principle of dialectical indifference, in the following way:

(PRINCIPLE OF DIALECTICAL INDIFFERENCE) When considering a given object o and the reference class E associated with it, from the angle of duality A/Ā, all things being equal, it should be given equal weight to the viewpoint of the A pole and the viewpoint of the Ā pole.

This principle is formulated in terms of a principle of indifference: if we consider an object o under the angle of an A/Ā duality, there is no reason to favour the viewpoint from A with regard to the viewpoint from Ā, and unless otherwise resulting from the context, we must weigh equally the viewpoints A and Ā. A direct consequence of this principle is that if one considers the perspective of the A pole, one also needs to take into consideration the standpoint of the opposite pole Ā (and vice versa). The need to consider both points of view, the one resulting from the A pole and the other associated with the Ā pole, meets the need of analysing the object o and the reference class associated with it from an objective point of view. This goal is achieved, as far as possible, by taking into account the complementary points of view which are those of the poles A and Ā. Each of these viewpoints has indeed, with regard to a given duality A/Ā, an equal relevance. Under such circumstances, when only the A pole or (exclusively) the pole Ā is considered, it consists then of a one-sided perspective. Conversely, the viewpoint which results from the synthesis of the standpoints corresponding to both poles A and Ā is of a two-sided type. Basically, this approach proves to be dialectical in essence. In effect, the step consisting of successively analysing the complementary views relative to a given reference class, is intended to allow, in a subsequent step, a final synthesis, which results from the joint consideration of the viewpoints corresponding to both poles A and Ā. In the present construction, the process of confronting the different perspectives relevant to an A/Ā duality is intended to build cumulatively, a more objective and comprehensive standpoint than the one, necessarily partial, resulting from taking into account those data that stem from only one of the two poles.

The definition of the dialectical principle of indifference proposed here refers to a reference class E, which is associated with the object o. The reference class2 is constituted by a number of phenomena or objects. Several examples can be given: the class of human beings who ever lived, the class of future events in the life of a person, the class of body parts of a given person, the class of ravens, etc.. We shall consider in what follows, a number of examples. Mention of such a reference class has its importance because its very definition is associated with the above-mentioned duality A/Ā. In effect, the reference class can be defined either from the viewpoint of A or from the viewpoint of Ā. Such a feature needs to be emphasized and will be useful in defining the bias which is associated with the very definition of the principle of dialectical indifference: the one-sidedness bias.

3. Characterisation of the one-sidedness bias

The previous formulation of the principle of dialectical indifference suggests straightforwardly an error of reasoning of a certain type. Informally, such a fallacy consists in focusing on a given standpoint when considering a given object, and of neglecting the opposite view. More formally, in the context described above, such a fallacy consists, when considering an object o and the reference class associated with it, in taking into account the viewpoint of the A pole (respectively Ā), while completely ignoring the viewpoint corresponding to its dual pole Ā (respectively A) to define the reference class. We shall term one-sidedness bias such type of fallacy. The conditions of this type of bias, in violation of the principle of dialectical indifference, needs however to be clarified. Indeed, in this context, we can consider that there are some cases where the two-sidedness with respect to a given duality A/Ā is not required. Such is the case when the elements of the context do not presuppose conditions of objectivity and exhaustiveness of views. Thus, a lawyer who would only emphasise the evidence in defence of his/her client, while completely ignoring the evidence against him/her does not commit the above-mentioned type of error of reasoning. In such a circumstance, in fact, the lawyer would not commit a faulty one-sidedness bias, since it is his/her inherent role. The same would go in a trial for the prosecutor, who conversely, would only focus on the evidence against the same person, by completely ignoring the exculpatory elements. In such a situation also the resulting one-sidedeness bias would not be inappropriate, because it follows well from the context that it consists well of the limited role assigned to the prosecutor. By contrast, a judge who would only take into account the evidence against the accused, or who would commit the opposite error, namely of only considering the exculpatory against the latter, would well commit an inappropriate one-sidedness bias because the mere role of the judge implies that he/she takes into account the two types of elements, and that his/her judgement is the result of the synthesis which is made.

In addition, as hinted at above, the mention of a reference class associated with the object o proves to be important. In effect, as we will have the opportunity to see it with the analysis of the following examples, the definition itself is associated with an A/Ā duality. And the reference class can be defined either from the viewpoint of A, or from the viewpoint of Ā. Such feature has the consequence that all objects are not likely to give rise to a one-sidedness bias. In particular, the objects that are not associated with a reference class that is itself likely to be envisaged in terms of an A/Ā duality, do not give rise to any such one-sidedness bias.

Before illustrating the present construction with the help of several practical examples, it is worth considering, at this stage, the one-sidedness bias which has been just defined, and which results from the very definition of the principle of dialectical indifference, in the light of several similar concepts. In a preliminary way, we can observe that a general description of this type of error of reasoning had already been made, in similar terms, by John Stuart Mill (On Liberty, II):

He who knows only his own side of the case, knows little of that. His reasons may be good, and no one may have been able to refute them. But if he is equally unable to refute the reasons on the opposite side; if he does not so much know what they are, he has no ground for preferring either opinion.

In the recent literature, some very similar concepts have also been described. It consists in particular of the dialectic bias notably described by Douglas Walton (1999). Walton (999, pp. 76-77) places then himself in the framework of the dialectical theory of bias, which opposes one-sided to two-sided arguments:

The dialectical theory of bias is based on the idea […] that an argument has two sides. […] A one-sided argument continually engages in pro-argumentation for the position supported and continually rejects the arguments of the opposed side in a dialogue. A two-sided (balanced) argument considers all arguments on both sides of a dialogue. A balanced argument weights each argument against the arguments that have been opposed to it.

Walton describes thus the dialectical bias as a one-sided perspective that occurs during the course of the argument. Walton emphasizes, though, that dialectic bias, which is universally common in human reasoning, does not necessarily constitute an error of reasoning. In line with the distinction between “good” and “bad” bias due to Antony Blair (1988), Walton considers that the dialectic bias is incorrect only under certain conditions, especially if it occurs in a context that is supposed to be balanced, that is to say where the two sides of the corresponding reasoning are supposed to be mentioned (p. 81):

Bad bias can be defined as “pure (one-sided) advocacy” in a situation where such unbalanced advocacy is normatively inappropriate in argumentation.

A very similar notion of one-sidedness bias is also described by Peter Suber (1998). Suber describes indeed a fallacy that he terms one-sidedness fallacy. He describes it as a fallacy which consists in presenting one aspect of the elements supporting a judgement or a viewpoint, by completely ignoring the other aspect of the relevant elements relating to the same judgement:

The fallacy consists in persuading readers, and perhaps ourselves, that we have said enough to tilt the scale of evidence and therefore enough to justify a judgment. If we have been one-sided, though, then we haven’t yet said enough to justify a judgment. The arguments on the other side may be stronger than our own. We won’t know until we examine them.

The error of reasoning consists then in taking only into account one viewpoint relating to the judgement in question, whereas the other viewpoint could as well prove to be decisive with regard to the conclusion to be drawn. Suber also undertakes to provide a characterization of the one-sidedness fallacy and notes in particular that the fallacy of one-sidedness constitutes a valid argument. For its conclusion is true if its premises are true. Moreover, Suber notes, it appears that the argument is not only valid but sound. For when the premises are true, the conclusion of the argument can be validly inferred. However, as hinted at by Suber, the argument is defective due to the fact that a number of premises are lacking. This is essential because if the missing premises are restored within the argument, the resulting conclusion can be radically different.

4. An instance of the one-sidedness bias

To illustrate the above concepts, it is worth at this stage providing an example of the one-sidedness bias. To this end, consider the following instance, which is a form of reasoning, mentioned by Philippe Boulanger (2000, p. 3)3, who attributes it to the mathematician Stanislaw Ulam. The one-sidedness bias shows up in a deductive form. Ulam estimates that if a company were to achieve a level of workforce large enough, its performance would be paralysed by the many internal conflicts that would result. Ulam estimates that the number of conflicts between people would increase according to the square of the number n of employees, while the impact on the work that would result would only grow as a function of n. Thus, according to this argument, it is not desirable that the number of employees within a company becomes important. However, it turns out that Ulam’s reasoning is fallacious, as Boulanger points it out, for it focuses exclusively on the conflictual relations between employees. But the n2 relationships among the company employees can well be confrontational, but may include as well collaborative relationships that are quite beneficial for the company. And so there is no reason to favour conflictual relationships with respect to collaborative ones. And when among n2 relationships established between the company employees, some are genuine collaborative relationships, the effect is, instead, of improving business performance. Therefore, we can not legitimately conclude that it is not desirable that the workforce of a company reaches a large size.

For the sake of clarity, it is worth formalizing the above reasoning. It turns out thus that Ulam’s reasoning can be described as follows:

(D1Ā ) if <a company has a large workforce>

(D2Ā ) then <n2 conflictual relationships will result>

(D3Ā ) then negative effects will result

(D4Ā ) the fact that <a company has a large workforce> is bad

This type of reasoning has the structure of a one-sidedness bias, since it focuses only on conflicting relationships (the dissociation pole of the association/dissociation duality), by ignoring a parallel argument with the same structure that could legitimately be raised, focusing on collaborative relationships (the association pole), which is the other aspect relevant to this particular topic. This parallel argument goes as follows:

(D1A) if <a company has a large workforce>

(D2A) then <n2 collaborative relationships will result>

(D3A) then positive effects will result

(D4A) the fact that <a company has a large workforce> is good

This finally casts light on how the two formulations of the argument lead to conflicting conclusions, i.e. (D4Ā) and (D4A). At this point, it is worth noting the very structure of the conclusion of the above reasoning, which is as follows:

(D5Ā ) the situation s is bad from the viewpoint of Ā (dissociation)

while the conclusion of the parallel reasoning is as follows:

(D5A) the situation s is good from the viewpoint of A (association)

But if the reasoning had been complete, by taking into account the two points of view, a different conclusion would have ensued:

(D5Ā ) the situation s is bad from the viewpoint of Ā (dissociation)

(D5A) the situation s is good from the viewpoint of A (association)

(D6A/Ā) the situation s is bad from the viewpoint of Ā (dissociation) and good from the viewpoint of A (association)

(D7A/Ā) the situation s is neutral from the viewpoint of the duality A/Ā (association/dissociation)

And such a conclusion turns out to be quite different from that resulting from (D5Ā ) and (D5A).

Finally, we are in a position to replace the one-sidedness bias which has just been described in the context of the present model: the object o is the above reasoning, the reference class is that of the relationships between the employees of a business, and the corresponding duality – allowing to define the reference class – is the dissociation/association duality.

5. Dichotomic analysis and meta-philosophy

The aforementioned principle of dialectic indifference and its corollary – one-sidedness bias – is likely to find applications in several domains4. We shall focus, in what follows, on its applications at a meta-philosophical level, through the analysis of several contemporary philosophical paradoxes. Meta-philosophy is that branch of philosophy whose scope is the study of the nature of philosophy, its purpose and its inherent methods. In this context, a specific area within meta-philosophy is the method to use to attach oneself to resolve, or make progress towards the resolution of philosophical paradoxes or problems. It is within this specific area that falls the present construction, in that it offers dichotomous analysis as a tool that may be useful to assist in the resolution of paradoxes or philosophical problems.

The dichotomous analysis as a methodology that can be used to search for solutions to some paradoxes and philosophical problems, results directly from the statement of the principle of dialectical indifference itself. The general idea underlying the dichotomous approach to paradox analysis is that two versions, corresponding to one and the other pole of a given duality, can be untangled within a philosophical paradox. The corresponding approach then is to find a reference class which is associated with the given paradox and the corresponding duality A/Ā, as well as the two resulting variations of the paradox that apply to each pole of this duality. Nevertheless, every duality is not suitable for this, as for many dualities, the corresponding version of the paradox remains unchanged, regardless of the pole that is being considered. In the dichotomous method, one focuses on finding a reference class and a relevant associated duality, such that the viewpoint of each of its poles actually lead to two structurally different versions of the paradox , or the disappearance of paradox from the point of view of one of the poles. Thus, when considering the paradox in terms of two poles A and Ā, and if it has no effect on the paradox itself, the corresponding duality A/Ā reveals itself therefore, from this point of view, irrelevant.

The dichotomous analysis is not by far a tool that claims to solve all philosophical problems, but only constitutes a methodology that is susceptible of shedding light on some of them. In what follows, we shall try to illustrate through several works of the author, how dichotomous analysis can be applied to progress towards the resolution of three contemporary philosophical paradoxes: Hempel’s paradox, the surprise examination paradox and the Doomsday argument.

In a preliminary way, we can observe here that in the literature, there is also an example of dichotomous analysis of a paradox in David Chalmers (2002). Chalmers attempts then to show how the two-envelope paradox leads to two fundamentally distinct versions, one of which corresponds to a finite version of the paradox and the other to an infinite version. Such an analysis, although conceived of independently of the present construction can thus be characterized as a dichotomous analysis based on the finite/infinite duality.

The dual poles in David Chalmers’ analysis of the two-envelope paradox

6. Application to the analysis of the philosophical paradoxes

Karl Hempel

At this point, it is worth applying the foregoing to the analysis of concrete problems. We shall illustrate this through the analysis of several contemporary philosophical paradoxes: Hempel’s paradox, the surprise examination paradox and the Doomsday argument. We will endeavour to show how a problem of one-sidednessn bias associated with a problem of definition of a reference class can be found in the analysis of the aforementioned philosophical paradoxes. In addition, we will show how the very definition of the reference class associated with each paradox is susceptible of being qualified with the help of the dual poles A and Ā of a given duality A/Ā as they have just been defined.

6.1. Application to the analysis of Hempel‘s paradox

Hempel’s paradox is based on the fact that the two following assertions:

(H) All ravens are black

(H*) All non-black things are non-ravens

are logically equivalent. By its structure (H*) presents itself indeed as the contrapositive form of (H). It follows that the discovery of a black raven confirms (H) and also (H*), but also that the discovery of a non-black thing that is not a raven such as a red flame or even a grey umbrella, confirms (H*) and therefore (H). However, this latter conclusion appears paradoxical.

We shall endeavour now to detail the dichotomous analysis on which is based the solution proposed in Franceschi (1999). The corresponding approach is based on finding a reference class associated with the statement of the paradox, which may be defined with the help of an A/Ā duality. If we scrutinise the concepts and categories that underlie propositions (H) and (H*), we first note that there are four categories: ravens, black objects, non-black objects and non- ravens. To begin with, a raven is precisely defined within the taxonomy in which it inserts itself. A category such as that of the ravens can be considered well-defined, since it is based on a precise set of criteria defining the species corvus corax and allowing the identification of its instances. Similarly, the class of black objects can be accurately described, from a taxonomy of colours determined with respect to the wave lengths of light. Finally, we can see that the class of non-black objects can also be a definition that does not suffer from ambiguity, in particular from the specific taxonomy of colours which has been just mentioned.

However, what about the class of non-ravens? What does constitute then an instance of a non-raven? Intuitively, a blue blackbird, a red flamingo, a grey umbrella and even a natural number, are non-ravens. But should we consider a reference class that goes up to include abstract objects? Should we thus consider a notion of non-raven that includes abstract entities such as integers and complex numbers? Or should we limit ourselves to a reference class that only embraces the animals? Or should we consider a reference class that encompasses all living beings, or even all concrete things, also including this time the artefacts? Finally, it follows that the initial proposition (H*) is susceptible of giving rise to several variations, which are the following:

(H1*) All that is non-black among the corvids is a non-raven

(H2*) All that is non-black among the birds is a non-raven

(H3*) All that is non-black among the animals is a non-raven

(H4*) All that is non-black among the living beings is a non-raven

(H5*) All that is non-black among the concrete things is a non-raven

(H6*) All that is non-black among the concrete and abstract objects is a non-raven

Thus, it turns out that the statement of Hempel’s paradox and in particular of proposition (H*) is associated with a reference class, which allow to define the non-ravens. Such a reference class can be assimilated to corvids, birds, animals, living beings, concrete things, or to concrete and abstract things, etc.. However, in the statement of Hempel’s paradox, there is no objective criterion for making such a choice. At this point, it turns out that one can choose such a reference class restrictively, by assimilating it for example to corvids. But in an equally legitimate manner, we can choose a reference class more extensively, by identifying it for example to the set of concrete things, thus notably including umbrellas. Why then choose such or such reference class defined in a restrictive way rather than another one extensively defined? Indeed, we are lacking a criterion allowing to justify the choice of the reference class, whether we proceed by restriction or by extension. Therefore, it turns out that the latter can only be defined arbitrarily. But the choice of such a reference class proves crucial because depending on whether you choose such or such class reference, a given object such as a grey umbrella will confirm or not (H*) and therefore (H). Hence, if we choose the reference class by extension, thus including all concrete objects, a grey umbrella will confirm (H). On the other hand, if we choose such a reference class by restriction, by assimilating it only to corvids, a grey umbrella will not confirm (H). Such a difference proves to be essential. In effect, if we choose a definition by extension of the reference class, the paradoxical effect inherent to Hempel’s paradox ensues. By contrast, if we choose a reference class restrictively defined, the paradoxical effect disappears.

The dual poles in the reference class of the non-ravens within Hempel’s paradox

The foregoing permits to describe accurately the elements of the preceding analysis of Hempel’s paradox in terms of one-sidedness bias such as it has been defined above: to the paradox and in particular to proposition (H*) are associated the reference class of non-ravens, which itself is susceptible of being defined with regard to the extension/restriction duality. However, for a given object such as a grey umbrella, the definition of the reference class by extension leads to a paradoxical effect, whereas the choice of the latter by restriction does not lead to such an effect.

6.2. Application to the analysis of the surprise examination paradox

The classical version of the surprise examination paradox (Quine 1953, Sorensen 1988) goes as follows: a teacher tells his students that an examination will take place on the next week, but they will not know in advance the precise date on which the examination will occur. The examination will thus occur surprisingly. The students reason then as follows. The examination cannot take place on Saturday, they think, otherwise they would know in advance that the examination would take place on Saturday and therefore it could not occur surprisingly. Thus, Saturday is eliminated. In addition, the examination can not take place on Friday, otherwise the students would know in advance that the examination would take place on Friday and so it could not occur surprisingly. Thus, Friday is also ruled out. By a similar reasoning, the students eliminate successively Thursday, Wednesday, Tuesday and Monday. Finally, every day of the week is eliminated. However, this does not preclude the examination of finally occurring by surprise, say on Wednesday. Thus, the reasoning of the students proved to be fallacious. However, such reasoning seems intuitively valid. The paradox lies here in the fact the students’ reasoning is apparently valid, whereas it finally proves inconsistent with the facts, i.e. that the examination can truly occur by surprise, as initially announced by the professor.

In order to introduce the dichotomous analysis (Franceschi 2005) that can be applied to the surprise examination paradox, it is worth considering first two variations of the paradox that turn out to be structurally different. The first variation is associated with the solution to the paradox proposed by Quine (1953). Quine considers then the student’s final conclusion that the examination can not take place surprisingly on any day of the week. According to Quine, the student’s error lies in the fact of not having envisaged from the beginning that the examination could take place on the last day. Because the fact of considering precisely that the examination will not take place on the last day finally allows the examination to occur by surprise on the last day. If the student had also considered this possibility from the beginning, he would not have been committed to the false conclusion that the examination can not occur surprisingly.

The second variation of the paradox that proves interesting in this context is the one associated with the remark made ​​by several authors (Hall 1999, p. 661, Williamson 2000), according to which the paradox emerges clearly when the number n of units is large. Such a number is usually associated with a number n of days, but we may as well use hours, minutes, seconds, etc.. An interesting feature of the paradox is indeed that it emerges intuitively more significantly when large values ​​of n are involved. A striking illustration of this phenomenon is thus provided by the variation of the paradox that corresponds to the following situation, described by Timothy Williamson (2000, p 139).

Advance knowledge that there will be a test, fire drill, or the like of which one will not know the time in advance is an everyday fact of social life, but one denied by a surprising proportion of early work on the Surprise Examination. Who has not waited for the telephone to ring, knowing that it will do so within a week and that one will not know a second before it rings that it will ring a second later?

The variation described by Williamson corresponds to the announcement made to someone that he/she will receive a phone call during the week, but without being able to determine in advance at what exact second the latter event will occur. This variation highlights how surprise may occur, in a quite plausible way, when the value of n is high. The unit of time considered here by Williamson is the second, in relation with a time duration that corresponds to one week. The corresponding value of n here is very high and equal to 604800 (60 x 60 x 24 x 7) seconds. However, it is not necessary to take into account a value as large of n, and a value of n equal to 365, for example, should also be well-suited.

The fact that two versions of the paradox that seem a priori quite different coexist suggests that two structurally different versions of the paradox could be inextricably intertwined within the surprise examination paradox. In fact, if we analyse the version of the paradox that leads to Quine’s solution, we find that it has a peculiarity: it is likely to occur for a value of n equal to 1. The corresponding version of the professor’s announcement is then as follows: “An examination will take place tomorrow, but you will not know in advance that this will happen and therefore it will occur surprisingly.” Quine’s analysis applies directly to this version of the paradox for which n = 1. In this case, the student’s error resides, according to Quine, in the fact of having only considered the hypothesis: (i) “the examination will take place tomorrow and I predict that it will take place.” In fact, the student should also have considered three cases: (ii) “the examination will not take place tomorrow, and I predict that it will take place” (iii) “the examination will not take place tomorrow and I do not predict that it will take place” (iv) “the examination will take place tomorrow and I do not predict that it will take place.” And the fact of having envisaged hypothesis (i), but also hypothesis (iv) which is compatible with the professor’s announcement would have prevented the student to conclude that the examination would not finally take place. Therefore, as Quine stresses, it is the fact of having only taken into account the hypothesis (i) that can be identified as the cause of the fallacious reasoning.

As we can see it, the very structure of the version of the paradox on which Quine’s solution is based has the following features: first, the non-surprise may actually occur on the last day, and second, the examination may also occur surprisingly on the last day. The same goes for the version of the paradox where n = 1: the non-surprise and the surprise may occur on day n. This allows to represent such structure of the paradox with the following matrix S[k, s] (where k denotes the day on which the examination takes place and S[k, s] denotes whether the corresponding case of non-surprise (s = 0) or surprise (s = 1) is possible (in this case, S[k, i] = 1) or not (in this case, S[k, i] = 0)):

daynon-surprisesurprise
111
211
311
411
511
611
711

Matrix structure of the version of the paradox corresponding to Quine’s solution for n = 7 (one week)

daynon-surprisesurprise
111

Matrix structure of the version of the paradox corresponding to Quine’s solution for n = 1 (one day)

Given the structure of the corresponding matrix which includes values that are equal to 1 in both cases of non-surprise and of surprise, for a given day, we shall term joint such a matrix structure.

If we examine the above-mentioned variation of the paradox set by Williamson, it presents the particularity, in contrast to the previous variation, of emerging neatly when n is large. In this context, the professor’s announcement corresponding for example to a value of n equal to 365, is the following: “An examination will take place in the coming year but the date of the examination will be a surprise.” If such a variation is analysed in terms of the matrix of non-surprise and of surprise, it turns out that this version of the paradox has the following properties: the non-surprise cannot occur on the first day while the surprise is possible on this very first day; however, on the last day, the non-surprise is possible whereas the surprise is not possible.

daynon-surprisesurprise
101
36510

Matrix structure of the version of the paradox corresponding to Williamson’s variation for n = 365 (one year)

The foregoing allows now to identify precisely what is at fault in the student’s reasoning, when applied to this particular version of the paradox. Under these circumstances, the student would then have reasoned as follows. The surprise cannot occur on the last day but it can occur on day 1, and the non-surprise can occur on the last day, but cannot occur on the first day. These are proper instances of non-surprise and of surprise, which prove to be disjoint. However, the notion of surprise is not captured exhaustively by the extension and the anti-extension of the surprise. But such a definition is consistent with the definition of a vague predicate, which is characterized by an extension and an anti-extension which are mutually exclusive and non-exhaustive. Thus, the notion of surprise associated with a disjoint structure is that of a vague notion. Thus, the student’s error of reasoning at the origin of the fallacy lies in not having taken into account the fact that the surprise is in the case of a disjoint structure, a vague concept and includes therefore the presence of a penumbra corresponding to borderline cases between non-surprise and surprise. Hence, the mere consideration of the fact that the surprise notion is here a vague notion would have prohibited the student to conclude that S[k, 1] = 0, for all values ​​of k, that is to say that the examination can not occur surprisingly on any day of the period.

Finally, it turns out that the analysis leads to distinguish between two independent variations with regard to the surprise examination paradox. The matrix definition of the cases of non-surprise and of surprise leads to two variations of the paradox, according to the joint/disjoint duality. In the first case, the paradox is based on a joint definition of the cases of non-surprise and of surprise. In the second case, the paradox is grounded on a disjoint definition. Both of these variations lead to a structurally different variation of the paradox and to an independent solution. When the variation of the paradox is based on a joint definition, the solution put forth by Quine applies. However, when the variation of the paradox is based on a disjoint definition, the solution is based on the prior recognition of the vague nature of the concept of surprise associated with this variation of the paradox.

The dual poles in the class of the matrices associated with the surprise examination paradox

As we finally see it, the dichotomous analysis of the surprise examination paradox leads to consider the class of the matrices associated with the very definition of the paradox and to distinguish whether their structure is joint or disjoint. Therefore, it follows an independent solution for each of the resulting two structurally different versions of the paradox.

6.3. Application to the analysis of the Doomsday Argument

The Doomsday argument, attributed to Brandon Carter, was described by John Leslie (1993, 1996). It is worth recalling preliminarily its statement. Consider then proposition (A):

(A) The human species will disappear before the end of the XXIst century

We can estimate, to fix ideas, to 1 on 100 the probability that this extinction will occur: P(A) = 0.01. Let us consider also the following proposition:

(Ā) The human species will not disappear at the end of the XXIst century

Let also E be the event: I live during the 2010s. We can also estimate today to 60 billion the number of humans that ever have existed since the birth of humanity. Similarly, the current population can be estimated at 6 billion. One calculates then that one human out of ten, if event A occurs, will have known of the 2010s. We can then estimate accordingly the probability that humanity will be extinct before the end of the twenty-first century, if I have known of the 2010s: P(E, A) = 6×109/6×1010 = 0.1. By contrast, if humanity passes the course of the twenty-first century, it is likely that it will be subject to a much greater expansion, and that the number of human will be able to amount, for example to 6×1012. In this case, the probability that humanity will not be not extinct at the end of the twenty-first century, if I have known of the 2010s, can be evaluated as follows: P(E, Ā) = 6×109/6×1012 = 0,001. At this point, we can assimilate to two distinct urns – one containing 60 billion balls and the other containing 6,000,000,000,000 – the total human populations that will result. This leads to calculate the posterior probability of the human species’ extinction before the end of the XXIst century, with the help of Bayes’ formula: P'(A) = [P(A) x P(E, A)] / [P(A) x P(E, A) + P(Ā) x P(E, Ā )] = (0.01 x 0.1) / (0.01 x 0.1 + 0.99 x 0.001) = 0.5025. Thus, taking into account the fact that I am currently living makes pass the probability of the human species’ extinction before 2150 from 1% to 50.25 %. Such a conclusion appears counter-intuitive and is in this sense, paradoxical.

It is worth now describing how a dichotomous analysis (Franceschi, 1999, 2009) can be applied to the Doomsday Argument. We will endeavour, first, to point out how the Doomsday Argument has an inherent reference class5 problem definition linked to a duality A/Ā. Consider then the following statement:

(A) The human race will disappear before the end of the XXIst century

Such a proposition presents a dramatic, apocalyptic and tragic connotation, linked to the imminent extinction of the human species. It consists here of a prediction the nature of which is catastrophic and quite alarming. However, if we scrutinise such a proposition, we are led to notice that it conceals an inaccuracy. If the time reference itself – the end of the twenty-first century – proves to be quite accurate, the term “human species” itself appears to be ambiguous. Indeed, it turns out that there are several ways to define it. The most accurate notion in order to define the“’human race” is our present scientific taxonomy, based on the concepts of genus, species, subspecies, etc.. Adapting the latter taxonomy to the assertion (A), it follows that the ambiguous concept of “human species” is likely to be defined in relation to the genus, the species, the subspecies, etc.. and in particular with regard to the homo genus, the homo sapiens species, the homo sapiens sapiens subspecies, etc.. Finally, it follows that assertion (A) is likely to take the following forms:

(Ah) The homo genus will disappear before the end of the XXIst century

(Ahs) The homo sapiens species will disappear before the end of the XXIst century

(Ahss) The homo sapiens sapiens subspecies will disappear before the end of the XXIst century

At this stage, reading these different propositions leads to a different impact, given the original proposition (A). For if (Ah) presents well in the same way as (A) a quite dramatic and tragic connotation, it is not the case for (Ahss). Indeed, such a proposition that predicts the extinction of our current subspecies homo sapiens sapiens before the end of the twenty-first century, could be accompanied by the replacement of our present human race with a new and more advanced subspecies than we could call homo sapiens supersapiens. In this case, the proposition (Ahss) would not contain any tragic connotation, but would be associated with a positive connotation, since the replacement of an ancient race with a more evolved species results from the natural process of evolution. Furthermore, by choosing a reference class even more limited as that of the humans having not known of the computer (homo sapiens sapiens antecomputeris), we get the following proposition:

(Ahsss) The infra-subspecies homo sapiens sapiens antecomputeris will disappear before the end of the XXIst century

which is no longer associated at all with the dramatic connotation inherent to (A) and proves even quite normal and reassuring, being devoid of any paradoxical or counterintuitive nature. In this case, in effect, the disappearance of the infra-subspecies homo sapiens sapiens antecomputeris is associated with the survival of the much-evolved infra-subspecies homo sapiens sapiens postcomputeris. It turns out then that a restricted class of reference coinciding with an infra-subspecies goes extinct, but a larger class corresponding to a subspecies (homo sapiens sapiens) survives. In this case, we observe well the Bayesian shift described by Leslie, but the effect of this shift proves this time to be quite innocuous.

Thus, the choice of the reference class for proposition (A) proves to be essential for the paradoxical nature of the conclusion associated with the Doomsday Argument. If one chooses then an extended reference class for the very definition of humans, associated with e.g. the homo genus, one gets the dramatic and disturbing nature associated with proposition (A). By contrast, if one chooses such a reference class restrictively, by associating it for example with the infra-subspecies homo sapiens sapiens antecomputeris, a reassuring and normal nature is now associated with the proposition (A) underlying the Doomsday Argument.

Finally, we are in a position to replace the foregoing analysis in the present context. The very definition of the reference class of the “humans” associated with the proposition (A) inherent to the Doomsday Argument is susceptible of being made according to the two poles of the extension/restriction duality. An analysis based on a two-sided perspective leads to the conclusion that the choice by extension leads to a paradoxical effect, whereas the choice by restriction of the reference class makes this paradoxical effect disappear.

The dual poles within the reference class of “humans” in the Doomsday Argument

The dichotomous analysis, however, as regards the Doomsday argument, is not limited to this. Indeed, if one examines the argument carefully, it turns out that it contains another reference class which is associated with another duality. This can be demonstrated by analysing the argument raised by William Eckhardt (1993, 1997) against the Doomsday argument. According to Eckhardt, the human situation corresponding to DA is not analogous to the two-urn case described by Leslie, but rather to an alternative model, which can be termed the consecutive token dispenser. The consecutive token dispenser is a device that ejects consecutively numbered balls at regular intervals: “(…) suppose on each trial the consecutive token dispenser expels either 50 (early doom) or 100 (late doom) consecutively numbered tokens at the rate of one per minute.” Based on this model, Eckhardt (1997, p. 256) emphasizes that it is impossible to make a random selection, where there are many individuals who are not yet born within the corresponding reference class: “How is it possible in the selection of a random rank to give the appropriate weight to unborn members of the population?”. The strong idea of Eckhardt underlying this diachronic objection is that it is impossible to make a random selection when there are many members in the reference class who are not yet born. In such a situation, it would be quite wrong to conclude that a Bayesian shift in favour of the hypothesis (A) ensues. However, what can be inferred rationally in such a case is that the initial probability remains unchanged.

At this point, it turns out that two alternative models for modelling the analogy with the human situation corresponding to the Doomsday argument are competing: first, the synchronic model (where all the balls are present in the urn when the draw takes place) recommended by Leslie and second, Eckhardt’s diachronic model, where the balls can be added in the urn after the draw. The question that arises is the following: is the human situation corresponding to the Doomsday argument in analogy with (i) the synchronic urn model, or with (ii) the diachronic urn model? In order to answer, the following question arises: does there exist an objective criterion for choosing, preferably, between the two competing models? It appears not. Neither Leslie nor Eckhardt has an objective motivation allowing to justify the choice of their own favourite model, and to reject the alternative model. Under these circumstances, the choice of one or the other of the two models – whether synchronic or diachronic – proves to be arbitrary. Therefore, it turns out that the choice within the class of the models associated with the Doomsday argument is susceptible of being made according to the two poles of the synchronic/diachronic duality. Hence, an analysis based on a two-sided viewpoint leads to the conclusion that the choice of the synchronic model leads to a paradoxical effect, whereas the choice of the diachronic model makes this latter paradoxical effect disappear.

The dual poles within the models’ class of the Doomsday Argument

Finally, given the fact that the above problem related to the reference class of the humans and its associated choice within the extension/restriction duality only concerns the synchronic model, the structure of the dichotomous analysis at two levels concerning the Doomsday Argument can be represented as follows:

Structure of embedded dual poles Diachronic/Synchronic and Extension/Restriction for the Doomsday Argument

As we can see it, the foregoing developments implement the form of dialectical contextualism that has been described above by applying it to the analysis of three contemporary philosophical paradoxes. In Hempel’s paradox, the reference class of the non-ravens is associated with proposition (H*), which itself is susceptible of being defined with regard to the extension/restriction duality. However, for a given object x such as a grey umbrella, the definition of the reference class by extension leads to a paradoxical effect , whereas the choice of the latter reference class by restriction eliminates this specific effect. Secondly, the matrix structures associated with the surprise examination paradox are analysed from the angle of the joint/disjoint duality, thus highlighting two structurally distinct versions of the paradox , which themselves admit of two independent resolutions. Finally, at the level of the Doomsday argument, a double dichotomic analysis shows that the class of humans is related to the extension/restriction duality, and that the paradoxical effect that is evident when the reference class is defined by extension, dissolves when the latter is defined by restriction. It turns out, second, that the class of models can be defined according to the synchronic/diachronic duality; a paradoxical effect is associated with the synchronic view, whereas the same effect disappears if we place ourselves from the diachronic perspective.

Acknowledgements

This text is written starting from some entirely revised elements of my habilitation to direct research work report, presented in 2006. The changes introduced in the text, comprising in particular the correction of a conceptual error, follow notably from the comments and recommendations that Pascal Engel had made to me at that time.

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Beck,AT. (1964) Thinking and depression: Theory and therapy, Archives of General Psychiatry, 10, 561-571.

Blair, J. Anthony (1988) What Is Bias?” in Selected Issues in Logic and Communication, ed. Trudy Govier [Belmont, CA: Wadsworth, 1988], 101-102).

Boulanger, P. (2000) Culture et nature, Pour la Science, 273, 3.

Chalmers, D. (2002) The St. Petersburg two-envelope paradox, Analysis, 62: 155-157.

Eckhardt, W. (1993) Probability Theory and the Doomsday Argument, Mind, 102, 483-488.

Eckhardt, W. (1997) A Shooting-Room view of Doomsday, Journal of Philosophy, 94, 244-259.

Ellis, A. (1962) Reason and Emotion in Psychotherapy, Lyle Stuart, New York.

Franceschi, P. (1999). Comment l’urne de Carter et Leslie se déverse dans celle de Carter, Canadian Journal of Philosophy, 29, 139-156.

Franceschi, P. (2002) Une classe de concepts, Semiotica, 139 (1-4), 211-226.

Franceschi, P. (2005) Une analyse dichotomique du paradoxe de l’examen surprise, Philosophiques, 32-2, 399-421.

Franceschi, P. (2007) Compléments pour une théorie des distorsions cognitives, Journal de Thérapie Comportementale et Cognitive, 17-2, 84-88. Preprint in English: www.cogprints.org/5261/

Franceschi, P. (2009) A Third Route to the Doomsday Argument, Journal of Philosophical Research, 34, 263-278.

Hall, N. (1999) How to Set a Surprise Exam, Mind, 108, 647-703.

Leslie, J. (1993) Doom and Probabilities, Mind, 102, 489-491.

Leslie, J. (1996) The End of the World: the science and ethics of human extinction, London: Routledge

Quine, W. (1953) On a So-called Paradox, Mind, 62, 65-66.

Sorensen, R. A. (1988) Blindspots, Oxford : Clarendon Press.

Stuart Mill, J. (1985) On Liberty, London: Penguin Classics, original publication in 1859.

Suber, E. (1998). The One-Sidedness Fallacy. Manuscript, https://www.earlham.edu/~peters/courses/inflogic/onesided.htm. Retrieved 11/25/2012

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1Such notion is central to the concept of matrices of concepts introduced in Franceschi (2002), of which we can consider that it constitutes the core, or a simplified form. In this paper that bears more specifically on the elements of dialectical contextualism and their application for solving philosophical paradoxes, merely presenting the dual poles proves to be sufficient.

2The present construction also applies to objects that are associated with several classes of reference. We shall limit ourselves here, for the sake of simplicity, to one single reference class.

3Philippe Boulanger says (personal correspondence) that he heard Stanislaw Ulam develop this particular point in a conference at the University of Colorado.

4An application of the present model to the cognitive distortions introduced by Aaron Beck (1963, 1964) in the elements of cognitive therapy, is provided in Franceschi (2007). Cognitive distortions are conventionally defined as fallacious reasoning that play a key role in the emergence of a number of mental disorders. Cognitive therapy is based in particular on the identification of these cognitive distortions in the usual reasoning of the patient, and their replacement by alternative reasoning. Traditionally, cognitive distortions are described as one of the twelve following methods of irrational reasoning: 1. Emotional reasoning 2. Hyper-generalization 3. Arbitrary inference 4. Dichotomous reasoning. 5. Should statements (Ellis 1962) 6. Divination or mind reading 7. Selective abstraction 8. Disqualifying the positive 9. Maximization and minimization 10. Catastrophism 11. Personalisation 12. Labelling.

5The analysis of the Doomsday Argument from the perspective of the reference class problem is performed in detail by Leslie (1996). But Leslie’s analysis aims at showing that the choice of the reference class, by extension or restriction does not affect the conclusion of the argument itself.


A Logical Defence of Maher’s Model of Polythematic Delusions

English translation of a paper published in French in Philosophiques, autumn 2008, under the title “Une défense logique du modèle de Maher pour les délires polythématiques”.

In this paper, we proceed to describe a model for the formation and maintenance of polythematic delusions encountered in schizophrenia, which is in adequation with Brendan Maher’s account of delusions. Polythematic delusions are considered here as the conclusions of arguments triggered by apophenia that include some very common errors of reasoning such as post hoc fallacy and confirmation bias. We describe first the structure of reasoning which leads to delusions of reference, of telepathy and of influence, by distinguishing between the primary, secondary, tertiary and quaternary types of delusional arguments. These four levels of arguments correspond to a stage the nature of which is respectively instantial, inductive, interpretative at a monothematic level and interpretative at a polythematic level. We also proceed to identify accurately the fallacious steps in the corresponding reasoning. We expose then the role of apophenia in the elaboration of delusional ideas. Lastly, we describe the role played by the hallucinations in the present model.


This article is cited in:

  • How delusion is formed? Park J.S., Kang U.G., Medical Hypotheses, 24 Dec 2015, 87: pages 61-65

A Logical Defence of Maher’s Model of Polythematic Delusions

Paul FRANCESCHI

Classically, the term of “delusion” applies to two fundamentally distinct forms: monothematic and polythematic delusions (Davies & Coltheart 2000; Bortolotti 2005). Monothematic delusions present an unique topic and are usually associated with cerebral lesions. Among the latter, one can mention Capgras’ delusion (by virtue of which the patient thinks that one of his/her fellows has been replaced by an impostor), Fregoli’s delusion (when the patient is persuaded that he/she is followed by one or several persons whom he cannot identify because they are dressed up) or Cotard’s delusion (when the patient is persuaded that he/she is died). Conversely, polythematic delusions have numerous topics, which are most often interconnected and usually linked to psychotic disturbances. Among polythematic delusions, one can notably mention: delusion of reference, delusion of grandeur, delusion of influence, delusion of persecution, delusion of control, delusion of telepathy.

In what follows, I will set out to introduce a new, as far as I know, model of the mechanism which leads to the formation of polythematic delusionsi met in schizophrenia. This model, which takes place in the recent development of psycho-pathological philosophy (Faucher, 2006), offers to describe the mechanism which leads, on the one hand, to the formation of delusional ideas and on the other hand, to their maintenance. In this model, delusions are the result of the patient’s cognitive activity in response to a specific form of abnormal perception. Even though the corresponding reasoning appears essentially normal, it includes however the repetition of some typical errors of reasoning. This leads to consider polythematic delusions as the conclusions of fallacious arguments, worked out in response to a particular type of abnormal perception, among which one can distinguish between primary, secondary, tertiary and quaternary delusional arguments. These four levels of arguments correspond, as we shall see it, to some functional stages the nature of which is respectively instantial (based on several instances), inductive (generalising the conclusion of each of the instances), interpretative at a monothematic level and finally, interpretative at a polythematic level.

It is worth mentioning, moreover, that the notion of delusion has important philosophical underpinnings. In particular, the understanding of delusions proves to be essential for the notions of belief (Engel 2001, Bayne & Pacherie 2005), of justification of beliefs, of knowledge, of rationality (Campbell 2001, Bortolotti 2005) and also of self-deception.

1. Cognitive models of delusions

Edvard Munch: The Scream.

Before describing in detail the present model, it is worth presenting the main cognitive models for delusions encountered in the literature. Some authors described then a cognitive model for delusional ideas observed in schizophrenia. As Chadwick & al. (1996) underline it, as well as Fowler & al. (1995) who set out to review these types of models, there does not truly exist a unique cognitive model for delusions, for it rather consists of a family of models.

A first cognitive model was described by Chadwick & al. (1996), who set out to introduce an application to delusions of Albert Ellis’ ABC-analysis. The original model described by Ellis (1962) consists of a diagram which plays a preponderant role in the emergence of mental disorders. Ellis distinguishes thus between three types of events: A, B and C. The As (for Activating event) are external facts or events of the patient’s internal life, such as thoughts or bodily feelings. The Bs (for Belief) are thoughts relating to the same events, which themselves are possibly rational in nature or not. Moreover, the corresponding cognitive process can be more or less conscious. Finally, the Cs (for Consequence) are emotional states such as anger, sadness, joy, frustration, etc. which can be of a positive or negative nature. Thus, the Cs that follow irrational thoughts are most often of a negative nature. The succession of events of type A, B and C plays a predominant role in the emergence of mental disorders: activating events trigger irrational thoughts, which themselves engender negative emotions. The type of therapy recommended by Ellis consists precisely in rendering the patient aware of this mechanism and in controlling the corresponding process. Adapting Ellis’ original model to psychosis, Chadwick & al. point out thus how the As constitute events that are external or internal to the patient, the Bs are his/her thoughts and the Cs are constituted by the emotions triggered by the patient’s thoughts. This specific framework allows to provide a cognitive ABC-analysis of the main types of delusions. For example, the delusion of persecution finds its origin in the external fact that the patient hears the noise of a car’s horn; this triggers in the patient the thought according to which his/her enemies come to kill him/her; it follows then in the patient’s a negative emotion of fright and of anxiety.

A second cognitive model of delusions was put forth by Brendan Maher (1974; 1988; 1999). Maher did suggest that delusions are the result – in the context of schizophrenia of paranoid subtype – of a broadly normal interpretation of the abnormal phenomena undergone by the patient (hallucinations, loss of audition, impairment in the intensity of perception, etc.). According to Maher, it is not therefore by his/her reasoning that the patient differs from a normal individual, but by his/her own altered perceptions. Delusional ideas are thus conceived of as a response to abnormal and emotionally disturbing phenomena experienced by the patient. Such disturbing phenomena lead the patient to search an explanation, which is at the origin of the delusional construction. According to Maher, the patient’s intellectual process is a product of normal reasoning and does not differ significantly from the one which is shown in every individual, or even in every scientist, when confronted with an unexplained phenomenon: “It is the core of the present hypothesis that the explanations (i.e. the delusions) of the patient are derived by cognitive activity that is essentially indistinguishable from that employed by non-patients, by scientists and by people generally.” (Maher 1974, 103). By normal reasoning, Maher means here a basically logical reasoning, but which occasionally includes some errors of reasoning of a common type. It is worth mentioning here that Maher’s model has led to several objections. Above all, this model was criticised on the grounds that it did not allow to account for the fact that delusions can also develop in seemingly normal conditions (Chapman & Chapman 1988). Secondly, it was objected to Maher’s model (Davies & Coltheart 2000, p. 8) that it did not explain how delusional beliefs are adopted and maintained in spite of their implausible nature. It is worth mentioning lastly that an important philosophical problem emerges within Maher’s model. It is what Pascal Engel termed the “paradox of delusions” (Engel 2001), and that can be formulated as follows: if the delusional construction is underlain by essentially normal reasoning and therefore by rationality, how it is possible to explain that the delusions’ conclusions are so manifestly wrong and contrary to evidence. The present analysis takes place in such context, and aims thus at proposing a solution to this paradox of delusions.

While Maher’s conception mentions abnormal perception as a unique factor at the origin of delusional ideas, another approach, notably put forth by Martin Davies and Max Coltheart (2000, 2001), describe two factors in the genesis and the maintenance of delusions. The first factor resides, as well as for Maher, in abnormal perception. And the second factor resides, according to Davies and Coltheart, in the patient’s disability to reject a hypothesis given its incoherent and implausible nature with regard to the patient’s rest of knowledge. Davies and Coltheart criticises thus Maher’s model by notably pointing out that it does not allow to provide an account of the maintenance of delusions, even though their conclusion turns out to be completely implausible.

It is worth also mentioning a third type of cognitive model, which stresses that several cognitive biases (Garety & al. 2001) can be observed in the thinking mode of patients suffering from schizophrenia. Among these biases is the patient’s tendency to jump-to-conclusions (Garety & Hemsley 1994). Experimental studies (Fear & Healy 1997; Garety & al. 1991) showed thus that patients had a more marked tendency than others to conclude very fast, starting from a limited group of information at their disposal. A second type of cognitive bias is an attribution or externalist bias which consists for the patient to attribute an external motive to events of a negative type which he/she undergoes. The patient favours then arbitrarily an external reason with regard to an internal and personal motive, when he/she sets out to determine the origin of an event which relates to him/her. Such conclusion notably results from the work of Bentall & Kaney (1989), and Kaney & al. (1989), who noticed that patients suffering from delusions of persecution were more prone than normal individuals to attribute both an external cause to negative events which they undergo and an internal motive to positive events which concerned them. This type of cognitive bias is also related to an attentional bias, which was noticed (Bentall & al. 1989; Kaney & al. 1989) in patients suffering from schizophrenia of paranoid subtype, who demonstrate as well a more marked tendency than others to turn their attention to menacing elements, among a group of stimuli, especially if the latter are related with themselves. Lastly, Aaron Beck (2002) also underlined how the reasoning of patients suffering from schizophrenia revealed an egocentric bias. This type of cognitive bias leads the patient to relate most external events with the elements of his/her personal life. Noise, sounds, smells, and generally facts and external phenomena, are thus bearing for the patient a hidden or explicit meaning, and which concerns him/her directly.

Finally, it is worth mentioning a cognitive model which sets out to define accurately the nature of delusions, by especially emphasising the fact that the latter do not constitute beliefs in the usual sense or, possibly, constitute beliefs of a special type. Such model made the subject of recent developments (Sass,1994; Young, 1999; Engel, 2001; Stephens & Graham, 2006) tending to question the classical definition of delusions, based on abnormal belief.

2. Apophenia

I will now endeavour to describe the present model and to expose accurately the mechanism which leads to the formation and maintenance of delusional ideas. In Maher’s model, delusions constitute a cognitive development elaborated by the patient in response to abnormal perception. The present model inserts itself within such conception: polythematic delusions constitute then conclusions of fallacious arguments worked out by the patient, in response to a particular type of abnormal perception: apophenia. Finally, although the reasoning which leads to delusions proves to be fallacious, it can however be considered as normal, because it includes errors of reasoning which turn out to be very common.

Before describing the structure of reasoning which leads to polythematic delusions, it is worth considering preliminarily the apophenia. One finds then mention, especially in the observations made by some patients in remission relating their psychotic experience (Stanton & David 2000), of a very specific feeling, which can be characterised as a feeling of interconnection with the ambient world. Such feeling is not felt in normal conditions and truly presents a bizarre nature. Schneider had already (1930) pointed out how in an individual suffering from schizophrenia, a meaningful interconnection was created between facts that are internal to the patient and external events (“Meaningful connections are created between temporary coincident external impressions … or perceptions with thoughts that happen to be present, or events and recollections happening to occur in consciousness at the same time”). Isabel Clarke (2000) also mentions in the patient a very particular feeling of fusion and of universal interconnection (“an exhilarating feeling of unity and interconnectedness”). Brundage (1983) also evokes a phenomenon of the same nature which manifests itself by a particular feeling of connection with all events that occur in the surroundings (“there is a connection to everything that happens”) as well as the feeling that the slightest things have a meaning (“every single thing means something”). It appears then that the patient experiences a strong feeling of interconnection between on the one hand, his/her internal phenomena and on the other hand, the external phenomena.

The role of such feeling of interconnection was recently underlined notably by Peter Brugger (2001). Brugger terms then apophenia the tendency to see connections between objects or ideas which are a priori without any relationship (“the propensity to see connections between seemingly unrelated objects or ideas”) and attributes the creation of this term to K. Conrad (1958). In the present context, one can consider a slightly more restrictive definition of apophenia, for it suffices here to characterise apophenia as the feeling in the patient that there is a narrow relationship between his/her internal phenomena (thought, feelings, emotions, acts) and external phenomena.

One can notice here that Maher does not mention explicitly apophenia when he enumerates abnormal perceptions which are susceptible of being experienced by the patient. However, he mentions a phenomenon which appears as closely related to apophenia. Among the abnormal perceptions undergone by the patient, Maher mentions indeed (Maher 1999) that it can consist, for example, of the fact that the patient perceives as salient some normally insignificant sensory data, of unrecognised defects in the sensory system of the patient such as a loss of audition, of temporary impairments in the intensity of perception, of hallucinations, of difficulties of concentration of neurological origin, etc. Maher includes then among abnormal perceptions the fact that the patient perceives as salient some ordinarily insignificant sensory data, what can be considered as closely related to apophenia.

At this step, it is worth describing more accurately the cognitive mechanism which, in relation with apophenia, leads to the formation of polythematic delusions. This will allow to cast more light on the role and the nature of apophenia itself.

3. Formation and maintenance of delusional ideas

In the present context, the reasoning that leads to delusional ideas is considered as a specific case of fallacious argument, i.e. as reasoning the conclusion of which is not logically justified by its premises, which are worked out in response to a particular type of abnormal perception: apophenia. In general, this type of reasoning leads to an erroneous conclusion. But it can happen very exceptionally that the resulting conclusion turns out to be true (for example if a patient suffering from schizophrenia with delusion of persecution was mistakenly spied on notably because he/she had been confused with a high diplomat). Another feature of the type of fallacious reasoning which leads to polythematic delusions is that it includes errors of reasoning of a normal type, i.e. very common. Finally, it is worth pointing out that in spite of their patently false conclusion, the task which consists in diagnosing accurately the fallacious steps in the reasoning which leads to delusional ideas proves to be far from easy.

The fallacious reasoning which leads to polythematic delusions presents a particular structure, as we will see it, within which it is worth distinguishing several functional steps, which take place successively within the elaboration of delusions ideas: primary, secondary, tertiary and quaternary steps. The primary step, first, is of an instantial nature, in the sense that it is based on some instances. The secondary step presents an inductive nature, which proceeds by generalisation of the conclusions resulting from each of the preceding instances. And the tertiary step is of an interpretative nature at a monothematic level. Finally, the quaternary step has an interpretative function, but this time at a polythematic level. The distinction of these four successive steps is of interest in the understanding of the mechanism which leads to the formation of delusional ideas, for it allows to describe its intrinsic structure, at the level of both its formation and maintenance. On the other hand, as we shall see it later, cognitive therapy of psychosis can apply differently to each of these specific steps.

In what follows, we shall especially be concerned with delusional ideas of reference, of telepathy, of influence and of grandeur, which correspond to polythematic delusions commonly met in schizophrenia. It is also worth mentioning that the corresponding model can be easily extended to other types of delusional ideas, especially to thought-broadcasting delusions or delusions of persecution. At this stage, it worth drawing a distinction between the mechanism which leads to the formation of delusional ideas, and the one which concurs to their maintenance.

3.1 Formation of delusional ideas

Classically, one distinguishes in schizophrenia the following types of delusions: delusion of reference, delusion of influence, delusion of control, delusion of telepathy, delusion of grandeur, delusion of persecution. The present model will set out first to describe the mechanism which leads to the formation of these main types of delusions, by setting out to introduce a reconstruction of the specific cognitive process in a patient at the beginning of psychosis.

Let us begin with delusions of reference. Let us consider the following argument, which leads the patient to conclude that television speaks about him/her, and therefore to delusional ideas of reference (T1 and T2 denote here two successive temporal positions, with a very short time interval between T1 and T2; the symbol  denotes the conclusion; and R is taken for reference):

(R1) in T1 I was drinking an aperitif

(R2) in T2 the presenter of the show said: “Stop drinking!”

(R3) ∴ in T2 the presenter said: “Stop drinking!” because in T1 I was drinking an aperitif

(R4) in T3 I was upset and anxious

(R5) in T4 the presenter of the show said “Stop stressing”

(R6) ∴ in T4 the presenter of the show said “Stop stressing!” because in T3 I was upset and anxious

(R7) in T5 I was smoking a cigarette

(R8) in T6 I heard the presenter saying “That is not good !“

(R9) ∴ in T6 the presenter said “That is not good !” because in T5 I was smoking a cigarette

(R10) in T7 I felt fine and lucid and I was relaxed

(R11) in T8 the presenter of the show said: “We are in great form!

(R12) ∴ in T8 the presenter said “We are in great form!” because in T7 I felt fine and lucid and I was relaxed

(R…) …

(R13) ∴ the presenters of the shows speak according to what I do or what I feel

(R14) ∴ television speaks about me

One can distinguish within the structure of this reasoning several parts the function of which turns out to be specific. These different parts correspond respectively to primary delusional arguments (it consists of the steps (R1)-(R3), (R4)-(R6), (R7)-(R9) and (R10)-(R12)), to secondary delusional arguments (the steps (R3), (R6), (R9), (R12) and (R13)) and tertiary delusional arguments (the steps (R13) and (R14)). It is worth considering in turn each of these arguments. Let us begin with primary delusional arguments, that correspond to an instantial step, in the sense that it is made up of several different instances. Primary delusional arguments are constituted here by four different instances, i.e. the steps (R1)-(R3), (R4)-(R6), (R7)-(R9) and (R10)-(R12). These four primary delusional arguments lead the patient to conclude that at a given time, the TV presenters spoke according to his/her acts or to what he/she felt.

Let us consider now the following stage (R13), which constitutes the conclusion of a secondary delusional argument, and is of a different nature. Its premises are the conclusion (R3), (R6), (R9), (R12) of the four previous instances of primary delusional arguments of reference. The patient generalises from the latter to the conclusion that the TV presenters speak according to what he/she is doing or to what he/she is feeling. The overall structure of this type of secondary delusional argument is then as follows:

(R3) in T2 the presenter of the show spoke according to what I was doing

(R6) in T4 the presenter of the show spoke according to what I was feeling

(R9) in T6 the presenter of the show spoke according to what I was doing

(R12) in T8 the presenter of the show spoke according to what I felt

(R…) …

(R13) ∴ the presenters of the shows speak according to what I do or feel

On can then term inductive this type of secondary delusional argument because it has the form of an enumerative induction, where the patient constructs his/her conclusion by generalising, in an inductive way, from the conclusions of several instances of primary delusional arguments. Thus, secondary delusional arguments correspond to a step the nature of which proves to be inductive.

At this stage, it is also worth mentioning the third step, which leads to delusion of reference. It consists of the tertiary delusional argument of reference, constituted by steps (R13) and (R14), the premise (R13) of which being the conclusion of the secondary delusional argument of reference:

(R13) ∴ the presenters of the shows speak according to what I do or feel

(R14) ∴ television speaks about me

In such argument, the patient takes into account the conclusion of the inductive step that the presenters of the shows speak according to his/her acts or to his/her internal state, and interprets it by concluding that television speaks about him/her. It consists, as we did see it, of a step the function of which is merely interpretative, in the sense that it aims at making sense of the inductive conclusion which results from the secondary delusional argument. Tertiary delusional arguments are therefore the product of an interpretative step, which situates itself at a monothematic level (here, the specific topic is that of delusion of reference).

A structurally identical mechanism leads to delusional ideas of telepathy. Several instances of primary delusional arguments of influence lead first the patient to conclude that his/her own thoughts are at the origin of acts that are accomplished by other persons. By an inductive step, the patient is then led to the conclusion that people act according to his/her thoughts. Finally, in an interpretative step, the patient concludes that other people can read his/her thoughts (or that they can hear them). It consists there, in the patient’s mind, of an attempt at explaining the very disturbing conclusion which results from the inductive step according to which other persons act according to his/her thoughts.

The same mechanism also engenders the formation of delusional ideas of influence. In that case, several instances of primary delusional arguments of influence lead the patient to conclude that his/her own thoughts are at the origin of annoyances caused to other persons. An inductive step leads then the patient to the conclusion that people react negatively in function of his/her thoughts. Finally, an interpretative step leads the patient to conclude that he/she perturbs and disturbs other people.

Moreover, such mechanism leads to the formation of delusional ideas of control. They find their origin in the instances of primary delusional arguments of control. Such instances have the same structure as that of the instances of primary delusional arguments of reference, of telepathy or of influence, with however this difference that the temporal order of both types of events – internal and external, with regard to the patient – is now reversed. Within the primary delusional arguments of reference, of telepathy or of influence, an internal event with regard to the patient precedes an external event, whereas it is the opposite with regard to a primary delusional argument of control: the external event precedes then the internal event. Thus, several instances of primary delusional arguments of control lead the patient to conclude inductively that some external events have an effect on his/her thoughts, his/her emotions or his/her acts. The interpretative step leads then the patient to think that he/she is controlled by external beings or objects such as robots or a satellite.

Finally, it is worth specifying the role played by quaternary delusional arguments. The premises of the latter arguments are conclusions of tertiary delusional arguments. Quaternary delusional arguments are more general arguments, which present, as well as tertiary delusional arguments, an interpretative nature. But unlike tertiary delusional arguments which turn out to be interpretative at a monothematic level, quaternary delusional arguments are interpretative at a polythematic level. They indeed take into account jointly the conclusions of tertiary delusional arguments of reference, of telepathy, of influence, etc. by striving to make sense, globally,of them and to interpret them. The reasoning below constitutes then a quaternary delusional argument leading to ideas of grandeur:

(R15) television and the media speak about me

(T16) people can read my thoughts

(I17) I influence other people’s behaviour

(18) ∴ I am someone exceptional

(19) ∴ I am an extraterrestrial

At a quaternary level, the patient takes then into account the different conclusions resulting from tertiary delusional arguments, the function of which is interpretative at the level of a given delusional topic and attempts this time to interpret the set constituted by the latter. The resulting conclusion constitutes veritably, for the patient, an overall theory the function of which is to make sense and to explain all the abnormal phenomena which he/she experiences.

3.2 Maintenance of delusional ideas

It is worth considering now the mechanism which leads to the maintenance of delusional ideas. Let us place ourselves at the level of secondary delusional arguments which, at the level of the formation of delusional ideas, are of an inductive nature. Consider then especially the form that take secondary delusional arguments of reference, at the stage of the maintenance. At this step, the conclusion (R13) which results from secondary delusional arguments, in virtue of which the presenters of the shows speak according to what the patient makes or feels, was already established at the stage of the formation of delusional ideas. And the corresponding reasoning takes then into account a new instance of primary delusional argument (R20) of reference, in the following way:

(R20) in T100 the presenter of the show spoke according to what I was doing

(R21) ∴ this confirms that television speaks according to what I do

On can notice here that the inductive generalisation (R13) was already established at the stage of the formation of the secondary delusional argument, and that the new instance of primary delusional argument constitutes then, in the patient’s mind, a case of confirmation of the latter generalisation. As we can see, the role of the new instance of primary delusional argument is to confirm and therefore to reinforce, at the stage of the maintenance, a generalisation which was already established at the previous stage of the formation of delusional ideas.

4. Analysis of delusional arguments

At this stage, it is worth analysing in detail the structure of the type of reasoning which has been just described, in order to identify accurately the fallacious steps and to determine the role played by apophenia. Let us consider in turn primary, secondary, tertiary and quaternary delusional arguments. Let us scrutinise first the following instance of primary delusional argument of telepathy, which manifests itself at the level of the formation of delusions:

(T1) in T1 I thought of Michael “What an idiot!”

(T2) in T2 I heard Michael shout

(T3) ∴ in T2 I heard Michael shout because in T1 I thought of him “What an idiot!”

It appears here that the two premises (T1) and (T2) constitute genuine facts and therefore turn out to be true. Conversely, the conclusion (T3) that concludes to the existence of a relation of causality between the two consecutive facts F1 (in T1 I thought of Michael “What an idiot!”) and F2 (in T2 I heard Michael shout) is it justified? It appears not. Indeed, both premises are only establishing a relation of anteriority between the two consecutive facts F1 and F2. And the conclusion (T3) which deducts from it a causality relationship turns out therefore to be too strong. The corresponding reasoning presents then a fallacious nature. The corresponding error of reasoning, which concludes to a relation of causality while there is only a simple relation of anteriority, is traditionally called post hoc fallacy, according to the Latin sentence “Post hoc, ergo propter hoc” (thereafter, hence because of it). This is a very common type of fallacious reasoning, which is notably at the root of many superstitions (Martin 1998; Bressan 2002). David Hemsley (1992) notably mentions such type of reasoning in clinical observation: “A patient of the present author, recalling his psychotic experiences noted that the co-occurrence of two events often led immediately to an assumption of a causal relationship between them”. Finally, one can notice that in the context of cognitive distortions, the type of error of reasoning corresponding to post hoc fallacy can be considered a specific case of arbitrary inference.

Let us also proceed to analyse the type of reasoning which underlies secondary delusional arguments, and that presents at the stage of the formation of delusional ideas, as we did see it, the following inductive structure:

(T22) in T2 Michael spoke according to my thoughts

(T23) in T4 the neighbour spoke according to my thoughts

(T24) in T6 the radio presenter spoke according to my thoughts

(…)

(T25) ∴ people act according to my thoughts

Such type of reasoning appears prima facie completely correct. It consists here of a reasoning based on an inductive generalisation, in which the patient simply builds a more general conclusion from several instances. Such reasoning is completely correct, for its conclusion can be considered as true, inasmuch as its premises are true. However, a scrutiny reveals that the patient only takes into account here a limited number of instances, i.e. those instances that are based on the concordance of both premises, at the stage of primary delusional arguments. The patient then directs his/her attention exclusively to those instances that include two premises of which the internal event (premise 1) and the external event (premise 2) turn out to be concordant and render thus plausible a relation of causality. The corresponding turn of mind can be described as a concordance bias. In effect, the patient does not take into account at this stage those instances which could possibly be composed of two discordant premises. The latter are likely to come under two different forms. An instance of the first form is as follows:

(T1) in T1 I thought of Michael “What an idiot!”

(26) in T2 Michel was quiet

And an instance of the second form is :

(27) in T1 I didn’t think of Michael

(T2) in T2 I heard Michael shout

In these two types of cases, one can notice a discordance between the two premises, which goes directly contrary to the idea of causality between the two events. As we see it finally, the flaw in the patient’s reasoning resides essentially in the fact of only taking into account those instances where the concordance between an internal event and an external event renders plausible a causality relationship. But if the patient had taken into account at the same time the concordant and the discordant instances, he/she would have been led to conclude that the concordant instances represent only a small part of the set constituted by the class of relevant instances, and are only therefore the result of a random process. In such context, as we see it, the concordant instances in fact constitute but mere coincidences.

If one places oneself now at the stage of the maintenance of delusional ideas, one can observe the presence of a mechanism of the same nature. At the stage of the emergence of delusional ideas, secondary delusional arguments present, as we did see it, an inductive nature. On the other hand, at the stage of the maintenance of delusional ideas, the latter come under the form of arguments which lead to the confirmation of an inductive generalisation. Consider then the following instance, where the conclusion (T25) according to which people act according to the patient’s thoughts results from a secondary delusional argument and was already established at the stage of the formation of delusional ideas:

(T28) in T100 my sister spoke according to my thoughts

(T29) ∴ this confirms that people act according to my thoughts

This type of argument appears completely valid, for the conclusion results directly from its premises. However, the latter argument is also at fault by default, for it does not take into account some premises, which turn out to be as much as relevant as the instance (T28). As we can see, the error of reasoning consists then in taking only into account those instances which confirm the generalisation (T25), while ignoring those instances which disconfirm the latter. Hence, this type of argument reveals a confirmation bias, i.e. a tendency to favour those instances which confirm a generalisation, whereas it would be necessary to take into consideration at the same time those which confirm and those which disconfirm it. One can notice however that such type of cognitive bias presents a very common nature (Nickerson 1998, Jonas et al. 2001).

It is worth considering, third, tertiary delusional arguments. Consider then the following tertiary delusional argument of telepathy (a similar analysis also applies to tertiary delusional arguments of reference and of influence):

(T30) ∴ people act in function of my thoughts

(T31) ∴ people can read my thoughts (people can hear my thoughts)

One can notice here that if premise (T30) is true, then the conclusion (T31) constitutes a credible explanation. This type of argument presents then an interpretative nature and the conclusion (T31) according to which people can read the patient’s thoughts appears finally plausible, inasmuch as it is considered as true that people act according to his/her thoughts. As we can see, such argument is motivated by the patient’s concern of explaining and of interpreting the disturbing generalisation which results from the repetition of the many concordant above-mentioned instances.

Finally, the following quaternary delusional argument aims, in the same way, at making sense of the conclusions which result from the conjunction of conclusions of different tertiary delusional argument:

(R15) television and the media speak about me

(T16) people can read my thoughts

(I17) I influence people’s behaviour

(18) ∴ I am someone exceptional

(19) ∴ I am an extraterrestrial

As we can see it, the conclusion (18) results here directly from the three premises (R15), (T16) and (I17) and the corresponding reasoning which leads the patient to conclude that he/she is someone exceptional can also be considered as valid. On the other hand, the conclusion (19) appears here too strong with regard to premise (18).

Given what precedes, it appears that a number of steps in the reasoning which leads to delusional ideas in schizophrenia are characterised by a reasoning which appears mainly normal. By normal reasoning, one intends here a broadly logical and rational reasoning, but also including some errors of logic of a very common type. Such viewpoint corresponds to the one put forth by Maher (1988; 1999) who considers, as we did see it, that the delusional construction in schizophrenia is nothing else than normal reasoning worked out by the patient to try to explain the abnormal phenomena which he/she experiences.

However, one can notice that in the above-mentioned structure of reasoning, one part of the reasoning cannot a priori be truly considered as normal. It consists here of the different instances of primary delusional arguments. The latter are based, as we did see it, on errors of reasoning corresponding to post hoc fallacies. This type of error of reasoning arguably turns out to be extremely common. However, the instances of primary delusional arguments mentioned above present an unusual nature, in the sense that they put in relationship the patient’s thoughts (or his/her emotions, feelings or actions) with external phenomena. Prima facie, such type of reasoning cannot be considered as normal. For why is the patient led to put his/her thoughts in relationship with external phenomena? One can formulate the question more generally as follows: why does the patient put in relationship the phenomena of his/her internal and personal life (his/her thoughts, emotions, feelings, etc.) with mere external phenomena? This distinguishes itself indeed significantly from the behaviour of a normal person, for whom it exists a very clear-cut intuitive separation between on one hand, his/her own internal world, and on the other hand, the external phenomena.

The answer to the previous question can be found here in the role of apophenia. Due to apophenia, the feeling indeed imposes itself to the patient that his/her internal world is closely linked up with the external world. So, his/her thoughts, emotions, feelings and acts appear to him/her to be closely linked up with the external phenomena that he/she perceives, such as ambient noise and dialogues, the words of the presenters of television or of radio, the dialogues of the characters of comic strips, the movements of the wings of a butterfly or of a bird, the natural phenomena such as the wind or the rain, etc. In the context which results from apophenia, the repeated instances that constitute primary delusional arguments can then take place naturally. For since the patient lives with a permanent feeling of interconnection between events that relate to him/her specifically and those which occur in the world which surrounds him, he/she is then led to observe many concordances between events related to him/her and external facts. In such context, primary delusional arguments can then take place naturally.

In the present context, the role of apophenia can be considered as fundamental. And this leads to suggest that considering its specificity and considering the leading role that it plays in the development of primary delusional arguments and therefore of all the characteristic delusional ideas of schizophrenia, apophenia could be counted among the criteria of the illnessii.

As we can see it, the process which gives rise to delusional arguments from the phenomenological experience constituted by apophenia proves finally to be in line with Maher’s account. And one finds here a clear explanation of delusions as the patient’s response to the abnormal phenomena which he/she experiences, among which one can then mention apophenia, as well as hallucinations.

Given what precedes, polythematic delusions can be defined as conclusions of arguments triggered by apophenia and that include some very common errors of reasoning such as post hoc fallacy and confirmation bias. Hence, apophenia and a normal reasoning including the type of aforementioned errors of reasoning turn out to be necessary and sufficient conditions for the development of polythematic delusions. This double condition notably explains why we are not all delusional. For if errors in reasoning based on post hoc fallacy and confirmation bias turn out indeed to be very common, they only trigger primary delusional arguments when they are associated with the abnormal perception which consists in apophenia. It is worth pointing out, moreover, that such model leaves also room for more stronger conditions. For if apophenia constitutes one of the two sufficient conditions for the development of polythematic delusions, the latter can also take place in conditions where abnormal perception is constituted not only by apophenia, but also by other abnormal perceptions such as hallucinations. And also, whereas the second condition which is sufficient for delusions identifies itself with normal reasoning including post hoc fallacy and confirmation bias, it proves that the development of delusions can also be made by means of a reasoning which deviates more or less from normal reasoning. But the essential characteristic of the present model resides in the fact that apophenia and normal reasoning including the aforementioned very common errors, constitute necessary and sufficient conditions for the development of polythematic delusions.

5. The role of the hallucinations

At this stage, it is worth highlighting the role played by hallucinations, the other major symptom of schizophrenia, in the process which has just been described. I will set out to describe in more detail here the role played by auditory hallucinations – given that the corresponding analysis can be easily extended to hallucinations relating to other sensory modalities, i.e. visual, tactile, olfactory and gustatory.

Auditory hallucinations are susceptible, first, of playing a role at the level of primary delusional arguments. In this type of case, the primary delusional argument presents the same structure as the one described above, with the only difference that an auditory hallucination – in place of a real external event – constitutes then the second premise of the primary delusional argument. The following instance constitutes then an example of primary delusional argument of reference, but it is there an auditory hallucination, by which the patient hears the voice of the presenter of the show saying “Clumsy!” while he/she watches TV, that constitutes the support of the second premise of the argument:

(32) in T1 I dropped my pen

(33) in T2 I heard the voice of the presenter of the show saying “Clumsy!”

(34) ∴ in T2 the presenter of the show said “Clumsy!” because in T1 I dropped my pen

In the same way, the following instance constitutes a case of primary delusional argument of telepathy. In that case, it is an auditory hallucination, by which the patient hears the voice of his neighbour saying “Calm down!”, that serves as a basis for the second premise of the argument:

(35) in T1 I was very upset

(36) in T2 I heard the voice of my neighbour saying “Calm down!”

(37) ∴ in T2 my neighbour said “Calm down!” because in T1 I was very upset

It is worth mentioning, second, the role that can be played by auditory hallucinations at the level of secondary delusional arguments. In such case, the corresponding generalisations develop from instances of primary delusional arguments which also include auditory hallucinations. In the example below, the patient generalises from the conclusions of three instances of primary delusional arguments of reference. But while the two latter instances (39) and (40) are based on real external phenomena, the first instance (38) is founded on hallucinated content, by which the patient heard the TV presenter saying “Clumsy!”:

(38) in T2 the TV presenter said “Clumsy!” because in T1 I dropped my pen

(39) in T4 the presenteress said “Calm down!” because in T3 I was upset

(40) in T6 the presenter of the show said “Thank you” because in T5 I thought “I love this presenter”

(…) …

(41) ∴ the TV presenters speak according to what I do or feel

As we see it, auditory hallucinations contribute in this way to increase the number of primary delusional arguments, by creating thus additional instances which add up themselves to the different types of standard instances previously defined. This gives then more weight to the inductive generalisations made by the patient from multiple instances of primary delusional arguments. Besides, it has also the effect of reinforcing the coherence of the patient’s delusional system and of rendering it then more resistant to contrary argumentation.

It is worth mentioning, lastly, another type of role which can be played by auditory hallucinations. Such is notably the case when the content of the hallucinations proves to be consistent with the conclusions that result from secondary, tertiary or quaternary arguments. Auditory hallucinations have then the effect of reinforcing the latter conclusions. The instance below constitutes a case where an auditory hallucination comes to reinforce the conclusion of a tertiary delusional argument of telepathy. In that case, the hallucinated content resides in the fact that the patient hears the voice of his friend Joseph saying “I know the slightest of your thoughts”:

(42) ∴ in T50 I thought that people know of my thoughts

(43) in T100 I heard Joseph saying: “I know the slightest of your thoughts”

(44) ∴ this confirms that people know of my thoughts

In a similar way, the following instance has the effect of reinforcing the conclusion which results from a quaternary delusional argument, where the hallucinated content consist of a voice heard by the patient that says: “You come from the planet Mars”:

(19) ∴ in T50 I thought I was an extraterrestrial

(45) in T100 I heard a voice saying : “You come from the planet Mars”

In a general way, we see here how hallucinations constitute an element which has the effect of reinforcing considerably the conclusions resulting from delusional arguments. The hallucinations have then the effect of reinforcing the strength and the consistency of the beliefs’ system of the patient, thus contributing to its maintenance, and rendering then his/her ideas more resistant to contrary argumentation.

6. Comparison with other cognitive models of delusions

The present model, as we can see it, mainly emphasises a cognitive approach of delusions encountered in schizophrenia. This model introduces a fundamental cognitive element, but also leaves room to a neurophysiological element (at the origin of apophenia), the role of which proves to be essential. One can notice finally that the model which has just been described turns out to be compatible with some other accounts of delusional ideas met in schizophrenia.

The present analysis, to begin with, is susceptible of inserting itself as part of the adaptation of Albert Ellis’ ABC-analysis described by Chadwick et al. (1996). In this context, the internal and external events with regard to the patient, that are the premises of primary delusional arguments, constitute the As. The primary, secondary, tertiary and quaternary delusional arguments, can also be assimilated to the Bs. Lastly, the negative emotions (anger, anxiety, frustration, etc.) felt by the patient, that result there from the conclusions of delusional tertiary and quaternary arguments, constitute the Cs. As we can see it, the present analysis leads, in comparison with the standard ABC-analysis, to distinguish several steps at the level of the Bs. This distinction is important, since it allows to distinguish several steps whose function is different, within the reasoning that leads to delusional ideas. Thus, the B1s (primary delusional arguments) are instances that lead to the attribution of a causality relationship between internal and external (to the patient) phenomena; the B2s (secondary delusional arguments) result from a generalisation of inductive nature; the B3s (tertiary delusional arguments) correspond to an interpretative step at a monothematic level; finally, the B4s (quaternary delusional arguments) are characteristic of a step of interpretation at a polythematic level, the conclusion of which truly constitutes a global explicative theory of the abnormal phenomena undergone by the patient. On the other hand, we are led there to distinguish between those parts of the patient’s reasoning which are globally valid (the B2s, B3s and B4s) and the part which is invalid (the B1s, based on post hoc fallacy). Such nuanced point of view should be likely to preserve – what constitutes one of the key points of cognitive and behaviour therapy – the therapeutic alliance, i.e., the relation of collaboration between the patient and the therapist aiming at shared objectives in the struggle against the illness. As we can see it, the present analysis leads to especially emphasise post hoc fallacy, which constitutes the weakness in the patient’s reasoning, but the repeated instances of which, triggered by apophenia, truly constitute the building block of the delusional construction.

The present model also has number of affinity with the approaches which are at the root of cognitive therapy of schizophrenia (Kingdon & Turkington 1994; Kingdon & Turkington 2005; Chadwick & al. 1996; Beck & Rector 2000). In this type of approach, the therapist sets out to reduce progressively the patient’s degree of belief in his/her delusional polythematic ideas. To this end, the therapist suggests the patient, in a spirit of dialogue of Socratic inspiration, to elaborate alternative hypotheses; he also teaches the patient the approach which consists in searching elements likely to confirm or to disconfirm his/her own hypotheses, as well as to build out alternative hypotheses. The contribution of the present analysis with regard to cognitive therapy of schizophrenia is likely to manifest itself in several ways. It proves to be useful to specify then, for the clinician, what could be such contribution, and to also provide a specific framework in which the present model will possibly be tested. The distinction of different steps in the development of delusions allows first to distinguish between different hypotheses corresponding to the conclusions of primary, secondary, ternary and quaternary arguments. The degree of belief associated with each of these levels of hypotheses is also susceptible of being evaluated separately, by notably allowing to determinate then at which level resides the strongest degree of conviction. In the same way, each of the conclusions of the primary, secondary, tertiary or quaternary arguments, will possibly be tested (confirmed/infirmed) and give room for the elaboration of alternative hypotheses (David Kingdon, personal communication). For example, at the level of primary delusional arguments, it will be possible to consider the belief according to which the presenter said in T2: “You should not drink!” because the patient was drinking an aperitif in T1; this hypothesis will possibly give rise to a search for evidence, and then confronted with an alternative hypothesis such as: the presenter said in T2: “You should not drink!” because it was scheduled in the script of the television program. At the level of tertiary delusional arguments, the hypothesis according to which television speaks about the patient will also possibly be the object of a search for evidence, etc.

The interest for the clinician of the present approach resides, second, in the fact that it provides the patient with an alternative global explanation of the abnormal phenomena that he/she undergoes. The delusional construction of the patient constitutes, as we did see it, a theory which allows him/her to explain all the abnormal phenomena that he/she experiences. On can assume in this respect, that the fact for the patient to get a satisfactory theory explaining all abnormal phenomena which he/she experiences, is also likely to play an important role in the maintenance of his/her delusional system. In this context, the present analysis allows to propose to the patient an alternative explicative theory, grounded on apophenia and the different steps of reasoning which result from it. Such theory distinguishes itself from the explicative theory with which the patient is usually confronted (according to the common elliptical point of view, the latter is “mad”) and proves to be less stigmatising, since the reasoning which leads to delusions is notably considered here as normal. For this reason, one can assume that the patient could be more willing to adhere to the present alternative theory, as a global explanation of the abnormal phenomena which he/she experiences.

As we did see it, the present model conforms mainly with the one developed by Brendan Maher (1974; 1988; 1999), based on the fact that delusions result from a broadly normal interpretation of the abnormal phenomena undergone by the patient. The present analysis also specifies with regard to Maher’s model that apophenia (eventually associated with hallucinations) constitutes an abnormal perception which is enough for giving rise to delusional polythematic ideas met in schizophrenia. It has then been objected, as we did see it, to Maher’s model, that it did not allow to account for the fact that delusions can also take place in seemingly normal conditions, especially in a patient not suffering from hallucinations. But the present model points out that such conditions are not normal, since apophenia is present in such a patient. Since apophenia leads to abnormal perceptions, the essential factor described by Maher at the origin of delusional ideas, is therefore present as well. On the second hand, the present model also provides some elements of response with regard to the second objection, formulated against Maher’s model, by Davies and Coltheart (2000), according to which it does not allow to describe how delusional beliefs are adopted and maintained in spite of their implausible nature. The present model, however, sets out first to describe step-by-step the type of reasoning which leads to the adoption of polythematic delusions. By its structure, this type of reasoning appears mainly normal. It proceeds by enumeration of some instances, and then by generalisation and lastly, by interpretation. The present model also provides, as far as I can see, an answer to the criticism raised by Davies and Coltheart with regard to Maher’s model, who blame the latter for not describing how delusional beliefs are maintained in the patient’s belief system, in spite of their implausible nature. In the present model, as we did see it, it is the fact that new instances are generated every day which explains that beliefs are maintained. For when delusional beliefs are established in the patient’s belief system at the end of the stage of their formation, they are then maintained because apophenia continues to trigger every dayiii new instances of primary delusional argument. The latter come, in the patient’s mind, to confirm the conclusions of delusional arguments at a secondary, tertiary and quaternary level, already established at the stage of the formation of delusional ideas. From this point of view, there is no essential difference in the present model in the way that the formation and the maintenance delusional ideas take place. For as we did see it, the building block of the delusional construction is constituted there by the instances of primary delusional arguments, triggered by apophenia. And these instances which concur to the formation of delusional ideas, also ensure their maintenance every day, by confirming the conclusions of secondary, tertiary and quaternary arguments, which are already established at the stage of the formation of delusional ideasiv.

Finally, the model which has just been described provides, as far as I can see, in comparison with Maher’s model, an element which proves to be necessary in the context of an explicative model of polythematic delusions. This element consists of an answer to the question of why the content of delusional ideas in schizophrenia identifies itself most often with delusions of reference, of telepathy, of thought insertion, of influence and of control. As it was exposed above, the answer provided by the present model is that a mechanism of the same nature, grounded on post hoc fallacy, leads to the development of these different delusional topics. In primary delusional arguments of reference, of telepathy or of influence, an event which is internal to the patient precedes an external event. And in the case of a primary delusional argument of control, this structure is simply reversed: it is an external event which precedes an event of the patient’s internal life.

We see it finally, the preceding analysis allows to justify and to reinforce Maher’s initial model. In this context, one can notice that one of the consequences of the present model is that the sole apophenia, associated with normal reasoning, turns out to be sufficient to give rise to the emergence and the maintenance of a delusional systemv.


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i Monothematic delusions are not included into the scope of the present study.

ii One can notice that a neurophysiological explanation on the origin of apophenia is provided by Manfred Spitzer (1995, p. 100). He describes then how the latter is linked to the level of activity of dopamine and of norepinephrine, which have an influence on the value of the signal/noise ratio that is at the root of the activation of neural circuits: “if the signal to noise ratio is too high, (…) small environmental signals (i.e. perceptions to which we would normally pay little or no attention at all) may become amplified to a degree that is much higher than usual. This could result in experiences of “significant events” when merely ordinary events were in fact happening”. Spitzer shows then how apophenia can be the consequence of an imbalance at the dopamine level. By placing normally insignificant events (among which the patient’s thoughts) in the foreground, the modification of the signal/noise ratio allows then the particular feeling of interconnection that constitutes apophenia to occur. Under these conditions, one can notably conceive of how the patient’s thoughts can appear to him/her as prominent so as to be put on the same plan, then put in relationship with external facts such as the words pronounced by a TV presenter.

iii As an anonymous referee for Philosophiques suggests, it would be necessary to quantify precisely the frequency of these instances. This could be made in a separate study.

iv These elements of response with regard to the way the maintenance of polythematic delusions takes place need to be supplemented, especially as regards the way the conclusions of quaternary delusional arguments are put in coherence with the rest of the patient’s beliefs, takes place. Such analysis, which requires subsequent work, is however beyond the scope of the present study.

v I am very grateful to David Kingdon, Albert Ellis, Eugen Fischer, Robert Chapman and two anonymous referees for Philosophiques for very helpful comments on ancestor versions and earlier drafts.

On a Class of Concepts

Posprint in English (with additional illustrations) of a paper published in French in Semiotica, vol. 139 (1-4), 2002, 211-226, under the title “Une Classe de Concepts”.

This article describes the construction, of philosophical essence, of the class of the matrices of concepts, whose structure and properties present an interest in several fields. The paper emphasises the applications in the field of paradigmatic analysis of the resulting taxonomy and proposes it as an alternative to the semiotic square put forth by Greimas.

This paper is cited in:


On a Class of Concepts

Classically, in the discussion relating to polar opposites1, one primarily directs his interest to the common and lexicalized concepts, i.e. for which there exists a corresponding word in the vocabulary inherent to a given language. This way of proceeding tends to generate several disadvantages. One of them resides in the fact (i) that such concepts are likely to vary from one language to another, from one culture to another. Another (ii) of the resulting problems is that certain lexicalized concepts reveal a nuance which is either meliorative or pejorative, with degrees in this type of nuances which prove difficult to appreciate. Finally, another problem (iii) lies in the fact that certain concepts, according to semiotic analysis2 are regarded as marked with regard to others concepts which are unmarked, the status of unmarked concept conferring a kind of precedence, of pre-eminence to the concepts in question.

In my view, all the above-mentioned disadvantages arise from the fact that one traditionally works primarily, from the lexicalized concepts. The methodology implemented in the present study is at the opposite of this way of proceeding. Indeed, one will begin here to construct concepts in an abstract way, without consideration of whether these concepts are lexicalized or not. This construction being performed, one will then be able to verify that some of the concepts thus constructed correspond indeed to lexicalized concepts, whereas some others cannot be put in correspondence with any existing word in the common language. This latter methodology allows, I think, to avoid the above-mentioned disadvantages.

It will finally appear that the construction described below will make it possible to propose a taxonomy of concepts which constitutes an alternative to the one based on the semiotic square which has been proposed by Greimas.

1. Dualities

Let us consider the class of dualities, which is made up of concepts corresponding to the intuition that these latter:

(i) are different one from the other

(ii) are minimal or irreducible, i.e. can no more reduce themselves to some other more simple semantic elements

(iii) present themselves under the form of pairs of dual concepts or contraries

(iv) are predicates

Each of the concepts composing a given duality will be termed a pole. I shall present here a list, which does not pretend to be exhaustive, and could if necessary, be supplemented. Consider then the following enumeration of dualities3:

Analytic/Synthetic, Animate/Inanimate, Exceptional/Normal, Antecedent/Consequent, Existent/Inexistent, Absolute/Relative, Abstract/Concrete, Accessory/Principal, Active/Passive, Aleatory/Certain, Discrete/Continuous, Deterministic/Indeterministic, Positive/Negative, True/False, Total/Partial, Neutral/Polarized, Static/Dynamic, Unique/Multiple, Container/Containing, Innate/Acquired (Nature/Nurture), Beautiful/Ugly, Good/Ill, Temporal/Atemporal, Extended/Restricted, Precise/Vague, Finite/Infinite, Simple/Composed, Attracted/Repulsed, Equal/Different, Identical/Contrary, Superior/Inferior, Internal/External, Individual/Collective, Quantitative/Qualitative, Implicit/Explicit4, …

At this step, it should be observed that certain poles present a nuance which is either meliorative (beautiful, good, true), or pejorative (ugly, ill, false), or simply neutral (temporal, implicit).

Let us denote by A/Ā a given duality. If words of the common language are used to denote the duality, capital letters will be then used to distinguish the concepts used here from the common concepts. For example: the Abstract/Concrete, True/False dualities.

It should be noted lastly that several questions5 immediately arise with regard to dualities. Do dualities exist (i) in a finite or infinite number? In the same way, does there exist (ii) a logical construction which makes it possible to provide an enumeration of the dualities?

2. Canonical poles

The positive canonical poles

Starting from the class of the dualities, we are now in a position to construct the class of the canonical poles. At the origin, the lexicalized concepts corresponding to each pole of a duality reveal a nuance6 which is respectively either meliorative, neutral, or pejorative. The class of the canonical poles corresponds to the intuition that, for each pole  of a given duality A/Ā, one can construct 3 concepts: a positive, a neutral and a negative concept. In sum, for a given duality A/Ā, one thus constructs 6 concepts, thus constituting the class of the canonical poles. Intuitively, positive canonical poles respond to the following definition: positive, meliorative form of ; neutral canonical poles correspond to the neutral, i.e. neither meliorative nor pejorative form of ; and negative canonical poles correspond to the negative, pejorative form of . It should be noted that these 6 concepts are exclusively constructed with the help of logical concepts. The only notion which escapes at this step to a logical definition is that of duality or base.

The neutral canonical poles

For a given duality A/Ā, we have thus the following canonical poles: {A+, A0, A, Ā+, Ā0, Ā}, that we can also denote respectively by (A/Ā, 1, 1), (A/Ā, 1, 0) , (A/Ā, 1, -1) , (A/Ā, -1, 1) , (A/Ā, -1, 0) , (A/Ā, -1, -1).

The negative canonical poles

A capital letter for the first letter of a canonical pole will be used, in order to distinguish it from the corresponding lexicalized concept. If one wishes to refer accurately to a canonical pole whereas the usual language lacks such a concept or well appears ambiguous, one can choose a lexicalized concept, to which the exponent corresponding to the chosen neutral or polarized state will be added. To highlight the fact that one refers explicitly to a canonical pole – positive, neutral or negative – the notations A+, A0 et A will be used. We have thus for example the concepts Unite+, Unite0, Unite etc. Where Unite+ = Solid, Undivided, Coherent and Unite = Monolithic. In the same way, Rational0 designates the neutral concept corresponding to the term rational of the common language, which reveals a slightly meliorative nuance. In the same way, Irrationnal0 designates the corresponding neutral state, whereas the common word irrational reveals a pejorative nuance. One will proceed in the same way, when the corresponding lexicalized word proves ambiguous. One distinctive feature of the present construction is that one begins by constructing the concepts logically, and puts them afterwards in adequacy with the concepts of the usual language, insofar as these latter do exist.

The constituents of a canonical pole are:

– a duality (or base) A/Ā

– a contrary component c  {-1, 1}

– a canonical polarity p  {-1, 0, 1}

A canonical pole presents the form: (A/Ā, c, p).

Furthermore, it is worth distinguishing, at the level of each duality A/Ā, the following derived classes:

– the positive canonical poles: A+, Ā+

– the neutral canonical poles: A0, Ā0

– the negative canonical poles: A, Ā

– the canonical matrix consisting of the 6 canonical poles: {A+, A0, A, Ā+, Ā0, Ā}. The 6 concepts constituting the canonical matrix can also be denoted under the form of a 3 x 2 matrix.

A canonical matrix

Let also  be a canonical pole, one will denote by ~ its complement, semantically corresponding to non-. We have thus the following complements: ~A+, ~A0, ~A, ~Ā+, ~Ā0, ~Ā. The notion of a complement entails the definition of a universe of reference U. Our concern will be thus with the complement of a given canonical pole in regard to the corresponding matrix7. It follows then that: ~A+ = {A0, A, Ā+, Ā0, Ā}. And a definition of comparable nature for the complements of the other concepts of the matrix ensues.

It should be noted lastly that the following questions arise with regard to canonical poles. The construction of the matrix of the canonical poles of the Positive/Negative duality: {Positive+, Positive0, Positive, Negative+, Negative0, Negative} ensues. But do such concepts as Positive0, Negative0 and especially Positive, Negative+ exist (i) without contradiction?

In the same way, at the level of the Neutral/Polarized duality, the construction of the matrix {Neutral+, Neutral 0, Neutral, Polarized+, Polarized0, Polarized} ensues. But do Neutral+, Neutral exist (ii) without contradiction? In the same way, does Polarized0 exist without contradiction?

This leads to pose the question in a general way: does any neutral canonical pole admit (iii) without contradiction a corresponding positive and negative concept? Is there a general rule for all dualities or well does one have as many specific cases for each duality?

3. Relations between the canonical poles

Among the combinations of relations existing between the 6 canonical poles (A+, A0, A, Ā+, Ā0, Ā) of a same duality A/Ā, it is worth emphasizing the following relations (in addition to the identity relation, denoted by I).

Two canonical poles 1(A/Ā, c1, p1) and 2(A/Ā, c2, p2) of a same duality are dual or antinomical or opposites if their contrary components are opposite and their polarities are opposite8.

Complementarity

Two canonical poles 1(A/Ā, c1, p1) and 2(A/Ā, c2, p2) of a same duality are complementary if their contrary components are opposite and their polarities are equal9.

Two canonical poles 1 (A/Ā, c1, p1) et 2(A/Ā, c2, p2) of a same duality are corollary if their contrary components are equal and their polarities are opposite10.

Two canonical poles 1 (A/Ā, c1, p1) and 2(A/Ā, c2, p2) of a same duality are connex if their contrary components are equal and the absolute value of the difference in their polarities is equal to 1 11.

Two canonical poles 1 (A/Ā, c1, p1) and 2(A/Ā, c2, p2) of a same duality are anti-connex if their contrary components are opposite and the absolute value of the difference in their polarities is equal to 1.12, 13

The following questions then arise, with regard to the relations between the canonical poles. Does there exist (i) one (or several) canonical pole which is its own opposite? A priori, it is not possible without contradiction for a positive pole or a negative pole. But the question remains for a neutral pole.

In the same way, does there exist (ii) one (or several) canonical pole which is its own complementary? The following two questions then ensue: does there exist a positive canonical pole which is its own complementary? And also: does there exist a negative canonical pole which is its own complementary?

The questions (i) and (ii) can be formulated in a more general way. Let R be a relation such that R  {I, c, , j, g, }. Does there exist (iii) one (or several) canonical pole a verifying a = Ra?

4. Degrees of duality

One constructs the class of the degrees of duality, from the intuition that there is a continuous succession of concepts from A+ to Ā, from A0 to Ā0 and from A to Ā+. The continuous component of a degree of duality corresponds to a degree in the corresponding dual pair. The approach by degree is underlied by the intuition that there is a continuous and regular succession of degrees, from a canonical pole Ap to its contrary Ā-p.14 One is thus led to distinguish 3 classes of degrees of duality: (i) from A+ to Ā (ii) from A0 to Ā0 (iii) from A to Ā+.

A degree of duality presents the following components:

– a dual pair Ap-p (corresponding to one of the 3 cases: A+, A00 or A+)

– a degree d Î [-1; 1] in this duality

A degree of duality  has thus the form: (A+, d), (A00, d) or (A+, d).

On the other hand, let us call neutral point a concept pertaining to the class of the degrees of duality, whose degree is equal to 0. Let us denote by 0 such a concept, which is thus of the form (Ap-p, 0) with d[0] = 0. Semantically, a neutral point 0 corresponds to a concept which responds to the following definition: neither Ap nor Ā-p. For example, (True/False, 0) corresponds to the definition: neither True nor False. In the same way (Vague/Precise, 0) corresponds the following definition: neither Vague nor Precise. Lastly, when considering the Neutral/Polarized and Positive/Negative dualities, one has then: Neutral0 = (Negative0/Positive0, 0) = (Neutral0/Polarized0, 1).

It is worth noting that this construction does not imply that the neutral point thus constructed is the unique concept which corresponds to the definition neither Ap nor Ā-p. It will appear on the contrary that several concepts and even hierarchies of concepts can correspond to this latter definition.

The following property of the neutral points then ensue, for a given duality A/Ā: (A+, 0) = (A00, 0) = (A+, 0).

At this point, it is worth also taking into account the following derived classes:

– a discrete and truncated class, built from the degrees of duality, including only those concepts whose degree of duality is such that d  {-1, -0.5, 0, 0.5, 1}.

– the class of the degrees of complementarity, the degrees of corollarity, etc. The class of the degrees of duality corresponds to the relation of antinomy. But it is worth considering, in a general way, as many classes as there exists relations between the canonical poles of a same duality. This leads to as many classes of comparable nature for the other relations, corresponding respectively to degrees of complementarity, corollarity, connexity and anti-connexity.

It is worth noting finally the following questions, with regard to degrees of duality and neutral points. Does there exist (i) one (or several) canonical pole which is its own neutral point? A priori, it is only possible for a neutral pole.

Does any duality A/Ā admit (ii) a neutral point or trichotomic zero? One can call this question the problem of the general trichotomy. Is it a general rule15 or well does there exists some exceptions? It seems a priori that the Abstract/Concrete duality does not admit a neutral point. It appears to be the same for the Finite/Infinite or the Precise/Vague duality. Intuitively, these latter dualities do not admit an intermediate state.

Does the concept corresponding to the neutral point (Neutral0/Polarized0, 0) and responding to the definition: neither neutral nor polarized exist (iii) without contradiction in the present construction?

5. Relations between the canonical poles of a different duality: includers

It is worth also considering the relation of includer for the canonical poles. Consider the following pairs of dual canonical poles: A+ and Ā+, A0 and Ā0, A and Ā. We have then the following definitions: a positive includer+ is a concept such that it is itself a positive canonical pole and corresponds to the definition + = A+  Ā+. A neutral includer0is a neutral canonical pole such that 0 = A0  Ā0. And a negative includer is a negative canonical pole such that  = A  Ā. Given these definitions, it is clear that one assimilates here the includer to the minimum includer. Examples: Determinate0 is an includer for True0/False0. And Determinate0 is also a pole for the Determinate0/Indeterminate0 duality. In the same way, Polarized0is an includer for Positive0/Negative0.

More generally, one has the relation of n-includer (n > 1) when considering the hierarchy of (n + 1) matrices. One has also evidently, the reciprocal relation of includer and of n-includer.

Let us also consider the following derived classes:

matricial includers: they consist of concepts including the set of the canonical poles of a same duality. They respond to the definition: 0 = A+  A0  A  Ā+  Ā0  Ā.

mixed includers: they consist of concepts responding to the definition 1 = A+  Ā or well 2 = A  Ā+

It is worth also considering the types of relations existing between the canonical poles of a different duality. Let A and E be two matrices whose canonical poles are respectively {A+, A0, A, Ā+, Ā0, Ā} and {E+, E0, E, Ē+, Ē0, Ē} and such that E is an includer for A/Ā i.e. such that E+ = A+  Ā+, E0 = A0  Ā0 and E = A  Ā. One extends then the just-defined relations between the canonical poles of a same matrix, to the relations of comparable nature between two matrices presenting the properties of A and E. We has then the relations of 2-antinomy, 2-complementarity, 2-corollarity, 2-connexity, 2-anti-connexity16. Thus, for example, A0 is 2-contrary (or trichotomic contrary) to Ē0, 2-connex (or trichotomic connex) to E+ and E and 2-anti-connex (or trichotomic anti-connex) to Ē+ and Ē. In the same way, A+ and Ā+ are 2-contrary to Ē, 2-complementary to Ē+, 2-corollary to E, 2-connex to E0 and 2-anti-connex to Ē0, etc.

Let us consider also the following property of neutral points and includers. Let A and E be two matrices, such that one of the neutral poles of E is an includer for the neutral dual pair of a: E0 = A0  Ā0. We has then the following property: the canonical pole Ē0 for the matrix E is a neutral point for the duality A00. Thus, the neutral point for the duality A00 is the dual of the includer E0 of A0 and Ā0. Example: Determinate0 = True0  False0. Here, the neutral point for the True/False duality corresponds to the definition: neither True nor False. And we have then (True0/False0, 0) = (Determinate0/Indeterminate0, -1).

This last property can be generalized to a hierarchy of matrices A1, A2, A3, …, An, such that one of the poles 2 of A2 of polarity p is an includer for a dual pair of A1, and that one of the poles 3 of A3 is an includer for a dual pair of A2, …, and that one of the poles n of An is an includer for a dual pair of An-1. It follows then an infinite construction of concepts.

One also notes the emergence of a hierarchy, beyond the sole neutral point of a given duality. It consists of the hierarchy of the neutral points of order n, constructed in the following way from the dual canonical poles A0 and Ā0:

– A0, Ā0

– A1 = neither A0 nor Ā0

– A21 = neither A0 nor A1

– A22 = neither Ā0 nor A1

– A31 = neither A0 nor A21

– A32 = neither A0 nor A22

– A33 = neither A0 nor A21

– A34 = neither Ā0 nor A22

– …

One can also consider the emergence of this hierarchy under the following form17:

– A0, Ā0

– A1 = neither A0 nor Ā0

– A2 = neither A0 nor Ā0 nor A1

– A3 = neither A0 nor Ā0 nor A1 nor A2

– A4 = neither A0 nor Ā0 nor A1 nor A2 nor A3

– A5 = neither A0 nor Ā0 nor A1 nor A2 nor A3 nor A4

– …

Classically, one constructs this infinite hierarchy for True/False by considering I1 (Indeterminate), I2, etc. It should be noticed that in this last construction, no mention is made of the includer (Determinate) of True/False. Neither does one make mention of the hierarchy of includers.

The notion of a complement of a canonical pole  corresponds semantically to non-. One has the concept of a 2-complement of a canonical pole , defined with regard to a universe of reference U that consists of the 2-matrix of . One has then for example: ~A+ = {A0, A, Ā+, Ā0, Ā, Ē+, Ē0, Ē}. And also, ~A+ = {Ā+, E0, E, Ē+, Ē0, Ē}, etc. More generally, one has then the notion of a n-complement (n > 0) of a canonical pole with regard to the corresponding n-matrix.

The following questions finally arise, concerning includers. For certain concepts, does there exist (i) one maximum includer or well does one have an infinite construction for each duality? Concerning the True/False duality in particular, the analysis of the semantic paradoxes has led to the use of a logic based on an infinite number of truth-values18.

Does any duality admit (ii) one neutral includer? Certain dualities indeed seem not to admit of an includer: such is in particular the case for the Abstract/Concrete or Finite/Infinite duality. It seems that Abstract constitutes a maximum element. Admittedly, one can well construct formally a concept corresponding to the definition neither Abstract nor Concrete, but such a concept appears very difficult to justify semantically.

Does there exist (iii) a canonical pole which is its own minimum includer?

Does there exist (iv) a canonical pole which is its own non-minimum includer? One can formulate this problem equivalently as follows. At a given level, does one not encounter a canonical pole which already appeared somewhere in the structure? It would then consist of a structure comprising a loop. And in particular, does one not encounter one of the poles of the first duality?

6. Canonical principles

Let  be a canonical pole. Intuitively, the class of the canonical principles corresponds to the concepts which respond to the following definition: principle corresponding to what is . Examples: Precise  Precision; Relative  Relativity; Temporal  Temporality. The canonical principles can be seen as 0-ary predicates, whereas the canonical poles are n-ary predicates (n > 0). The lexicalized concepts corresponding to canonical principles are often terms for which the suffix –ity (or –itude) has been added to the radical corresponding to a canonical pole. For example: Relativity0, Beauty+, Activity0, Passivity0, Neutrality0, Simplicity0, Temporality0, etc. A list (necessarily non-exhaustive) of the canonical principles is the following:

Analysis0/Synthesis0, [Animate0]/[Inanimate0], [Exceptional0]/Normality0, [Antecedent0]/[Consequent0], Existence0/Inexistence0, Absolute0/Relativity0, Abstraction0/[Concrete], [Accessory0]/[Principal0], Activity0/Passivity0, [Random0]/Certainty0, [Discrete0]/[Continuous0], Determinism0/Indeterminism0, [Positive0]/[Negative0], Truth0/Falsity0, Attraction0/Repulsion0, Neutrality0/Polarization0, [Static0]/Dynamic0, Unicity0/Multiplicity0, Contenance0/[Containing0], Innate0/Acquired0, Beauty+/Ugliness, Good+/Evil, Identity0/Contrary0, Superiority0/Inferiority0, Extension0/Restriction0, Precision0/Vagueness0, Finitude0/Infinitude0, Simplicity0/Complexity0, [Internal0]/[External0], Equality0/Difference0, Whole0/Part0, Temporality0/Atemporality0, Individuality0/Collectivity0, Quantity0/Quality0, [Implicit0]/[Explicit0], …

It should be noticed that a certain number of canonical principles are not lexicalized. The notations A+, A0, A will be used to denote without ambiguity a canonical principle which is respectively positive, neutral or negative. One could also use the following notation:  being a canonical pole, then -ity (or -itude) is a canonical principle. The following notation could then be used: Abstract0ity, Absolute0ity, Acessory0ity, etc. or as above [Abstract0], [Absolute0], etc.

The constituents of the canonical principles are the same ones as for the class of the canonical poles.

It is worth distinguishing finally the following derived classes:

positive canonical principles

neutral canonical principles

negative canonical principles

polarized canonical principles

with some obvious definitions19.

7. Meta-principles

Let a0 be a neutral canonical principle20. The class of the meta-principles corresponds to a disposition of the mind directed towards what is a0, to an interest with regard to what is a0. Intuitively, a meta-principle corresponds to a point of view, a perspective, an orientation of the human mind. Thus, the attraction for Abstraction0, the interest for Acquired0, the propensity to place oneself from the viewpoint of Unity0, etc. constitute meta-principles. It should be noted that this construction makes it possible in particular to construct some concepts which are not lexicalized. This has the advantage of a better exhaustiveness and leads to a better and richer semantics.

Let a0 be a neutral canonical principle. Let us also denote by p a meta-principle (p  {-1, 0, 1}). One denotes thus a positive meta-principle by +, a neutral meta-principle by 0 and a negative meta-principle by . We have then the enumeration of the meta-principles, for a given duality: {A+, A0, A, Ā+, Ā0, Ā}. Moreover, one will be able to denote by a-ism a meta-principle. Example: Unite  Unite-ism. We have then Internalism, Externalism, Relativism, Absolutism, etc. which correspond in particular to dispositions of the mind. A capital letter will preferably be used here to distinguish the meta-principles from the lexicalized concepts, and in particular to differentiate them from the corresponding philosophical doctrines, which often have very different meanings. It will be however possible to make use of the classical terms when they exist to designate the corresponding meta-principle. Thus All-ism corresponds to Holism.

One can term Ultra-a-ism or Hyper-a-ism the concept corresponding to . This latter form corresponds to an exclusive, excessive, exaggerated use of the viewpoint corresponding to a given principle. One has thus for example: Externalism = Hyper-externalism.

The constituents of the meta-principles are:

– a polarity p Î {-1, 0, 1}

– a neutral canonical principle composed of:

– a duality (or base) A/Ā

– a contrary component c  {-1, 1}

– a neutral polarity q = 0

The positive, neutral, negative canonical meta-principles are respectively of the form ((A/Ā, c, 0), 1), ((A/Ā, c, 0), 0), ((A/Ā, c, 0), -1).

Between the canonical meta-principles of a same duality, one has the same relations as for the canonical poles.

One has lastly the derived classes consisting in:

– the positive meta-principles (p > 0)

– the neutral meta-principles (p = 0)

– the negative meta-principles (p < 0)

– the polarized meta-principles which include the positive and negative meta-principles

– the matrix of the canonical meta-principles, consisting of 6 meta-principles applicable to a given duality{A+, A0, A, Ā+, Ā0, Ā}.

– the degrees of canonical meta-principles. Intuitively, such concepts are more or less positive or negative. The polarity is regarded here as a degree of polarity. These concepts are such that p Î [-1; 1].

– the class of the behavioral principles. Intuitively, the class of the behavioral principles constitutes an extension of that of the meta-principles. While the meta-principle constitutes a disposition of the human mind, the concepts concerned here are those which aim to describe, in a more general way, the tendencies of the human behavior21. Among the lexicalized concepts corresponding to the behavioral principles, one can mention: courage, prudence, pessimism, rationality, avarice, fidelity, tendency to analysis, instability, objectivity, pragmatism, etc. A first analysis reveals (i) that a certain number of them reveal a meliorative nuance: courage, objectivity, pragmatism; that (ii) others, by contrast, present a pejorative, unfavorable connotation: cowardice, avarice, instability; and finally (iii) that certain concepts present themselves under a form which is neither meliorative nor pejorative: tendency to analysis22. One has here the same classes as for the meta-principles, and in particular the degrees of behavioral principles. Example: coward is more negative than apprehensive; in the same way, bravery is more positive than courage.

Conclusion

The concepts constructed with the help of the present theory need to be distinguished in several regards from those resulting from the application of the semiotic square described by Greimas (1977, p. 25). This last theory envisages in effect four concepts: S1, S2, ~S1, ~S2. On the one hand, it appears that the semiotic square is based on two lexicalized concepts S1 and S2 that constitute a dual pair. It does not distinguish, when considering the dual concepts, whether these latter are positive, neutral or negative. By contrast, the present theory considers six concepts, lexicalized or not.

On the other hand, the present analysis differs from the semiotic square by a different definition of the complement-negation. Indeed, the semiotic square comprises two concepts corresponding to the complement-negation: non-S1 and non-S2. By contrast, in the present context, the negation is defined with regard to a universe of reference U, which can be defined with regard to the corresponding matrix, or well to the 2-matrix…, to the n -matrix. For each canonical pole, there is thus a hierarchy of concepts corresponding to non-S1 and non-S2.

One sees it, the present taxonomy of concepts differs in several respects from the one conceived of by Greimas. Implemented from the dualities and the logical concepts, the present theory has the advantage of applying itself to lexicalized concepts or not, and also of being freed [affranchie] from the definitions of concepts inherent to a given culture. In this context, the classification which has been just described constitutes an alternative to the one based on the semiotic square which has been proposed by Greimas.

References

FINE, Kit (1975). Vagueness, Truth and Logic. Synthese 30: 265-300
GREIMAS, A. J. (1977). Elements of a Narrative Grammar, Diacritics 7: 23-40
JAKOBSON, Roman (1983). Dialogues, Cambridge MA: MIT Press
PEACOCKE, C. A. B. (1981). Are Vague Predicates Incoherent?. Synthese 46: 121-141
RESCHER, Nicholas (1969). Many-Valued Logic, New York: McGraw Hill

1 Or polar contraries.

2 Cf. Jakobson (1983).

3 In the same way, it would have been possible to define a more restricted class, including only half of the semantic poles, by retaining only one of the two dual predicates, and by constructing the others with the contrary relation. However, the choice of either of the dual poles would have been arbitrary, and I have preferred to avoid it. The following construction would have then resulted. Let Contrary be the semantic pole and a whatever semantic pole, not necessarily distinct from Contrary; the concept resulting from the composition of Contrary and a is a semantic pole. It should also be noted that this type of construction would have led to:

Contrary° Contrary = Identical.

Contrary° Identical = Contrary.

Contraryn = Identical (for n even).

Contraryn = Contrary (for n odd).

In this context, it is worth noting that Contrary constitutes a specific case. In effect, if one seeks to build a class of the canonical poles which is minimal, it is worth noting that one can dispense oneself from Identical, whereas one cannot dispense oneself from Contrary. There is here an asymmetry. In effect, one can construct Identical with the help of Contrary, by using the property of involution: Contrary° Contrary = Identical. For other dualities, one can indifferently choose either of the concerned semantic poles.

4 It is worth noting that one could have drawn here a distinction between unary and binary poles, by considering that they consist of predicates. But a priori, such a distinction does not prove very useful for the resulting construction.

5 In what follows, the questions relating to the various classes are only mentioned. It goes without saying that they require an in-depth treatment which goes far beyond the present study.

6 With variable degrees in the nuance.

7 When it is defined with regard to a dual pair, the complement of the pole  of a given duality identifies itself with the corresponding dual pole.

8 Formally c1 = –c2, p1 = – p2 ® 1(A/Ā, c1, p1) = 2(A/Ā, c2, p2).

9 Formally c1 = – c2, p1 = p2 ® 1(A/Ā, c1, p1) = f2(A/Ā, c2, p2).

10 Formally c1 = c2, p1 = – p2 ® 1(A/Ā, c1, p1) = c2(A/Ā, c2, p2).

11 Formally c1 = c2, |p1p2| = 1 ® 1(A/Ā, c1, p1) = g2(A/Ā, c2, p2).

12 Formally c1 = – c2, |p1p2| = 1 ® 1(A/Ā, c1, p1) = b2(A/Ā, c2, p2).

13 We have then the following properties, with regard to the above-mentioned relations. The relation of identity constitutes a relation of equivalence. Antinomy, complementarity and corollarity are symmetrical, anti-reflexive, non-associative, involutive.

The operation of composition on the relations {identity, corollarity, antinomy, complementarity} defines an abelian group of order 4. With G = {I, c, , j}:

°IcjIIcjccIjjIcjjcI

where for all A Î G, A-1 = A, and A ° I = A, I being the neutral element. It should be noted that the group properties make it possible in particular to give straightforwardly a valuation to any propositions of the form: the contrary concept of the complementary of a1 is identical to the corollary of the complementary of a2.

14 This construction of concepts can be regarded as an application of the degree theory. Cf. in particular Fine (1975), Peacocke (1981). The present theory however is not characterized by the preferential choice of the degree theory, but considers simply this latter theory as one of the methods of construction of concepts.

15 Some common trichotomies are: {past, present, future}, {right, center, left}, {high, center, low}, {positive, neutral, negative}.

16 There is a straightforward generalization to n matrices (n > 1) of this construction with the relations of n-antinomy, n-complementarity, n-corollarity, n-connexity, n-anti-connexity.

17 One can assimilate the two just-described hierarchies to only one single hierarchy. It suffices to proceed to the following assimilation:

– A2 = A21 or A22

– A3 = A31 or A32 or A33 or A34

– A4 = A41 or A42 or A43 or A44 or A45 or A46 or A47 or A48

– …

18 Infinite-valued logics. Cf. Rescher (1969).

19 Furthermore, it should be noted that some other concepts can be thus constructed. Let also  be a canonical pole. We have then the classes of concepts responding to the following definition: to render  (Example: Unite  Unify; Different  Differentiate); action of rendering  (Unite  Unification; Different  Differentiation); that it is possible to render  (Unite  Unitable; Different  Differentiable), etc. These concepts are not however of interest in the present context.

20 It should be observed that we could have taken alternatively as a basis for the definition of the meta-principles a canonical principle, without distinguishing whether this latter is positive, neutral or negative. But it seems that such a definition would have engendered more complexity, without giving in return a genuine semantic interest.

21 This particular class would require however a much finer analysis than the one which is summarily presented here. I am only concerned here with showing that a many concepts pertaining to this category can be the subject of a classification whose structure is that of the meta-principles.

22 One can consider the following – necessarily partial – enumeration corresponding to the behavioral principles, in the order (A+), (A0), (A), (Ā+), (Ā0), (Ā):

firmness, propensity to repress, severity, leniency, propensity to forgive, laxism

defense, refusal, violence, pacifism, acceptance, weakness

pride, self-esteem, hyper-self-esteem, modesty, withdrawal of the ego, undervaluation of self

expansion, search of quantity, excess, perfectionism, search of quality, hyper-selectivity

delicacy, sensitivity, sentimentality, coolness, impassibility, coldness

objectivity, to be neutral being, impersonality, to be partisan, parti pris

uprightness, to act in a direct way, brusqueness, tact, to act in an indirect way, to flee the difficulties

combativeness, disposition to attack, aggressiveness, protection, disposition to defense, tendency to retreat

receptivity, belief, credulity, incredulity, doubt, excessive skepticism

expansion, oriented towards oneself, selfishness, altruism, oriented towards others, to render dependent

sense of economy, propensity to saving, avarice, generosity, propensity to expenditure, prodigality

mobility, tendency to displacement, instability, stability, tendency to stay at the same place, sedentariness

logical, rationality, hyper-materialism, imagination, irrationality, inconsistency

sense of humour, propensity to play, lightness, serious, propensity to the serious activity, hyper-serious

capacity of abstraction, disposition to the abstract, dogmatism, pragmatism, disposition to the concrete, prosaicness

audacity, tendency to risk, temerity, prudence, tendency to avoid the risks, cowardice

discretion, to keep for oneself, inhibition, opening, to make public, indiscretion

optimism, to apprehend the advantages, happy optimism, mistrust, to see the disadvantages, pessimism

sense of the collective, to act like the others, conformism, originality, to demarcate oneself from others, eccentricity

resolution, tendency to keep an opinion, pertinacity, flexibility of spirit, tendency to change opinion, fickleness

idealism, tendency to apprehend the objectives, quixotism, realism, tendency to apprehend the means, prosaicness

taste of freedom, to be freed, indiscipline, obedience, to subject oneself to a rule, servility

reflexion, interiorization, inhibition, sociability, exteriorisation, off-handednes

spontaneousness, tendency to react immediately, precipitation, calm, tendency to differ one’s reaction, slowness

eclecticism, multidisciplinarity, dispersion, expertise, mono-disciplinarity, bulk-heading

revival, propensity to change, rupture, safeguarding of the assets, propensity to maintenance, conservatism

motivation, passion, fanaticism, moderation, reason, tepidity

width of sights, tendency to synthesis, overflight, precision, tendency to analysis, to lose oneself in the details

availability, propensity to leisure, idleness, activity, propensity to work, overactivity

firmness, tendency not to yield, intransigence, diplomacy, tendency to make concessions, weakness

causticity, tendency to criticism, denigration, valorization, tendency to underline qualities, angelism

authority, propensity to command, authoritarianism, docility, propensity to obey, servility

love, tendency to be attracted, exaggerate affection, tendency to know to take one’s distances, repulsion, hatred

conquest, greed, bulimia, sobriety, to have the minimum, denudement

A Brief Introduction to N-universes

Preprint.  I present in this paper the basic elements of the n-universes, from an essentially pragmatic standpoint, i.e. by describing accurately the step-by-step process which leads to the modelling of a thought experiment.

A Brief Introduction to N-universes

Paul Franceschi

The n-universes are a methodological tool whose scope proves to be general and which find to apply in the thought experiments underlying the philosophical problems. The n-universes have been introduced in Franceschi (2001) in the context of the study of Goodman’s paradox and described in more detail in Franceschi (2002), in the context of their application to the paradoxes related to the Doomsday argument. I propose here to present the basic elements of n-universes, from a fundamentally practical standpoint, i.e. by describing the step-by-step process which leads to the modelisation of a thought experiment.

Presentation of n-universes

The n-universes are simplified models of the physical universe corresponding to a real situation put forth in a thought experiment. Making application of Occam’s razor, the n-universes thus make it possible to model a real situation with the help of the simplest model of universe, while nevertheless preserving the intrinsic structure of the corresponding real situation. Let us describe then the fundamental elements of the n-universes, from a basically operational standpoint. When one proceeds thus to model a concrete situation with a n-universe, one can determine its structure by means of the successive answers to the following questions:

1. does the n-universe have constants or variables?

The first task consists in determining what are the criteria of the n-universe corresponding to a given situation. These latter criteria include both constants and variables. Among the most common criteria, one can thus mention the temporal (Time) and spatial (Loc) criteria, but also the criteria of colour (Col), shape, temperature, polarisation, etc. Usually, the n-universe corresponding to a given thought experiment comprises at least the criteria of time and space. This can be illustrated through the following examples:

– the ΩObj0Time0Loc0: a n-universe comprising a unique object, a temporal constant and a spatial constant

– the ΩObjColTime0Loc: a n-universe comprising multiple objects, a colour variable, a temporal constant and a space variable

2. does the n-universe comprises a unique or multiple objects?

It is also worth drawing a distinction according to whether the given n-universe comprises either one single (ΩObj0) or multiple objects (ΩObj). To cite a few examples:

– the ΩObj0TimeLoc0: a n-universe comprising a unique object, a temporal variable and a spatial constant

– the ΩObjTimeLoc: a n-universe comprising multiple objects, a temporal variable and a space variable.

3. is a given variable-criterion demultiplied or not?

This distinction only relates to the variable-criterions of a given n-universe, and does not apply to constant-criterions. A given variable-criterion  (time, space, colour, etc.) of a n-universe can be demultiplied (*) or not (). If the variable-criterion  is demultiplied, an object in this type of n-universe can exemplify several taxa of the criterion . In our physical universe, objects have a property of temporal persistence: they exemplify thus several successive temporal positions. The corresponding model is a n-universe where the objects are demultiplied with regard to the temporal criterion (ΩObjTime*). To take an example:

– the ΩObjTime*ObjLoc0: a n-universe with multiple objects comprising a temporal variable and a spatial constant; the objects are also demultiplied with regard to the temporal criterion, so that a given object can thus exemplify several different temporal positions.

4. are the multiple objects in a one-one or in a many-one relation with regard to a given criterion?

This last distinction does not apply to the n-universes comprising one single object and only relates to those comprising multiple objects. Among these latter n-universes, one can then distinguish two cases. Firstly, when the same taxon of a given criterion  is exemplified by several objects, these latter are in many-one relation with the criterion  (ΩObj*). By contrast, when every taxon of a given criterion  is only exemplified by one single object, the objects are in relation one-one with this last criterion (ΩObj). To give a few examples:

– the ΩObj*Col0Obj*Time0ObjLoc: this type of n-universe with multiple objects comprises a colour constant (Col0), a temporal constant (Time0) and a spatial variable (Loc); moreover, the objects are in a many-one relation with the colour constant, so that all objets share the same colour; in addition, the objects are in a many-one relation with the time constant, so that several objects can exist at a the unique temporal position; on the other hand, the objects are in relation one-one with the space criterion, so that only one object can exist at a given space location (Fig. 1)

– the ΩObjColObj*Time0ObjLoc: this type of n-universe with multiple objects comprises a colour variable (Col), a temporal constant (Time0) and a spatial variable (Loc); moreover, the objects are in a one-one relation with the colour variable, so that all objects can have a different colour; in addition, the objects are in a many-one relation with the time constant, so that several objects can exist at a the unique temporal position; on the other hand, the objects are in relation one-one with the space criterion, so that only one object can exist at a given space location (Fig. 2)

– the ΩObj*ColObj*Time0ObjLoc: this type of n-universe with multiple objects comprises a colour variable (Col), a temporal constant (Time0) and a spatial variable (Loc); moreover, the objects are in a many-one relation with the colour variable, so that several objets can have the same colour; in addition, the objects are in a many-one relation with the time constant, so that several objects can exist at a the unique temporal position; lastly, the objects are in relation one-one with the space criterion, so that only one object can exist at a given space location (Fig. 3)

– the ΩObj*Col0Obj*Time0Obj*Loc: this type of n-universe with multiple objects comprises a colour constant (Col0), a temporal constant (Time0) and a spatial variable (Loc); moreover, the objects are in a many-one relation with the colour constant, so that several objets can share the same colour; in addition, the objects are in a many-one relation with the time constant, so that several objects can exist at a the unique temporal position; on the other hand, the objects are in relation many-one with the space criterion, so that only several objects can occupy the same space location (Fig. 4)

First steps with n-universes

At this step, we are in a position to illustrate what precedes through a concrete example. Consider then the following experiment, described by John Leslie (1996, p. 191):

You develop amnesia in a windowless room. Where should you think yourself more likely to be: in Little Puddle with a tiny situation, or in London? Suppose you remember that Little Puddle’s population is fifty while London’s is ten million, and suppose you have nothing but those figures to guide you. (…) Then you should prefer to think yourself in London. For what if you instead saw no reason for favouring the belief that you were in the larger of the two places? Forced to bet on the one or on the other, suppose you betted you were in Little Puddle. If everybody in the two places developed amnesia and betted as you had done, there would be ten million losers and only fifty winners. So, it would seem, betting on London is far more rational. The right estimate of your chances of being there rather than in Little Puddle, on the evidence on your possession, could well be reckoned as ten million to fifty.

Let us proceed now to model the situation corresponding to the Little Puddle/London experiment in terms of n-universes. What are thus the criteria of the corresponding n-universe? It appears first that the corresponding situation characterises itself with the presence of multiple individuals: there are indeed 50 inhabitants in Little Puddle and 10 million in London. Consequently, the corresponding n-universe comprises multiple objects (ΩObj). It also appears that the Little Puddle/London experiment takes place at one single temporal position. Thus, the corresponding n-universe has a constant-time (ΩTime0). Moreover, several inhabitants exist simultaneously at the unique temporal position Time0. Hence, the objects are in relation many-one with the temporal constant (ΩObj*Time0). Moreover, two space locations are explicitly distinguished: Little Puddle (Loc1) and London (Loc2). The corresponding situation can thus be modelled in a n-universe comprising a space variable (ΩLoc) which includes two different locations: Loc1 and Loc2. Furthermore, it proves that each inhabitant is either in Little Puddle or in London, so as a given inhabitant cannot occupy several space locations at the same time. Thus the space criterion is not demultiplied. Lastly, one can observe that several people can be at the same time at a given space location: there are thus 50 inhabitants in Little Puddle and 10 million in London. Consequently, the objects are in relation many-one with the space variable (ΩObj*Loc). Taking into account what precedes, it follows that the situation corresponding to the Little Puddle/London experiment can be modelled in a ΩObj*Time0Obj*Loc, a n-universe with multiple objects, comprising a temporal constant and a space variable, where the objects are in relation many-one with the time-constant and the space variable.

Conclusion

The n-universes, as we have just seen through the above illustration, make it possible to model the situations described in thought experiments, by simplifying their intrinsic elements in virtue of Occam’s razor. The n-universes then often allow to remove the inherent ambiguity and complexity which renders more difficult the reasoning applied to thought experiments.

Works cited

Franceschi, Paul. 2001. A Solution to Goodman’s paradox, e-print Philsci: 602, English translation of Une solution pour le paradoxe de Goodman, Dialogue 40: 99-123.
Franceschi, Paul. 2002. Une application des n-univers à l’argument de l’Apocalypse et au paradoxe de Goodman. doctoral dissertation, Corté: University of Corsica
Leslie, John. 1996. The End of the World: the science and ethics of human extinction, London: Routledge.

A Third Route to the Doomsday Argument

A paper published (2009) in English in the Journal of Philosophical Research, vol. 34, pages 263-278 (with significant changes with regard to the preprint).

In this paper, I present a solution to the Doomsday argument based on a third type of solution, by contrast with, on the one hand, the Carter-Leslie view and on the other hand, the Eckhardt et al. analysis. I begin by strengthening both competing models by highlighting some variations of their ancestors models, which renders them less vulnerable to several objections. I describe then a third line of solution, which incorporates insights from both Leslie and Eckhardt’s models and fits more adequately with the human situation corresponding to the Doomsday argument. I argue then that the resulting two-sided analogy casts new light on the reference class problem. This leads finally to a novel formulation of the argument that could well be more consensual than the original one.

This paper is cited in:

  • Alasdair Richmond, The Doomsday Argument, Philosophical Books Vol. 47 No. 2 April 2006, pp. 129–142
  • Robert Northcott, A Dilemma for the Doomsday Argument, Ratio, Volume29-3, September 2016, pages 268-282
  • William Poundstone, How to Predict Everything: The Formula Transforming What We Know About Life and the Universe, 2019, Oneworld

A Third Route to the Doomsday Argument

In what follows, I will endeavor to present a solution to the problem arising from the Doomsday argument (DA). The solution thus described constitutes a third way out, compared to, on the one hand, the approach of the promoters of DA (Leslie 1993 and 1996) and on the other hand, the solution recommended by its detractors (Eckhardt 1993 and 1997, Sowers 2002).i

I. The Doomsday Argument and the Carter-Leslie model

For the sake of the present discussion, it is worth beginning with a brief presentation of DA. This argument can be described as reasoning which leads to a Bayesian shift, starting from an analogy between what was has been called the two-urn caseii and the corresponding human situation.

Let us consider first the two-urn case experiment (adapted from Bostrom 1997):

The two-urn case experiment An opaque urniii is in front of you. You know that it contains either 10 or 1000 numbered balls. A fair coin has been tossed at time T0 and if the coin landed tails, then 10 balls were placed in the urn; on the other hand, if the coin landed heads, 1000 balls were placed in the urn. The balls are numbered 1,2,3,…. You formulate then the assumptions Hfew (the urn contains only 10 balls) and Hmany (the urn contains 1000 balls) with the initial probabilities P (Hfew) = P (Hmany) = 1/2.

Informed of all the preceding, you randomly draw a ball at time T1 from the urn. You get then the ball #5. You endeavor to estimate the number of balls that were contained at T0 in the urn. You conclude then to an upward Bayesian shift in favor of the Hfew hypothesis.

The two-urn case experiment is an uncontroversial application of Bayes’ theorem. It is based on the two following concurrent assumptions:

(H1few)the urn contains 10 balls
(H2many)the urn contains 1000 balls

and the corresponding initial probabilities: P (H1) = P (H2) = 1/2. By taking into account the fact that E denotes the evidence according to which the randomly drawn ball carries the #5 and that P (E|H1) = 1/10 and P (E|H2) = 1/1000, an upward Bayesian shift follows, by a straightforward application of Bayes’ theorem. Consequently, the posterior probabilities are such that P'(H1) = 0.99 and P'(H2) = 0.01.

Let us consider, on the second hand, the human situation corresponding to DA. While being interested in the total number of humans that humankind will finally count, it is worth considering the two following concurrent hypotheses:

(H3few)the total number of humans having ever lived will amount to 1011 (Apocalypse near)
(H4many)the total number of humans having ever lived will amount to 1014 (Apocalypse far)

It appears now that every human being has his own birth rank, and that yours, for example, is about 60×109. Let us also assume, for the sake of simplicity, that the initial probabilities are such as P(H3) = P(H4) = 1/2. Now, according to Carter and Leslie, the human situation corresponding to DA is analogous to the two urn case.iv If we denote by E the fact that our birth rank is 60×109, an application of Bayes’ theorem, by taking into account the fact that P(E|H3) = 1/1011 and that P(E|H4) = 1/1014, leads to an important Bayesian shift in favor of the hypothesis of a near Apocalypse, i.e., P'(H3) = 0.999. The importance of the Bayesian shift which results from this reasoning, associated with a very worrying situation related to the future of humankind, from the only recognition of our birth rank, appears counter-intuitive. This intrinsic problem requires that we set out to find it a solution.

In such context, it appears that a solution to DA has to present the following characteristics. On the one hand, it must point out in which ways the human situation corresponding to DA is similar to the two-urn case or possibly, to an alternative model, the characteristics of which are to be specified. On the second hand, such solution to DA must point out in which ways one or several models on analogy with the human situation corresponding to DA are associated with a frightening situation for the future of humankind.

In what follows, I will endeavor to present a solution to DA. In order to develop it, it will be necessary first to build up the space of solutions for DA. Such a construction is a non-trivial task that requires the consideration of not only several objections that have been raised against DA, but also the reference class problem. Within this space of solutions, the solutions advocated by the supporters as well as critics of DA, will naturally be placed. I will finally show that within the space of solutions thus established, there is room for a third way out, which is in essence a different solution from that offered by the proponents and opponents of DA.

II. Failure of an alternative model based on the incremental objection of Eckhardt et al.

DA is based on the matching of a probabilistic model – the two-urn case – with the human situation corresponding to DA. In order to build the space of solutions for DA, it is necessary to focus on the models that constitute an alternative to the two-urn case, which can also be put in correspondence with the human situation corresponding to DA. Several alternative models have been described by the opponents to DA. However, for reasons that will become clearer later, not all these models can be accepted as valid alternative models to the two-urn case, and take a place within the space of solutions for DA. It is therefore necessary to distinguish among these models proposed by the detractors of DA, between those which are not genuine alternative models, and those which can legitimately be included within the space of solutions for DA.

A certain number of objections to DA were formulated first by William Eckhardt (1993, 1997). For the sake of the present discussion, it is worth distinguishing between two objections, among those which were raised by Eckhardt, and that I will call respectively: the incremental objection and the diachronic objection. With each one of these two objections is associated an experiment intended to constitute an alternative model to the two-urn case.

Let us begin with the incremental objection mentioned in Eckhardt (1993, 1997) and the alternative model associated with it. Recently, George Sowers (2002) and Elliott Sober (2003) have echoed this objection. According to this objection, the analogy with the urn that is at the root of DA, is ungrounded. Indeed, in the two-urn case experiment, the number of the balls is randomly chosen. However, these authors emphasize, in the case of the human situation corresponding to DA, our birth rank is not chosen at random, but is indeed indexed on the corresponding time position. Therefore, Eckhardt stresses, the analogy with the two-urn case is unfounded and the whole reasoning is invalidated. Sober (2003) develops a similar argument,v by stressing that no mechanism designed to randomly assign a time position to human beings, can be highlighted. Finally, such an objection was recently revived by Sowers. The latter focused on the fact that the birth rank of every human being is not random because it is indexed to the corresponding time position.

According to the viewpoint developed by Eckhardt et al., the human situation corresponding to DA is not analogous to the two-urn case experiment, but rather to an alternative model, which may be called the consecutive token dispenser. The consecutive token dispenser is a device, originally described by Eckhardtvi, that ejects consecutively numbered balls at regular intervals: “(…) suppose on each trial the consecutive token dispenser expels either 50 (early doom) or 100 (late doom) consecutively numbered tokens at the rate of one per minute”. A similar device – call it the numbered balls dispenser – is also mentioned by Sowers, where the balls are ejected from the urn and numbered in the order of their ejection, at the regular interval of one per minute:vii

There are two urns populated with balls as before, but now the balls are not numbered. Suppose you obtain your sample with the following procedure. You are equipped with a stopwatch and a marker. You first choose one of the urns as your subject. It doesn’t matter which urn is chosen. You start the stopwatch. Each minute you reach into the urn and withdraw a ball. The first ball withdrawn you mark with the number one and set aside. The second ball you mark with the number two. In general, the nth ball withdrawn you mark with the number n. After an arbitrary amount of time has elapsed, you stop the watch and the experiment. In parallel with the original scenario, suppose the last ball withdrawn is marked with a seven. Will there be a probability shift? An examination of the relative likelihoods reveals no.

Thus, under the terms of the viewpoint defended by Eckhardt et al., the human situation corresponding to DA is not analogous with the two-urn case experiment, but with the numbered balls dispenser. And this last model leads us to leave the initial probabilities unchanged.

The incremental objection of Eckhardt et al. is based on a disanalogy. Indeed, the human situation corresponding to DA presents a temporal nature, for the birth ranks are successively attributed to human beings depending on the time position corresponding to their appearance on Earth. Thus, the corresponding situation takes place, for example, from T1 to Tn, where 1 and n are respectively the birth ranks of the first and of the last humans. However, the two-urn case experiment appears atemporal, because when the ball is drawn at random, all the balls are already present within the urn. The two-urn case experiment takes place at a given time T0. It appears thus that the two-urn case experiment is an atemporal model, while the situation corresponding to DA is a temporal model. And this forbids, as Eckhardt et al. underscore, considering the situation corresponding to DA and the two-urn case as isomorphic.viii

At this stage, it appears that the atemporal-temporal disanalogy is indeed a reality and it cannot be denied. However, this does not constitute an insurmountable obstacle for DA. As we shall see, it is possible indeed to put in analogy the human situation corresponding to DA, with a temporal variation of the two-urn case. This can be done by considering the following experiment, which can be termed the incremental two-urn case (formally, the two-urn case++):

The two-urn case++. An opaque urn in front of you. You know that it contains either 10 or 1000 numbered balls. A fair coin has been tossed at time T0 and if the coin landed tails, then the urn contains only 10 balls, while if the coin landed heads, then the urn contains the same 10 balls plus 990 extra balls, i.e. 1000 balls in total. The balls are numbered 1, 2, 3, …. You formulate then the Hfew (the box contains only 10 balls) and Hmany (the box contains 1000 balls) hypotheses with initial probabilities P(Hfew) = P(Hmany) = 1/2. At time T1, a device will draw a ball at random, and will eject then every second a numbered ball in increasing order, from the ball #1 until the number of the randomly drawn ball. At that very time, the device will stop.

You are informed of all the foregoing, and the device expels then the ball #1 at T1, the ball #2 at T2, the ball #3 at T3, the ball #4 at T4, and the ball #5 at T5. The device then stops. You wish to estimate the number of balls that were contained at T0 in the urn. You conclude then to an upward Bayesian shift in favor of the Hfew hypothesis.

As we can see, such a variation constitutes a mere adaptation of the original two-urn case, with the addition of an incremental mechanism for the expulsion of the balls. The novelty with this variationix is that the experience has now a temporal feature, because the random selection is made at T1 and the randomly drawn ball is finally ejected, for example at T5.

At this stage, it is also worth analyzing the consequences of the two-urn case++ for the analysis developed by Eckhardt et al. Indeed, in the two-urn case++, the number of each ball ejected from the device is indexed on the range of its expulsion. For example, I draw the ball #60000000000. But I also know that the previous ball was the ball #59999999999 and that the penultimate ball was the ball #59999999998, and so on. However, this does not prevent me from thinking in the same manner as in the original two-urn case and from concluding to a Bayesian shift in favor of the Hfew hypothesis. In this context, the two-urn case++ experiment leads to the following consequence: the fact of being indexed with regard to time does not mean that the number of the ball is not randomly chosen. This can now be confronted with the main thesis of the incremental objection raised by Eckhardt et al., i.e. that the birth rank of each human being is not randomly chosen, but is rather indexed on the corresponding time position. Sowers especially believes that the cause of DA is that the number corresponding to the birth rank is time-indexed.x But what the two-urn case++ experiment and the corresponding analogy demonstrates is that our birth rank can be time-indexed and nevertheless be determined randomly in the context of DA. For this reason, the numbered balls dispenser model proposed by Eckhardt and Sowers can not be considered as an alternative model to the two-urn case, within the space of solutions for DA.

III. Success of an alternative model grounded on William Eckhardt’s diachronic objection

William Eckhardt (1993, 1997) also describes another objection to DA, which we shall call, for the sake of the present discussion, the diachronic objection. This latter objection, as we shall see it, is based on an alternative model to the two-urn case, which is different from the one that corresponds to the incremental objection. Eckhardt highlights the fact that it is impossible to perform a random selection, when there exists many yet unborn individuals within the corresponding reference class: “How is it possible in the selection of a random rank to give the appropriate weight to unborn members of the population?” (1997, p. 256).

This second objection is potentially stronger than the incremental objection. In order to assess its scope accurately, it is worth translating now this objection in terms of a probabilistic model. It appears that the model associated with Eckhardt’s diachronic objection can be built from the two-urn case’s structure. The corresponding variation, which can be termed the diachronic two-urn case, goes as follows:

The diachronic two-urn case. An opaque urn in front of you. You know that it contains either 10 or 1000 numbered balls. A fair coin has been tossed at time T0. If the coin fell tails, 10 balls were then placed in the urn, while if the coin fell heads, 10 balls were also placed in the urn at time T0, but 990 supplementary balls will be also added to the urn at time T2, bringing up the total number of balls finally contained in the urn to 1000. The balls are numbered 1, 2, 3, …. You then formulate Hfew (the urn finally contains only 10 balls) and Hmany (the urn finally contains1000 balls) hypotheses with the initial probabilities P (Hfew) = P (Hmany) = 1 / 2.

Informed of all the above, you randomly draw at time T1 a ball from the urn. You get then the ball #5. You wish to estimate the number of balls that ultimately will be contained in the urn at T2. You conclude then that the initial probabilities remain unchanged.

At this stage, it appears that the protocol described above does justice to Eckhardt’s strong idea that it is impossible to perform a random selection where there are many yet unborn members in the reference class. In the diachronic two-urn case, the 990 balls, which are possibly (if the coin falls heads) added in T2 account for these members not yet born. In such a situation, it would be quite erroneous to conclude to a Bayesian shift in favor of the Hfew hypothesis. But what can be inferred rationally in such a case is that the prior probabilities remain unchanged.

We can also see that the structure of the protocol of the diachronic two-urn case is quite similar to the original two-urn case experiment (which we shall now term, by contrast, the synchronic two-urn case). This will allow now for making easy comparisons. So we see that if the coin lands tails: the situation is the same in both experiments, synchronic and diachronic. However, the situation is different if the coin lands heads: in the synchronic two-urn case, the 990 balls are already present in the urn at T0; on the other hand, in the model of the diachronic two-urn case, 990 extra balls are added to the urn later, namely at T2. As we can see, the diachronic two-urn case based on Eckhardt’s diachronic objection deserves completely to take a place within the space of solutions for DA.

IV. Construction of the preliminary space of solutions

In light of the foregoing, we are now in a position to appreciate how much the analogy underlying DA is appropriate. It appears indeed that two alternative models to model the analogy with the human situation corresponding to DA are in competition: on the one hand, the synchronic two-urn case advocated by the promoters of DA and, on the other hand, the diachronic two-urn case, based on Eckhardt’s diachronic objection. It turns out that these two models share a common structure, which allows for making comparisons.xi

At this step, the question that arises is the following: is the human situation corresponding to DA in analogy with (i) the synchronic two-urn case, or (ii) the diachronic two-urn case? In response, the next question follows: is there an objective criterion that allows one to choose, preferentially, between the two competing models? It appears not. Indeed, neither Leslie nor Eckhardt do provide objective reasons for justifying the choice of their favorite model, and for rejecting the alternative model. Leslie, first, defends the analogy of the human situation corresponding to DA with the lottery experiment (here, the synchronic two-urn case). At the same time, Leslie acknowledges that DA is considerably weakened if our universe is of an indeterministic nature, i.e. if the total number of people who will ever exist has not yet been settled.xii But it turns out that such indeterministic situation corresponds completely with the diachronic two-urn case. For the protocol of this experiment takes into account the fact that the total number of balls which will ultimately be contained in the urn, is not known at the time when the random drawing is performed. We see it finally, Leslie liberally accepts that the analogy with the synchronic two-urn case may not prevail in certain indeterministic circumstances, where, as we have seen, the diachronic two-urn case would apply.

Otherwise, a weakness in the position defended by Eckhardt is that he rejects the analogy with the lottery experiment (in our terminology, the synchronic two-urn case) in all cases. But how can we be certain that an analogy with the synchronic two-urn case does not prevail, at least for a given situation? It appears here that we lack the evidence allowing us to reject such an hypothesis with absolute certainty.

To sum now. Within the space of solutions for DA resulting from the foregoing, it follows now that two competing models may also be convenient to model the human situation corresponding to DA: Leslie’s synchronic two-urn case or Eckhardt’s diachronic two-urn case. At this stage, however, it appears that no objective criterion allows for preferring one or the other of these two models. In these circumstances, in the lack of objective evidence to make a choice between the two competing models, we are led to apply a principle of indifference, which leads us to retain both models as roughly equiprobable. We attribute then (Figure 1), applying a principle of indifference, a probability P of 1/2 to the analogy with the synchronic two-urn case (associated with a terrifying scenario), and an identical probability of 1/2 to the analogy with the diachronic two-urn case (associated with a reassuring scenario).

CaseModelT0T2PNature of the scenario
1synchronic two-urn case1/2terrifying
2diachronic two-urn case1/2reassuring

Figure 1.

However, it appears that such an approach is of a preliminary nature, for in order to assign a probability to each specific situation inherent in DA, it is necessary to take into account all the elements underlying DA. But it appears that a key element of DA has not yet been taken into account. It is the notoriously awkward reference class problem.

V. The reference class problem

Let us begin by recalling the reference class problem.xiii Basically, it is the problem of the correct definition of “humans”. More accurately, the problem can be stated as follows: how can the reference class be objectively defined in the context of DA? For a more or less extensive or restrictive definition of the reference class can be used. An extensively defined reference class would include, for example, the somewhat exotic varieties corresponding to a future evolution of humankind, with for example an average IQ equal to 200, a double brain or backward causation abilities. On the other hand, a restrictively defined reference class would only include those humans whose characteristics are exactly those of – for example – our subspecies Homo sapiens sapiens. Such a definition would exclude the extinct species such as Homo sapiens neandertalensis, as well as a possible future subspecies such as Homo sapiens supersapiens. To put this in line with our current taxonomy, the reference class can be set at different levels, which correspond to the Superhomo super-genus, the Homo genus, the Homo sapiens species, the Homo sapiens sapiens subspecies, etc. At this stage, it appears that we lack an objective criterion allowing to choose the corresponding level non-arbitrarily.

The solution to the reference class problem proposed by Leslie’s, which is exposed in the response made to Eckhardt (1993) and in The End of the World (1996), goes as follows: one can choose the reference class more or less as one wishes, i.e. at any level of extension or of restriction. Once this choice has been made, it suffices to adjust accordingly the initial probabilities, and DA works again. The only reservation mentioned by Leslie is that the reference class should not be chosen at an extreme level of extension or restriction.xiv According to Leslie, the fact that every human being can belong to different classes, depending on whether they are restrictively or extensively defined, is not a problem, because the argument works for each of those classes. In this case, says Leslie, a Bayesian shift follows for whatever class reference, chosen at a reasonable level of extension or of restriction. And Leslie illustrates this point of view by an analogy with a multi-color urn, unlike the one-color urn of the original two-urn case experiment. He considers an urn containing balls of different colors, for example red and green. A red ball is drawn at random from the urn. From a restrictive viewpoint, the ball is a red ball and then there is no difference with the two-urn case. But from a more extensive viewpoint, the ball is also a red-or-green ball.xv According to Leslie, although the initial probabilities are different in each case, a Bayesian shift results in both cases.xvi As we can see, the synchronic two-urn case can be easily adapted to restore the essence of Leslie’s multi-color model. It suffices in effect to replace the red balls of the original synchronic two-urn case with red-or-green balls. The resulting two-color model is then in all respects identical to the original synchronic two-urn case experiment, and leads to a Bayesian shift of the same nature.

At this stage, in order to incorporate properly the reference class problem into the space of solutions for DA, we still need to translate the diachronic two-urn case into a two-color variation.

A. The two-color diachronic two-urn case

In the one-color original experiment which corresponds to the diachronic two-urn case, the reference class is that of the red balls. It appears here that one can construct a two-color variation, which is best suited for handling the reference class problem, where the relevant class is that of red-or-green balls. The corresponding two-color variation is in all respects identical with the original diachronic two-urn case, the only difference being that the first 10 balls (#1 to #10) are red and the other 990 balls (#11 to #1000) are green. The corresponding variation runs as follows:

The two-color diachronic two-urn case. An opaque urn in front of you. You know it contains either 10 or 1000 numbered balls (consisting of 10 red balls and 990 green balls). The red balls are numbered #1, #2, …, #9, #10 and the green ones #11, #12, .., #999, #1000. A fair coin has been tossed at time T0. If the room fell tails, 10 balls were then placed in the urn, while if the coin fell heads, 10 red balls were also placed in the urn at time T0, but 990 green balls will be then added to the urn at time T2, bringing thus the total number of balls in the urn to 1000. You formulate then the hypotheses Hfew (the urn contains finally only 10 red-or-green balls) and Hmany (the box finally contains 1000 red-or-green balls) with the prior probabilities P(Hfew) = P(Hmany) = 1/2.

After being informed of all the above, you draw at time T1 a ball at random from the urn. You get the red ball #5. You proceed to estimate the number of red-or-green balls which will ultimately be contained in the urn at T2. You conclude that the initial probabilities remain unchanged.

As we can see, the structure of this two-color variation is in all respects similar to that of the one-color version of the diachronic two-urn case. In effect, we can considered here the class of red-or-green balls, instead of the original class of red balls. And in this type of situation, it is rational to conclude in the same manner as in the original one-color version of the diachronic two-urn case experiment that the prior probabilities remain unchanged.

B. Non-exclusivity of the synchronic one-color model and of the diachronic two-color model

With the help of the machinery at hand to tackle the reference class problem, we are now in a position to complete the construction of the space of solutions for DA, by incorporating the above elements. On a preliminary basis, we have assigned a probability of 1/2 to each of the one-color two-urn case – synchronic and diachronic – models, by associating them respectively with a terrifying and a reassuring scenario. But what is the situation now, with the presence of two-color models, which are better suited for handling the reference class problem?

Before evaluating the impact of the two-color model on the space of solutions for DA, it is worth defining first how to proceed in putting the two-color models and our present human situation into correspondence. For this, it suffices to assimilate the class of red balls to our current subspecies Homo sapiens sapiens and the class of red-or-green balls to our current species Homo sapiens. Similarly, we shall assimilate the class of green balls to the subspecies Homo sapiens supersapiens, a subspecies more advanced than our own, which is an evolutionary descendant of Homo sapiens sapiens. A situation of this type is very common in the evolutionary process that governs living species. Given these elements, we are now in a position to establish the relationship of the probabilistic models with our present situation.

At this stage it is worth pointing out an important property of the two-color diachronic model. It appears indeed that the latter model is susceptible of being combined with a one-color synchronic two-urn case. Suppose, then, that a one-color synchronic two-urn case prevails: 10 balls or 1000 red balls are placed in the urn at time T0. But this does not preclude green balls from being also added in the urn at time T2. It appears thus that the one-color synchronic model and the diachronic two-color model are not exclusive of one another. For in such a situation, a synchronic one-color two-urn case prevails for the restricted class of red balls, whereas a diachronic two-color model applies to the extended class of red-or-green balls. At this step, it appears that we are on a third route, of pluralistic essence. For the fact of matching the human situation corresponding to DA with the synchronic or the (exclusively) diachronic model, are well monist attitudes. In contrast, the recognition of the joint role played by both synchronic and diachronic models, is the expression of a pluralistic point of view. In these circumstances, it is necessary to analyze the impact on the space of solutions for DA of this property of non-exclusivity which has just been emphasized.

In light of the foregoing, it appears that four types of situations must now be distinguished, within the space of solutions for DA. Indeed, each of the two initial one-color models – synchronic and diachronic – can be associated with a two-color diachronic two-urn case. Let us begin with the case (1) where the synchronic one-color model applies. In this case, one should distinguish between two types of situations: either (1a) nothing happens at T2 and no green ball is added to the urn at T2, or (1b) 990 green balls are added in the urn at T2. In the first case (1a) where no green ball is added to the urn at T2, we have a rapid disappearance of the class of red balls. Similarly, we have a disappearance of the corresponding class of red-or-green balls, since it identifies itself here with the class of red balls. In such a case, the rapid extinction of Homo sapiens sapiens (the red balls) is not followed by the emergence of Homo sapiens supersapiens (the green balls). In such a case, we observe the rapid extinction of the sub-species Homo sapiens sapiens and the correlative extinction of the species Homo sapiens (the red-or-green balls). Such a scenario, admittedly, corresponds to a form of Doomsday that presents a very frightening nature.

Let us consider now the second case (1b), where we are always in the presence of a synchronic one-color model, but where now green balls are also added in the urn at T2. In this case, 990 green balls are added at T2 to the red balls originally placed in the urn at T0. We have then a rapid disappearance of the class of red balls, which accompanies, however, the survival of the class of red-or-green balls given the presence of green balls at T2. In this case (1b), one notices that a synchronic one-color model is combined with a diachronic two-color model. Both models prove to be compatible, and non-exclusive of one another. If we translate this in terms of the third route, one notices that, according to the pluralistic essence of the latter, the synchronic one-color model applies to the class, narrowly defined, of red balls, while a two-color diachronic model also applies to the class, broadly defined, of red-or-green balls. In this case (1b), the rapid extinction of Homo sapiens sapiens (the red balls) is followed by the emergence of the most advanced human subspecies Homo sapiens supersapiens (the green balls). In such a situation, the restricted class Homo sapiens sapiens goes extinct, while the more extended class Homo sapiens (red-or-green balls) survives. While the synchronic one-color model applies to the restricted class Homo sapiens sapiens, the diachronic two-color model prevails for the wider class Homo sapiens. But such an ambivalent feature has the effect of depriving the original argument of the terror which is initially associated with the one-color synchronic model. And finally, this has the effect of rendering DA innocuous, by depriving it of its originally associated terror. At the same time, this leaves room for the argument to apply to a given class reference, but without its frightening and counter-intuitive consequences .

As we can see, in case (1), the corresponding treatment of the reference class problem is different from that advocated by Leslie. For on Leslie’s view, the synchronic model applies irrespective of the chosen reference class. But the present analysis leads to a differential treatment of the reference class problem. In case (1a), the synchronic model prevails and a Bayesian shift applies, as well as in Leslie’s account, both to the class of red balls and to the class of red-or-green balls. But in case (1b), the situation goes differently. Because if a one-color synchronic model applies to the restricted reference class of red balls and leads to a Bayesian shift, it appears that a diachronic two-color model applies to the extended reference class of red-or-green balls, leaving the initial probability unchanged. In case (1b), as we can see, the third route leads to a pluralistic treatment of the reference class problem.

Let us consider now the second hypothesis (2) where the diachronic one-color model prevails. In this case, 10 red balls are placed in the urn at T0, and 990 other red balls are added to the urn at T2. Just as before, we are led to distinguish two situations. Either (2a) no green ball is added to the urn at T2, or (2b) 990 green balls are also added to the urn at T2. In the first case (2a), the diachronic one-color model applies. In such a situation (2a), no appearance of a much-evolved human subspecies such as Homo sapiens supersapiens occurs. But the scenario in this case is also very reassuring, since our current subspecies Homo sapiens sapiens survives. In the second case (2b), where 990 green balls are added to the urn at T2, a diachronic two-color model adds up to the initial diachronic one-color model. In such a case (2b), it follows the emergence of the most advanced subspecies Homo sapiens supersapiens. In this case, the scenario is doubly reassuring, since it leads both to the survival of Homo sapiens sapiens and of Homo sapiens supersapiens. As we can see, in case (2), it is the diachronic model which remains the basic model, leaving the prior probability unchanged.

At this step, we are in a position to complete the construction of the space of solutions for DA. Indeed, a new application of a principle of indifference leads us here to assign a probability of 1/4 to each of the 4 sub-cases: (1a), (1b), (2a), (2b). The latter are represented in the figure below:

CaseT0T2P
11a1/4
1b1/4
22a1/4
2b● ○1/4

Figure 2.

It suffices now to determine the nature of the scenario that is associated with each of the four sub-cases just described. As has been discussed above, a worrying scenario is associated with hypothesis (1a), while a reassuring scenario is associated with the hypotheses (1b), (2a) and (2b):

CaseT0T2PNature of the scenarioP
11a1/4terrifying1/4
1b1/4reassuring
22a1/4reassuring3/4
2b● ○1/4reassuring

Figure 3.

We see it finally, the foregoing considerations lead to a novel formulation of DA. For it follows from the foregoing that the original scope of DA should be reduced, in two different directions. It should be acknowledged, first, that either the one-color synchronic model or the diachronic one-color model applies to our current subspecies Homo sapiens sapiens. A principle of indifference leads us then to assign a probability of 1/2 to each of these two hypotheses. The result is a weakening of DA, as the Bayesian shift associated with a terrifying assumption no longer concerns but one scenario of the two possible scenarios. A second weakening of DA results from the pluralist treatment of the reference class problem. For in the case where the one-color synchronic model (1) applies to our subspecies Homo sapiens sapiens, two different situations must be distinguished. Only one of them, (1a) leads to the extinction of both Homo sapiens sapiens and Homo sapiens and corresponds thus to a frightening Doomsday. In contrast, the other situation (1b) leads to the demise of Homo sapiens sapiens, but to the correlative survival of the most advanced human subspecies Homo sapiens supersapiens, and constitutes then a quite reassuring scenario. At this stage, a second application of the principle of indifference leads us to assign a probability of 1/2 to each of these two sub-cases (see Figure 3). In total, a frightening scenario is henceforth associated with a probability of no more than 1/4, while a reassuring scenario is associated with a probability of 3/4.

As we can see, given these two sidesteps, a new formulation of DA ensues, which could prove to be more plausible than the original one. Indeed, the present formulation of DA can now be reconciled with our pretheoretical intuition. For the fact of taking into account DA now gives a probability of 3/4 for all reassuring scenarios and a probability of no more than 1/4 for a scenario associated with a frightening Doomsday. Of course, we have not completely eliminated the risk of a frightening Doomsday. And we must, at this stage, accept a certain risk, the scope of which appears however limited. But most importantly, it is no longer necessary now to give up our pretheoretical intuitions.

Finally, the preceding highlights a key facet of DA. For in a narrow sense, it is an argument related to the destiny of humankind. And in a broader sense (the one we have been concerned with so far) it emphasizes the difficulty of applying probabilistic models to everyday situations,xvii a difficulty which is often largely underestimated. This opens the path to a wide field which presents a real practical interest, consisting of a taxonomy of probabilistic models, the philosophical importance of which would have remained hidden without the strong and courageous defense of the Doomsday argument made by John Leslie.xviii

References

Bostrom, Nick. 1997. “Investigations into the Doomsday argument.” preprint.

———. 2002. Anthropic Bias: Observation Selection Effects in Science and Philosophy New York: Routledge.

Chambers, Timothy. 2001. “Do Doomsday’s Proponents Think We Were Born Yesterday?” Philosophy 76: 443-450.

Delahaye, Jean-Paul. 1996. “Recherche de modèles pour l’argument de l’apocalypse de Carter-Leslie.” manuscrit.

Eckhardt, William. 1993. “Probability Theory and the Doomsday Argument.” Mind 102: 483-488.

———. 1997. “A Shooting-Room view of Doomsday.” Journal of Philosophy 94: 244-259.

Franceschi, Paul. 1998. “Une solution pour l’argument de l’Apocalypse.” Canadian Journal of Philosophy 28: 227-246.

———. 1999. “Comment l’urne de Carter et Leslie se déverse dans celle de Hempel.” Canadian Journal of Philosophy 29: 139-156, English translation under the title “The Doomsday Argument and Hempel’s Problem” .

———. 2002. “Une application des n-univers à l’argument de l’Apocalypse et au paradoxe de Goodman.” Corté: University of Corsica, doctoral dissertation.

Hájek, Alan. 2002. “Interpretations of Probability.” The Stanford Encyclopedia of Philosophy, E. N. Zalta (ed.), http://plato.stanford.edu/archives/win2002/entries/probability-interpret.

Korb, Kevin. & Oliver, Jonathan. 1998. “A Refutation of the Doomsday Argument.” Mind 107: 403-410.

Leslie, John. 1993. “Doom and Probabilities.” Mind 102: 489-491.

———. 1996. The End of the World: the science and ethics of human extinction London: Routledge.

Sober, Eliott. 2003.An Empirical Critique of Two Versions of the Doomsday Argument – Gott’s Line and Leslie’s Wedge.” Synthese 135-3: 415-430.

Sowers, George. 2002. “The Demise of the Doomsday Argument.” Mind 111: 37-45.

i The present analysis of DA is an extension of Franceschi (2002).

ii Cf. Korb & Oliver (1998).

iii The original description by Bostrom of the two-urn case refers to two separate urns. For the sake of simplicity, we shall refer here equivalently to one single urn (which contains either 10 or 1000 balls).

iv More accurately, Leslie considers an analogy with a lottery experiment.

v Cf (2003: 9): “But who or what has the propensity to randomly assign me a temporal location in the duration of the human race? There is no such mechanism.” But Sober is mainly concerned with providing evidence with regard to the assumptions used in the original version of DA and with broadening the scope of the argument by determining the conditions of its application to real-life situations.

vi Cf. (1997: 251).

vii Cf. (2002: 39).

viii I borrow this terminology from Chambers (2001).

ix Other variations of the two-urn case++ can even be envisaged. In particular, variations of this experiment where the random process is performed diachronically and not synchronically (i.e. at time T0) can even be imagined.

x Cf. Sowers (2002: 40).

xi Both synchronic and diachronic two-urn case experiments can give rise to an incremental variation. The incremental variant of the (synchronic) two-urn case has been mentioned earlier: it consists of the two-urn case++. It is also possible to build a similar incremental variation of the diachronic two-urn case, where the ejection of the balls is made at regular time intervals. At this stage it appears that both models can give rise to such incremental variations. Thus, the fact of considering incremental variations of the two competing models – the synchronic two-urn case++ and the diachronic two-urn case++ – does not provide any novel elements with regard to the two original experiments. Similarly, we might consider some variations where the random sampling is done not at T0, but gradually, or some variants where a quantum coin is used, and so on. But in any case, such variations are susceptible to be adapted to each of the two models.

xii Leslie (1993: 490) evokes thus: “(…) the potentially much stronger objection that the number of names in the doomsday argument’s imaginary urn, the number of all humans who will ever have lived, has not yet been firmly settled because the world is indeterministic”.

xiii The reference class problem in probability theory is notably mentioned in Hájek (2002: s. 3.3). For a treatment of the reference class problem in the context of DA, see Eckhardt (1993, 1997), Bostrom (1997, 2002: ch. 4 pp. 69-72 & ch. 5), Franceschi (1998, 1999). The point emphasized in Franceschi (1999) can be construed as a treatment of the reference class problem within confirmation theory.

xiv Cf. 1996: 260-261.

xv Cf. Leslie (1996: 259).

xvi Cf. Leslie (1996: 258-259): “The thing to note is that the red ball can be treated either just as a red ball or else as a red-or-green ball. Bayes’s Rule applies in both cases. […] All this evidently continues to apply to when being-red-or-green is replaced by being-red-or-pink, or being-red-or-reddish”.

xvii This important aspect of the argument is also underlined in Delahaye (1996). It is also the main theme of Sober (2003).

xviii I thank Nick Bostrom for useful discussion on the reference class problem, and Daniel Andler, Jean-Paul Delahaye, John Leslie, Claude Panaccio, Elliott Sober, and an anonymous referee for the Journal of Philosophical Research, for helpful comments on earlier drafts.

A Solution to Goodman’s Paradox

English Posprint (with additional illustrations) of a paper published in French in Dialogue Vol. 40, Winter 2001, pp. 99-123 under the title “Une Solution pour le Paradoxe de Goodman”.
In the classical version of Goodman’s paradox, the universe where the problem takes place is ambiguous. The conditions of induction being accurately described, I define then a framework of n-universes, allowing the distinction, among the criteria of a given n-universe, between constants and variables. Within this framework, I distinguish between two versions of the problem, respectively taking place: (i) in an n-universe the variables of which are colour and time; (ii) in an n-universe the variables of which are colour, time and space. Finally, I show that each of these versions admits a specific resolution.


This paper is cited in:

  • Alasdair Richmond, The Doomsday Argument, Philosophical Books Vol. 47 No. 2 April 2006, pp. 129–142

A Solution to Goodman’s Paradox

Paul FRANCESCHI

a paper originally published in Dialogue, winter 2001, vol. 40, pp. 99-123

ABSTRACT: In the classical version of Goodman’s paradox, the universe where the problem takes place is ambiguous. The conditions of induction being accurately described, I define then a framework of n-universes, allowing the distinction, among the criteria of a given n-universe, between constants and variables. Within this framework, I distinguish between two versions of the problem, respectively taking place: (i) in an n-universe the variables of which are colour and time; (ii) in an n-universe the variables of which are colour, time and space. Finally, I show that each of these versions admits a specific resolution.

1. The problem

Nelson Goodman

Goodman’s Paradox (thereafter GP) has been described by Nelson Goodman (1946).i Goodman exposes his paradox as follows.ii Consider an urn containing 100 balls. A ball is drawn each day from the urn, during 99 days, until today. At each time, the ball extracted from the urn is red. Intuitively, one expects that the 100th ball drawn from the urn will also be red. This prediction is based on the generalisation according to which all the balls in the urn are red. However, if one considers the property S “drawn before today and red or drawn after today and non-red”, one notes that this property is also satisfied by the 99 instances already observed. But the prediction which now ensue, based on the generalisation according to which all the balls are S, is that the 100th ball will be non-red. And this contradicts the preceding conclusion, which however conforms with our intuition.iii

Goodman expresses GP with the help of an enumerative induction. And one can model GP in terms of the straight rule (SR). If one takes (D) for the definition of the “red” predicate, (I) for the enumeration of the instances, (H) for the ensuing generalisation, and (P) for the corresponding prediction, one has then:

(D) R = red

(I) Rb1·Rb2·Rb3·…·Rb99

(H) Rb1·Rb2·Rb3·…·Rb99·Rb100

∴ (P) Rb100

And also, with the predicate S:

(D*) S = red and drawn before T or non-red and drawn after T

(I*) Sb1·Sb2·Sb3·…·Sb99

(H*) Sb1·Sb2·Sb3·…·Sb99·Sb100 that is equivalent to:

(H’*) Rb1·Rb2·Rb3·…·Rb99·~Rb100

∴ (P*) Sb100 i. e. finally:

∴ (P’*) ~Rb100

The paradox resides here in the fact that the two generalisations (H) and (H*) lead respectively to the predictions (P) and (P’*), which are contradictory. Intuitively, the application of SR to (H*) appears erroneous. Goodman also gives in Fact, Fiction and Forecastiv a slightly different version of the paradox, applied in this case to emeralds.v This form is very well known and based on the predicate “grue” = green and observed before T or non-green and observed after T.

The predicate S used in Goodman (1946) presents with “grue”, a common structure. P and Q being two predicates, this structure corresponds to the following definition: (P and Q) or (~P and ~Q). In what follows, one will designate by grue a predicate having this particular structure, without distinguishing whether the specific form used is that of Goodman (1946) or (1954).

2. The unification/differentiation duality

The instances are in front of me. Must I describe them by stressing their differences? Or must I describe them by emphasising their common properties? I can proceed either way. To stress the differences between the instances, is to operate by differentiation. Conversely, to highlight their common properties, is to proceed by unification. Let us consider in turn each of these two modes of proceeding.

Consider the 100 balls composing the urn of Goodman (1946). Consider first the case where my intention is to stress the differences between the instances. There, an option is to apprehend the particular and single moment, where each of them is extracted from the urn. The considered predicates are then: red and drawn on day 1, red and drawn on day 2, …, red and drawn on day 99. There are thus 99 different predicates. But this prohibits applying SR, which requires one single predicate. Thus, what is to distinguish according to the moment when each ball is drawn? It is to stress an essential difference between each ball, based on the criterion of time. Each ball thus is individualised, and many different predicates are resulting from this: drawn at T1, drawn at T2, …, drawn at T99. This indeed prevents then any inductive move by application of SR. In effect, one does not have then a common property to allow induction and to apply SR. Here, the cause of the problem lies in the fact of having carried out an extreme differentiation.

Alternatively, I can also proceed by differentiation by operating an extremely precisevi measurement of the wavelength of the light defining the colour of each ball. I will then obtain a unique measure of the wavelength for each ball of the urn. Thus, I have 100 balls in front of me, and I know with precision the wavelength of the light of 99 of them. The balls respectively have a wavelength of 722,3551 nm, 722,3643 nm, 722,3342 nm, 722,3781 nm, etc. I have consequently 99 distinct predicates P3551, P3643, P3342, P3781, etc. But I have no possibility then to apply SR, which requires one single predicate. Here also, the common properties are missing to allow to implement the inductive process. In the same way as previously, it proves here that I have carried out an extreme differentiation.

What does it occur now if I proceed exclusively by unification? Let us consider the predicate R corresponding to “red or non-red”. One draws 99 red balls before time T. They are all R. One predicts then that the 100th ball will be R after T, i.e. red or non-red. But this form of induction does not bring any information here. The resulting conclusion is empty of information. One will call empty induction this type of situation. In this case, one observes that the process of unification of the instances by the colour was carried out in a radical way, by annihilating in this respect, any step of differentiation. The cause of the problem lies thus in the implementation of a process of extreme unification.

If one considers now the viewpoint of colour, it appears that each case previously considered requires a different taxonomy of colours. Thus, it is made use successively:

– of our usual taxonomy of colours based on 9 predicates: purple, indigo, blue, green, yellow, orange, red, white, black

– of a taxonomy based on a comparison of the wavelengths of the colours with the set of the real numbers (real taxonomy)

– of a taxonomy based on a single predicate (single taxon taxonomy): red or non-red

But it proves that each of these three cases can be replaced in a more general perspective. Indeed, multiple taxonomies of colours are susceptible to be used. And those can be ordered from the coarser (single taxon taxonomy) to the finest (real taxonomy), from the most unified to the most differentiated. We have in particular the following hierarchy of taxonomies:

– TAX1 = {red or non-red} (single taxon taxonomy)

– TAX2 = {red, non-red} (binary taxonomy)

– …

– TAX9 = {purple, indigo, blue, green, yellow, orange, red, white, black} (taxonomy based on the spectral colours, plus white and black)

– …

– TAX16777216 = {(0, 0, 0), …, (255, 255, 255)} (taxonomy used in computer science and distinguishing 256 shades of red/green/blue)

– …

– TAXR = {370, …, 750} (real taxonomy based on the wavelength of the light)

Within this hierarchy, it appears that the use of extreme taxonomies such as the one based on a single taxon, or the real taxonomy, leads to specific problems (respectively extreme unification and extreme differentiation). Thus, the problems mentioned above during the application of an inductive reasoning based on SR occur when the choice in the unification/differentiation duality is carried out too radically. Such problems relate to induction in general. This invites to think that one must rather reason as follows: I should privilege neither unification, nor differentiation. A predicate such as “red”, associated with our usual taxonomy of colours (TAX9)vii, corresponds precisely to such a criterion. It corresponds to a balanced choice in the unification/differentiation duality. This makes it possible to avoid the preceding problems. This does not prevent however the emergence of new problems, since one tries to implement an inductive reasoning, in certain situations. And one of these problems is naturally GP.

Thus, it appears that the stake of the choice in the duality unification/differentiation is essential from the viewpoint of induction, because according to whether I choose one way or the other, I will be able or not to use SR and produce valid inductive inferences. Confronted with several instances, one can implement either a process of differentiation, or a process of unification. But the choice that is made largely conditions the later success of the inductive reasoning carried out on those grounds. I must describe both common properties and differences. From there, a valid inductive reasoning can take place. But at this point, it appears that the role of the unification/differentiation duality proves to be crucial for induction. More precisely, it appears at this stage that a correct choice in the unification/differentiation duality constitutes one of the conditions of induction.

3. Several problems concerning induction

The problems which have been just mentioned constitute the illustration of several difficulties inherent to the implementation of the inductive process. However, unlike GP, these problems do not generate a genuine contradiction. From this point of view, they distinguish from GP. Consider now the following situation. I have drawn 99 balls respectively at times T1, T2, …, T99. The 100th ball will be drawn at T100. One observes that the 99 drawn balls are red. They are thus at the same time red and drawn before T100. Let R be the predicate “red” and T the predicate “drawn before T100“. One has then:

(I) RTb1, RTb2, …, RTb99

(H) RTb1, RTb2, …, RTb99, RTb100

∴ (P) RTb100

By direct application of SR, the following prediction ensue: “the 100th ball is red and drawn before T100“. But this is in contradiction with the data of the experiment in virtue of which the 100th ball is drawn in T100. There too, the inductive reasoning is based on a formalisation which is that of SR. And just as for GP, SR leads here to a contradiction. Call 2 this problem, where two predicates are used.

It appears that one can easily build a form of 2 based on one single predicate. A way of doing that is to consider the unique predicate S defined as “red and drawn before T100” in replacement of the predicates R and T used previously. The same contradiction then ensues.

Moreover, it appears that one can highlight another version (1) of this problem comprising only one predicate, without using the “red” property which appears useless here. Let indeed T be the predicate drawn before T100. One has then:

(I) Tb1, Tb2, …, Tb99

(H) Tb1, Tb2, …, Tb99, Tb100

∴ (P) Tb100

Here also, the conclusion according to which the 100th ball is drawn before T100 contradicts the data of the experiment according to which the 100th ball is drawn at T100. And one has then a contradictory effect, analogous to that of GP, without the structure of “grue” being implemented. Taking into account the fact that only the criterion of time is used to build this problem, it will be denoted in what follows by 1-time.

It appears here that the problems such as 1-time and 2 lead just as GP to a contradiction. Such is not the case for the other problems related to induction previously mentionedviii, which involve either the impossibility of carrying out induction, or a conclusion empty of information. However, it proves that the contradiction encountered in 1-time is not of the same nature as that observed in GP. Indeed in GP, one has a contradiction between the two concurrent predictions (P) and (P*). On the other hand, in 1-time, the contradiction emerges between on the one hand the conditions of the experiment (T  100) and on the other hand the prediction resulting from generalisation (T < 100).

Anyway, the problems which have been just encountered suggest that the SR formalism does not capture the whole of our intuitions related to induction. Hence, it is worth attempting to define accurately the conditions of induction, and adapting consequently the relevant formalism. However, before carrying out such an analysis, it is necessary to specify in more detail the various elements of the context of GP.

4. The universe of reference

Let us consider the law (L1) according to which “diamond scratches the other solids”. A priori, (L1) strikes us as an undeniable truth. Nevertheless, it proves that at a temperature higher than 3550°C, diamond melts. Therefore in last analysis, the law (L1) is satisfied at a normal temperature and in any case, when the temperature is lower than 3550°C. But such a law does not apply beyond 3550°C. This illustrates how the statement of the conditions under which the law (L1) is verified is important, in particular with regard to the conditions of temperature. Thus, when one states (L1), it proves necessary to specify the conditions of temperature in which (L1) finds to apply. This is tantamount to describing the type of universe in which the law is satisfied.

Let also (P1) be the following proposition: “the volume of the visible universe is higher than 1000 times that of the solar system”. Such a proposition strikes us as obvious. But there too, it appears that (P1) is satisfied at modern time, but that it proves to be false at the first moments of the universe. Indeed, when the age of our universe was 10-6 second after the big-bang, its volume was approximately equal to that of our solar system. Here also, it thus appears necessary to specify, at the same time as the proposition (P1) the conditions of the universe in which it applies. A nonambiguous formulation of (P1) thus comprises a more restrictive temporal clause, such as: “at our time, the volume of the visible universe is higher than 1000 times that of the solar system”. Thus, generally, one can think that when a generalisation is stated, it is necessary to specify the conditions of the universe in which this generalisation applies. The precise description of the universe of reference is fundamental, because according to the conditions of the universe in which one places oneself, the stated law can appear true or false.

One observes in our universe the presence of both constants and variables. There are thus constants, which constitute the fundamental constants of the universe: the speed of light: c = 2,998 x108 m/s; Planck’s constant: h = 6,626 x 10-34 J.s; the electron charge; e = 1,602 x 10-19 C; etc. There are on the other hand variables. Among those, one can mention in particular: temperature, pressure, altitude, localisation, time, presence of a laser radiation, presence of atoms of titanium, etc.

One often tends, when a generalisation is stated, not to take into account the constants and the variables which are those of our universe envisaged in its totality. Such is the case for example when one considers the situation of our universe on 1 January 2000, at 0h. One places then oneself explicitly in what constitutes a section, a slice of our universe. In effect, time is not regarded then a variable, but well as a constant. Consider also the following: “the dinosaurs had hot blood”ix. Here, one places oneself explicitly in a sub-universe of our where the parameters of time and space have a restricted scope. The temporal variable is reduced to the particular time of the Earth history which knew the appearance of the dinosaurs: the Triassic and the Cretaceous. And similarly, the space parameter is limited to our planet: Earth. Identically, the conditions of temperature are changing within our universe, according to whether one is located at one site or another of it: at the terrestrial equator, the surface of Pluto, the heart of Alpha Centauri, etc. But if one is interested exclusively in the balloon being used for the experimentation within the laboratory of physics, where the temperature is maintained invariably at 12°C, one can then regard valuably the temperature as a constant. For when such generalisations are expressed, one places oneself not in our universe under consideration in his totality, but only in what veritably constitutes a specific part, a restriction of it. One can then assimilate the universe of reference in which one places oneself as a sub-universe of our. It is thus frequent to express generalisations which are only worth for the present time, or for our usual terrestrial conditions. Explicitly or not, the statement of a law comprises a universe of reference. But in the majority of the cases, the variables and the constants of the considered sub-universe are distinct from those allowing to describe our universe in its totality. For the conditions are extremely varied within our universe: the conditions are very different according to whether one places oneself at the 1st second after the big-bang, on Earth at the Precambrian epoch, in our planet in year 2000, inside the particle accelerator of the CERN, in the heart of our Sun, near a white dwarf, or well inside a black hole, etc.

One can also think that it is interesting to be able to model universes the constants of which are different from the fundamental constants of our universe. One can thus wish to study for example a universe where the mass of the electron is equal to 9,325 x10-31 kg, or well a universe where the electron charge is equal to 1,598 x 10-19 C. And in fact, the toy-universes, which take into account fundamental constants different from those of our familiar universe, are studied by the astrophysicists.

Lastly, when one describes the conditions of a thought experiment, one places oneself, explicitly or not, under the conditions which are related to those of a sub-universe. When one considers for example 100 balls extracted from an urn during 100 consecutive days, one places then oneself in a restriction of our universe where the temporal variable is limited to one period of 100 days and where the spatial location is extremely reduced, corresponding for example to a volume approximately equal to 5 dm3. On the other hand, the number of titanium or zirconium atoms possibly present in the urn, the possible existence of a laser radiation, the presence or the absence of a sound source of 10 db, etc. can be omitted and ignored. In this context, it is not necessary to take into account the existence of such variables. In this situation, it is enough to mention the variables and the constants actually used in the thought experiment. For one can think indeed that the number of variables in our universe is so large that it is impossible to enumerate them all. And consequently, it does not appear possible to characterise our universe in function of all its variables, because one can not provide an infinite enumeration of it. It appears sufficient to describe the considered sub-universe, by mentioning only the constants and the variables which play an effective role in the experiment. Thus, in such situations, one will describe the considered sub-universe by mentioning only the effective criteria necessary to the description of the experiment.

What precedes encourages to think that generally, in order to model the context in which the problems such as GP take place, it is convenient to describe a given universe in terms of variables and constants. This leads thus to define a n-universe (n 0) as a universe the criteria of which comprise m constants, and n variables, where the m constants and n variables constitute the criteria of the given universe. Within this particular framework, one defines a temporal 1-universe1T) as a universe comprising only one criterion-variable: time. In the same way, one defines a coloured 1-universe1C) as a universe comprising only one criterion-variable: colour. One will define also a coloured and temporal 2-universe2CT) as a universe comprising two criterion-variables: time and colour. Etc. In the same way, a universe where all the objects are red, but are characterised by a different localisation will be modelled by a localised 1-universe1L) a criterion-constant (red) of which is colour.

It should be noted incidentally that the n-universe framework makes it possible in particular to model several interesting situations. Thus, a temporal universe can be regarded as a n-universe one of the variables of which is a temporal criterion. Moreover, a universe where one single moment T0 is considered, deprived of the phenomenon of succession of time, can be regarded as a n-universe where time does not constitute one of the variables, but where there is a constant-time. In the same way, an atemporal universe corresponds to a n-universe no variable of which corresponds to a temporal criterion, and where there is not any time-constant.

In the context which has been just defined, what is it now to be red? Here, being “red” corresponds to two different types of situations, according to the type of n-universe in which one places oneself. It can be on the one hand a n-universe one of the constants of which is colour. In this type of universe, the colour of the objects is not susceptible to change, and all the objects are there invariably red.

The fact of being “red” can correspond, on the second hand, to a n-universe one of the criterion-variables of which is constituted by colour. There, an object can be red or non-red. Consider the case of a Ω1C. In such a universe, an object is red or non-red in the absolute. No change of colour is possible there, because no other criterion-variable exists, of which can depend such a variation. And in a Ω2CT, being red is being red at time T. Within such a universe, being red is being red relatively to time T. Similarly, in a coloured, temporal and localised 3-universe (Ω3CTL), being red is being red at time T and at place L. Etc. In some such universe, being red is being red relatively to other criterion-variables. And the same applies to the n-universes which model a universe such as our own.

At this step arises the problem of the status of the instances of an object of a given type. What is it thus to be an instance, within this framework? This problem has its importance, because the original versions of GP are based on instances of balls (1946) and emeralds (1954). If one takes into account the case of Goodman (1946), the considered instances are 100 different balls. However, if one considers a unique ball, drawn at times T1, T2, …, T100, one notices that the problem inherent to GP is always present. It suffices indeed to consider a ball whose colour is susceptible to change during the course of time. One has drawn 99 times the ball at times T1, T2, …, T99, and one has noted each time that the ball was red. This leads to the prediction that the ball will be red at T100. However, this last prediction proves to be contradictory with an alternative prediction based on the same observations, and the projection of the predicate S “red and drawn before T100 or non-red and drawn at T100x.

The present framework must be capable of handling the diversity of these situations. Can one thus speak of an instantiated and temporal 1-universe, or well of an instantiated and coloured 1-universe? Here, one must observe that the fact of being instantiated, for a given universe, corresponds to an additional criterion-variable. For, on the contrary, what makes it possible to distinguish between the instances? If no criterion distinguishes them, it is thus only one and the same thing. And if they are distinct, it is thus that a criterion makes it possible to differentiate them. Thus, an instantiated and temporal 1-universe is in fact a 2-universe, whose 2nd criterion, which makes it possible to distinguish the instances between them, is in fact not mentioned nor explicited. By making explicit this second criterion-variable, it is thus clear that one is placed in a 2-universe. In the same way, an instantiated and coloured 1-universe is actually a 2-universe one of the criteria of which is colour and the second criterion exists but is not specified.

Another aspect which deserves mention here, is the question of the reduction of a given n-universe to another. Is it not possible indeed, to logically reduce a n-universe to a different system of criteria? Consider for example a Ω3CTL. In order to characterise the corresponding universe, one has 3 criterion-variables: colour, time and localisation. It appears that one can reduce this 3-universe to a 2-universe. That can be carried out by reducing two of the criteria of the 3-universe to one single criterion. In particular, one will reduce both criteria of colour and time to a single criterion of tcolour* (shmolorxi). And one will only preserve two taxa of tcolour*: G and ~G. Consider then a criterion of color comprising two taxa (red, non-red) and a criterion of time comprising two taxa (before T, after T). If one associates the taxa of colour and time, one obtains four new predicates: red before T, red after T, non-red before T, non-red after T, which one will denote respectively by RT, R~T, ~RT and ~R~T. Several of these predicates are compatible (RT and R~T, RT and ~R~T, ~RT and R~T, ~RT and ~R~T) whereas others are incompatible (RT and ~RT, R~T and ~R~T). At this stage, one has several manners (16)xii of grouping the compatible predicates, making it possible to obtain two new predicates G and ~G of tcolour*:

0123456789101112131415
RT  R~TXXXXXXXX
RT  ~R~TXXXXXXXX
~RT  R~TXXXXXXXX
~RT  ~R~TXXXXXXXX

In each of these cases, it results indeed a new single criterion of tcolour* (Z), which substitutes itself to the two preceding criteria of colour and time. One will denote by Zi (0 i 15) the taxa of tcolour* thus obtained. If it is clear that Z15 leads to the empty induction, it should be observed that several cases corresponding to the situation where the instances are RT lead to the problem inherent to GP. One will note thus that Z2, i.e. grue2 (by assimilating the Zi to gruei and the Z15-i to bleeni) is based on the definition: grue2 = red before T and non-red after T. It appears here as a conjunctive interpretation of the definition of “grue”. In the same way, grue7 corresponds to a definition of “grue” based on an exclusive disjunction. Lastly, grue12 is based on the traditional definition: grue12 = red before T or non-red after T, where the disjunction is to be interpreted as an inclusive disjunction.

Similarly, it also proves that a Ω2CT can be reduced to a tcoloured* 1-universe (Ω1Z). And more generally, a n-universe is thus reducible to an (n-1)-universe (for n > 1). Thus, if one considers a given universe, several characterisations in terms of n-universe can valuably be used. One can in particular apprehend a same universe like a Ω3CTL, or like a Ω2ZL. In the same way, one can represent a Ω2CT like a Ω1Z. At this stage, none of these views appears fundamentally better than the other. But each of these two characterisations constitute alternative ways to describe a same reality. This shows finally that a n-universe constitutes in fact an abstract characterisation of a real or an imaginary universe. A n-universe constitutes thus a system of criteria, comprising constants and variables. And in order to characterise a same real or imaginary given universe, one can resort valuably to several n-universes. Each of them appears finally as a different characterisation of the given universe, simply based on a different set of primitives.

5. Conditions of induction

The fact that the SR formalism involves the GP effect suggests that the intuition which governs our concept of induction is not entirely captured by SR. It is thus allowed to think that if the formal approach is necessary and useful to be used as support to induction, it does not constitute however a sufficient step. For it appears also essential to capture the intuition which governs our inductive reasoning. Therefore it proves necessary to supplement the formal approach of induction by a semantic approach. Goodman himself provides us with a definition of inductionxiii. He defines induction as the projection of characteristics of the past through the future, or more generally, as the projection of characteristics corresponding to a given aspect of an object through another aspect. This last definition corresponds to our intuition of induction. One can think however that it is necessary to supplement it by taking into account the preceding observationsxiv concerning the differentiation/unification duality. In that sense, it has been pointed out that induction consists of an inference from instances presenting both common properties and differences. Let the instances-source (instances-S) be the instances to which relate (I) or (I*) and the instance-destination (instance-D) that which is the subject of (P) or (P*). The common properties relate to the instances-S and the differentiated properties are established between the instances-S and the instance-D. The following definition ensues: induction consists precisely in the fact that the instance-Dxv also presents the property that is common to the instances-S, whereas one does vary the criterion (criteria) on which the differences between the instances-S and the instance-D is (are) based. The inductive reasoning is thus based on the constant nature of a property, whereas such other property is variable.

From this definition of induction arise straightforwardly several conditions of induction. I shall examine them in turn. The first two conditions are thus the following ones:

(C1) the instances-S must present some common properties

(C2) the instances-S and the instance-D must present some distinctive properties

This has for consequence that one cannot apply induction in two particular circumstances: firstly (i) when the instances do not present any common property. One will call such a situation a total differentiation of the instances. The problems corresponding to this particular circumstance have been mentioned abovexvi. And secondly (ii) when the instances do not present any distinctive property. One will call such a situation total unification. The problems encountered in this type of situation have also been mentioned previouslyxvii.

It should also be noted that it is not here a question of intrinsic properties of the instances, but rather of the analysis which is carried out by the one who is on the point of reasoning by induction.

Taking into account the definition of induction which has been given, a third condition can be thus stated:

(C3) a criterion-variable is necessary for the common properties of the instances-S and another criterion-variable for the distinctive properties

This refers to the structure of the considered universe of reference. Consequently, two criterion-variables are at least necessary, in the structure of the corresponding universe of reference. One will call that the minimalcondition of induction. Hence, a 2-universe is at least necessary in order that the conditions of induction can be satisfied. Thus, a 2CT will be appropriate. In the same way, a temporal and localised 2-universe (2TL) will also satisfy the conditions which have been just defined, etcxviii.

It should be noted that another way of stating this condition is as follows: the criterion-variable for the common properties and the criterion-variable for the differentiated properties must be distinct. One should not have confusion between the two. One can call that the condition of separation of the common properties and the distinctive properties. Such a principle appears as a consequence of the minimal condition for induction: one must have two criteria to perform induction, and these criteria must be different. If one chooses a same criterion for the common properties and the differentiated properties, one is brought back in fact to one single criterion and the context of a 1-universe, itself insufficient to perform induction.

Lastly, a fourth condition of induction results from the preceding definition:

(C4) one must project the common properties of the instances-S (and not the distinctive properties)

The conditions of induction which have been just stated make it possible from now on to handle the problems involved in the use of SR mentioned abovexix. It follows indeed that the following projectionsxx are correct: C°T in a Ω2CT, C°L in a Ω2CL, Z°L in a Ω2ZL, etc. Conversely, the following projections are incorrect: T°T in a Ω1T, Z°Z in a Ω1Z. In particular, one will note here that the projection T°T in the Ω1T is that of 1-time. 1-time takes indeed place in a Ω1T, whereas induction requires at the same time common properties and distinctive properties. Thus, a 2-universe is at least necessary. Usually, the criterion of time is used for differentiation. But here, it is used for unification (“drawn before T”). That can be done, but provided that one uses a distinct criterion for the differentiated properties. However, whereas common properties results here from that, the differentiated properties are missing. It thus misses a second criterion – corresponding to the differentiated properties – in the considered universe, to perform induction validly. Thus 1-time finds its origin in a violation of the minimal condition of induction. One can formulate this solution equivalently, with regard to the condition of separation. In effect, in 1-time, a same temporal criterion (drawn before T/drawn after T) is used for the common properties and the differentiated properties, whereas two distinct criteria are necessary. It can be thus analysed as a manifest violation of the condition of separation.

Lastly, the conditions of induction defined above lead to adapt the formalism used to describe GP. It proves indeed necessary to distinguish between the common and the distinctive property(ies). One will thus use the following formalism in replacement of the one used above:

(I) RT1·RT2·RT3·…·RT99

(H) RT1·RT2·RT3·…·RT99·RT100

where R denotes the common property and the Ti a distinctive property. It should be noted here that it can consist of a single object, or alternatively, of instances which are distinguished by a given criterion (which is not concerned by the inductive process) according to n-universe in which one places oneself. Thus, one will use in the case of a single instance , the colour of which is susceptible to change according to time:

(I) RT1·RT2·RT3·…·RT99

or in the case where several instances 1, 2, …, 99, 100 existxxi:

(I) RT11·RT22·RT33·…·RT9999

6. Origin of the paradox

Given the conditions of induction and the framework of n-universes which have been just defined, one is now in a position to proceed to determine the origin of GP. Preliminarily it is worth describing accurately the conditions of the universe of reference in which GP takes place. Indeed, in the original version of GP, the choice of the universe of reference is not defined accurately. However one can think that it is essential, in order to avoid any ambiguity, that this last is described precisely.

The universe of reference in which Goodman (1946) places himself is not defined explicitly, but several elements of the statement make it possible to specify its intrinsic nature. Goodman thus mentions the colours “red” and “non-red”. Therefore, colour constitutes one of the criterion-variables of the universe of reference. Moreover, Goodman distinguishes the balls which are drawn at times T1, T2, T3, …, T100. Thus, time is also a criterion-variable of the considered universe. Consequently, one can describe the minimal universe in which Goodman (1946) places himself as a Ω2CT. Similarly, in Goodman (1954), the criterion-variables of colour (green/non-green) and time (drawn before T/drawn after T) are expressly mentioned. In both cases, one thus places oneself implicitly within the minimal framework of a Ω2CT.

Goodman in addition mentions instances of balls or emeralds. Is it necessary at this stage to resort to an additional criterion-variable making it possible to distinguish between the instances? It appears that not. On the one hand indeed, as we have seen previouslyxxii, it proves that one has well a version of GP by simply considering a Ω2CT and a single object, the colour of which is susceptible to change during the course of time. On the other hand, it appears that if the criterion which is used to distinguish the instances is not used in the inductive process, it is then neither useful as a common criterion, nor as a differentiated criterion. It follows that one can dispense with this 3rd additional criterion. Thus, it proves that the fact of taking into account one single instance or alternatively, several instances, is not essential in the formulation of GP. In what follows, one will be able thus to consider that the statement applies, indifferently, to a single object or several instances that are distinguished by a criterion which is not used in the inductive process.

At this step, we are in a position to replace GP within the framework of n-universes. Taking into account the fact that the context of GP is that of a minimalΩ2CT, one will consider successively two situations: that of a Ω2CT, and then that of a Ω3CT (where  denotes a 3rd criterion).

6.1 “Grue” in the coloured and temporal 2-universe

Consider first the hypothesis of a Ω2CT. In such a universe, being “red” is being red at time T. One has then a criterion of colour for the common properties and a criterion of time for the differentiated properties. Consequently, it appears completely legitimate to project the common property of colour (“red”) into the differentiated time. Such a projection proves to be in conformity with the conditions of induction stated above.

Let us turn now to the projection of “grue”. One has observed previouslyxxiii that the Ω2CT was reducible to a Ω1Z. Here, the fact of using “grue” (and “bleen”) as primitives, is characteristic of the fact that the system of criteria used is that of a Ω1Z. What is then the situation when one projects “grue” in the Ω1Z? In such a universe of reference, the unique criterion-variable is the tcolour*. An object is there “grue” or “bleen” in the absolute. Consequently, if one has well a common criterion (the tcolour*), it appears that the differentiated criterion is missing, in order to perform induction validly. And the situation in which one is placed is that of an extreme differentiation. Thus, such a projection is carried out in violation of the minimal condition of induction. Consequently, it proves that GP cannot take place in the Ω2CT and is then blocked at the stage of the projection of “grue”.

But are these preliminary remarks sufficient to provide, in the context of a Ω2CT, a satisfactory solution to GP? One can think that not, because the paradox also arises in it in another form, which is that of the projection of tcolour* through time. One can formalise this projection Z°T as follows:

(I*) GT1·GT2·GT3·…·GT99

(H*) GT1·GT2·GT3·…·GT99·GT100 that is equivalent to:

(H’*) RT1·RT2·RT3·…·RT99·~RT100

(P*) GT100 that is equivalent to:

(P’*) ~RT100

where it is manifest that the elements of GP are still present.

Fundamentally in this version, it appears that the common properties are borrowed from the system of criteria of the Ω1Z, whereas the differentiated properties come from the Ω2CT. A first analysis thus reveals that the projection of “grue” under these conditions presents a defect which consists in the choice of a given system of criteria for the common properties (tcolour*) and of a different system of criteria for the differentiated properties (time). For the selection of the tcolour* is characteristic of the choice of a Ω1Z, whereas the use of time is revealing of the fact that one places oneself in a Ω2CT. But one must choose one or the other of the reducible systems of criteria to perform induction. On the hypotheses envisaged previously, the choice of the criteria for the common and differentiated properties was carried out within the same system of criteria. But here, the choice of the criteria for the common properties and the differentiated properties is carried out within two different (and reducible) systems of criteria. Thus, the common and differentiated criteria selected for induction are not genuinely distinct. And this appears as a violation of the condition of separation. Consequently, one of the conditions of induction is not respected.

However, the projection Z°T has a certain intuitive support, because it is based on the fact that the notions of “grue before T” and “grue after T” have a certain intuitive meaning. Let us then disregard the violation of the conditions of the induction which has been just mentioned, and consider thus this situation in more detail. In this context, GP is always present, since one observes a contradiction between (P) and (P’*). It is with this contradiction that it is worth from now on being interested. Consider the particular step of the equivalence between (H*) and (H’*). One conceives that “grue before T” is assimilated here to RT, because the fact that the instances-S are red before T results clearly from the conditions of the experiment. On the other hand, it is worth being interested by the step according to which (P*) entails (P’*). According to the classical definitionxxiv: “grue” = {RT  R~T, RT  ~R~T, ~RT  ~R~T }. What is it then to be “grue after T”? There, it appears that a “grue” object can be R~T (this corresponds to the case RT  R~T) or ~R~T (this correspond to the cases RT  ~R~T and ~RT  ~R~T). In conclusion, the object can be either R~T or ~R~T. Thus, the fact of knowing that an object is “grue after T” does not make it possible to conclude that this object is ~R~T, because this last can also be R~T. Consequently, the step according to which (P*) involves (P’*) appears finally false. From where it ensues that the contradiction between (P) and (P’*) does not have any more a raison d’etre.

One can convince oneself that this analysis does not depend on the choice of the classical definition of “grue” (grue12) which is carried out, by considering other definitions. Consider for example the definition based on grue9: “grue” = {RT  ~R~T, ~RT  ~R~T} and “bleen” = {RT  R~T, ~RT  R~T}. But in this version, one notes that one does not have the emergence of GP, because the instances-S, which are RT, can be at the same time “grue” and ” bleen”. And the same applies if one considers a conjunctive definition (grue2) such as “grue” = {RT  ~R~T}. In such a case indeed, the instances-S are “grue” only if they are RT but also ~R~T. However this does not correspond to the initial conditions of GP in the 2CT where one ignores if the instances-S are ~R~T.

One could also think that the problem is related to the use of a taxonomy of tcolour* based on two taxa (G and ~G). Consider then a taxonomy of tcolour* based on 4 taxa: Z0 = RT  R~T, Z1 = RT  ~R~T, Z2 = ~RT  R~T, Z3 = ~RT  ~R~T. But on this hypothesis, it appears clearly that since the instances-S are for example Z1, one finds himself replaced in the preceding situation.

The fact of considering “grue after T”, “grue before T”, “bleen before T”, “bleen after T” can be assimilated with an attempt of expressing “grue” and ” bleen” with the help of our own criteria, and in particular that of time. It can be considered here as a form of anthropocentrism, underlain by the idea to express the Ω1Z with the help of the taxa of the Ω2CT. Since one knows the code defining the relations between two reducible n-universes – the Ω1Z and the Ω2CT – and that one has partial data, one can be tempted to elucidate completely the predicates of the foreign n-universe. Knowing that the instances are GT, G~T, ~GT, ~G~T, I can deduce that they are respectively {RT, ~RT}, {R~T, ~R~T}, {~RT}, {R~T}. But as we have seen, due to the fact that the instances are GT and RT, I cannot deduce that they will be ~R~T.

The reasoning in this version of GP is based on the apparently inductive idea that what is “grue before T” is also “grue after T”. But in the context which is that of the Ω1Z, when an object is “grue”, it is “grue” in the absolute. For no additional criterion exists which can make its tcolour* vary. Thus, when an object is GT, it is necessarily G~T. And from the information according to which an object is GT, one can thus conclude, by deduction, that it is also G~T.

From what precedes, it ensues that the version of GP related to the Z°T presents the apparent characters of induction, but it does not constitute an authentic form of this type of reasoning. Z°T thus constitutes a disguised form of induction for two principal reasons: first, it is a projection through the differentiated criterion of time, which constitutes the standard mode of our inductive practice. Second, it is based on the intuitive principle according to which everything that is GT is also G~T. But as we have seen, it consists here in reality of a deductive form of reasoning, whose true nature is masked by an apparent inductive move. And this leads to conclude that the form of GP related to Z°T analyses itself in fact veritably as a pseudo-induction.

6.2 “Grue” in the coloured, temporal and localised 3-universe

Consider now the case of a Ω3CT. This type of universe of reference also corresponds to the definition of a minimal Ω2CT, but it also comprises one 3rd criterion-variablexxv. Let us choose for this last a criterion such as localisationxxvi. Consider then a Ω3CTL. Consider first (H) in such a 3-universe. To be “red” in the Ω3CTL, is to be red at time T and at location L. According to the conditions of GP, colour corresponds to the common properties, and time to the differentiated properties. One has then the following projection C°TL:

(I) RT1L1·RT2L2·RT3L3·…·RT99L99

(H) RT1L1·RT2L2·RT3L3·…·RT99L99·RT100L100

∴ (P) RT100L100

where taking into account the conditions of induction, it proves to be legitimate to project the common property (“red”) of the instances-S, into differentiated time and location, and to predict that the 100th ball will be red. Such a projection appears completely correct, and proves in all points in conformity with the conditions of induction mentioned above.

What happens now with (H*) in the Ω3CTL? It has been observed that the Ω3CTL could be reduced to a Ω2ZL. In this last n-universe, the criterion-variables are tcolour* and localisation. The fact of being “grue” is there relative to location: to be “grue”, is to be “grue” at location L. What is then projected is the tcolour*, i.e. the fact of being “grue” or “bleen”. There is thus a common criterion of tcolour* and a differentiated criterion of localisation. Consequently, if it is considered that the instances-S are “grue”, one can equally well project the property common “grue” into a differentiated criterion of localisation. Consider then the projection Z°L in the Ω2ZL:

(I*) GL1·GL2·GL3·…·GL99

(H*) GL1·GL2·GL3·…·GL99·GL100

∴ (P*) GL100

Such a projection is in conformity with the conditions mentioned above, and constitutes consequently a valid form of induction.

In this context, one can project valuably a predicate having a structure identical to that of “grue”, in the case of emeralds. Consider the definition “grue” = green before T or non-green after T, where T = 10 billion years. It is known that at that time, our Sun will be extinct, and will become gradually a dwarf white. The conditions of our atmosphere will be radically different from what they currently are. And the temperature will rise in particular in considerable proportions, to reach 8000°. Under these conditions, the structure of many minerals will change radically. It should normally thus be the case for our current emeralds, which should see their colour modified, due to the enormous rise in temperature which will follow. Thus, I currently observe an emerald: it is “grue” (for T = 10 billion years). If I project this property through a criterion of location, I legitimately conclude from it that the emerald found in the heart of the Amazonian forest will also be “grue”, in the same way as the emerald which has been just extracted from a mine from South Africa.

At this stage, one could wonder whether the projectibility of “grue” is not intrinsically related to the choice of a definition of “grue” based on inclusive disjunction (grue12)? Nevertheless, one easily checks by using an alternative definition of “grue” that its projection remains validxxvii.

It should be noticed that one has here the expression of the fact that the taxonomy based on the tcolour* is coarser than that based on time and colour. In effect, the former only comprises 2 taxa (grue/bleen), whereas the latter presents 4 of them. By reducing the criteria of colour and time to a single criterion of tcolor*, one has replaced 4 taxa (RT  R~T, RT  ~R~T, ~RT  R~T, ~RT  ~R~T) by 2. Thus, “grue” constitutes from this point of view a predicate coarser than “red”. The universe which is described did not change, but the n-universes which are systems of criteria describing these universes are different. With the tcolour* thus defined, one has less predicates at its disposal to describe a same reality. The predicates “grue” and “bleen” are for us not very informative, and are less informative in any case that our predicates “red”, “non-red”, “before T”, etc. But that does not prevent however “grue” and “bleen” to be projectibles.

Whereas the projection of “grue” appears valid in the Ω2ZL, it should be noticed however that one does not observe in this case the contradiction between (P) and (P’*). For here (I*) is indeed equivalent to:

(I’*) RT1L1·RT2L2·RT3L3·…·RT99 L99

since, knowing according to the initial data of GP that the instances-S are RT, one valuably replaces the GLi by the RTiLi (i < 100). But it appears that on this hypothesis, (P*) does not involve:

(P’*) ~RT100L100

because one does not have an indication relating to the temporality of the 100th instance, due to the fact that only the localisation constitutes here the differentiated criterion. Consequently, one has well in the case of the Ω3CTL a version built with the elements of GP where the projection of “grue” is carried out valuably, but which does not present a paradoxical nature.

7. Conclusion

In the solution to GP proposed by Goodman, a predicate is projectible or nonprojectible in the absolute. And one has in addition a correspondence between the entrenchedxxviii/non-entrenched and the projectible/nonprojectible predicates. Goodman in addition does not provide a justification to this assimilation. In the present approach, there is no such dichotomy, because a given predicate P reveals itself projectible in a given n-universe, and nonprojectible in another n-universe. Thus, P is projectible relatively to such universe of reference. There is thus the projectible/nonprojectible relative to such n-universe distinction. And this distinction is justified by the conditions of induction, and the fundamental mechanism of induction related to the unification/differentiation duality. There are thus n-universes where “green” is projectible and others where it is not. In the same way, “grue” appears here projectible relative to certain n-universes. Neither green nor grue are projectible in the absolute, but only relative to such given universe. Just as of some other predicates, “grue” is projectible in certain universes of reference, but nonprojectible in othersxxix.

Thus, it proves that one of the causes of GP resides in the fact that in GP, one classically proceeds to operate a dichotomy between the projectible and the nonprojectible predicates. The solutions classically suggested to GP are respectively based on the distinction temporal/nontemporal, local/non-local, qualitative/nonqualitative, entrenched/non-entrenched, etc. and a one-to-one correspondence with the projectible/nonprojectible distinction. One wonders whether a given predicate P* having the structure of “grue” is projectible, in the absolute. This comes from the fact that in GP, one has a contradiction between the two concurrent predictions (P) and (P*). One classically deduces from it that one of the two predictions must be rejected, at the same time as one of the two generalisations (H) or (H*) on which these predictions are respectively based. Conversely, in the present analysis, whether one places himself in the case of the authentic projection Z°L or in the case of the pseudo-projection Z°T, one does not have a contradiction between (P) and (P’*). Consequently, one is not constrained any more to reject either (H) or (H*). And the distinction between projectible/nonprojectible predicates does not appear indispensable any morexxx.

How is the choice of our usual n-universe carried out in this context? N-universes such as the Ω2CT, the Ω3CTL, the Ω2ZL etc. are appropriate to perform induction. But we naturally tend to privilege those which are based on criteria structured rather finely to allow a maximum of combinations of projections. If one operates from the criteria Z and L in the Ω2ZL, one restricts oneself to a limited number of combinations: Z°L and L°Z. Conversely, if one retains the criteria C, T and L, one places oneself in the Ω3CTL and one has the possibility of projections C°TL, T°CL, L°CT, CT°Lxxxi, CL°T, TL°C. One has thus a maximum of combinations. This seems to encourage to prefer the Ω3CTL to the Ω2ZL. Of course, pragmatism seems to have to play a role in the choice of the best alternative of our criteria. But it seems that it is only one of the multiple factors which interact to allow the optimisation of our criteria to carry out the primitive operations of grouping and differentiation, in order to then be able to generalise, classify, order, make assumptions or forecastxxxii. Among these factors, one can in particular mention: pragmatism, simplicity, flexibility of implementation, polyvalencexxxiii, economy in means, powerxxxiv, but also the nature of our real universe, the structure of our organs of perception, the state of our scientific knowledge, etcxxxv. Our usual n-universes are optimised with regard to these various factors. But this valuably leaves room for the choice of other systems of criteria, according to the variations of one or the other of these parametersxxxvi.

i Nelson Goodman, “A Query On Confirmation”, Journal of Philosophy, vol. 43 (1946), p. 383-385; in Problems and Projects, Indianapolis, Bobbs-Merrill, 1972, p. 363-366.

ii With some minor adaptations.

iii See Goodman “A Query On Confirmation”, p. 383: “Suppose we had drawn a marble from a certain bowl on each of the ninety-nine days up to and including VE day and each marble drawn was red. We would expect that the marble drawn on the following day would also be red. So far all is well. Our evidence may be expressed by the conjunction “Ra1·Ra2·…·Ra99” which well confirms the prediction Ra100.” But increase of credibility, projection, “confirmation” in any intuitive sense, does not occur in the case of every predicate under similar circumstances. Let “S” be the predicate “is drawn by VE day and is red, or is drawn later and is non-red.” The evidence of the same drawings above assumed may be expressed by the conjunction “Sa1·Sa2·…·Sa99“. By the theories of confirmation in question this well confirms the prediction “Sa100“; but actually we do not expect that the hundredth marble will be non-red. “Sa100” gains no whit of credibility from the evidence offered.”

iv Nelson Goodman, Fact, Fiction and Forecast, Cambridge, MA, Harvard University Press, 1954.

v Ibid., p. 73-4: “Suppose that all emeralds examined before a certain time t are green. At time t, then, our observations support the hypothesis that all emeralds are green; and this is in accord with our definition of confirmation. […] Now let me introduce another predicate less familiar than “green”. It is the predicate “grue” and it applies to all things examined before t just in case they are green but to other things just in case they are blue. Then at time t we have, for each evidence statement asserting that a given emerald is green, a parallel evidence statement asserting that that emerald is grue.”

vi For example with an accuracy of 10-4 nm.

vii Or any taxonomy which is similar to it.

viii See §2 above.

ix This assertion is controversial.

x Such a remark also applies to the statement of Goodman, Fact, Fiction and Forecast.

xi As J.S. Ullian mentions it, “More one ‘Grue’ and Grue”, Philosophical Review, vol. 70 (1961), p. 386-389, in p. 387.

xii I. e. C(0, 4)+C(1, 4)+C(2, 4)+C(3, 4)+C(4, 4) = 24, where C(p, q) denotes the number of combinations of q elements taken p times.

xiii See Goodman, “A Query On Confirmation”, p. 383: “Induction might roughly be described as the projection of characteristics of the past into the future, or more generally of characteristics of one realm of objects into another.”

xiv See §2 above.

xv One can of course alternatively take into account several instances-D.

xvi See §2 above.

xvii Ibid.

xviii For the application of this condition, one must take into account the remarks mentioned above concerning the problem of the status of the instances. Thus, one must actually compare an instantiated and temporal 1-universe to a 2-universe one of the criteria of which is temporal, and the second criterion is not explicitly mentioned. Similarly, an instantiated and coloured 1-universe is assimilated in fact to a 2-universe one of the criteria of which is temporal, and the second criterion is not specified.

xix See §3 above.

xx With the notations C (colour), T (time), L (localisation) and Z (tcolour*).

xxi However, since the fact that there exists one or more instances is not essential in the formulation of the given problem, one will obviously be able to abstain from making mention of it.

xxii See §4.

xxiii Ibid.

xxiv It is the one based on the inclusive disjunction (grue12).

xxv A same solution applies, of course, if one considers a number of criterion-variables higher than 3.

xxvi All other criterion distinct from colour or time, would also be appropriate.

xxvii In particular, it appears that the projection of a conjunctive definition (grue2) is in fact familiar for us. In effect, we do not proceed otherwise when we project the predicate “being green before maturity and red after maturity” applicable to tomatoes, through a differentiated criterion of location: this is true of the 99 instance-S observed in Corsica and Provence, and is projected validly to a 100th instance located in Sardinia. One can observe that such a type of projection is in particular regarded as nonproblematic by Jackson (Franck Jackson, “‘Grue'”, Journal of Philosophy, vol. 72 (1975), p. 113-131): “There seems no case for regarding ‘grue’ as nonprojectible if it is defined this way. An emerald is grue1 just if it is green up to T and blue thereafter, and if we discovered that all emeralds so far examined had this property, then, other things being equal, we would probably accept that all emeralds, both examined and unexamined, have this property (…).” If one were to replace such a predicate in the present analysis, one should then consider that the projection is carried out for example through a differentiated criterion of localisation (p. 115).

xxviii Goodman, Fact, Fiction and Forecast.

xxix The account presented in J Holland, K Holyoak, R. Nisbett and P. Thagard (Induction, Cambridge, MA; London, MIT Press, 1986) appears to me to constitute a variation of Goodman’s solution, directed towards the computer-based processing of data and based on the distinction integrated/non-integrated in the default hierarchy. But Holland’s solution presents the same disadvantages as that of Goodman: what justification if not anthropocentric, does one have for this distinction? See p. 235: “Concepts such as “grue”, which are of no significance to the goals of the learner, will never be generated and hence will not form part of the default hierarchy. (…) Generalization, like other sorts of inference in a processing system, must proceed from the knowledge that the system already has”.

The present analysis also distinguishes from the one presented by Susan Haack (Evidence and Inquiry, Oxford; Cambridge, MA, Blackwell, 1993) because the existence of natural kinds does not constitute here a condition for induction. See p. 134: “There is a connection between induction and natural kinds. […] the reality of kinds and laws is a necessary condition of successful inductions”. In the present context, the fact that the conditions of induction (a common criterion, a distinct differentiated criterion, etc.) are satisfied is appropriate to perform induction.

xxx A similar remark is made by Franck Jackson in conclusion of his article (“‘Grue'”, p. 131): “[…] the SR can be specified without invoking a partition of predicates, properties or hypotheses into the projectible and the nonprojectible”. For Jackson, all noncontradictory predicates are projectible: “[…] all (consistent) predicates are projectible.” (p. 114). Such a conclusion appears however stronger than the one that results from the current analysis. Because for Jackson, all predicates are thus projectible in the absolute. However in the present context, there are no projectible or nonprojectible predicates in the absolute. It is only relative to a given n-universe, that a predicate P reveals projectible or nonprojectible.

More generally, the present analysis distinguishes fundamentally from that of Jackson in the sense that the solution suggested to GP does not rest on the counterfactual condition. This last appears indeed too related to the use of certain predicates (examined, sampled, etc.). On the other hand, in the present context, the problem is considered from a general viewpoint, independently of the particular nature of the predicates constituting the definition of grue.

xxxi Such a projection corresponds for example to the generalisation according to which “the anthropomorphic statue-menhirs are of the colour of granite and date from the Age of Bronze”.

xxxii As Ian Hacking underlines it, Le plus pur nominalisme, Combas, L’éclat, 1993, p. 9: “Utiliser un nom pour une espèce, c’est (entre autres choses) vouloir réaliser des généralisations et former des anticipations concernant des individus de cette espèce. La classification ne se limite pas au tri : elle sert à prédire. C’est une des leçons de la curieuse “énigme” que Nelson Goodman publia il y a quarante ans.” My translation: “To use a name for a species, it is (among other things) to want to carry out generalisations and to form anticipations concerning the individuals of this species. Classification is not limited to sorting: it is used to predict. It is one of the lessons of the strange “riddle” which Nelson Goodman published forty years ago.”

xxxiii The fact that a same criterion can be used at the same time as a common and a differentiated criterion (while eventually resorting to different taxa).

xxxiv I.e. the number of combinations made possible.

xxxv This enumeration does not pretend to be exhaustive. A thorough study of this question would be of course necessary.

xxxvi I thank the editor of Dialogue and two anonymous referees for very helpful comments on an earlier draft of this paper.

A Dichotomic Analysis of the Surprise Examination Paradox

English translation of a paper appeared in French in Philosophiques 2005, vol. 32, pages 399-421 (with minor changes with regard to the published version).

This paper proposes a new framework to solve the surprise examination paradox. I survey preliminary the main contributions to the literature related to the paradox. I introduce then a distinction between a monist and a dichotomic analysis of the paradox. With the help of a matrix notation, I also present a dichotomy that leads to distinguish two basically and structurally different notions of surprise, which are respectively based on a conjoint and a disjoint structure. I describe then how Quine’s solution and Hall’s reduction apply to the version of the paradox corresponding to the conjoint structure. Lastly, I expose a solution to the version of the paradox based on the disjoint structure.

A Dichotomic Analysis of the Surprise Examination Paradox

I shall present in what follows a new conceptual framework to solve the surprise examination paradox (henceforth, SEP), in the sense that it reorganizes, by adapting them, several elements of solution described in the literature. The solution suggested here rests primarily on the following elements: (i) a distinction between a monist and a dichotomic analysis of the paradox; (ii) the introduction of a matrix definition, which is used as support with several variations of the paradox; (iii) the distinction between a conjoint and a disjoint definition of the cases of surprise and of non-surprise, leading to two structurally different notions of surprise.

In section 1, I proceed to describe the paradox and the main solutions found in the literature. I describe then in section 2, in a simplified way, the solution to the paradox which results from the present approach. I also introduce the distinction between a monist and a dichotomic analysis of the paradox. I present then a dichotomy which makes it possible to distinguish between two basically and structurally different versions of the paradox: on the one hand, a version based on a conjoint structure of the cases of non-surprise and of surprise; in the other hand, a version based on a disjoint structure. In section 3, I describe how Quine’s solution and Hall’s reduction apply to the version of SEP corresponding to the conjoint structure of the cases of non-surprise and of surprise. In section 4, I expose the solution to SEP corresponding to the disjoint structure. Lastly, I describe in section 5, within the framework of the present solution, what should have been the student’s reasoning.

1. The paradox

The surprise examination paradox finds its origin in an actual fact. In 1943-1944, the Swedish authorities planned to carry out a civil defence exercise. They diffused then by the radio an announcement according to which a civil defence exercise would take place during the following week. However, in order to perform the latter exercise under optimal conditions, the announcement also specified that nobody could know in advance the date of the exercise. Mathematician Lennart Ekbom understood the subtle problem arising from this announcement of a civil defence exercise and exposed it to his students. A broad diffusion of the paradox throughout the world then ensued.

SEP first appeared in the literature with an article of D. O’ Connor (1948). O’ Connor presents the paradox under the form of the announcement of a military training exercise. Later on, SEP appeared in the literature under other forms, such as the announcement of the appearance of an ace in a set of cards (Scriven 1951) or else of a hanging (Quine 1953). However, the version of the paradox related to the professor’s announcement of a surprise examination has remained the most current form. The traditional version of the paradox is as follows: a professor announces to his/her students that an examination will take place during the next week, but that they will not be able to know in advance the precise day where the examination will occur. The examination will thus occur surprisingly. The students reason as follows. The examination cannot take place on Saturday, they think, for otherwise they would know in advance that the examination would take place on Saturday and thus it could not occur surprisingly. Thus, Saturday is ruled out. Moreover, the examination cannot take place on Friday, for otherwise the students would know in advance that the examination would take place on Friday and thus it could not occur surprisingly. Thus, Friday is also ruled out. By a similar reasoning, the students eliminate successively Thursday, Wednesday, Tuesday and Monday. Finally, all days of the week are then ruled out. However, this does not prevent the examination from finally occurring surprisingly, say, on Wednesday. Thus, the students’ reasoning proved to be fallacious. However, such a reasoning appears intuitively valid. The paradox lies here in the fact that the students’ reasoning seems valid, whereas it finally proves to be in contradiction with the facts, namely that the examination can truly occur surprisingly, in accordance with the announcement made by the professor.

In the literature, several solutions to SEP have been proposed. There does not exist however, at present time, a consensual solution. I will briefly mention the principal solutions which were proposed, as well as the fundamental objections that they raised.

A first attempt at solution appeared with O’ Connor (1948). This author pointed out that the paradox was due to the contradiction which resulted from the professor’s announcement and the implementation of the latter. According to O’ Connor, the professor’s announcement according to which the examination was to occur by surprise was in contradiction with the fact that the details of the implementation of the examination were known. Thus, the statement of SEP was, according to O’ Connor, self-refuting. However, such an analysis proved to be inadequate, because it finally appeared that the examination could truly take place under some conditions where it occurred surprisingly, for example on Wednesday. Thus, the examination could finally occur by surprise, confirming thus and not refuting, the professor’s announcement. This last observation had the effect of making the paradox re-appear.

Quine (1953) also proposed a solution to SEP. Quine considers thus the student’s final conclusion according to which the examination can occur surprisingly on no day of the week. According to Quine, the student’s error lies in the fact of having not considered from the beginning the hypothesis that the examination could not take place on the last day. For the fact of considering precisely that the examination will not take place on the last day makes it finally possible for the examination to occur surprisingly, on the last day. If the student had also taken into account this possibility from the beginning, he would not concluded fallaciously that the examination cannot occur by surprise. However, Quine’s solution has led to criticisms, emanating notably from commentators (Ayer 1973, Janaway 1989 and also Hall 1999) who stressed the fact that Quine’s solution did not make it possible to handle several variations of the paradox. Ayer imagines thus a version of SEP where a given person is informed that the cards of a set will be turned over one by one, but where that person will not know in advance when the ace of Spades will be issued. Nevertheless, the person is authorized to check the presence of the ace of Spades before the set of cards is mixed. The purpose of the objection to Quine’s solution based on such a variation is to highlight a situation where the paradox is quite present but where Quine’s solution does not find to apply any more, because the student knows with certainty, given the initial data of the problem, that the examination will take place as well.

According to another approach, defended in particular by R. Shaw (1958), the structure of the paradox is inherently self-referential. According to Shaw, the fact that the examination must occur by surprise is tantamount to the fact that the date of the examination cannot be deduced in advance. But the fact that the students cannot know in advance, by deduction, the date of the examination constitutes precisely one of the premises. The paradox thus finds its origin, according to Shaw, in the fact that the structure of the professor’s announcement is self-referential. According to the author, the self-reference which results from it constitutes thus the cause of the paradox. However, such an analysis did not prove to be convincing, for it did not make it possible to do justice to the fact that in spite of its self-referential structure, the professor’s announcement was finally confirmed by the fact that the examination could finally occur surprisingly, say on Wednesday.

Another approach, put forth by Richard Montague and David Kaplan (1960) is based on the analysis of the structure of SEP which proves, according to these authors, to be that of the paradox of the Knower. The latter paradox constitutes a variation of the Liar paradox. What thus ultimately proposes Montague and Kaplan, is a reduction of SEP to the Liar paradox. However, this last approach did not prove to be convincing. Indeed, it was criticized because it did not take account, on the one hand, the fact that the professor’s announcement can be finally confirmed and on the other hand, the fact that one can formulate the paradox in a non-self-referential way.

It is also worth mentioning the analysis developed by Robert Binkley (1968). In his article, Binkley exposes a reduction of SEP to Moore’s paradox. The author makes the point that on the last day, SEP reduces to a variation of the proposition ‘P and I don’t know that P’ which constitutes Moore’s paradox. Binkley extends then his analysis concerning the last day to the other days of the week. However, this approach has led to strong objections, resulting in particular from the analysis of Wright and Sudbury (1977).

Another approach also deserves to be mentioned. It is the one developed by Paul Dietl (1973) and Joseph Smith (1984). According to these authors, the structure of SEP is that of the sorites paradox. What then propose Dietl and Smith, is a reduction of SEP to the sorites paradox. However, such an analysis met serious objections, raised in particular by Roy Sorensen (1988).

It is worth lastly mentioning the approach presented by Crispin Wright and Aidan Sudbury (1977). The analysis developed by these authors1 results in distinguishing two cases: on the one hand, on the last day, the student is in a situation which is that which results from Moore’s paradox; in addition, on the first day, the student is in a basically different situation where he can validly believe in the professor’s announcement. Thus, the description of these two types of situations leads to the rejection of the principle of temporal retention. According to this last principle, what is known at a temporal position T0 is also known at a later temporal position T1 (with T0 < T1). However, the analysis of Wright and Sudbury appeared vulnerable to an argument developed by Sorensen (1982). The latter author presented indeed a version of SEP (the Designated Student Paradox) which did not rely on the principle of temporal retention, on which the approach of Wright and Sudbury rested. According to Sorensen’s variation, the paradox was quite present, but without the conditions of its statement requiring to rely on the principle of temporal retention. Sorensen describes thus the following variation of the paradox. Five students, A, B, C, D and E are placed, in this order, one behind the other. The professor then shows to the students four silver stars and one gold star. Then he places a star on the back of each student. Lastly, he announces to them that the one of them who has a gold star in the back has been designated to pass an examination. But, the professor adds, this examination will constitute a surprise, because the students will only know that who was designated when they break their alignment. Under these conditions, it appears that the students can implement a similar reasoning to that which prevails in the original version of SEP. But this last version is diachronic, whereas the variation described by Sorensen appears, by contrast, synchronic. And as such, it is thus not based on whatever principle of temporal retention.

Given the above elements, it appears that the stake and the philosophical implications of SEP are of importance. They are located at several levels and thus relate2 to the theory of knowledge, deduction, justification, the semantic paradoxes, self-reference, modal logic, and vague concepts.

2. Monist or dichotomic analysis of the paradox

Most analyses classically proposed to solve SEP are based on an overall solution which applies, in a general way, to the situation which is that of SEP. In this type of analysis, a single solution is presented, which is supposed to apply to all variations of SEP. Such type of solution has a unitary nature and appears based on what can be termed a monist theory of SEP. Most solutions to SEP proposed in the literature are monist analyses. Characteristic examples of this type of analysis of SEP are the solutions suggested by Quine (1953) or Binkley (1968). In a similar way, the solution envisaged by Dietl (1973) which is based on a reduction of SEP to the sorite paradox also constitutes a monist solution to SEP.

Conversely, a dichotomic analysis of SEP is based on a distinction between two different scenarios of SEP and on the formulation of an independent solution for each of the two scenarios. In the literature, the only analysis which has a dichotomic nature, as far as I know, is that of Wright and Sudbury mentioned above. In what follows, I will present a dichotomic solution to SEP. This solution is based on the distinction of two variations of SEP, associated with concepts of surprise that correspond to different structures of the cases of non-surprise and of surprise.

At this step, it proves to be useful to introduce the matrix notation. With the help of this latter, the various cases of non-surprise and of surprise be modelled with the following S[k, s] table, where k denotes the day where the examination takes place and S[k, s] denotes if the corresponding case of non-surprise (s = 0) or of surprise (s = 1) is made possible (S[k, s] = 1) or not (S[k, s] = 0) by the conditions of the announcement (with 1  kn).3 If one considers for example 7-SEP 4, S[7, 1] = 0 denotes the fact that the surprise is not possible on the 7th day, and conversely, S[7, 1] = 1 denotes the fact that the surprise is possible on the 7th day; in the same way, S[1, 0] = 0 denotes the fact that the non-surprise is not possible on the 1st day by the conditions of the announcement, and conversely, S[1, 0] = 1 denotes the fact that the non-surprise is possible on the 1st day.

The dichotomy on which rests the present solution results directly from the analysis of the structure which makes it possible to describe the concept of surprise corresponding to the statement of SEP. Let us consider first the following matrix, which corresponds to a maximal definition, where all cases of non-surprise and of surprise are made possible by the professor’s announcement (with ■ = 1 and □ = 0):

(D1)S[k, 0]S[k, 1]
S[7,s]
S[6,s]
S[5,s]
S[4,s]
S[3,s]
S[2,s]
S[1,s]

At the level of (D1), as we can see it, all values of the S[k, s] matrix are equal to 1, which corresponds to the fact that all the cases of non-surprise and of surprise are made possible by the corresponding version of SEP. The associated matrix can be thus defined as a rectangular matrix.

At this stage, it appears that one can conceive of some variations of SEP associated with more restrictive matrix structures, where certain cases of non-surprise and of surprise are not authorized by the announcement. In such cases, certain values of the matrix are equal to 0. It is now worth considering the structure of these more restrictive definitions. The latter are such that it exists at least one case of non-surprise or of surprise which is made impossible by the announcement, and where the corresponding value of the matrix S[k, s] is thus equal to 0. Such a condition leaves place [***room] with a certain number of variations, of which it is now worth studying the characteristics more thoroughly.

One can notice preliminarily that certain types of structures can be discarded from the beginning. It appears indeed that any definition associated with a restriction of (D1) is not adequate. Thus, there are minimal conditions for the emergence of SEP. In this sense, a first condition is that the base step be present. This base step is such that the non-surprise must be able to occur on the last day, that is to say S[n, 0] = 1. With the previously defined notation, it presents the general form n*n* and corresponds to 7*7* for 7-SEP. In the lack of this base step, there is no paradoxical effect of SEP. Consequently, a structure of matrix such as S[n, 0] = 0 can be discarded from the beginning.

One second condition so that the statement leads to a genuine version of SEP is that the examination can finally occur surprisingly. This renders indeed possible the fact that the professor’s announcement can be finally satisfied. Such a condition – let us call it the vindication step – is classically mentioned as a condition for the emergence of the paradox. Thus, a definition which would be such that all the cases of surprise are made impossible by the corresponding statement would also not be appropriate. Thus, the structure corresponding to the following matrix would not correspond either to a licit statement of SEP:

(D2)S[k, 0]S[k, 1]
S[7,s]
S[6,s]
S[5,s]
S[4,s]
S[3,s]
S[2,s]
S[1,s]

because the surprise is possible here on no day of the week (S[k, 1 ] = 0) and the validation step is thus lacking in the corresponding statement.

Taking into account what precedes, one is now in a position to describe accurately the minimal conditions which are those of SEP:

(C3) S[n, 0] = 1 (base step)

(C4) k (1  kn) such that S[k, 1] = 1 (validation step)

At this step, it is worth considering the structure of the versions of SEP based on the definitions which satisfy the minimal conditions for the emergence of the paradox which have just been described, i.e. which contain at the same time the basic step and the validation step. It appears here that the structure associated with the cases of non-surprise and of surprise corresponding to a variation with SEP can present two forms of a basically different nature. A first form of SEP is associated with a structure where the possible cases of non-surprise and of surprise are such that it exists during the n-period at least one day where the non-surprise and the surprise are simultaneously possible. Such a definition can be called conjoint. The following matrix constitutes an example of this type of structure:

(D5)S[k, 0]S[k, 1]
S[7,s]
S[6,s]
S[5,s]
S[4,s]
S[3,s]
S[2,s]
S[1,s]

because the non-surprise and the surprise are simultaneously possible here on the 7th, 6th, 5th and 4th days. However, it proves that one can also encounter a second form of SEP the structure of which is basically different, in the sense that for each day of the n-period, it is impossible to have simultaneously the surprise and the non-surprise.5 A definition of this nature can be called disjoint. The following matrix thus constitutes an example of this type of structure:

(D6)S[k, 0]S[k, 1]
S[7,s]
S[6,s]
S[5,s]
S[4,s]
S[3,s]
S[2,s]
S[1,s]

Consequently, it is worth distinguishing in what follows two structurally distinct versions of SEP: (a) a version based on a conjoint structure of the cases of non-surprise and of surprise made possible by the announcement; (b) a version based on a disjoint structure of these same cases. The need for making such a dichotomy finds its legitimacy in the fact that in the original version of SEP, the professor does not specify if one must take into account a concept of surprise corresponding to a disjoint or a conjoint structure of the cases of non-surprise and of surprise. With regard to this particular point, the professor’s announcement of SEP appears ambiguous. Consequently, it is necessary to consider successively two different concepts of surprise, respectively based on a disjoint or conjoint structure of the cases of non-surprise and of surprise, as well as the reasoning which must be associated with them.

3. The surprise notion corresponding to the conjoint structure

Let us consider first the case where SEP is based on a concept of surprise corresponding to a conjoint structure of the cases of non-surprise and of surprise. Let SEP(I) be the version associated with such a concept of surprise. Intuitively, this version corresponds to a situation where there exists in the n-period at least one day where the non-surprise and the surprise can occur at the same time. Several types of definitions are likely to satisfy this criterion. It is worth considering them in turn.

4.1 The definition associated with the rectangular matrix and Quine’s solution

To begin with, it is worth considering the structures which are such that all cases of non-surprise and of surprise are made possible by the statement. The corresponding matrix is a rectangular matrix. Let thus SEP(I□) be such a version. The definition associated with such a structure is maximal since all cases of non-surprise and of surprise are authorized. The following matrix corresponds thus to such a general structure:

(D7)S[k, 0]S[k, 1]
S[7,s]
S[6,s]
S[5,s]
S[4,s]
S[3,s]
S[2,s]
S[1,s]

and the associated professor’s announcement is the following:

(S7)An examination will occur in the next week but the date of the examination will constitute a surprise.

At this step, it appears that we also get a version of SEP for n = 1 which satisfies this definition. The structure associated with 1-SEP(I□) is as follows:

(D8)S[1, 0]S[1, 1]
S[1,s]

which corresponds to the following professor’s announcement:

(S8)An examination will occur on tomorrow but the date of the examination will constitute a surprise.

Thus, 1-SEP(I□) is the minimal version of SEP which satisfies not only the above condition, but also the base step (C3) according to which the non-surprise must possibly occur on the last day, as well as the validation step (C4) in virtue of which the examination can finally occur by surprise. Moreover, it is a variation which excludes, by its intrinsic structure, the emergence of the version of SEP based on a concept of surprise corresponding to a disjoint structure. For this reason, (D8) can be regarded as the canonical form of SEP(I□). Thus, it is the genuine core of SEP(I□) and in what follows, we will thus endeavour to reason on 1-SEP(I□).

At this stage, it is worth attempting to provide a solution to SEP(I□). For that purpose, let us recall first Quine’s solution. The solution to SEP proposed by Quine (1953) is well-known. Quine highlights the fact that the student eliminates successively the days n, n -1…, 1, by a reasoning based on backward-induction and concludes then that the examination will not take place during the week. The student reasons as follows. On day n, I will predict that the examination will take place on day n, and consequently the examination cannot take place on day n; on day n -1, I will predict that the examination will take place on day n-1, and consequently the examination cannot take place on day n -1; …; on day 1, I will predict that the examination will take place on day 1, and consequently the examination cannot take place on day 1. Finally, the student concludes that the examination will take place on no day of the week. But this last conclusion finally makes it possible to the examination to occur surprisingly, including on day n. According to Quine, the error in the student’s reasoning lies precisely in the fact of not having taken into account this possibility since the beginning, which would then have prevented the fallacious reasoning.6

Quine, in addition, directly applies his analysis to the canonical form 1-SEP(I□), where the corresponding statement is that of (S8). In this case, the error of the student lies, according to Quine, in the fact of having considered only the single following assumption: (a) “the examination will take place tomorrow and I will predict that it will take place”. In fact, the student should have also considered three other cases: (b) “the examination will not take place tomorrow and I will predict that it will take place”; (c) “the examination will not take place tomorrow and I will not predict that it will take place”; (d) “the examination will take place tomorrow and I will not predict that it will take place”. And the fact of considering the assumption (a) but also the assumption (d) which is compatible with the professor’s announcement would have prevented the student from concluding that the examination would not finally take place.7 Consequently, it is the fact of having taken into account only the hypothesis (a) which can be identified as the cause of the fallacious reasoning. Thus, the student did only take partially into account the whole set of hypotheses resulting from the professor’s announcement. If he had apprehended the totality of the relevant hypotheses compatible with the professor’s announcement, he would not have concluded fallaciously that the examination would not take place during the week.

At this stage, it proves to be useful to describe the student’s reasoning in terms of reconstitution of a matrix. For one can consider that the student’s reasoning classically based on backward-induction leads to reconstitute the matrix corresponding to the concept of surprise in the following way:

(D9)S[1, 0]S[1, 1]
S[1,s]

In reality, he should have considered that the correct way to reconstitute this latter matrix is the following :

(D8)S[1, 0]S[1, 1]
S[1,s]

4.2 The definition associated with the triangular matrix and Hall’s reduction

As we have seen, Quine’s solution applies directly to SEP(I□), i.e. to a version of SEP based on a conjoint definition of the surprise and a rectangular matrix. It is now worth being interested in some variations of SEP based on a conjoint definition where the structure of the corresponding matrix is not rectangular, but which satisfies however the conditions for the emergence of the paradox mentioned above, namely the presence of the base step (C3) and the validation step (C4). Such matrices have a structure that can be described as triangular. Let thus SEP(I∆) be the corresponding version.

Let us consider first 7-SEP, where the structure of the possible cases of non-surprise and of surprise corresponds to the matrix below:

(D10)S[k, 0]S[k, 1]
S[7,s]
S[6,s]
S[5,s]
S[4,s]
S[3,s]
S[2,s]
S[1,s]

and to the following announcement of the professor

(S10)An examination will occur in the next week but the date of the examination will constitute a surprise. Moreover, the fact that the examination will take place constitutes an absolute certainty.

Such an announcement appears identical to the preceding statement to which the Quine’s solution applies, with however an important difference: the student has from now on the certainty that the examination will occur. And this has the effect of preventing him/her from questioning the fact that the examination can take place, and of making thus impossible the surprise to occur on the last day. For this reason, we note S[7, 1] = 0 in the corresponding matrix. The general structure corresponding to this type of definition is:

(D11)S[k, 0]S[k, 1]
S[n,s]
S[n-1,s]
………………………………

And similarly, one can consider the following canonical structure (from where the denomination of triangular structure finds its justification), which is that of SEP(I∆) and which corresponds thus to 2-SEP(I∆):

(D12)S[k, 0]S[k, 1]
S[2,s]
S[1,s]

Such a structure corresponds to the following announcement of the professor:

(S12)An examination will occur on the next two days, but the date of the examination will constitute a surprise. Moreover, the fact that the examination will take place constitutes an absolute certainty.

As we see it, the additional clause of the statement according to which it is absolutely certain that the examination will occur prevents here the surprise of occurring on the last day. Such a version corresponds in particular to the variation of SEP described by A. J. Ayer. The latter version corresponds to a player, who is authorized to check, before a set of playing cards is mixed, that it contains the ace, the 2, 3…, 7 of Spades. And it is announced that the player that he will not be able to envisage in advance justifiably, when the ace of Spades will be uncovered. Finally the cards, initially hidden, are uncovered one by one. The purpose of such a version is to render impossible, before the 7th card being uncovered, the belief according to which the ace of Spades will not be uncovered. And this has the effect of forbidding to Quine’ solution to apply on the last day.

It is now worth presenting a solution to the versions of SEP associated with the structures corresponding to (D11). Such a solution is based on a reduction recently exposed by Ned Hall, of which it is worth beforehand highlighting the context. In the version of SEP under consideration by Quine (1953), it appears clearly that the fact that the student doubts that the examination will well take place during the week, at a certain stage of the reasoning, is authorized. Quine thus places himself deliberately in a situation where the student has the faculty of doubting that the examination will truly occur during the week. The versions described by Ayer (1973), Janaway (1989) but also Scriven (1951) reveal the intention to prevent this particular step in the student’s reasoning. Such scenarios correspond, in spirit, to SEP(I∆). One can also attach to it the variation of the Designated Student Paradox described by Sorensen (1982, 357)8, where five stars – a gold star and four silver stars – are attributed to five students, given that it is indubitable that the gold star is placed on the back of the student who was designated.

However, Ned Hall (1999, 659-660) recently exposed a reduction, which tends to refute the objections classically raised against Quine’s solution. The argumentation developed by Hall is as follows:

We should pause, briefly, to dispense with a bad – though oft-cited – reason for rejecting Quine’s diagnosis. (See for example Ayer 1973 and Janaway 1989). Begin with the perfectly sound observation that the story can be told in such a way that the student is justified in believing that, come Friday, he will justifiably believe that an exam is scheduled for the week. Just add a second Iron Law of the School : that there must be at least one exam each week. (…) Then the first step of the student’s argument goes through just fine. So Quine’s diagnosis is, evidently, inapplicable.

Perhaps – but in letter only, not in spirit. With the second Iron Law in place, the last disjunct of the professor’s announcement – that E5 & J(E5) – is, from the student’s perspective, a contradiction. So, from his perspective, the content of her announcement is given not by SE5 but by SE4 : (E1 & J1(E1))  …  (E4 & J4(E4)). And now Quine’s diagnosis applies straightforwardly : he should simply insist that the student is not justified in believing the announcement and so, come Thursday morning, not justified in believing that crucial part of it which asserts that if the exam is on Friday then it will come as a surprise – which, from the student’s perspective, is tantamount to asserting that the exam is scheduled for one of Monday through Thursday. That is, Quine should insist that the crucial premise that J4(E1  E2  E3  E4) is false – which is exactly the diagnosis he gives to an ordinary 4-day surprise exam scenario. Oddly, it seems to have gone entirely unnoticed by those who press this variant of the story against Quine that its only real effect is to convert an n-day scenario into an n-1 day scenario.

Hall puts then in parallel two types of situations. The first corresponds to the situation in which Quine’s analysis finds classically to apply. The second corresponds to the type of situation under consideration by the opponents to Quine’s solution and in particular by Ayer (1973) and Janaway (1989). On this last hypothesis, a stronger version of SEP is taken into account, where one second Iron Law of the School is considered and it is given that the examination will necessarily take place during the week. The argumentation developed by Hall leads to the reduction of a version of n-SEP of the second type to a version of (n-1)-SEP of the quinean type. This equivalence has the effect of annihilating the objections of the opponents to Quine’s solution.9 For the effect of this reduction is to make it finally possible to Quine’s solution to apply in the situations described by Ayer and Janaway. In spirit, the scenario under consideration by Ayer and Janaway corresponds thus to a situation where the surprise is not possible on day n (i.e. S[n, 1] = 0). This has indeed the effect of neutralizing Quine’s solution based on n-SEP(I□). But Hall’s reduction then makes it possible to Quine’s solution to apply to (n-1)-SEP(I□). The effect of Hall’s reduction is thus of reducing a scenario corresponding to (D11) to a situation based on (D8). Consequently, Hall’s reduction makes it possible to reduce n-SEP(I∆) to (n-1)-SEP(I□). It results from it that any version of SEP(I∆) for one n-period reduces to a version of SEP(I□) for one (n-1)-period (formally n-SEP(I∆)  (n-1)-SEP(I□) for n > 1). Thus, Hall’s reduction makes it finally possible to apply Quine’s solution to SEP(I∆).10

4. The surprise notion corresponding to the disjoint structure

It is worth considering, second, the case where the notion of surprise is based on a disjoint structure of the possible cases of non-surprise and of surprise. Let SEP(II) be the corresponding version. Intuitively, such a variation corresponds to a situation where for a given day of the n-period, it is not possible to have at the same time the non-surprise and the surprise. The structure of the associated matrix is such that one has exclusively on each day, either the non-surprise or the surprise.

At this step, it appears that a preliminary question can be raised: can’t Quine’s solution apply all the same to SEP(II)? However, the preceding analysis of SEP(I) shows that a necessary condition in order to Quine’s solution to apply is that there exists during the n-period at least one day when the non-surprise and the surprise are at the same time possible. However such a property is that of a conjoint structure and corresponds to the situation which is that of SEP(I). But in the context of a disjoint structure, the associated matrix, in contrast, verifies k S[k, 0] + S[k, 1] = 1. Consequently, this forbids Quine’s solution to apply to SEP(II).

In the same way, one could wonder whether Hall’s reduction wouldn’t also apply to SEP(II). Thus, isn’t there a reduction of SEP(II) for a n-period to SEP(I) for a (n – 1)-period? It also appears that not. Indeed, as we did see it, Quine’s solution cannot apply to SEP(II). However, the effect of Hall’s reduction is to reduce a given scenario to a situation where Quine’s solution finally finds to apply. But, since Quine’s solution cannot apply in the context of SEP(II), Hall’s reduction is also unable to produce its effect.

Given that Quine’s solution does not apply to SEP(II), it is now worth attempting to provide an adequate solution to the version of SEP corresponding to a concept of surprise associated with a disjoint structure of the cases of non-surprise and of surprise. To this end, it proves to be necessary to describe a version of SEP corresponding to a disjoint structure, as well as the structure corresponding to the canonical version of SEP(II).

In a preliminary way, one can observe that the minimal version corresponding to a disjoint version of SEP is that which is associated with the following structure, i.e. 2-SEP(II):

(D13)S[1, 0]S[1, 1]
S[2,s]
S[1,s]

However, for reasons that will become clearer later, the corresponding version of SEP(II) does not have a sufficient degree of realism and of plausibility to constitute a genuine version of SEP, i.e. such that it is susceptible of inducing in error our reasoning.

In order to highlight the canonical version of SEP(II) and the corresponding statement, it is first of all worth mentioning the remark, made by several authors11, according to which the paradox emerges clearly, in the case of SEP(II), when n is large. An interesting characteristic of SEP(II) is indeed that the paradox emerges intuitively in a clearer way when great values of n are taken into account. A striking illustration of this phenomenon is thus provided to us by the variation of the paradox which corresponds to the following situation, described by Timothy Williamson (2000, 139):

Advance knowledge that there will be a test, fire drill, or the like of which one will not know the time in advance is an everyday fact of social life, but one denied by a surprising proportion of early work on the Surprise Examination. Who has not waited for the telephone to ring, knowing that it will do so within a week and that one will not know a second before it rings that it will ring a second later?

The variation suggested by Williamson corresponds to the announcement made to somebody that he will receive a phone call during the week, without being able however to determine in advance at which precise second the phone call will occur. This variation underlines how the surprise can appear, in a completely plausible way, when the value of n is high. The unit of time considered by Williamson is here the second, associated with a period which corresponds to one week. The corresponding value of n is here very high and equals 604800 (60 x 60 x 24 x 7) seconds. This illustrates how a great value of n makes it possible to the corresponding variation of SEP(II) to take place in both a plausible and realistic way. However, taking into account such large value of n is not indeed essential. In effect, a value of n which equals, for example, 365, seems appropriate as well. In this context, the professor’s announcement which corresponds to a disjoint structure is then the following:

(S14)An examination will occur during this year but the date of the examination will constitute a surprise.

The corresponding definition presents then the following structure :

(D14)S[1, 0]S[1, 1]
S[365,s]
………………………………
S[1,s]

which is an instance of the following general form :

(D15)S[1, 0]S[1, 1]
S[n,s]
………………………………
S[1,s]

This last structure can be considered as corresponding to the canonical version of SEP(II), with n large. In the specific situation associated with this version of SEP, the student predicts each day – in a false way but justified by a reasoning based on backward-induction – that the examination will take place on no day of the week. But it appears that at least one case of surprise (for example if the examination occurs on the first day) makes it possible to validate, in a completely realistic way, the professor’s announcement..

The form of SEP(II) which applies to the standard version of SEP is 7-SEP(II), which corresponds to the classical announcement:

(S7)An examination will occur on the next week but the date of the examination will constitute a surprise.

but with this difference with the standard version that the context is here exclusively that of a concept of surprised associated with a disjoint structure.

At this stage, we are in a position to determine the fallacious step in the student’s reasoning. For that, it is useful to describe the student’s reasoning in terms of matrix reconstitution. The student’s reasoning indeed leads him/her to attribute a value for S[k, 0] and S[k, 1]. And when he is informed of the professor’s announcement, the student’s reasoning indeed leads him/her to rebuild the corresponding matrix such that all S[k, 0] = 1 and all S[k, 1] = 0, in the following way (for n = 7):

(D16)S[k, 0]S[k, 1]
S[7,s]
S[6,s]
S[5,s]
S[4,s]
S[3,s]
S[2,s]
S[1,s]

One can notice here that the order of reconstitution proves to be indifferent. At this stage, we are in a position to identify the flaw which is at the origin of the erroneous conclusion of the student. It appears indeed that the student did not take into account the fact that the surprise corresponds here to a disjoint structure. Indeed, he should have considered here that the last day corresponds to a proper instance of non-surprise and thus that S[n, 0] = 1. In the same way, he should have considered that the 1st day12 corresponds to a proper instance of surprise and should have thus posed S[1, 1] = 1. The context being that of a disjoint structure, he could have legitimately added, in a second step, that S[n, 1] = 0 and S[1, 0] = 0. At this stage, the partially reconstituted matrix would then have been as follows:

(D17)S[k, 0]S[k, 1]
S[7,s]
S[6,s]
S[5,s]
S[4,s]
S[3,s]
S[2,s]
S[1,s]

The student should then have continued his reasoning as follows. The proper instances of non-surprise and of surprise which are disjoint here do not capture entirely the concept of surprise. In such context, the concept of surprise is not captured exhaustively by the extension and the anti-extension of the surprise. However, such a definition is in conformity with the definition of a vague predicate, which characterizes itself by an extension and an anti-extension which are mutually exclusive and non-exhaustive13. Thus, the surprise notion associated with a disjoint structure is a vague one.

What precedes now makes it possible to identify accurately the flaw in the student’s reasoning, when the surprise notion is a vague notion associated with a disjoint structure. For the error which is at the origin of the student’s fallacious reasoning lies in lack of taking into account the fact that the surprise corresponds in the case of a disjoint structure, to a vague concept, and thus comprises the presence of a penumbral zone corresponding to borderline cases between the non-surprise and the surprise. There is no need however to have here at our disposal a solution to the sorites paradox. Indeed, whether these borderline cases result from a succession of intermediate degrees, from a precise cut-off between the non-surprise and the surprise whose exact location is impossible for us to know, etc. is of little importance here. For in all cases, the mere fact of taking into account the fact that the concept of surprise is here a concept vague forbids to conclude that S[k, 1] = 0, for all values of k.

Several ways thus exist to reconstitute the matrix in accordance with what precedes. In fact, there exists as many ways of reconstituting the latter than there are conceptions of vagueness. One in these ways (based on a conception of vagueness based on fuzzy logic) consists in considering that there exists a continuous and gradual succession from the non-surprise to the surprise. The corresponding algorithm to reconstitute the matrix is then the one where the step is given by the formula 1/(np) when p corresponds to a proper instance of surprise. For p = 3, we have here 1/(7-3) = 0,25, with S[3, 1] = 1. And the corresponding matrix is thus the following one:

(D18)S[k, 0]S[k, 1]
S[7,s]10
S[6,s]0,750,25
S[5,s]0,50,5
S[4,s]0,250,75
S[3,s]01
S[2,s]01
S[1,s]01

where the sum of the values of the matrix associated with a day given is equal to 1. The intuition which governs SEP (II) is here that the non-surprise is total on day n, but that there exists intermediate degrees of surprise si (0 < si < 1), such as the more one approaches the last day, the higher the effect of non-surprise. Conversely, the effect of surprise is total on the first days, for example on days 1, 2 and 3.

One can notice here that the definitions corresponding to SEP (II) which have just been described, are such that they present a property of linearity (formally, k (for 1 < kn), S[k, 0]  S[k-1, 0]). It appears indeed that a structure corresponding to the possible cases of non-surprise and of surprise which would not present such a property of linearity, would not capture the intuition corresponding to the concept of surprise. For this reason, it appears sufficient to limit the present study to the structures of definitions that satisfy this property of linearity.

An alternative way to reconstitute the corresponding matrix, based on the epistemological conception of vagueness, could also have been used. It consists of the case where the vague nature of the surprise is determined by the existence of a precise cut-off between the cases of non-surprise and of surprise, of which it is however not possible for us to know the exact location. In this case, the matrix could have been reconstituted, for example, as follows:

(D19)S[k, 0]S[k, 1]
S[7,s]
S[6,s]
S[5,s]
S[4,s]
S[3,s]
S[2,s]
S[1,s]

At this stage, one can wonder whether the version of the paradox associated with SEP(II) cannot be assimilated with the sorites paradox. The reduction of SEP to the sorites paradox is indeed the solution which has been proposed by some authors, notably Dietl (1973) and Smith (1984). The latter solutions, based on the assimilation of SEP to the sorites paradox, constitute monist analyses, which do not lead, to the difference of the present solution, to two independent solutions based on two structurally different versions of SEP. In addition, with regard to the analyses suggested by Dietl and Smith, it does not clearly appear whether each step of SEP is fully comparable to the corresponding step of the sorites paradox, as underlined by Sorensen.14 But in the context of a conception of surprise corresponding to a disjoint structure, the fact that the last day corresponds to a proper instance of non-surprise can be assimilated here to the base step of the sorites paradox.

Nevertheless, it appears that such a reduction of SEP to the sorites paradox, limited to the notion of surprise corresponding to a disjoint structure, does not prevail here. On the one hand, it does not appear clearly if the statement of SEP can be translated into a variation of the sorites paradox, in particular for what concerns 7-SEP(II). Because the corresponding variation of the sorites paradox would run too fast, as already noted by Sorensen (1988).15 It is also noticeable, moreover, as pointed out by Scott Soames (1999), than certain vague predicates are not likely to give rise to a corresponding version of the sorites paradox. Such appears well to be the case for the concept of surprise associated with 7-SEP(II). Because as Soames16 points out, the continuum which is semantically associated with the predicates giving rise to the sorites paradox, can be fragmented in units so small that if one of these units is intuitively F, then the following unit is also F. But such is not the case with the variation consisting in 7-SEP(II), where the corresponding units (1 day) are not fine enough with regard to the considered period (7 days).

Lastly and overall, as mentioned above, the preceding solution to SEP(II) applies, whatever the nature of the solution which will be adopted for the sorites paradox. For it is the ignorance of the semantic structure of the vague notion of surprise which is at the origin of the student’s fallacious reasoning in the case of SEP(II). And this fact is independent of the solution which should be provided, in a near or far future, to the sorites paradox – whether this approach be of epistemological inspiration, supervaluationnist, based on fuzzy logic…, or of a very different nature.

5. The solution to the paradox

The above developments make it possible now to formulate an accurate solution to the surprise examination paradox. The latter solution can be stated by considering what should have been the student’s reasoning. Let us consider indeed, in the light of the present analysis, how the student should have reasoned, after having heard the professor’s announcement:

– The student: Professor, I think that two semantically distinct conceptions of surprise, which are likely to influence the reasoning to hold, can be taken into account. I also observe that you did not specify, at the time of your announcement, to which of these two conceptions you referred. Isn’t it?

– The professor: Yes, it is exact. Continue.

– The student: Since you refer indifferently to one or the other of these conceptions of surprise, it is necessary to consider each one of them successively, as well as the reasoning to be held in each case.

– The professor: Thus let us see that.

– The student: Let us consider, on the one hand, the case where the surprise corresponds to a conjoint definition of the cases of non-surprise and of surprise. Such a definition is such that the non-surprise and the surprise are possible at the same time, for example on the last day. Such a situation is likely to arise on the last day, in particular when a student concludes that the examination cannot take place on this same last day, since that would contradict the professor’s announcement. However, this precisely causes to make it possible for the surprise to occur, because this same student then expects that the examination will not take place. And in a completely plausible way, as put forth by Quine, such a situation corresponds then to a case of surprise. In this case, the fact of taking into account the possibility that the examination can occur surprisingly on the last day, prohibits eliminating successively the days n, n-1, n-2, …, 2, and 1. In addition, the concept of surprise associated with a conjoint structure is a concept of total surprise. For one faces on the last day either the non-surprise or the total surprise, without there existing in this case some intermediate situations.

– The professor: I see that. You did mention a second case of surprise…

– The student: Indeed. It is also necessary to consider the case where the surprise corresponds to a disjoint definition of the cases of non-surprise and of surprise. Such a definition corresponds to the case where the non-surprise and the surprise are not possible on the same day. The intuition on which such a conception of the surprise rests corresponds to the announcement made to students that they will undergo an examination in the year, while being moreover unaware of the precise day where it will be held. In such a case, it results well from our experience that the examination can truly occur surprisingly, on many days of the year, for example on whatever day of the first three months. It is an actual situation that can be experienced by any student. Of course, in the announcement that you have just made to us, the period is not as long as one year, but corresponds to one week. However, your announcement also leaves place to such a conception of surprise associated with a disjoint structure of the cases of non-surprise and of surprise. Indeed, the examination can indeed occur surprisingly, for example on the 1st day of the week. Thus, the 1st day constitutes a proper instance of surprise. In parallel, the last day constitutes a proper instance of non-surprise, since it results from the announcement that the examination cannot take place surprisingly on this day. At this stage, it also appears that the status of the other days of the corresponding period is not determined. Thus, such a disjoint structure of the cases of non-surprise and of surprise is at the same time disjoint and non-exhaustive. Consequently, the concept of corresponding surprise presents here the criteria of a vague notion. And this casts light on the fact that the concept of surprise associated with a conjoint structure is a vague one, and that there is thus a zone of penumbra between the proper instances of non-surprise and of surprise, which corresponds to the existence of borderline cases. And the mere existence of these borderline cases prohibits to eliminate successively, by a reasoning based on backward-induction, the days n, n-1, n-2, …, 2, and then 1. And I finally notice, to the difference of the preceding concept of surprise, that the concept of surprise associated with a conjoint structure leads to the existence of intermediate cases between the non-surprise and the surprise.

– The professor: I see. Conclude now.

– The student: Finally, the fact of considering successively two different concepts of surprise being able to correspond to the announcement which you have just made, resulted in both cases in rejecting the classical reasoning which results in eliminating successively all days of the week. Here, the motivation to reject the traditional reasoning appears different for each of these two concepts of surprise. But in both cases, a convergent conclusion ensues which leads to the rejection of the classical reasoning based on backward-induction.

6. Conclusion

I shall mention finally that the solution which has been just proposed also applies to the variations of SEP mentioned by Sorensen (1982). Indeed, the structure of the canonical forms of SEP(I□), SEP(I∆) or SEP(II) indicates that whatever the version taken into account, the solution which applies does not require to make use of whatever principle of temporal retention. It is also independent of the order of elimination and can finally apply when the duration of the n-period is unknown at the time of the professor’s announcement.

Lastly, it is worth mentioning that the strategy adopted in the present study appears structurally similar to the one used in Franceschi (1999): first, establish a dichotomy which makes it possible to divide the given problem into two distinct classes; second, show that each resulting version admits of a specific resolution.17 In a similar way, in the present analysis of SEP, a dichotomy is made and the two resulting categories of problems lead then to an independent resolution. This suggests that the fact that two structurally independent versions are inextricably entangled in philosophical paradoxes could be a more widespread characteristic than one could think at first glance and could also partly explain their intrinsic difficulty.18

REFERENCES

AYER, A. J. 1973, “On a Supposed Antinomy”, Mind 82, pp. 125-126.
BINKLEY, R. 1968, “The Surprise Examination in Modal Logic”, Journal of Philosophy 65, pp. 127-136.
CHALMERS, D. 2002, “The St. Petersburg two-envelope paradox”, Analysis 62, pp. 155-157.
CHOW, T. Y. 1998, “The Surprise Examination or Unexpected Hanging Paradox”, The American Mathematical Monthly 105, pp. 41-51.
DIETL, P. 1973, “The Surprise Examination”, Educational Theory 23, pp. 153-158.
FRANCESCHI, P. 1999, “Comment l’urne de Carter et Leslie se déverse dans celle de Hempel”, Canadian Journal of Philosophy 29, pp. 139-156. English translation.
HALL, N. 1999, “How to Set a Surprise Exam”, Mind 108, pp. 647-703.
HYDE, D. 2002 “Sorites Paradox”, The Stanford Encyclopedia of Philosophy (Fall 2002 Edition), E. N. Zalta (ed.), http ://plato.stanford.edu/archives/fall2002/entries/sorites-paradox.
JANAWAY, C. 1989, “Knowing About Surprises : A Supposed Antinomy Revisited”, Mind 98, pp. 391-410.
MONTAGUE, R. & KAPLAN, D. 1960, “A Paradox Regained”, Notre Dame Journal of Formal Logic 3, pp. 79-90.
O’ CONNOR, D. 1948, “Pragmatic paradoxes”, Mind 57, pp. 358-359.
QUINE, W. 1953, “On a So-called Paradox”, Mind 62, pp. 65-66.
SAINSBURY, R. M. 1995, Paradoxes, 2ème édition, Cambridge : Cambridge University Press.
SCRIVEN, M. 1951, “Paradoxical announcements”, Mind 60, pp. 403-407.
SHAW, R. 1958, “The Paradox of the Unexpected Examination”, Mind 67, pp. 382-384.
SMITH, J. W. 1984, “The surprise examination on the paradox of the heap”, Philosophical Papers 13, pp. 43-56.
SOAMES, S. 1999, Understanding Truth, New York, Oxford : Oxford University Press.
SORENSEN, R. A. 1982, “Recalcitrant versions of the prediction paradox”, Australasian Journal of Philosophy 69, pp. 355-362.
SORENSEN, R. A. 1988, Blindspots, Oxford : Clarendon Press.
WILLIAMSON, T. 2000, Knowledge and its Limits, London & New York : Routledge.
WRIGHT, C. & SUDBURY, A. 1977, “The Paradox of the Unexpected Examination”, Australasian Journal of Philosophy 55, pp. 41-58.

1 I simplify here considerably.

2 Without pretending to being exhaustive.

3 In what follows, n denotes the last day of the term corresponding to the professor’s announcement.

4 Let 1-SEP, 2-SEP,…, n-SEP be the problem for respectively 1 day, 2 days,…, n days.

5 The cases where neither the non-surprise nor the surprise are made possible on the same day (i.e. such that S[k, 0] + S[k, 1] = 0) can be purely and simply ignored.

6 Cf. (1953, 65) : ‘It is notable that K acquiesces in the conclusion (wrong, according to the fable of the Thursday hanging) that the decree will not be fulfilled. If this is a conclusion which he is prepared to accept (though wrongly) in the end as a certainty, it is an alternative which he should have been prepared to take into consideration from the beginning as a possibility.’

7 Cf. (1953, 66) : ‘If K had reasoned correctly, Sunday afternoon, he would have reasoned as follows : “We must distinguish four cases : first, that I shall be hanged tomorrow noon and I know it now (but I do not) ; second, that I shall be unhanged tomorrow noon and do not know it now (but I do not) ; third, that I shall be unhanged tomorrow noon and know it now ; and fourth, that I shall be hanged tomorrow noon and do not know it now. The latter two alternatives are the open possibilities, and the last of all would fulfill the decree. Rather than charging the judge with self-contradiction, let me suspend judgment and hope for the best.”‘

8 ‘The students are then shown four silver stars and one gold star. One star is put on the back of each student.’.

9 Hall refutes otherwise, but on different grounds, the solution proposed by Quine.

10 Hall’s reduction can be easily generalised. It is then associated with a version of n-SEP(I∆) such that the surprise will not possibly occur on the m last days of the week. Such a version is associated with a matrix such that (a) m (1  m < n) and S[nm, 0] = S[nm, 1] = 1 ; (b) p > nm S[p, 0] = 1 and S[p, 1] = 0 ; (c) q < nm S[q, 0] = S[q, 1] = 1. In this new situation, a generalised Hall’s reduction applies to the corresponding version of SEP. In this case, the extended Hall’s reduction leads to : n-SEP(I∆)  (nm)-SEP(I□).

11 Cf. notably Hall (1999, 661), Williamson (2000).

12 It is just an example. Alternatively, one could have chosen here the 2nd or the 3rd day.

13 This definition of a vague predicate is borrowed from Soames. Considering the extension and the anti-extension of a vague predicate, Soames (1999, 210) points out thus: “These two classes are mutually exclusive, though not jointly exhaustive”.

14 Cf. Sorensen (1988, 292-293) : ‘Indeed, no one has simply asserted that the following is just another instance of the sorites.

i. Base step : The audience can know that the exercise will not occur on the last day.

ii. Induction step : If the audience can know that the exercise will not occur on day n, then they can also know that the exercise will not occur on day n – 1

iii. The audience can know that there is no day on which the exercise will occur.

Why not blame the whole puzzle on the vagueness of ‘can know’? (…) Despite its attractiveness, I have not found any clear examples of this strategy.’

15 Cf. (1988, 324): ‘One immediate qualm about assimilating the prediction paradox to the sorites is that the prediction paradox would be a very ‘fast’ sorites. (…) Yet standard sorites arguments involve a great many borderline cases.’

16 Cf. Soames (1999, 218): ‘A further fact about Sorites predicates is that the continuum semantically associated with such a predicate can be broken down into units fine enough so that once one has characterized one item as F (or not F), it is virtually irresistible to characterize the same item in the same way’.

17 One characteristic example of this type of analysis is also exemplified by the solution to the two-envelope paradox described by David Chalmers (2002, 157) : ‘The upshot is a disjunctive diagnosis of the two-envelope paradox. The expected value of the amount in the envelopes is either finite or infinite. If it is finite, then (1) and (2) are false (…). If it is infinite, then the step from (2) to (3) is invalid (…)’.

18 I am grateful toward Timothy Chow, Ned Hall, Claude Panaccio and the anonymous referees for very useful comments concerning previous versions of this paper.

Theory of Cognitive Distortions: Over-generalisation and Labeling

English translation of a paper published in French in the Journal de Thérapie Comportementale et Cognitive, 2009, 19-4, pages 136-140 under the title “Théorie des distorsions cognitives : la sur-généralisation et l’étiquetage”.

In a previous article (Complements to a theory of cognitive distorsions, Journal de Thérapie Comportementale et Cognitive, 2007), we introduced some elements aimed at contributing to a general theory of cognitive distortions. Based on the reference class, the duality and the system of taxa, these elements allow to define the general cognitive distortions as well as the specific cognitive distortions. This model is extended here to the description of two other classical cognitive distortions: over-generalisation and mislabelling. The definition of the two latter cognitive distortions is based on prior differentiation between three levels of reasoning: primary, secondary and ternary pathogenic arguments. The latter analysis also leads to define two other cognitive distortions which insert themselves into this framework: ill-grounded inductive projection and confirmation bias.

This article is cited in:

  • Richard-Lepouriel H., Bipolar disorder, self-stigma and cognitive restructuring: A first attempt to support, Journal de Therapie Comportementale et Cognitive, Volume 27, 2017
  • Juliette Marty, Et si vous étiez trop perfectionniste ? Eyrolles, 2021

Theory of Cognitive Distortions: Over-generalisation and Labeling

In Franceschi (2007), we set out to introduce several elements aimed at contributing to a general theory of cognitive distortions. These elements are based on three fundamental notions: the reference class, the duality and the system of taxa. With these three elements, we could define within the same conceptual framework the following general cognitive distortions: dichotomous reasoning, disqualification of one pole, minimisation and maximisation, requalification in the other pole and omission of the neutral. We could also describe as specific cognitive distortions: disqualification of the positive, selective abstraction and catastrophism. In the present article, we offer to define and to situate, within the same conceptual framework, two other classical cognitive distortions: over-generalisation and mislabelling.

Over-generalisation and mislabelling constitute two of the twelve traditionally defined cognitive distortions: emotional reasoning; over-generalisation; arbitrary inference; dichotomous reasoning; should statements; divination or mental reading; selective abstraction; disqualification of the positive; maximisation/minimisation; catastrophising; personalisation; mislabelling (Beck 1964, Ellis 1962). Over-generalisation is classically defined as a rough and ill-grounded generalisation, usually including either of the quantifiers “all”, “none”, “never”, “always”. Moreover, it is often described as a cognitive distortion including four subcategories: dichotomous reasoning, selective abstraction, maximisation/minimisation, and disqualification of the positive. Mislabelling is also classically defined as an extreme form of over-generalisation, consisting in the apposition of a label with a strong negative and emotional connotation to oneself or to an external subject.

1. Primary, secondary and ternary pathogenic arguments

Before setting out to define over-generalisation and mislabelling in the present context, it is worth describing preliminarily a structure of pathogenic reasoning (in the etymological sense: engendering suffering), with a general scope, susceptible of being found in some disorders of a very different nature, such as depression, generalised anxiety disorder, body dismorphic disorder, scrupulosity or intermittent explosive disorder. Such structure of reasoning includes several levels of arguments: primary, secondary and ternary. In a simplified way, primary pathogenic arguments are constituted by an enumeration of instances. Secondary pathogenic arguments consist of a generalisation from the latter instances. Lastly, pathogenic ternary arguments are constituted by an interpretation of the latter generalisation. Such reasoning as a whole presents an inductive structure.

At this stage, it is worth mentioning several instances of this type of reasoning. A first instance, susceptible to be found in depression (Beck 1967, 1987), is the following (the  symbol denotes the conclusion):

(11)I gave my ankle a wrench last Januarypremise1
(12)I lost my job last Februarypremise2
(13)Fifteen days ago, I had an influenza with feverpremise3
(14)I got into an argument with Lucy last monthpremise4
(…)(…)
(110)Today, my horoscope is not goodpremise10
(2)∴ Everything that occurs to me is badfrom (11)-(110)
(3)∴ I am a complete failure!from (2)

The patient enumerates first some events of his/her past and present life (11)-(110), that he/she qualifies as negative, through a primary stage which consists of an enumeration of instances. Then he/she performs a generalisation (2) from the previous enumeration, which presents the following structure:

(2)∴ All events that occur to me are negativefrom (11)-(110)

Lastly, the patient interprets (3) the latter conclusion by concluding “I am a complete failure!”. Such instance applies then to the reference class of the present and past events of the patient’s life and to the Positive/Negative duality.

One can also mention a reasoning that presents an identical structure, which is susceptible to be met in body dysmorphic disorder (Veale 2004, Rabinowitz & al. 2007). The patient enumerates then different parts of his/her body, which he/she qualifies as ugly. He/she generalises then by concluding that all parts of his/her body are ugly. Finally, he/she adds: “I am ugly!”. The corresponding reasoning applies then to the Beautiful/Ugly duality and to the reference class of the parts of the patient’s body.

In the same way, in a reasoning of identical structure, susceptible to be met in scrupulosity (Teak & Ulug 2001, Miller & Edges 2007), the patient enumerates several instances corresponding to some acts which he/she made previously or recently, and which he/she considers as morally bad. He/she concludes then: “Everything I do is bad, morally reprehensible”, and he/she further interprets it by concluding: “I am a horrible sinner!”. Such conclusion is likely to trigger an intense feel of guilt and a compulsive practice of religious rituals. The corresponding instance applies here to the duality Good/Evil and to the reference class of the present and past actions of the patient’s life.

Lastly, an instance of this structure of reasoning can contribute to the development of hostility, of a potentially aggressive attitude toward other people. In that case, the patient concludes regarding an external subject: “All acts that he committed toward me are bad”. He/she concludes then: “He is a bastard!”. Such conclusion can then play a role in intermittent explosive disorder (Coccaro & al. 1998, Galovski & al. 2002). In such case, the over-generalisation applies to the Good/Evil duality and to the reference class of the actions of an external subject with regard to the patient.

At this step, it is worth describing in more detail each of the three stages – primary, secondary and ternary – which compose this type of reasoning.

Primary pathogenic arguments

The first step in the aforementioned type of reasoning, consists for the patient to enumerate some instances. The general structure of each instance is as follows:

(1i)The object xi of the class of reference E has property Ā (in the duality A/Ā)premisei

In the aforementioned example applied to depression, the patient enumerates some events of his/her present and past life, which he/she qualifies as negative, under the form:

(1i)The event Ei of negative nature occurred to mepremisei

Different instances corresponding to this cognitive process can be described under the form of a primary pathogenic argument, the structure of which is the following:

(1a)The event E1 occurred to mepremise
(1b)The event E1 was of a negative naturepremise
(1)∴ The event E1 of a negative nature occurred to mefrom (1a), (1b)

By such cognitive process, the patient is led to the conclusion according to which some negative event did occur to him/her.

From a deductive point of view, this type of argument proves to be completely valid (the conclusion is true if the premises are true) since the very event presents well, objectively, a negative nature. However, this type of primary argument can turn out to be fallacious, when the very event presents, objectively, a positive or neutral nature. The flaw in the reasoning resides then in the fact that the premise (1b) turns then out to be false. Such can be case for example if the patient makes use of a specific cognitive distortion such as requalification in the negative. In such case, the patient considers as negative an event the nature of which is objectively positive.

Secondary pathogenic arguments

At the level of the above-mentioned reasoning, secondary pathogenic arguments are constituted by the sequence which proceeds by generalisation, from the instances (11) to (110), according to the following structure:

(2)∴ All elements xi of the class of reference E have property Āfrom (11)-(110)

Such over-generalisation leads then to the conclusion “All events that occur to me are bad” (depression); “All parts of my body are ugly” (body dysmorphic disorder); “All my acts are morally reprehensible” (scrupulosity); “All acts that he committed toward me are bad” (intermittent explosive disorder).

From a deductive point of view, such generalisation may constitute a completely valid argument. Indeed, the resulting generalisation constitutes a correct deductive reasoning, if the premises (11)-(110) are true. However, it often proves to be that the premises of the argument are false. Such is notably the case when the patient counts among the elements having property Ā, some elements which objectively have the opposite property A. The flaw in the argument resides then in a requalification in the other pole related to some elements and the enumeration of instances includes then some false premises, thus invalidating the resulting generalisation. In such case, secondary pathogenic argument turns out to be ungrounded, because of the falseness of some premises.

In other cases, the secondary pathogenic argument turns out to be fallacious from an inductive standpoint. For some positive (or neutral) events can well have been omitted in the corresponding enumeration of instances. Such omission can result from the use of general cognitive distortions, such as the omission of the neutral or disqualification of the positive. In such case, the elements of the relevant class of reference are only partly taken into account, thus biasing the resulting generalisation. The corresponding reasoning remains then logically valid and sound, but fundamentally incorrect of an inductive point of view, because it does only take partly into account the relevant instances within the reference class. Such feature of over-generalisation – a conclusion resulting from a valid reasoning from a deductive point of view, but inductively wrong – allows to explain how it notably succeeds in deceiving patients whose level of intelligence can otherwise prove to be high.

Ternary pathogenic arguments

It is worth mentioning, lastly, the role played by pathogenic ternary arguments which consist, at the level of the aforementioned reasoning, of the following sequence:

(2)All events that occur to me are of a negative naturepremise
(3)∴ I am a complete failure!from (2)

In such argument, the premise is constituted by the conclusion (2) of the secondary pathogenic argument, of which, in an additional stage (3), the patient aims at making sense by interpreting it. It consists here of a case of mislabelling. At the stage of a ternary pathogenic argument, mislabelling can thus take the following forms: “I am a complete failure!” (depression); “I am ugly!” (bodily dysmorphic disorder); “I am a horrible sinner!” (scrupulosity); “He is a bastard! “ (intermittent explosive disorder). In the present context, mislabelling proves to be an invalid argument, which constitutes a rough and unjustified interpretation of the over-generalisation (2).

2. Over-generalisation

At this stage, we are in a position to give a definition of over-generalisation, by drawing a distinction between general and specific over-generalisations. A general over-generalisation applies to any duality and to any reference class. It can be analysed as the ill-grounded conclusion of a secondary pathogenic argument, the premises of which include some general cognitive distortions: dichotomous reasoning, disqualification of one pole, arbitrary focus, minimisation/maximisation, omission of the neutral or requalification in the other pole. It consists of an ungrounded inductive reasoning, because the resulting generalisation is based on an incorrect counting of the corresponding instances. In the same way, a specific over-generalisation consists of an instance of a general over-generalisation, applied to a given duality and reference class. Thus, the specific over-generalisation “All events which occur to me are of a negative nature” (depression, generalised anxiety disorder) applies to the Positive/Negative duality and to the class of the events of the patient’s life. In the same way, “All parts of my body are ugly” (body dysmorphic disorder) is a specific over-generalisation that applies to the reference class of the parts of the patient’s body and to the Beautiful/Ugly duality.

3. Ungrounded inductive projection

At this step, it proves to be useful to describe another error of reasoning, which is likely to manifest itself at the stage of secondary pathogenic arguments. It consists of an ill-grounded inductive projection. The latter concludes, from the preceding over-generalisation (2), that a new instance will occur in the near future. Such instance is susceptible to be met in depression (Miranda & al. 2008), as well as in generalised anxiety disorder (Franceschi 2008). In the context of depression, such inductive projection presents the following form:

(2)All events that occur to me are of a negative naturepremise
(111a)The future event E11 of a negative nature may occurpremise
(111b)∴ The future event E11 of a negative nature will occurfrom (2), (111a)

The corresponding conclusion is susceptible of contributing to depression, notably by triggering the patient’s feeling of despair. Other instances of this type of conclusion are: “My next action will be morally reprehensible” (scrupulosity), or “The next act that he will commit toward me will be bad” (intermittent explosive disorder).

4. Confirmation bias

The cognitive process which has just been described illustrates how over-generalisation contributes to the formation of pathogenic ideas. However, a process of the same nature is also likely to concur to their maintenance. For once the over-generalisation (2) has been established by means of the above reasoning, its maintenance is made as soon as an instance occurs that confirms the generalisation according to which all elements xi of the reference class E have property Ā. This constitutes a confirmation bias, for the patient does only count those elements which present the property Ā, without taking into account those which have the opposite property A, thus disconfirming generalisation (2). Hence, in depression or generalised anxiety disorder, when a new negative event occurs, the patient concludes from it that it confirms that all events which occur to him/her are of a negative nature.

We see it finally, the above developments suggest a classification of cognitive distortions, depending on whether they manifest themselves at the level of primary, secondary or ternary pathogenic arguments. Thus, among the cognitive distortions which arise at the stage of primary pathogenic arguments, one can distinguish: on the one hand, the general cognitive distortions (dichotomous reasoning, disqualification of one pole, minimisation/maximisation, requalification into the other pole, omission of the neutral) and on the other hand, the specific cognitive distortions (disqualification of the positive, requalification into the negative, selective abstraction, catastrophising). Morevoer, among the cognitive distortions which manifest themselves at the stage of secondary pathogenic arguments, one can mention over-generalisation (at the stage of the formation of pathogenic ideas), ill-grounded inductive projection, and confirmation bias (at the stage of the maintenance of pathogenic ideas). Mislabelling, finally, is susceptible to occur at the level of ternary pathogenic arguments.

References

Beck A. Thinking and depression: Theory and therapy. Archives of General Psychiatry, 1964, 10, 561-571.

Beck, A. Depression: Clinical, experimental, and theoretical aspects, Harper & Row, New York, 1967.

Beck, A. Cognitive models of depression. Journal of Cognitive Psychotherapy, 1, 1987, 5-37.

Coccaro E., Richard J., Kavoussi R., Mitchell E., Berman J., Lish J. Intermittent explosive disorder-revised: Development, reliability, and validity of research criteria. Comprehensive Psychiatry, 39-6, 1998, 368-376.

Eckhardt C., Norlander B., Deffenbacher J., The assessment of anger and hostility: a critical review, Aggression and Violent Behavior, 9-1, 2004, 17-43.

Ellis A. Reason and Emotion in Psychotherapy, Lyle Stuart, New York, 1962.

Franceschi P. Compléments pour une théorie des distorsions cognitives. Journal de Thérapie Comportementale et Cognitive, 2007, 17-2, 84-88. English translation.

Franceschi P. Théorie des distorsions cognitives : application à l’anxiété généralisée. Journal de Thérapie Comportementale et Cognitive, 2008, 18, 127-131. English translation.

Galovski T., Blanchard E., Veazey C. Intermittent explosive disorder and other psychiatric co-morbidity among court-referred and self-referred aggressive drivers. Behaviour Research and Therapy, 40-6, 2002, 641-651.

Miller C., Hedges D. Scrupulosity disorder: An overview and introductory analysis. Journal of Anxiety Disorders, 2007, 22-6, 1042-1048.

Miranda R., Fontes M., Marroquín B. Cognitive content-specificity in future expectancies: Role of hopelessness and intolerance of uncertainty in depression and GAD symptoms. Behaviour Research and Therapy, 46-10, 2008, 1151-1159.

Tek C., Ulug B. Religiosity and religious obsessions in obsessive–compulsive disorder. Psychiatry Research, 2001, 104-2, 99-108.

Rabinowitz D., Neziroglu F., Roberts M. Clinical application of a behavioral model for the treatment of body dysmorphic disorder. Cognitive and Behavioral Practice, 2007, 14-2, 231-237.

Veale D. Advances in a cognitive behavioural model of body dysmorphic disorder. Body Image, 2004, 1, 113-125.

Theory of Cognitive Distortions: Personalisation

English translation of a paper published in French in the Journal de Thérapie Comportementale et Cognitive, 2010, 20-2, pages 51-55 under the title “Théorie des distorsions cognitives : la personnalisation”.

In a previous paper (Complements to a theory of cognitive distorsions, Journal de Thérapie Comportementale et Cognitive, 2007), we did present some elements aimed at contributing to a general theory of cognitive distortions. Based on the reference class, the duality and the system of taxa, these elements led to distinguish between the general cognitive distortions (dichotomous reasoning, disqualification of one pole, minimisation, maximisation) and the specific cognitive distortions (disqualifying the positive, selective abstraction, catastrophism). By also distinguishing between three levels of reasoning – the instantiation stage, the interpretation stage and the generalisation stage – we did also define two other cognitive distortions: over-generalisation and mislabelling (Théorie des distorsions cognitives : la sur-généralisation et l’étiquetage, Journal de Thérapie Comportementale et Cognitive, 2009). We currently extend this model to another classical cognitive distortion: personalisation.

This article is cited in:

  • Juliette Marty, Et si vous étiez trop perfectionniste ? Eyrolles, 2021

Theory of Cognitive Distortions: Personalisation

In Franceschi (2007), we set out to introduce several elements intended to contribute to a general theory of cognitive distortions. These elements are based on three fundamental notions: the reference class, the duality and the system of taxa. With the aid of these three elements, we could define within the same conceptual framework the general cognitive distortions such as dichotomous reasoning, disqualification of one pole, minimisation and maximisation, as well as requalification in the other pole and omission of the neutral. In the same way, we could describe as specific cognitive distortions: disqualification of the positive, selective abstraction and catastrophising. In Franceschi (2009), we introduced three levels of reasoning – the instantiation stage, the interpretation stage and the generalisation stage, which allowed to define within the same conceptual framework, two other classical cognitive distortions: over-generalisation and mislabelling. In the present paper, we set out to define and to situate in this conceptual framework another classical cognitive distortion: personalisation.

Personalisation constitutes one of but twelve classically defined cognitive distortions: emotional reasoning; over-generalisation; arbitrary inference; dichotomous reasoning; should statements; divination or mind-reading; selective abstraction; disqualification of the positive; maximisation/minimisation; catastrophising; personalisation; mislabelling (Ellis 1962, Beck 1964). Personalisation is usually defined as the fact of attributing unduly to oneself the cause of an external event. For example, seeing a person who laughs, the patient thinks that it is because of his/her physical appearance. Also, the patient makes himself/herself responsible for a negative event, in an unjustified way. If his/her companion then failed his/her examination, the patient estimates that is due to the fact that he/she is depressed. In what follows, we propose first to clarify the definition of personalisation and to situate it in the context of the theory of cognitive distortions (Franceschi 2007, 2009). Secondly, we set out to clarify the relationships existing between personalisation and several close notions mentioned in the literature: personalising bias (Langdon et al. 2006), ideas of reference (Startup & Startup 2005, Meyer & Lenzenweger 2009) and delusions of reference.

Personalisation and post hoc fallacy

We will set out first to highlight the mere structures of the cases of personalisation. Let us consider the aforementioned example where the patient sees a person who laughs and thinks that this one laughs because of the patient’s physical appearance. This constitutes an instance of personalisation. We can describe more accurately the reasoning which underlies such instance (in what follows, the symbol  denotes the conclusion):

(P11)in T1 I went for a walkpremiss1
(P12)in T2 the peasant started to laughpremiss2
(P13)∴ in T2 the peasant started to laugh because he saw that in T1 I went for a walkconclusion1
(P14)∴ in T2 the peasant made fun of meconclusion2

The patient puts here in relationship an internal event (“I went for a walk”) with an external event (“the peasant started to laugh”). He/she concludes then that the internal event is the cause of the external event. In this stage, the patient “personalises” an external event, which he/she considers to be the effect of an internal event, while this external event is in reality devoid of any relationship with the patient himself/herself. In a subsequent stage (P14), the patient interprets the previous conclusion (P13) by considering that the peasant made fun of him.

At this stage, it is worth wondering about the specific nature of the patient’s error of reasoning. It appears here that both premises (P11) and (P12) constitute genuine facts and therefore turn out to be true. On the other hand, the conclusion (P13) which concludes to the existence of a relation of causality between two consecutive events E1 (“In T1 I went for a walk”) and E2 (“In T2 the peasant started to laugh”) appears to be unjustified. Indeed, both premises are only establishing a relation of anteriority between the two consecutive facts E1 and E2. And the conclusion (P13) which deducts from it a relation of causality turns out therefore to be too strong. The argument proves here to be invalid and the corresponding reasoning is then fallacious. The corresponding error of reasoning, which concludes to a relation of causality whereas there is only a mere relation of anteriority, is classically termed post hoc fallacy, according to the Latin sentence “Post hoc, ergo propter hoc” (after this therefore because of this). It consists here of a very common error of reasoning, which is notably at the root of many superstitions (Martin 1998, Bressan 2002).

In this context, we can point out that the case of post hoc fallacy which has just been described as an argument of personalisation, also constitutes a case of arbitrary inference, another classically defined cognitive distortion.

Steps of instantiation, of interpretation and of generalisation at the level of the arguments of personalisation

At this step, it proves to be useful to draw a distinction between the levels of arguments that lead to personalisation as cognitive distortion. This leads to differentiate three levels within the arguments of personalisation, among the reasoning’ stages. The latter correspond respectively to three different functions: it consists of the successive stages of instanciation, of interpretation and of generalisation. To this end, it is useful to describe the whole reasoning which underlies the arguments of personalisation and which includes the three aforementioned stages:

(P11)in T1 I went for a walkpremiss11
(P12)in T2 the peasant started to laughpremiss12
(P13)∴ en T2 the peasant started to laugh because he saw that in T1 I went for a walkconclusion11
(P14)∴ in T2 the peasant made fun of meconclusion12
(P21)in T3 I was leafing through a magazine in the librarypremiss21
(P22)in T4, the librarian smirkedpremiss22
(P23)∴ en T4 the librarian smirked because in T3 I was leafing through a magazine in the libraryconclusion21
(P24)∴ in T4, the librarian made fun of meconclusion22
(P31)in T5 I did enter in the show-roompremiss31
(P32)in T6, my colleagues started to laughpremiss32
(P33)∴ in T6, my colleagues started to laugh because in T5 I did enter in the show-roomconclusion31
(P34)∴ in T6, my colleagues were laughing at meconclusion32
(…)
(P105)∴ people make fun of mefrom (P14)-(P104)

Here, the instances of the previous arguments (P11)-(P13), (P21)-(P23), (P31)-(P33), etc. constitute primary stages of arguments of personalisation, by which the patient considers that an event related to him/her is the cause of an external event. This type of argument corresponds to the stage of instantiation. As mentioned earlier, such argument is fallacious since it is based on post hoc fallacy. In a subsequent stage the function of which is interpretative, and that is aimed at making sense of the conclusions (P13), (P23), (P33), … of the instances of arguments of the previous type, the patient interprets it by concluding that some people made fun of him. Such conclusions (P14), (P24), (P34) appear to be grounded, inasmuch as the premisses (P13), (P23), (P33) are true. Finally, in a subsequent stage of generalisation, the patient enumerates some instances or circumstances where he/she thinks that people laughed or made fun of him/her ((P14), (P24), (P34), …) and generalises then to the conclusion (P105) according to which people make fun of him/her. This last stage is of an inductive nature, and corresponds to an enumerative induction, the structure of which is the following:

(P14)in T2 the peasant made fun of meconclusion12
(P24)in T4, the librarian made fun of meconclusion22
(P34)in T6, my colleagues were laughing at meconclusion32
(…)
(P105)∴ people make fun of mefrom (P14)-(P104)

Given what precedes, we can from now on provide a definition of personalisation. The preceding analysis leads then to distinguish between three stages in arguments of personalisation. At the level of primary arguments of personalisation (stage of instantiation), it consists of the tendency in the patient to establish an unjustified relation of causality between two events, among which one is external and the other one is internal to the patient. The patient personalises then, that is to say puts in relationship with himself/herself, an external event, which proves to be in reality devoid of any relation of causality. The mechanism which underlies such argument consists then of the erroneous attribution of a relation of causality, based on post hoc fallacy. At the level of secondary arguments of personalisation (stage of interpretation), the patient makes sense of the previous conclusion by concluding that at a given time, a person (or several persons) made fun of him, laughed at him, etc. Finally, at the level of arguments of ternary personalisation (stage of generalisation), the patient concludes that, in a general way, people make fun of him.

Personalisation and personalising bias

At this step, it proves to be useful to distinguish personalisation as cognitive distortion from personalising bias. The latter is defined as an attribution bias (“personalising attributional bias”), by whom the patient attributes to other persons rather than to circumstances the cause of a negative event (McKay & al. 2005, Langdon & al. 2006). Personalising bias is often related to polythematic delusions (Kinderman & Bentall 1997, Garety & Freeman 1999, McKay & al. 2005) met in schizophrenia.

Considering this definition, the difference between the two notions can be thus underlined: in personalisation as cognitive distortion, the patient attributes the cause of an external event to an event which concerns the patient himself/herself; on the other hand, in personalising bias the patient attributes the cause of an internal event to external persons. This allows to highlight several fundamental differences between the two notions. Firstly, in personalisation as cognitive distortion, the “person” is the patient himself/herself, while in personalising bias, it consist of external “persons”. Secondly, in the structure of personalisation, an internal event precedes an external event; by contrast, in the scheme of personalising bias, it is an external event which precedes an internal event. Finally, in personalisation as cognitive distortion, the internal event is indifferently of a positive, neutral or negative nature, whereas in personalising bias, the internal event is of a negative type. Hence, it finally proves to be that both notions appear fundamentally distinct.

Personalisation and ideas of reference

It appears also useful, for the sake of clarity, to specify the relationships between personalisation and ideas of reference. It is worth preliminary mentioning that one usually distinguishes between ideas of reference and delusions of reference (Dziegielewski 2002, p. 266). Ideas of reference characterise themselves by the fact that a patient considers that insignificant events relate to himself/herself, while is not the case in reality. For example, the patient hears several persons laugh, and considers, in an unjustified way, that the latter make fun of him/her. In parallel, delusions of reference constitute one of the most salient symptoms noticed in schizophrenia, and leads the patient to be persuaded that the media, television, or the radio speak about him/her or issue messages concerning him/her. Several criteria allow to draw a distinction between ideas of reference and delusions of reference. First, ideas of reference have much less impact on the patient’s life than reference delusions of reference. Second, the degree of conviction which is associated with ideas of reference is far lesser than with delusions of reference. Lastly, ideas of reference (“the neighbour made fun of me”) are related with beliefs the degree of plausibility of which is much stronger than the one which is inherent to delusions of reference (“newspapers speak about me”).

In this context, the aforementioned arguments of personalisation (P11)-(P14), (P21)-(P24), and (P31)-(P34), by whom the patient concludes that some people make fun of him, corresponds completely to the definition of ideas of reference. It appears then that personalisation, such as it was defined above as cognitive distortion, identifies itself with ideas of reference.

Personalisation and delusion of reference

One traditionally distinguishes at the level of polythematic delusions met in schizophrenia between: delusions of reference, delusions of influence, delusions of control, telepathy-like delusions, delusions of grandeur, and delusions of persecution. Delusions of reference leads for example the patient to believe with a very strong conviction that the media, the newspapers, the television speak about him/her.

It is worth describing here a mechanism which is susceptible to lead to the formation of delusions of reference. Such mechanism appears to be grounded on a reasoning (Franceschi 2008) which includes, as well as the above-mentioned primary instances of personalisation, a post hoc fallacy:

(DR11)in T1 I was drinking an appetizerpremiss11
(DR12)in T2 the presenter of the show said: “Stop drinking!”premiss12
(DR13)∴ in T2 the presenter of the show said: “Stop drinking!” because in T1 I was drinking an appetizerconclusion11
(DR14)∴ in T2 the presenter of the show spoke about meconclusion12

Consider also this second instance :

(DR21)in T3 I hardly got out of bedpremiss21
(DR22)in T4 the radio presenter said: “Be forceful:”premiss22
(DR23)∴ in T4 the radio presenter said: “Be forceful:” because in T3 I hardly got out of bedconclusion21
(DR24)∴ in T4 the radio presenter spoke about meconclusion22

At the level of the instantial step (DR11)-(DR13), (DR21)-(DR23), … the patient concludes here that an internal event is the cause of an external event. In a further interpretative stage, he/she interprets the conclusions (DR13), (DR23), … of the preceding arguments by considering that the presenters of radio or of television speak about him/her. Finally, in a generalisation step, of inductive nature, the patient enumerates the conclusions (DR14), (DR24), … of secondary arguments (interpretation stage) and generalises thus:

(DR14)∴ in T2 , the presenter of the show spoke about me
(DR24)∴ in T4, the radio presenter spoke about me
(…)
(DR105)∴ the media speak about meconclusion

It proves then that the structure of the mechanism which leads to the formation of delusions of reference thus described, is identical to that of the reasoning which leads to ideas of reference which is associated with personalisation as cognitive distortion.

Finally, it appears that the preceding developments allow to provide a definition of personalisation and to situate it in the context of cognitive distortions (Franceschi 2007, 2009). Personalisation is then likely to manifest itself at the level of primary, secondary or ternary pathogenic arguments, which correspond respectively to the stages of instantiation, of interpretation, and of generalisation. At the level of primary pathogenic arguments, corresponding to a function of instantiation, it consists of instances, the conclusions of which lead the patient to conclude in an unjustified way that some external events are caused by some of his/her actions. At the level of secondary pathogenic arguments, which correspond to a function of interpretation, personalisation takes the form of a reasoning by which the patient interprets the conclusion of primary pathogenic argument by concluding for example that people make fun of him/her. Finally, at the level of ternary pathogenic arguments, associated with a function of generalisation, the patient generalises from the conclusions of several secondary pathogenic arguments and concludes that, in a general way, people make fun of him/her.

Lastly, it appears that the previous definition of personalisation as cognitive distortion allows to describe precisely the relationships between personalisation and close notions such as personalising bias, ideas of reference and delusions of reference.

References

Beck A. Thinking and depression: Theory and therapy. Archives of General Psychiatry 1964; 10:561-571.

Bressan, P. The Connection Between Random Sequences, Everyday Coincidences, and Belief in the Paranormal. Applied Cognitive Psychology, 2002, 16, 17-34.

Dziegielewski, S. F. DSM-IV-TR in action, Wiley, New York, 2002.

Ellis A. Reason and Emotion in Psychotherapy, Lyle Stuart, New York, 1962.

Franceschi P. Compléments pour une théorie des distorsions cognitives. Journal de Thérapie Comportementale et Cognitive, 2007, 17-2, 84-88. English translation: http://cogprints.org/5261/

Franceschi P. Une défense logique du modèle de Maher pour les délires polythématiques. Philosophiques, 2008, 35-2, 451-475. English translation.

Franceschi P. Théorie des distorsions cognitives : la sur-généralisation et l’étiquetage. Journal de Thérapie Comportementale et Cognitive, 2009, 19-4. English translation.

Garety, P., Freeman, D., 1999. Cognitive approaches to delusions: a critical review of theories and evidence. British Journal of Clinical Psychology 38, 113-154.

Langdon R., Cornera T., McLarena J., Wardb P. & Coltheart M., 2006, Externalizing and personalizing biases in persecutory delusions: The relationship with poor insight and theory-of-mind, Behaviour Research and Therapy, 44:699-713

Kinderman, P., Bentall, R., 1997. Causal attributions in paranoia and depression: internal, personal, and situational attributions for negative events. Journal of Abnormal Psychology 106 (2), 341- 345.

Martin, B. Coincidences: Remarkable or random. Skeptical Inquirer, 1998, 22-5, 23-27.

McKay, R., Langdon, R. & Coltheart, 2005. M. Paranoia, persecutory delusions and attributional biases, Psychiatry Research, 136, 233–245

Meyer, E., Lenzenweger, M., 2009. The specificity of referential thinking: A comparison of schizotypy and social anxiety, Psychiatry Research, 165, 78-87.

Startup, M., Startup, S., 2005. On two kinds of delusion of reference, Psychiatry Research ,137, 87-92.

Theory of Cognitive Distortions: Application to Generalised Anxiety Disorder

English translation and postprint (with additional illustrations) of a paper published in French under the title “Théorie des distorsions cognitives : application à l’anxiété généralisée” in the Journal de Thérapie Comportementale et Cognitive, 2008, 18, pp. 127-131.
This article follows the contribution to the general theory of cognitive distortions exposed in “Complements to a theory of cognitive distorsions” (Journal de Thérapie Comportementale et Cognitive, 2007). The elements described, namely the reference class, the duality and the system of taxa, are applied here to generalised anxiety disorder. On the one hand, these elements allow to describe the cognitive distortions which are specific to generalised anxiety disorder, consistent with recent work emphasising the role played uncertain situations relative to future events. On the second hand, they allow to define a type of structured reasoning, of inductive nature, which leads to the formation and maintenance of anxious ideas.

This paper is cited in:

  • Étude des profils de distorsion cognitive en fonction des états anxieux et dépressifs chez des adultes tout-venant, Anita Robert, Nicolas Combalbert, Valérie Pennequin, Annales Médico-psychologiques, revue psychiatrique, Volume 176-3, 2018, pages 225-230
  • Deperrois Romain & Nicolas Combalbert, Links between cognitive distortions and cognitive emotion regulation strategies in non-clinical young adulthood, in Psychological Applications and Trends, Eb. by C. Pracana & M. Wang, InScience Press, 2019
  • Nawal Ouhmad, Nicolas Combalbert, Wissam El-Hage, Cognitive distortions and emotion regulation among post traumatic stress disorder victims, in Psychological Applications and Trends, Eb. by C. Pracana & M. Wang, InScience Press, 2019
  • A.Robert, N.Combalbert, V.Pennequin, R.Deperrois, N.Ouhmad, Création de l’Échelle de Distorsions Cognitives pour adultes (EDC-A) : étude des propriétés psychométriques en population générale et association avec l’anxiété et la dépression, Psychologie Française, 2021

Theory of Cognitive Distortions: Application to Generalised Anxiety Disorder

In Franceschi (2007), we set out to introduce several elements intended to contribute to a general theory of cognitive distortions. These elements are based on three basic notions: the reference class, the duality and the system of taxa. With the help of these three elements, we could define within the same conceptual framework the general cognitive distortions such as dichotomous reasoning, the disqualification of one pole, minimisation and maximisation, as well as the requalification in the other pole and the omission of the neutral. In addition, we could describe as specific cognitive distortions: the disqualification of the positive, selective abstraction and catastrophism.

In what follows, we offer to extend this work by applying it in a specific way to generalised anxiety disorder (GAD), in order to allow their use within cognitive therapy. The present study inserts itself in the context of recent work (Butler & Mathews 1983, 1987, Dalgleish et al. 1997), which notably underlined the major role played, in the context of GAD, by indeterminate situations, and especially by uncertain situations relating to future events. Recent developments, emphasising especially the intolerance with regard to indeterminate future situations, echoed this (Dugas et al. 2004, Canterbury et al. 2004, Carleton et al. 2007).

We shall be interested successively in two main forms of reasoning likely to occur in the context of GAD: on the one hand, the cognitive distortions which are specific to GAD; on the other hand, a structured argument relating to GAD and grounded on inductive logic, which is likely to include one or several of the aforementioned cognitive distortions.

Cognitive distortions in the context of generalized anxiety disorder

The optimal system of taxa

The conceptual framework defined in Franceschi (2007) is based on three fundamental elements: the duality, the reference class, and the system of taxa, which allow to define the general cognitive distortions. These three notions also allow to describe the specific cognitive distortions which are applicable to GAD. In this context, as we will see it, the reference class for the latter specific cognitive distortions identifies itself with the class of future events of the patient’s life. Moreover, the duality assimilates itself to the Positive/Negative duality. Finally, for the sake of the present discussion, we shall make use of the system of taxa (its choice is more or less arbitrary) described in Franceschi (2007), which includes 11 taxa, denoted by E1 to E11, where E6 denotes the neutral taxon. Such conceptual framework allows then to define the specific cognitive distortions in the context of GAD. We offer to examine them in turn.

Dichotomous reasoning

Dichotomous reasoning

An instance of dichotomous reasoning related to GAD consists for the patient to only consider future events from the viewpoint of the extreme taxa corresponding to each pole of the Positive/Negative duality. Hence, the patient only considers future events which present either a very positive, or a very negative nature. All other events, being either neutral, positive or negative to a lesser degree, are thus ignored. This type of reasoning can be analysed as an instance of dichotomous reasoning, applied to the class of the events of the patient’s future life and to the Positive/Negative duality.

Disqualification of one pole

The disqualification of one pole

An instance of the disqualification of one pole related to GAD consists for the patient to only envisage, among future events likely to occur, those which present a negative nature. The patient tends then to be unaware of positive future events that could happen, by considering that they do not count, for this or that reason. In the present context, this type of reasoning can be analysed as an instance of disqualification of one pole, applied to the reference class of the events of the patient’s future life and to the Positive/Negative duality, i.e. disqualification of the positive.

Arbitrary focus on a given modality

Arbitrary focus

In GAD, a typical instance of arbitrary focus, consists for the patient to focus on a possible future event, the nature of which turns out to be negative. This can be analysed as focusing on one of the taxa of the Positive/Negative duality, at the level of the class of the future events of the patient’s life.

Omission of the neutral

Omission of the neutral

A specific instance for GAD consists for the patient to be completely unaware of possible future events the nature of which is neutral, i.e. those which are neither positive nor negative.

Requalification into the other pole

Requalifcation into the other pole

In the context of GAD, the corresponding cognitive distortion consists in requalifying as negative a possible future event, whereas it should be considered objectively as positive. Such cognitive distortion consists of a requalification in the other pole applied to the reference class of the future events of the patient’s life and to the Positive/Negative duality, i.e. requalification in the negative.

Minimisation or maximisation

Maximisation and minimisation

A specific instance of minimisation applied to GAD consists for the patient to consider some possible future events as less positive than they truly are in reality. With maximisation, the patient considers some possible future events as more negative than they objectively are.

Primary, secondary and tertiary anxiogenous arguments

At this stage, it is worth also considering a certain type of reasoning, likely to be met in GAD, which can include several instances of the aforementioned cognitive distortions. This type of reasoning presents an anxiogenous nature, because it leads the patient to predict that a future event of negative nature is going to occur. Such reasoning is underlain by a structure which presents an inductive nature. Before analysing in detail the different steps of the corresponding reasoning, it is worth describing preliminarily its internal structure. The latter is the following (in what follows, the symbol ∴ denotes the conclusion):

(1) the event E1 of negative nature did occur to mepremiss
(2) the event E2 of negative nature did occur to mepremiss
(3) the event E3 of negative nature did occur to mepremiss
(…)premiss
(10) the event E10 of negative nature did occur to mepremiss
(11) all events that occur to me are of negative naturefrom (1)-(10)
(12) « I am always unlucky », « I am ill-fated »from (11)
(13) the future event E11 of negative nature may occurpremiss
(14) ∴ the future event E11 of negative nature will occurfrom (11), (13)

The essence of such reasoning is of a logically inductive nature. The patient enumerates then some events of his/her past or present life, the nature of which he/she considers as negative. He/she reaches then by generalisation the conclusion according to which all events that which occur to him/her are negative. From this generalisation, he/she infers a prediction relating to a future event, likely to happen, which he/she considers as negative. The patient is thus led to the anxiogenous conclusion that an event of negative nature is going to occur.

In such reasoning, it is worth pointing out that the reference class identifies itself with the class of past, present and future events, of the patient’s life. Typically, in this type of reasoning, the generalisation is grounded on present or past events, while a future event is the object of the corresponding inductive prediction. This is different from the reference class applicable to the cognitive distortions mentioned above, where the reference class identifies itself exclusively with the future events of the patient’s life.

At this stage, it proves to be necessary to identify the fallacious steps in the patient’s reasoning, to allow their use in cognitive therapy of GAD. To this end, we can differentiate several steps in the structure of the corresponding reasoning. It proves indeed that some steps are valid arguments (an argument is valid when its conclusion is true if its premises are true), while others are invalid. For this purpose, it is worth drawing within this type of reasoning, a distinction between primary, secondary or ternary anxiogenous arguments.

Primary anxiogenous arguments

The first step in the type of aforementioned reasoning, consists for the patient to think to a past negative event, in the following way:

(1) the event E1 of negative nature did occur to me

It is however possible to describe more accurately the corresponding cognitive process, under the form of an argument that we can term a primary anxiogenous argument, the structure of which is the following:

(1a) the event E1 did occur to me
(1b) the event E1 was of negative nature
(1) ∴ the event E1 of negative nature did occur to mefrom (1a), (1b)

By such cognitive process, the patient is led to the conclusion that some negative event did occur to him/her. This type of argument proves to be entirely valid inasmuch as the event in question presents well, objectively, a negative nature. However, it can also turn out to be invalid, if the event in question presents, objectively, a positive (or neutral) nature. What is then defective in this type of reasoning, is the fact that premise (1b) turns then out to be false. Such can notably be the case for example if the patient makes use of a cognitive distortion such as requalification in the negative. In such case, the patient considers then as negative an event the nature of which is objectively positive.

Secondary anxiogenous arguments

Anxiogenous secondary arguments are constituted, at the level of the above-mentioned reasoning, of the part that takes into account the instances (1)-(10) and proceeds then by generalisation. The patient counts thus some instances of events that did occur to him/her, the nature of which he/she considers as negative, and concludes that all events that did occur to him/her were negative, in the following way:

(1) the event E1 of negative nature did occur to me
(2) the event E2 of negative nature did occur to me
(…)
(10) the event E10 of negative nature did occur to me
(11) ∴all events that occur to me are of negative naturefrom (1)-(10)

Such generalisation may constitute a fully valid argument. For the resulting generalisation constitutes a fully correct inductive reasoning, if the premises (1)-(10) are true. However, such type of reasoning is most often defective from two different viewpoints, thus distorting the conclusion which results from it. Above all, as we have did just see it, some past events of positive nature can have been counted among the number of negative events, by the effect of a requalification in the negative. In that case, the enumeration of instances includes then some false premises, thus invalidating the resulting generalisation. Secondly, some past or present positive (or neutral) events can have been omitted in the corresponding enumeration. Such omission can result from the use of some cognitive distortions, such as disqualification of the positive. In such case, the relevant class of reference consisting in present and past events of the patient’s life is only taken into account in a partial or erroneous way. The corresponding reasoning remains then logically valid, but proves to be incorrect, since it takes into account only partly the relevant instances within the reference class, that of the present and past events of the patient’s life.

As we see it finally, the patient proceeds then to a reconstruction of the relevant reference class which proves to be erroneous, due to the use of the following specific cognitive distortions: requalification in the negative and disqualification of the positive (and possibly, omission of the neutral). The corresponding type of reasoning is illustrated on the figure below:

A series of events of the patient’s life, seen (objectively) from the optimal system of taxa
 After omission of the neutral
 After requalification in the negative
After disqualification of the positive Conclusion: «All events that occur to me are negative »
Incorrect construction of the reference class for induction, after omission of the neutral, requalification in the negative, and then disqualification of the positive

Such mechanism, as we did see it, illustrates how the formation of anxious ideas is made. However; a mechanism of the same nature is also likely to contribute to their maintenance. For once the generalisation (11) according to which all events which occur to the patient are of negative nature, has been established by means of the above reasoning, its maintenance is made as soon as an event occurs which confirms this latter generalisation. When a new negative event indeed happens, the patient concludes from it that it confirms generalisation (11). Such mechanism, at the stage of the maintenance of anxious ideas, constitutes a confirmation bias. For the patient only counts those events of negative nature related to him/her that confirm indeed the generalisation (11), but without taking into account those events of positive nature which occur to him/her and that would then disconfirm the idea according to which all events which occur to him/her are of negative nature.

Ternary anxiogenous arguments

Lastly, it is worth mentioning the role played by ternary anxiogenous arguments which consist, at the level of the aforementioned reasoning, in the following sequence:

(11) ∴all events that occur to me are of negative nature

(12) ∴« I am always unlucky », « I am ill-fated »

It consists here of an argument which follows the conclusion of the secondary anxiogenous argument (11), and which, by an additional step (12), aims at interpreting it, at making sense of it. The patient interprets here the fact that the events which occur to him/her are negative, due to the fact that he/she is unlucky, ill-fated.

As we did see it, the interest of drawing a distinction between three types of arguments resides in the fact that each of them has a specific function: the primary stage proceeds by enumerating the instances, the secondary stage operates by generalisation, and the ternary stage, lastly, proceeds by interpretation (Franceschi 2008).

The present study, as we see it, extends recent work (Butler and Mathews on 1987, Dalgleish et al. 1997) emphasising the role played, in GAD, by anticipations concerning indeterminate situations related to future events. In this context, the specific cognitive distortions as well as a reasoning of an inductive structure, contribute then to the vicious circle (Sgard et al. 2006), which results from the process of formation and maintenance of the anxious state.

References

Butler G & Matews A. Cognitive processes in anxiety. Advances in Behaviour Research and Therapy 1983 ; 5 : 51-62.

Butler G & Matews A. Anticipatory anxiety and risk perception. Cognitive Therapy and Research 1987 ; 11 : 551-565.

Carleton R, Norton M & Aslundson G. Fearing the unknown: A short version of the Intolerance of Uncertainty Scale. Journal of Anxiety Disorders 2007 ; 21-1 : 105-117.

Canterbury R, Golden A, Taghavi R, Neshat-Doost H, Moradi A & Yule W. Anxiety and judgements about emotional events in children and adolescents. Personality and Individual Differences 2004 ; 36 : 695-704.

Dalgleish T, Taghavi R, Neshat-Doost H, Moradi A, Yule W & Canterbury R. Information processing in clinically depressed and anxious children and adolescents. Journal of Child Psychology and Psychiatry 1997 ; 38 : 535-541.

Dugas M, Buhr K & Ladouceur R. The role of intolerance of uncertainty in etiology and maintenance. In R. Heimberg, C. Turk, & D. Mennin (Eds.). Generalized anxiety disorder: Advances in research and practice. Guilford, New York, 2004(143-163).

Franceschi P. Compléments pour une théorie des distorsions cognitives. Journal de Thérapie Comportementale et Cognitive 2007 ; 17-2 : 84-88. English translation.

Franceschi P. Une défense logique du modèle de Maher pour les délires polythématiques. Philosophiques 2008 ; 35-2 : 451-475. English translation.

Sgard F, Rusinek S, Hautekeete M & Graziani P. Biais anxieux de perception des risques. Journal de Thérapie Comportementale et Cognitive 2006 ; 16-1 : 12-15.

Probabilistic Situations for Goodmanian N-universes

A paper appeared (2006) in French in the Journal of Philosophical Research, vol. 31, pages 123-141, under the title “Situations probabilistes pour n-univers goodmaniens.”

I proceed to describe several applications of the theory of n-universes through several different probabilistic situations. I describe first how n-universes can be used as an extension of the probability spaces used in probability theory. The extended probability spaces thus defined allow for a finer modelling of complex probabilistic situations and fits more intuitively with our intuitions related to our physical universe. I illustrate then the use of n-universes as a methodological tool, with two thought experiments described by John Leslie. Lastly, I model Goodman’s paradox in the framework of n-universes while also showing how these latter appear finally very close to goodmanian worlds.


Probabilistic Situations for Goodmanian N-universes

The n-universes were introduced in Franceschi (2001, 2002) in the context of the study of the probabilistic situations relating to several paradoxes which are currently the object of intensive studies in the field of analytical philosophy: Goodman’s paradox and the Doomsday Argument. The scope of the present article is twofold: on one hand, to describe how modelling within the n-universes allows to extend the properties of the classical probability spaces used in probability theory, by providing at the same time a finer modelling of some probabilistic situations and a better support for intuition; on the other hand, to show how the use of n-universes allows to simplify considerably the study of complex probabilistic situations such as those which appear in the study of paradoxes.

When one models for example the situation corresponding to the drawing of a ball from an urn, one considers then a restricted temporal space, which limits itself to the few seconds that precede and follow the drawing. Events which took place the day before or one hour before, but also those who will happen for example the day after the drawing, can be purely and simply ignored. A very restricted interval of time, that it is possible to reduce to one or two discrete temporal positions, is then enough for characterising the corresponding situation. It suffices also to consider a restriction of our universe where the space variable is limited to the space occupied by the urn. For it is not useful to take into consideration the space corresponding to the neighbouring room and to the objects which are there. In a similar way, the number of atoms of copper or of molybdenum that are possibly present in the urn, the number of photons which are interacting with the urn at the time of the drawing, or the presence or absence of a sound source of 75 db, etc. can be omitted and ignored. In this context, it is not necessary to take into account the existence of such variables. In such situation, it is enough to mention the variables and constants really used in the corresponding probabilistic situation. For to enumerate all the constants and the variables which describe of our whole universe appears here as an extremely complicated and moreover useless task. In such context, one can legitimately limit oneself to describe a simplified universe, by mentioning only those constants and variables which play a genuine role in the corresponding probabilistic situation.

Let us consider the drawing of a ball from an urn which contains several balls of different colours. To allow the calculation of the likelihood of different events related to the drawing of one or several balls from the urn, probability theory is based on a modelling grounded on probability spaces. The determination of the likelihood of different events is then not based on the modelling of the physical forces which determine the conditions of the drawing, i.e. the mass and the dimensions of the balls, the material of which they are constituted, their initial spatio-temporal position, as well as the characteristics of the forces exercised over the balls to perform a random drawing. The modelling of random phenomena with the help of probability spaces does only retain some very simplified elements of the physical situation which corresponds to the drawing of a ball. These elements are the number and the colour of the balls, as well as their spatio-temporal position. Such methodological approach can be generalised in other probabilistic situations that involve random processes such as the drawing of one or several dices or of one or several cards. Such methodology does not constitute one of the axioms of probability theory, but it consists here of one important tenet of the theory, of which one can suggest that it would be worth being more formalized. It may also be useful to explain in more detail how the elements of our physical world are converted into probability spaces. In what follows, I will set out to show how the probability spaces can be extended, with the help of the theory of n-universes, in order to better restore the structure of the part of our universe which is so modelled.

1. Introduction to n-universes

It is worth describing preliminarily the basic principles underlying the n-universes. N-universes constitute a simplified model of the physical world which is studied in a probabilistic situation. Making use of Ockam’s razor, we set out then to model a physical situation with the help of the simplest universe’s model, in a compatible way however with the preservation of the inherent structure of the corresponding physical situation. At this stage, it proves to be necessary to highlight several important features of n-universes.

1.1. Constant-criteria and variable-criteria

The criteria of a given n-universe include both constants and variables. Although n-universes allow to model situations which do not correspond to our physical world, our concern will be here exclusively with the n-universes which correspond to common probabilistic situations, in adequacy with the fundamental characteristics of our physical universe. The corresponding n-universes include then at the very least one temporal constant or variable, as well as one constant or variable of location. One distinguishes then among n-universes: a T0L0 (a n-universe including a temporal constant and a location constant), a T0L (a temporal constant and a location variable), a TL0 (a temporal variable and a location constant), a TL (a temporal variable and a location variable). Other n-universes also include a constant or a variable of colour, of direction, etc.

1.2. N-universes with a unique object or with multiple objects

Every n-universe includes one or several objects. One distinguishes then, for example: a 0TL0 (n-world including a unique object, a temporal variable and a constant of location), a TL0 (multiple objects, a temporal variable and a location constant).

1.3. Demultiplication with regard to a variable-criterion

It is worth highlighting the property of demultiplication of a given object with regard to a variable-criterion of a given n-universe. In what follows, we shall denote a variable-criterion  with demultiplication by *. Whatever variable-criterion of a given n-universe can so be demultiplicated. The fact for a given object to be demultiplicated with regard to a criterion  is the property for this object to exemplify several taxa of criterion . Let us take the example of the time criterion. The fact for a given object to be demultiplicated with regard to time resides in the fact of exemplifying several temporal positions. In our physical world, an object 0 can exist at several (successive) temporal positions and finds then itself demultiplicated with regard to the time criterion. Our common objects have then a property of temporal persistence, which constitutes a special case of temporal demultiplication. So, in our universe of which one of the variable-criteria is time, it is common to note that a given object 0 which exists at T1 also exists at T2, …, Tn. Such object has a life span which covers the period T1-Tn. The corresponding n-universe presents then the structure 0T*L0 (T* with simplified notation).

1.4. Relation one/many of the multiple objects with a given criterion

At this stage, it proves to be necessary to draw an important distinction. It is worth indeed distinguishing between two types of situations. An object can thus exemplify, as we did just see it, several taxa of a given variable-criterion. This corresponds to the case of demultiplication which has just been described with regard to a given variable-criterion. But it is also worth taking into account another type of situation, which concerns only those n-universes with multiple objects. Indeed, several objects can instantiate the same taxon of a given criterion. Let us consider first the temporal criterion. Let us place ourselves, for example, in a n-universe with multiple objects including at the same time a temporal variable and a location constant L0. This can correspond to two types of different n-universes. In the first type of n-universe, there is one single object by temporal position. At some point in time, it is therefore only possible to have a unique object in L0 in the corresponding n-universe. We can consider in that case that every object of this n-universe is in relation one with the time taxa. We denote by T*L0 (with simplified notation T) such n-universe. Let us consider now a n-universe with multiple objects including a temporal variable and a location constant, but where several objects 1, 2, 3 can exist at the same time. In that case, the multiple objects are at a given temporal position in L0. The situation then differs fundamentally from the T*L0, because several objects can now occupy the same given temporal position. In other words, the objects can co-exist at a given time. In that case, one can consider that the objects are in relation many with the temporal taxa. We denote then by *T*L0 such n-universe (with simplified notation *T) .

Let us place ourselves now from the point of view of the location criterion. Let us consider a n-universe with multiple objects including at the same time a temporal variable and a variable of location, and where the objects are in relation many with the temporal criterion. It is also worth distinguishing here between two types of n-universes. In the first, a single object can find itself at a given taxon of the location criterion at the same time. There is then one single object by space position at a given time. This allows for example to model the situation which is that of the pieces of a chess game. Let us denote by *TL such n-universe (with simplified notation *TL). In that case, the objects are in relation one with the location criterion. On the other hand, in the second type of n-universe, several objects can find themselves in the same taxon of a location criterion at the same time. Thus, for example, the objects 1, 2, 3 are in L1 at T1. Such situation corresponds for example to an urn (which is thus assimilated with a given taxon of location) where there are several balls at a given time. We denote by *T*L such n-universe, where the objects are in relation many with the location taxa.

One can notice lastly that such differentiation is also worth for the variable-criterion of colour. One can then draw a distinction between: (a) a *T0*L0C (with simplified notation C) where several objects which can co-exist at the same time in a given space position present all necessarily a different colour, because the objects are in relation one with the colour criterion there; (b) a *T0*L0*C (with simplified notation *C) where several objects which can co-exist at the same time at a given space position can present the same colour, because the objects are in relation many with the colour criterion there.

1.5. Notation

At this stage, it is worth highlighting an important point which concerns the used notation. It was indeed made use in what precedes of an extended and of a simplified notation. The extended notation includes the explicit specification of all criteria of the considered n-universe, including at the same time the variable-criteria and the constant-criteria. By contrast, the simplified notation includes only the explicit specification of the variable-criteria of the considered n-universe. For constant-criteria of time and of location of the considered n-universe can be merely deduced from variable-criteria of the latter. This is made possible by the fact that the studied n-universes include, in a systematic way, one or several objects, but also a variable-criterion or a constant-criterion of time and of location.

Let us illustrate what precedes by an example. Consider first the case where we situate ourselves in a n-universe including multiple objects, a constant-criterion of time and a constant-criterion of location. In that case, it appears that the multiple objects exist necessarily at T0. As a result, in the considered n-universe, the multiple objects are in relation many with the constant-criterion of time. And also, there exist necessarily multiple objects at L0. So, the multiple objects are also in relation many with the constant-criterion of location. We place ourselves then in the situation which is that of a *T0*L0. But for the reasons which have just been mentioned, such n-universe can be denoted, in a simplified way, by .

The preceding remarks suggest then a simplification, in a general way, at the level of the used notation. Indeed, since a n-universe includes multiple objects and since it includes a constant-criterion of time, the multiple objects are necessarily in relation many with the constant-criterion of time. The n-universe is then a *T0. But it is possible to simplify the corresponding notation into . If a n-universe also includes multiple objects and a constant-criterion of location, the multiple objects are necessarily in relation many with the constant-criterion of location. The given n-universe is then a *L0, and it is possible to simplify the notation of the considered n-universe in . As a result, it is possible to simplify the notations *L0*T0 into , *L0T into T, *L0*T into *T, *L0*T* into *T*, etc.

2. Modelling random events with n-universes

The situations traditionally implemented in probability theory involve dices, coins, card games or else some urns that contain balls. It is worth setting out to describe how such objects can be modelled within the n-universes. It also proves to be necessary to model the notion of a “toss” in the probability spaces extended to n-universes. One can make use of the modellings that follow:1

2.1. Throwing a dice

How can we model a toss such as the result of the throwing of the dice is “5 “? We model here the dice as a unique object that finds itself at a space location L0 and which is susceptible of presenting at time T0 one discrete modality of space direction among {1,2,3,4,5,6}. The corresponding n-universe includes then a unique object, a variable of direction and a temporal constant. The unique object can only present one single direction at time T0 and is not with demultiplication with regard to the criterion of direction. The n-universe is a O (with extended notation 0T0L0O). Traditionally, we have the sample space  = {1,2,6} and the event {5}. The drawing of “5 ” consists here for the unique object to have direction 5 among {1,2,6} at time T0 and at location L0. We denote then the sample space by 0T0L0O{1,2,…,6} and the event by 0T0L0O{5}.2

How can we model two successive throws of the same dice, such as the result is “5” and then “1”? Traditionally, we have the sample space  = {1,2,…,6}2 and the event {5,1}. Here, it corresponds to the fact that the dice 0 has direction 5 and 1 respectively at T1 and T2. In the corresponding n-universe, we have now a time variable, including two positions: T1 and T2. Moreover, the time variable is with demultiplication because the unique object exists at different temporal positions. The considered n-universe is therefore a T*O (with extended notation 0T*L0O). We denote then the sample space by 0T*{1,2}L0O{1,2,…,6} and the event by {0T*{1}L0O{5}, 0T*{2}L0O{1}}.

2.2. Throwing a coin

How can we model the toss, for example of Tails, resulting from the flipping of a coin? We model here the coin as a unique object presenting 2 different modalities of direction among {P,F}. The corresponding n-universe is identical to the one which allows to model the dice, with the sole difference that the direction criterion includes only two taxa: {P,F}. The corresponding n-universe is therefore a O (with extended notation 0T0L0O). Classically, we have:  = {P,F} and {P}. Here, the Tails-toss is assimilated with the fact for the unique object to take direction {P} among {P,F} at time T0 and at location L0. The sample space is then denoted by 0T0L0O{P,F} and the event by 0T0L0O{P}.

How can we model two successive tosses of the same coin, such as the result is “Heads” and then “Tails”? Classically, we have the sample space  = {P,F}2 and the event {F,P}. As well as for the modelling of the successive throws of the same dice, the corresponding n-universe is here a T*O (with extended notation 0T*L0O). The sample space is then denoted by by 0T*{1,2}L0O{P,F} and the event by {0T*{1}L0O{F}, 0T*{2}L0O{P}}.

2.3. Throwing several discernible dices

How can we model the throwing of two discernible dices at the same time, for example the simultaneous toss of one “3” and of one “5”? The discernible dices are modelled here as multiple objects being each at a given space position and susceptible of presenting at time T0 one modality of space direction among {1,2,3,4,5,6}. The multiple objects co-exist at the same temporal position, so that the objects are in relation many with the temporal constant. In addition, the multiple objects can only present one single direction at time T0 and are not therefore with demultiplication with regard to the criterion of direction. The fact that both dices could have the same direction corresponds to the fact that objects are in relation many with the criterion of direction. There exists also a location variable, each of the dices 1 and 2 being at one distinct space position. We consider then that the latter property renders the dices discernible. The objects are here in relation one with the location criterion. In addition, the objects can only occupy one single space position at time T0 and are not therefore with demultiplication with regard to the location criterion. The n-universe is then a L*O (with extended notation *T0L*O). Classically, one has:  = {1,2,3,4,5,6}2 and {3,5}. Here, it corresponds to the fact that the dices 1 and 2 are to be found respectively at L1 and L2 and present a given direction among {1,2,6} at time T0. We denote then the sample space by {1,2}*T0L{1,2}*O{1,2,…,6} and the event by {{1}*T0L{1}*O{3}, {2}*T0L{2}*O{5}}.

2.4. Throwing several indiscernible dices

How can we model the throwing of two indiscernible dices, for example the toss of one “3” and one “5” at the same time? Both indiscernible dices are modelled as multiple objects being at space position L0 and susceptible of presenting at time T0 one modality of space direction among {1,2,3,4,5,6} at a given location. The multiple objects co-exist at the same temporal position, so that the objects are in relation many with the temporal constant. The multiple objects can only present one single direction at time T0 and are not therefore with demultiplication with regard to the criterion of direction. The fact that both dices are susceptible of having the same direction corresponds to the fact that the objects are in relation many with the criterion of direction. Both dices 1 and 2 are at the same location L0, what makes them indiscernible. In addition, the multiple objects are in relation many with the constant-criterion of location. Lastly, the objects can only be at one single space position at time T0 and are not therefore with demultiplication with regard to the location criterion. The corresponding n-universe is then a *O (with extended notation *T0*L0*O). Classically, we have:  = (i, j) with 1  ij  6 and {3,5}. Here, it corresponds to the fact that the dices 1 and 2 are both in L0 and present a given direction among {1,2,…,6} at T0. The sample space is then denoted by {1,2}*T0*L0*O{1,…,6} and the event by {{1}*T0*L0*O{3}, {2}*T0*L0*O{5}}.

2.5. Drawing a card

How can we model the drawing of a card, for example the card #13, in a set of 52 cards? Cards are modelled here as multiple objects presenting each a different colour among {1,2,…,52}. The cards’ numbers are assimilated here with taxa of colour, numbered from 1 to 52. Every object can have only one single colour at a given time. As a result, the multiple objects are not with demultiplication with regard to the colour criterion. In addition, a given card can only present one single colour at the same time. Hence, the objects are in relation one with the colour criterion. Moreover, the multiple objects can be at a given time at the same space location (to fix ideas, on the table). The objects are then in relation many with the location criterion. Lastly, the objects can co-exist at the same given temporal position. Thus, they are in relation many with the time criterion. The corresponding n-universe is then a C (with extended notation *T0*L0C). How can we model the drawing of a card? Classically, we have the sample space  = {1,2,…,52} and the event {13}. Here, the drawing of the card #13 is assimilated with the fact that the object the colour of which is #13 is at T0 at location L0. The sample space is then denoted by {1,2,…,52}*T0*L0C{1,2,…,52} and the event by {1}*T0*L0C{13}.

The drawing of two cards at the same time or the successive drawing of two cards are then modelled in the same way.

2.6 Drawing of a ball from an urn containing red and blue balls

How can we model the drawing of, for example, a red bowl, from an urn containing 10 balls among which 3 red balls and 7 blue balls? The balls are modelled here as multiple objects presenting each one colour among {R,B}. There exists then a colour variable in the corresponding n-universe. In addition, several objects can present the same colour. The objects are then in relation many with the variable-criterion of colour. Moreover, the objects are in relation many with regard to the constant-criteria of time and location. The corresponding n-universe is therefore a *T0**L0*C (with simplified notation *C). Classically, we have the sample space  = {R,R,R,B,B,B,B,B,B,B} and the event {R}. The sample space is then denoted by {1,2,…,10}*T0**L0*C{R,B} and the event by {{1}*T0**L0*C{R}}.

The drawing of two balls at the same time or the successive drawing of two balls are modelled in the same way.

3. Dimorphisms and isomorphisms

The comparison of the structures of the extended (to n-universes) sample spaces corresponding to two given probabilistic situations allows to determine if these situations are, from a probabilistic viewpoint, isomorphic or not. The examination of the structures of the sample spaces allows to determine easily the isomorphisms or, on the contrary, the dimorphisms. Let us give some examples.

Consider a first type of application where one wonders whether two probabilistic situations are of comparable nature. To this end, we model the two distinct probabilistic situations within the n-universes. The first situation is thus modelled in a *T0*L0*C (with simplified notation *C), and the second one in a *T0*L0C (with simplified notation C). One notices then a dimorphism between the n-universes that make it possible to model respectively the two probabilistic situations. Indeed, in the first situation, the multiple objects are in relation many with the colour criterion, corresponding thus to the fact that several objects can have an identical colour at a given moment and location. On the other hand, in the second situation, the multiple objects are in relation one with the colour criterion, what corresponds to the fact that each object has a different colour at a given time and location. The dimorphism observed at the level of the demultiplication of the variable-criterion of colour in the two corresponding n-universes makes it possible to conclude that the two probabilistic situations are not of a comparable nature.

It is worth considering now a second type of application. The throwing of two discernible dice is modelled, as we did see it, in a {1,2}T0*L{1,2}*O{1,…,6}. Now let us consider a headlight which can take at a given time one colour of 6 colours numbered from 1 to 6. If one considers now two headlights of this type, it appears that the corresponding situation can be modelled in a {1,2}T0*L{1,2}*C{1,…, 6}. In this last case, it appears that the variable-criterion of colour replaces the criterion of orientation. At this stage, it proves that the structure of such n-universe (with simplified notation L*C) is isomorphic to that of the n-universe in which the throwing of two discernible dice was modelled (with simplified notation L*O). This makes it possible to conclude that the two probabilistic situations are of a comparable nature.

Let us consider now a concrete example. John Leslie (1996, 20) describes in the following terms the Emerald case:

Imagine an experiment planned as follows. At some point in time, three humans would each be given an emerald. Several centuries afterwards, when a completely different set of humans was alive, five thousands humans would again each be given an emerald in the experiment. You have no knowledge, however, of whether your century is the earlier century in which just three people were to be in this situation, or the later century in which five thousand were to be in it. Do you say to yourself that if yours were the earlier century then the five thousand people wouldn’t be alive yet, and that therefore you’d have no chance of being among them? On this basis, do you conclude that you might just as well bet that you lived in the earlier century?

Leslie thus puts in parallel a real situation related to some emeralds and a probabilistic model concerning some balls in a urn. Let us proceed then to model the real, concrete, situation, described by Leslie, in terms of n-universes. It appears first that the corresponding situation is characterized by the presence of multiple objects: the emeralds. We find then ourselves in a n-universe with multiple objects. On the second hand, one can consider that the emeralds are situated at one single place: the Earth. Thus, the corresponding n-universe has a location constant (L0). Leslie also distinguishes two discrete temporal positions in the experiment: the one corresponding to a given time and the other being situated several centuries later. The corresponding n-universe comprises then a time variable with two taxa: T1 and T2. Moreover, it proves to be that the emeralds existing in T1 do not exist in T2 (and reciprocally). Consequently, the n-universe corresponding to the emerald case is a n-universe which is not with temporal demultiplication. Moreover, one can observe that several emeralds can be at the same given temporal position Ti: three emeralds exist thus in T1 and five thousand in T2. Thus, the objects are in relation many with the time variable. Lastly, several emeralds can coexist in L0 and the objects are thus in relation many with the location constant. Taking into account what precedes, it appears thus that the Emerald case takes place in a *T (with extended notation *T*L0), a n-universe with multiple objects, comprising a location constant and a time variable with which the objects are in relation many.

Compare now with the situation of the Little Puddle/London experiment, also described by Leslie (1996, 191):

Compare the case of geographical position. You develop amnesia in a windowless room. Where should you think yourself more likely to be: in Little Puddle with a tiny situation, or in London? Suppose you remember that Little Puddle’s population is fifty while London’s is ten million, and suppose you have nothing but those figures to guide you. (…) Then you should prefer to think yourself in London. For what if you instead saw no reason for favouring the belief that you were in the larger of the two places? Forced to bet on the one or on the other, suppose you betted you were in Little Puddle. If everybody in the two places developed amnesia and betted as you had done, there would be ten million losers and only fifty winners. So, it would seem, betting on London is far more rational. The right estimate of your chances of being there rather than in Little Puddle, on the evidence on your possession, could well be reckoned as ten million to fifty.

The latter experiment is based on a real, concrete, situation, to be put in relation with an implicit probabilistic model. It appears first that the corresponding situation characterises itself by the presence of multiple inhabitants: 50 in Little Puddle and 10 million in London. The corresponding n-universe is then a n-universe with multiple objects. It appears, second, that this experiment takes place at one single time: the corresponding n-universe has then one time constant (T0). Moreover, two space positions – Little Puddle and London – are distinguished, so that we can model the corresponding situation with the help of a n-universe comprising two space positions: L1 and L2. Moreover, each inhabitant is either in Little Puddle or in London, but but no one can be at the two places at the same time. The corresponding n-universe is then not with local demultiplication. Lastly, one can notice that several people can find themselves at a given space position Li: there are thus 50 inhabitants at Little Puddle (L1) and 10 million in London (L2). The objects are thus in a relation many with the space variable. And in a similar way, several inhabitants can be simultaneously either in Little Puddle, or in London, at time T0. Thus, the objects are in relation many with the time constant. Taking into account what precedes, it appears that the situation of the Little Puddle/London experiment takes place in a *L (with extended notation *T0*L), a n-universe with multiple objects, comprising a time constant and a location variable, with which the objects are in relation many.

As we can see it, the emerald case takes place in a *T, whereas the Little Puddle/London experiment situates itself in a *L. This makes it possible to highlight the isomorphic structure of the two n-universes in which the two experiments are respectively modelled. This allows first to conclude that the probabilistic model which applies to the one, is also worth for the other one. Moreover, it appears that both the *T and the *L are isomorphic with the *C. This makes it possible to determine straightforwardly the corresponding probabilistic model. Thus, the situation corresponding to both the emerald case and the Little Puddle/London experiment can be modelled by the drawing of a ball from an urn comprising red and blue balls. In the emerald case, it consists of an urn comprising 3 red balls and 5000 green balls. In the Little Puddle/London experiment, the urn includes thus 50 red balls and 107 green balls.

4. Goodman’s paradox

Another interest of the n-universes as a methodological tool resides in their use to clarify complex situations such as those which are faced in the study of paradoxes. I will illustrate in what follows the contribution of the n-universes in such circumstances through the analysis of Goodman’s paradox.3

Goodman’s paradox was described in Fact, Fiction and Forecast (1954, 74-75). Goodman explains then his paradox as follows. Every emeralds which were until now observed turned out to be green. Intuitively, we foresee therefore that the next emerald that will be observed will also be green. Such prediction is based on the generalisation according to which all emeralds are green. However, if one considers the property grue, that is to say “observed before today and green, or observed after today and not-green”,4 we can notice that this property is also satisfied by all instances of emeralds observed before. But the prediction which results from it now, based on the generalisation according to which all emeralds are grue, is that the next emerald to be observed will be not-green. And this contradicts the previous conclusion, which is conforms however with our intuition. The paradox comes here from the fact that the application of an enumerative induction to the same instances, with the two predicates green and grue, leads to predictions which turn out to be contradictory. This contradiction constitutes the heart of the paradox. One of the inductive inferences must then be fallacious. And intuitively, the conclusion according to which the next observed emerald will be not-green appears erroneous.

Let us set out now to model the Goodman’s experiment in terms of n-universes. It is necessary for it to describe accurately the conditions of the universe of reference in which the paradox takes place. Goodman makes thus mention of properties green and not-green applicable to emeralds. Colour constitutes then one of the variable-criteria of the n-universe in which the paradox takes place. Moreover, Goodman draws a distinction between emeralds observed before T and those which will be observed after T. Thus, the corresponding n-universe also includes a variable-criterion of time. As a result, we are in a position to describe the minimal universe in which Goodman (1954) situates himself as a coloured and temporal n-universe, i.e. a CT.

Moreover, Goodman makes mention of several instances of emeralds. It could then be natural to model the paradox in a n-universe with multiple objects, coloured and temporal. However, it does not appear necessary to make use of a n-universe including multiple objects. Considering the methodological objective which aims at avoiding a combinatorial explosion of cases, it is indeed preferable to model the paradox in the simplest type of n-universe, i.e. a n-universe with a unique object. We observe then the emergence of a version of the paradox based on one unique emerald the colour of which is likely to vary in the course of time. This version is the following. The emerald which I currently observe was green all times when I did observe it before. I conclude therefore, by induction, that it will be also green the next time when I will observe it. However, the same type of inductive reasoning also leads me to conclude that it will be grue, and therefore not-green. As we can see, such variation always leads to the emergence of the paradox. The latter version takes p lace in a n-universe including a unique object and a variable of colour and of time, i.e. a CT. At this step, given that the original statement of the paradox turns out to be ambiguous in this respect, and that the minimal context is that of a CT, we will be led to distinguish between two situations: the one which situates itself in a CT, and the one which takes place in a CT (where  denotes a third variable-criterion).

Let us place ourselves first in the context of a coloured and temporal n-universe, i;e. a CT. In such universe, to be green, is to be green at time T. In this context, it appears completely legitimate to project the shared property of colour (green) of the instances through time. The corresponding projection can be denoted by C°T. The emerald was green every time where I observed it before, and the inductive projection leads me to conclude that it will be also green next time when I will observe it. This can be formalized as follows (V denoting green):

(I1)VT1·VT2·VT3·…·VT99instances
(H2)VT1·VT2·VT3·…·VT99·VT100generalisation
(P3) VT100from (H2)

The previous reasoning appears completely correct and conforms to our inductive practice. But are we thus entitled to conclude from it that the green predicate is projectible without restriction in the CT? It appears not. For the preceding inductive enumeration applies indeed to a n-universe where the temporal variable corresponds to our present time, for example the period of 100 years surrounding our present epoch, that is to say the interval [-100, +100] years. But what would it be if the temporal variable extended much more far, by including for example the period of 10 thousand million years around our current time, that is to say the interval [-1010, +1010] years. In that case, the emerald is observed in 10 thousand million years. At that time, our sun is burned out, and becomes progressively a white dwarf. The temperature on our planet then warmed itself up in significant proportions to the point of attaining 8000°: the observation reveals then that the emerald – as most mineral – suffered important transformations and proves to be now not-green. Why is the projection of green correct in the CT where the temporal variable is defined by restriction in comparison with our present time, and incorrect if the temporal variable assimilates itself by extension to the interval of 10 thousand million years before or after our present time? In the first case, the projection is correct because the different instances of emeralds are representative of the reference class on which the projection applies. An excellent way of getting representative instances of a given reference class is then to choose the latter by means of a random toss. On the other hand, the projection is not correct in the second case, for the different instances are not representative of the considered reference class. Indeed, the 99 observations of emeralds come from our modern time while the 100th concerns an extremely distant time. So, the generalisation (H2) results from 99 instances which are not representative of the CT[-1010, +1010] and does not allow to be legitimately of use as support for induction. Thus green is projectible in the CT[-102, +102] and not projectible in the CT[-1010, +1010]. At this stage, it already appears that green is not projectible in the absolute but turns out to be projectible or not projectible relative to this or that n-universe.

In the light of what precedes, we are from now on in a position to highlight what proved to be fallacious in the projection of generalisation according to which “all swans are white”. In 1690, such hypothesis resulted from the observation of a big number of instances of swans in Europe, in America, in Asia and in Africa. The n-universe in which such projection did take place was a n-universe with multiple objects, including a variable of colour and of location. To simplify, we can consider that all instances had being picked at constant time T0. The corresponding inductive projection C°L led to the conclusion that the next observed swan would be white. However, such prediction turned out to be false, when occurred the discovery in 1697 by the Dutch explorer Willem de Vlamingh of black swans in Australia. In the n-universe in which such projection did take place, the location criterion was implicitly assimilating itself to our whole planet. However, the generalisation according to which “all swans are white” was founded on the observation of instances of swans which came only from one part of the n-universe of reference. The sample turned out therefore to be biased and not representative of the reference class, thus yielding the falseness of the generalisation and of the corresponding inductive conclusion.

Let us consider now the projection of grue. The use of the grue property, which constitutes (with bleen) a taxon of tcolour*, is revealing of the fact that the used system of criteria comes from the Z. The n-universe in which takes place the projection of grue is then a Z, a n-universe to which the CT reduces. For the fact that there exists two taxa of colour (green, not-green) and two taxa of time (before T, after T) in the CT determines four different states: green before T, not-green before T, green after T, not-green after T. By contrast, the Z only determines two states: grue and bleen. The reduction of the CT to the Z is made by transforming the taxa of colour and of time into taxa of tcolour*. The classical definition of grue (green before T or not-green after T) allows for that. In this context, it appears that the paradox is still present. It comes indeed under the following form: the emerald was grue every time that I did observe it before, and I conclude inductively that the emerald will also be grue and thus not-green the next time when I will observe it. The corresponding projection Z°T can then be formalized (G denoting grue):

(I4*)GT1·GT2·GT3·…·GT99instances
(H5*)GT1·GT2·GT3·…·GT99·GT100generalisation
(H5’*)VT1·VT2·VT3·…·VT99·~VT100from (H5*), definition
(P6*) GT100prediction
(P6’*) ~VT100from (P6*), definition

What is it then that leads to deceive our intuition in this specific variation of the paradox? It appears here that the projection of grue comes under a form which is likely to create an illusion. Indeed, the projection Z°T which results from it is that of the tcolor* through time. The general idea which underlies inductive reasoning is that the instances are grue before T and therefore also grue after T. But it should be noticed here that the corresponding n-universe is a Z. And in a Z, the only variable-criterion is tcolor*. In such n-universe, an object is grue or bleen in the absolute. By contrast, an object is green or not-green in the CT relative to a given temporal position. But in the Z where the projection of grue takes place, an additional variable-criterion is missing so that the projection of grue could be legitimately made. Due to the fact that an object is grue or bleen in the absolute in a Z, when it is grue before T, it is also necessarily grue after T. And from the information according to which an object is grue before T, it is therefore possible to conclude, by deduction, that it is also grue after T. As we can see it, the variation of the paradox corresponding to the projection Z°T presents a structure which gives it the appearance of an enumerative generalisation but that constitutes indeed a genuine deductive reasoning. The reasoning that ensues from it constitutes then a disguised form of induction, a pseudo-induction.

Let us envisage now the case of a coloured, temporal n-universe, but including an additional variable-criterion , i.e. a CT. A n-universe including variable-criteria of colour, of time and location,5 i.e. a CTL, will be suited for that. To be green in a CTL, is to be green at time T and at location L. Moreover, the CTL reduces to a ZL, a n-universe the variable-criteria of which are tcolor* and location. The taxa of tcolor* are grue and bleen. And to be grue in the ZL, is to be grue at location L.

In a preliminary way, one can point out here that the projections CTL and ZTL do not require a separate analysis. Indeed, these two projections present the same structure as those of the projections CT and ZT which have just been studied, except for an additional differentiated criterion of location. The conditions under which the paradox dissolves when one compares the projections CT and ZT apply therefore identically to the variation of the paradox which emerges when one relates the projections CTL and ZTL .

On the other hand, it appears here opportune to relate the projections CT°L and Z°L which respectively take place in the CTL and the ZL. Let us begin with the projection CT°L. The shared criteria of colour and of time are projected here through a differentiated criterion of location. The taxa of time are here before T and after T. In this context, the projection of green comes under the following form. The emerald was green before T in every place where I did observe it before, and I conclude from it that it will be also green before T in the next place where it will be observed. The corresponding projection C°TL can then be formalized as follows:

(I7)VTL1·VTL2·VTL3·…·VTL99instances
(H8)VTL1·VTL2·VTL3·…·VTL99·VTL100generalisation
(P9) VTL100prediction

At this step, it seems completely legitimate to project the green and before T shared by the instances, through a differentiated criterion of location, and to predict that the next emerald which will be observed at location L will present the same properties.

What is it now of the projection of grue in the CTL? The use of grue conveys the fact that we place ourselves in a ZL, a n-universe to which reduces the CTL and the variable-criteria of which are tcolour* and location. The fact of being grue is relative to the variable-criterion of location. In the ZL, to be grue is to be grue at location L. The projection relates then to a taxon of tcolour* ( grue or bleen) which is shared by the instances, through a differentiated criterion of location. Consider then the classical definition of grue (green before T or non-grue after T). Thus, the emerald was grue in every place where I did observe it before, and I predict that it will also be grue in the next place where it will be observed. If we take T = in 1010 years, the projection Z°L in the ZL appears then as a completely valid form of induction (V~T denoting green after T):

(I10*)GL1·GL2·GL3·…·GL99instances
(H11*)GL1·GL2·GL3·…·GL99·GL100generalisation
(H11’*)VT~V~TL1·VT~V~TL2·VT~V~TL3·…·VT~V~TL99·VT~V~TL100from (H11*), definition
(P12*) GL100prediction
(P12’*) VT~V~TL100from (P12*), definition

As pointed out by Franck Jackson (1975, 115), such type of projection applies legitimately to all objects which colour changes in the course of time, such as tomatoes or cherries. More still, one can notice that if we consider a very long period of time, which extends as in the example of emeralds until 10 thousand million years, such property applies virtually to all concrete objects. Finally, one can notice here that the contradiction between both concurrent predictions (P9) and (P12’*) has now disappeared since the emerald turns out to be green before T in L100 (VTL100) in both cases.

As we can see, in the present analysis, a predicate turns out to be projectible or not projectible in relative to this or that universe of reference. As well as green, grue is not projectible in the absolute but turns out to be projectible in some n-universes and not projectible in others. It consists here of a difference with several classical solutions offered to solve the Goodman’s paradox, according to which a predicate turns out to be projectible or not projectible in the absolute. Such solutions lead to the definition of a criterion allowing to distinguish the projectible predicates from the unprojectible ones, based on the differentiation temporal/non-temporal, local/non-local, qualitative/non-qualitative, etc. Goodman himself puts then in correspondence the distinction projectible/ unprojectible with the distinction entrenchedi/unentrenched, etc. However, further reflexions of Goodman, formulated in Ways of Worldmakingii, emphasize more the unabsolute nature of projectibility of green or of grue: “Grue cannot be a relevant kind for induction in the same world as green, for that would preclude some of the decisions, right or wrong, that constitute inductive inference”. As a result, grue can turn out to be projectible in a goodmanian world and not projectible in some other one. For green and grue belong for Goodman to different worlds which present different structures of categories.6 In this sense, it appears that the present solution is based on a form of relativism the nature of which is essentially goodmanian.

5. Conclusion

From what precedes and from Goodman’s paradox analysis in particular, one can think that the n-universes are of a fundamentally goodmanian essence. From this viewpoint, the essence of n-universes turns out to be pluralist, thus allowing numerous descriptions, with the help of different systems of criteria, of a same reality. A characteristic example, as we did see it, is the reduction of the criteria of colour and time in a CTL into a unique criterion of tcolour* in a ZL. In this sense, one can consider the n-universes as an implementation of the programme defined by Goodman in Ways of Worldmaking. Goodman offers indeed to construct worlds by composition, by emphasis, by ordering or by deletion of some elements. The n-universes allow in this sense to represent our concrete world with the help of different systems of criteria, which correspond each to a relevant point of view, a way of seeing or of considering a same reality. In this sense, to privilege this or that system of criteria, to the detriment of others, leads to a truncated view of this same reality. And the exclusive choice, without objective motivation, of such or such n-universe leads to engender a biased point of view.

However, the genuine nature of the n-universes turns out to be inherently ambivalent. For the similarity of the n-universes with the goodmanian worlds does not prove to be exclusive of a purely ontological approach. Alternatively, it is indeed possible to consider the n-universes from the only ontological point of view, as a methodological tool allowing to model directly this or that concrete situation. The n-universes constitute then so much universes with different properties, according to combinations resulting from the presence of a unique object or multiple objects, in relation one or many, with demultiplication or not, with regard to the criteria of time, location, colour, etc. In a goodmanian sense also, the n-universes allow then to build so much universes with different structures, which sometimes correspond to the properties of our real world, but which have sometimes some exotic properties. To name only the simplest of the latter, the L* is then a n-universe which includes only one ubiquitous object, presenting the property of being at several locations at the same time.7

At this stage, it is worth mentioning several advantages which would result from the use of the n-universes for modelling probabilistic situations. One of these advantages would be first to allow a better intuitive apprehension of a given probabilistic situation, by emphasising its essential elements and by suppressing its superfluous elements. By differentiating for example depending on whether the situation to model presents a constant or a time variable, a constant or a space variable, a unique object or several objects, etc. the modelling of concrete situations in the n-universes provides a better support to intuition. On the other hand, the distinction according to whether the objects are or not with demultiplication or in relation one/many with regard to the different criteria allows for a precise classification of the different probabilistic situations which are encountered.

One can notice, second, that the use of the notation of the probability spaces extended to the n-universes would allow to withdraw the ambiguity which is sometimes associated with classical notation. As we did see it, we sometimes face an ambiguity. Indeed, it proves to be that {1,2,…,6}2 denotes at the same time the sample space of a simultaneous throwing of two discernible dices in T0 and that of two successive throwing of the same dice in T1 and then in T2. With the use of the notation extended to n-universes, the ambiguity disappears. In effect, the sample space of the simultaneous throwing of two discernible dices in T0 is a {1,2}*T0L{1,2}*O{1,2,…,6}, whilst that of two successive throwing of the same dice in T1 and then in T2 is a 0T*{1,2}L0O{1,2,…,6}.

Finally, an important advantage, as we have just seen it, which would result from a modelling of probabilistic situations extended to n-universe is the easiness with which it allows comparisons between several probabilistic models and it highlights the isomorphisms and the corresponding dimorphisms. But the main advantage of the use of the n-universes as a methodological tool, as we did see it through Goodman’s paradox, would reside in the clarification of the complex situations which appear during the study of paradoxes.8

References

Franceschi, Paul. 2001. Une solution pour le paradoxe de Goodman. Dialogue 40: 99-123, English translation under the title The Doomsday Argument and Hempel’s Problem, http://cogprints.org/2172/. English translation
—. 2002. Une application des n-univers à l’argument de l’Apocalypse et au paradoxe de Goodman. Doctoral dissertation, Corté: University of Corsica. <http://www.univ-corse.fr/~franceschi/index-fr.htm> [retrievec Dec.29, 2003]
Goodman, Nelson. 1954. Fact, Fiction and Forecast. Cambridge, MA: Harvard University Press.
—. 1978. Ways of Worldmaking. Indianapolis: Hackett Publishing Company.
Jackson, Franck. 1975. “Grue”. The Journal of Philosophy 72: 113-131.
Leslie, John. 1996. The End of the World: The Science and Ethics of Human Extinction. London: Routledge.

1 Il convient de noter que ces différentes modélisations ne constituent pas une manière unique de modéliser les objets correspondants dans les n-univers. Cependant, elles correspondent à l’intuition globale que l’on a de ces objets.

2 De manière alternative, on pourrait utiliser la notation 0T0L0O5 en lieu et place de 0T0L0O{5}. Cette dernière notation est toutefois préférée ici, car elle se révèle davantage compatible avec la notation classique des événements.

3 Cette analyse du paradoxe de Goodman correspond, de manière simplifiée et avec plusieurs adaptations, à celle initalement décrite dans Franceschi (2001). La variation du paradoxe qui est considérée ici est celle de Goodman (1954), mais avec une émeraude unique.

4 P and Q being two predicates, grue presents the following structure: (P and Q) or (~P and ~Q).

5 Tout autre critère différent de la couleur et du temps tel que la masse, la température, l’orientation, etc. conviendrait également.

6 Cf. Goodman (1978, 11): “(…) a green emerald and a grue one, even if the same emerald (…) belong to worlds organized into different kinds”.

7 Les n-univers aux propriétés non standard nécessitent une étude plus détaillée, qui dépasse le cadre de la présente étude.

8 Je suis reconnaissant envers Jean-Paul Delahaye pour la suggestion de l’utilisation des n-univers en tant qu’espaces de probabilité étendus. Je remercie également Claude Panaccio et un expert anonyme pour le Journal of Philosophical Research pour des discussions et des commentaires très utiles.

i Entrenched.

ii Cf. Goodman (1978, 11).

Post-publication of a review of John Leslie, Infinite Minds

Review of John Leslie, Infinite Minds, Oxford, Oxford University Press, 2001, 234 pages.1

 Paul Franceschi

 University of Corsica

Post-publication of the review appeared in Philosophiques, Volume 30, number 2, Autumn 2003

Infinite Minds is the fourth book of John Leslie, which follows Value and Existence (1979), Universes (1989) and The End of the World (1996). Infinite Minds presents a very rich content, and covers a number of particularly varied subjects. Among these latter, one can notably mention: omniscience, the problem of Evil, the fine-tuning argument, observational selection effects, the identity of indiscernables, time, infiniteness, the nature of consciousness.

The book places itself clearly within the field of speculative philosophy. And Leslie is primarily concerned here with considerations not of rigorous demonstration, but rather of plausibility and of coherence. He thus does not hesitate sometimes to attribute a rather weak probability to certain assertions.

Some readers may be rebutted from the beginning by the counter-intuitive assertion that galaxies, planets, animals, but also each of us and our surrounding objects, are mere structures among divine thoughts. One can think that such an assertion has motivated the commentary placed on the book’s cover by a reader from Oxford University Press, according to which it may be difficult to believe that the universe is such that the author describes it. This was also my primary reaction. But if certain readers were to draw from that a hasty conclusion, they would miss then, I think, what constitutes the hidden treasure of the book. Because Infinite Minds resembles a sumptuous temple, whose access however is dissimulated by a gate which looks poorly attractive. Those who will not cross the door, rebutted by the aspect of this latter, will not have the occasion to contemplate the hidden treasures that the book contains. Because the book presents an overall deep structure and coherence, based on the consistency of the author’s pantheist conception of the universe with our current most advanced scientific views with regard to cosmology, physics, as well as with the solutions to several contemporary philosophical problems. To show synthetically how a pantheist vision of the world can cohere with our most recent views with regard to multiple universes, physics and quantum computer science, inasmuch as with relativity theory and recent discussions relating to omniscience, the problem of Evil, the fine-tuning argument, observational selection effects, etc. appears both an immense and deeply original task.

It should be observed here that Leslie is familiar with this type of wide-scale work. It suffices for that to consider his whole work relating to the Doomsday Argument. It is worth evaluating here the immense task which consists in defending point by point the Doomsday Argument against a good hundred different objections. But this vigorous defense of the Doomsday argument has stimulated in return the development of a rich literature, which continuously enlightens a number of fields hitherto ignored.

The variety of pantheism described by Leslie, inspired by Spinoza, characterizes itself by the fact that each of us is nothing but a structure of divine thoughts. Because the divine mind only exists. The galaxies, the planets, the mountains, the human beings that we are, the animals, the flowers are nothing but structures within divine minds. As Leslie points out, this is coherent with the way physicists themselves describe physical objects, by specifying their intrinsic properties. Nevertheless, Leslie is not committed to a conception of panpsychism where all beings and objects which are part of our universe, have mental properties. For according to the author, physical objects such as trees, rocks, sand, exist as structures within the divine mind, but without being equipped themselves with conscience or thought. Here, all things are not equipped with conscience, but are such however that a conscience of these latter things exists.

Moreover, universes in infinite number can exist as structures in the divine mind. The author’s theory appears thus compatible with recent cosmological theories based on the existence of multiple universes. One of these universes is thus our own, which presents such characteristics and an accurate tuning of its parameters (the ratio of the respective masses of the electron and proton, the electron charge, the gravitational constant, Planck’s constant, etc.), that it allows the emergence of an intelligent life.

Furthermore, Leslie suggests the existence not of a single divine mind, but as well of an infinity of divine minds. Each of them is absolutely identical to the others, but has however an autonomous conscience of its own existence.

What is then the status of abstract objects, such as natural integers, in this context? According to certain philosophers, abstract objects also constitute divine thoughts. Such is in particular the viewpoint put forth by Alvin Plantinga, according to which natural integers constitute divine thoughts. But Leslie adopts a different line of thought. Abstract objects such as natural integers have in Infinite Minds’ ontology a completely original status, which is not prima facie obvious, and which deserves a detailed mention. Abstract objects such as natural integers, the idea of an apple, or the idea that “2 + 2 = 4”, are of Platonic essence. And Leslie points out that such abstract objects do not result from our brains, which themselves constitute thoughts in the divine mind. Neither do such objects of Platonic nature result from the divine mind itself. The natural integers, the idea of an apple, or the idea that “2 + 2 = 4”, constitute eternal realities, which are independent of our existence as human beings, of our thoughts and of our language. Leslie explains clearly how the idea that “2 + 2 = 4”, i.e. the fact that “IF two sets of two apples exist, THEN four apples exist” (p. 160) constitutes a Platonic reality, independent of the thoughts of the divine mind and of the human beings that we are.

Leslie also develops the topic of omniscience. According to Leslie, God simply knows all that is worth knowning: (“God knows everything worth knowing”). This seems probably more plausible than the idea according to which God has any knowledge, which notably conflicts with the logically impossible existence, already noted by Patrick Grim, of the set which contains absolutely all truths. Our pretheoretical conception of an omniscient God could well appear naïve, as the author points out, because a many unimportant facts could appear undesirable knowledge there.

Lastly, Leslie develops the point of view according to which God exists by ethical need (“because of its ethical requiredness”). The existence of God and of the cosmos in his entirety is ethically necessary, from all eternity. This argument could well appear more convincing than certain ontological arguments. Because such an ethical need has, according to Leslie, an inherently creative power. But such creative capacity, of Platonic essence, does not proceed of any external cause. It is simply inherent by nature to the ethical necessity.

Leslie’s book also constitutes the courageous expression of a viewpoint. For such pantheist conception does not constitute a widespread opinion within contemporary analytical philosophy. Moreover, Leslie’s variety of pantheism also constitutes a variation of panpsychism. But the attitude of the author appears eminently constructive, because it constrains us to consider more attentively some doctrines than we would tend to reject too easily. One will or not adhere to the pantheist and panpsychist theory exposed in Infinite Minds. But for the majority of readers for whom we can suppose that they will not adhere to the variation of panpsychism thus described, Leslie’s work constitutes nevertheless an admirable and highly original synthesis, showing how an astonishing construction can be elaborated around the pantheist model, while bringing answers to many contemporary philosophical problems. The work will provide new arguments to the defenders of panpsychism. But Infinite Minds will be also prove to be essential to the detractors of panpsychism, who will find there a particularly strong and structured defense.

1This review only differs from the version published in Philosophiques with regard to the status of abstract objects. I thank John Leslie for very useful discussion on this topic.

The Problem of the Relationships of Love-Hate-Indifference

English translation of a paper published in French in Semiotica, vol. 150(1-4), 2004 under the title “Le problème des relations amour-haine-indifférence”.

In On a class of concepts (2002), I described a theory based on the matrices of concepts which aims at constituting an alternative to the classification proposed by Greimas, in the field of paradigmatic analysis. The problem of the determination of the relationships of love/hate/indifference arises in this construction. I state then the problem of the relationships of love/hate/indifference in a detailed way, and several solutions that have been proposed in the literature to solve it. I describe lastly a solution to this problem, based on an extension of the theory of matrices of concepts.


This paper is cited in:

  • Isis Truck, Nesrin Halouani, & Souhail Jebali (2016) Linguistic negation and 2-tuple fuzzy linguistic representation model : a new proposal, pages 81–86, in Uncertainty Modelling in Knowledge Engineering and Decision Making, The 12th International FLINS Conference on Computational Intelligence in Decision and Control, Eds. Xianyi Zeng, Jie Lu, Etienne E Kerre, Luis Martinez, Ludovic Koehl, 2016, Singapore: World Scientific Publishing.

The Problem of the Relationships of Love-Hate-Indifference

Paul Franceschi

I shall be concerned in this paper with presenting a problem related to the proper definition of the relationships of the following concepts: love, hate and indifference. I will describe first the problem in detail and some proposed solutions. Lastly, I will present my own solution to the problem.

1. The problem

The problem is that of the proper definition of the relationships of the concepts love, hate and indifference. Let us call it the LHI problem. What are then the accurate relationships existing between these three concepts? At first sight, the definition of the relation between love and hate is obvious. These concepts are contraries. The definition of such a relation should be consensual. Nevertheless, the problem arises when one considers the relationship of love and indifference, and of hate and indifference. In these latter cases, no obvious response emerges.

However, the issue needs clarifying. In this context, what should we expect of a solution to the LHI problem? In fact, a rigorous solution ought to define precisely the three relations R, S, T such that love R hate, love S indifference and hate T indifference. And the definitions of these relations should be as accurate as possible.

It is worth mentioning that several authors must be credited for having mentioned and investigated the LHI problem. In particular, it is worth stressing that the difficulties presented within propositional calculus by some assertions of the type x loves y, x hates y, or x is indifferent to y have been hinted at by Emile Benzaken (1990)1:

Nevertheless, the difficulty can arise from pairs of words where the one expresses the contrary (negation) of the other; ‘to hate’ can be considered as the strong negation of ‘to love’, whereas ‘to be indifferent’ would be its weak negation.

The author exposes then the problem of the relationships of love/hate/indifference and proposes his own solution: hate is the strong negation of love, and indifferent is the weak negation of love.

However, it turns out that Benzaken’s solution is unsatisfying for a logician, for the following reasons. On the one hand, this way of solving the problem defines the relations between love and hate (strong negation, according to the author) and between love and indifference (weak negation, on the author’s view), but it fails to define accurately the relations existing between indifference and hate. There is a gap, a lack of response at this step. And mentioned above, a satisfying solution should elucidate the nature of the relationships of the three concepts. On the other hand, the difference between weak negation and strong negation is not made fully explicit within the solution provided by Benzaken. For these reasons, Benzaken’s solution to the LHI problem proves to be unsatisfying.

In a very different context, Rick Garlikov (1998) stresses some difficulties of essentially the same nature as those underlined by Benzaken:

In a seminar I attended one time, one of the men came in all excited because he had just come across a quotation he thought very insightful – that it was not hate that was the opposite of love, but that indifference was the opposite of love, because hate was at least still an emotion. I chuckled, and when he asked why I was laughing, I pointed out to him that both hate and indifference were opposites of love, just in different ways, that whether someone hated you or was indifferent toward you, in neither case did they love you.

Garlikov describes in effect the problem of the relationships of love/hate/indifference and implicitly proposes a solution of a similar nature as that provided by Benzaken. For this reason, Galikov’s account suffers from the same defects as those presented by Benzaken’s solution.

In what follows, my concern will be with settling first the relevant machinery, in order to prepare a few steps toward a solution to the LHI problem.

2. The framework

I will sketch here the formal apparatus described in more detail in Franceschi (2002). To begin with, consider a given duality. Let us denote it by A/Ā. At this step, A and Ā are dual concepts. Moreover, A and Ā can be considered as concepts that are characterized by a contrary component c {-1, 1} within a duality A/Ā, such that c[A] = -1 and c[Ā] = 1. Let us also consider that A and Ā are neutral concepts that can be thus denoted by A0 and Ā0.

Figure 1

At this point, we are in a position to define the class of the canonical poles. Consider then an extension of the previous class {A0, Ā0}, such that A0 and Ā0 respectively admit of a positive and a negative correlative concept. Such concepts are intuitively appealing. Let us denote them respectively by {A+, A} and {Ā+, Ā}. At this step, for a given duality A/Ā, we get then the following concepts: {A+, A0, A, Ā+, Ā0, Ā}. Let us call them canonical poles. It should be noted that one could use alternatively the notation α(A/Ā, c, p) for a canonical pole.2 In all cases, the components of a canonical pole are a duality A/Ā, a contrary component c {-1, 1} and a canonical polarity p {-1, 0, 1}. This definition of the canonical poles leads to distinguish between the positive (A+, Ā+), neutral (A0, Ā0) and negative (A, Ā) canonical poles. Lastly, the class made up by the 6 canonical poles can be termed the canonical matrix: {A+, A0, A, Ā+, Ā0, Ā}.

Let us investigate now into the nature of the relations existing between the canonical poles of a given matrix. Among the combinations of relations existing between the 6 canonical poles (A+, A0, A, Ā+, Ā0, Ā) of a same duality A/Ā, it is worth emphasizing the following relations: duality, antinomy, complementarity, corollarity, connexity, and anti-connexity. Thus, two canonical poles α1(A/Ā, c1, p1) and α2(A/Ā, c2, p2) of a same matrix are:

(i) dual if their contrary components are opposite and their polarities are neutral3

(ii) contrary (or antinomical) if their contrary components are opposite and their polarities are non-neutral and opposite4

(iii) complementary if their contrary components are opposite and their polarities are non-neutral and equal5

(iv) corollary if their contrary components are equal and their polarities are non-neutral and opposite6

(v) connex if their contrary components are equal and the absolute value of the difference of their polarities equals 17

(vi) anti-connex if their contrary components are opposite and the absolute value of the difference of their polarities equals 18

To sum up: {A0, Ā0} are dual, {A+, Ā} and {A, Ā+} are contraries, {A+, Ā+} and {A, Ā} are complementary, {A+, A} and {Ā+, Ā} are corollary, {A0, A+}, {A0, A}, {Ā0, Ā+} and {Ā0, Ā} are connex, {A0, Ā+}, {A0, Ā}, {Ā0, A+} and {Ā0, A} are anti-connex.

I shall focus now on the types of relations existing, under certain circumstances between the canonical poles of different dualities. Let us define preliminarily the includer relation. Let a concept α be an includer for two other concepts β and γ if and only if α = β γ. Such a definition captures the intuition that α is the minimal concept whose semantic content includes that of β and γ. To give an example concerning truth-value, determinate is an includer for {true, false}.

Let now A and E be two matrices whose canonical poles are respectively {A+, A0, A, Ā+, Ā0, Ā} and {E+, E0, E, Ē+, Ē0, Ē}. These matrices are such that E+, E0, E are the respective includers for {A+, Ā+}, {A0, Ā0}, {A, Ā} i.e. the two matrices are such that E+ = A+ Ā+, E0 = A0 Ā0 and E = A Ā.9

Figure 2

Let us denote this relation by A < E. One is now in a position to extend the relations previously defined between the canonical poles of a same matrix, to the relations of a same nature between two matrices presenting the properties of A and E, i.e. such that A < E. The relations of 2-duality, 2-antinomy, 2-complementarity, 2-anti-connexity10 ensue then straightforwardly. Thus, two canonical poles α1(A/Ā, c1, p1) and α2(E/Ē, c2, p2) of two different matrices are:

(i’) 2-dual (or trichotomic dual) if their polarities are neutral and if the dual of 2 is an includer for 1

(ii’) 2-contrary11 (or trichotomic contrary) if their polarities are non-neutral and opposite and if the contrary of α2 is an includer for α1

(iii’) 2-complementary (or trichotomic complementary) if their polarities are non-neutral and equal and if the complementary of α2 is an includer for α1

(vi’) 2-anti-connex (or trichotomic anti-connex) if the absolute value of the difference of their polarities is equal to 1 and if the anti-connex of α2 is an includer for α1

To sum up now: {A0, Ē0} and {Ā0, Ē0} are 2-dual, {A+, Ē}, {A, Ē+}, {Ā+, Ē} and {Ā, Ē+} are 2-contrary, {A+, Ē+}, {A, Ē}, {Ā+, Ē+} and {Ā, Ē} are 2-complementary, {A0, Ē+}, {A0, Ē}, {Ā0, Ē+} and {Ā0, Ē} are 2-anti-connex.

Lastly, the notion of a complement of a canonical pole also deserves mention. Let α be a canonical pole. Let us denote by ~α its complement, semantically corresponding to nonα. In the present context, the notion of a complement entails the definition of a universe of reference. I shall focus then on the notion of a complement of a canonical pole defined with regard to the corresponding matrix. In this case, the universe of reference is equal to {A+, A0, A, Ā+, Ā0, Ā} and then ~α = {A+, A0, A, Ā+, Ā0, Ā} – α. One has thus for example ~A+ = {A0, A, Ā+, Ā0, Ā} and a similar definition for the complements of the other canonical poles of the matrix. Consider now two matrices such that A < E. Under these circumstances, the universe of reference12 is equal to {A+, A0, A, Ā+, Ā0, Ā, Ē+, Ē0, Ē}. Call it the 2-matrix of α. It ensues that ~α = {A+, A0, A, Ā+, Ā0, Ā, Ē+, Ē0, Ē} – α. We have then the notion of a 2-complement of a canonical pole α, defined with regard to a universe of reference consisting of the 2-matrix of α. More generally, one has the notion of a ncomplement (n > 0) of a canonical pole with regard to the corresponding n-matrix.

3. A solution

With the relevant machinery in place, we are now in a position to present a solution to the LHI problem. Let us now analyze the problem in the light of the above framework. To begin with, let us analyze the relevant concepts in more detail. The concept love has a positive connotation. It is a meliorative concept that can be denoted by love+. Conversely, the concept hate has a negative connotation. It is a pejorative concept that can be rendered by hate. Similarly, the concept indifference also has a negative connotation. It can be considered a pejorative notion that can be denoted by indifference.

At this step, a difficulty emerges. In effect, it should be stressed that the three concepts are either meliorative or pejorative at a certain degree. And such a degree might be different from one concept to another. For example hate might be pejorative at a 0.95 degree, while indifference might be pejorative at a lesser degree of 0.7. Moreover, it could be said that such a degree might vary from culture to culture, from a given language to another. In sum, the meliorative or pejorative degree of the three concepts, so the objection goes, could be culture-relative.

Nevertheless, such difficulties can be avoided in the present context, since our reasoning will not bear upon the concepts inherent to a specific culture or language, but rather on the canonical concepts described above. Accordingly, we shall replace our usual concepts by the corresponding canonical concepts. There is room for variation in degrees, from culture to culture in the usual concepts of love, hate and indifference. But this point does not affect the current line of reasoning, since it only focuses on canonical concepts. The passage from the non-canonical concepts to the canonical ones goes straightforwardly as follows. Let d[α] be the pejorative or meliorative degree of a concept α. Hence if d[α] ]0.5; 1] then p[α] = 1 else if d[α] [-1; -0.5[ then p[α] = -1. At this point, one can pose legitimately that p[Love] = 1, p[Hate] = -1 and p[Indifference] = -113. As a result, the three concepts can be denoted by Love+, Hate, Indifference.

Figure 3

As noted from the beginning, the relationship of love/hate is unproblematic and identifies itself with the relation of contrary. This applies straightforwardly to the relationship of the canonical concepts Love+/Hate. Hence, the corresponding matrix has the following structure: {Love+, A0, A, Ā+, Ā0, Hate}. Now the next step is the reconstitution of the complete matrix. This task can be accomplished with the help of the definition of the relations of the canonical poles, namely: A is corollary to Love+, Ā+ is corollary to Hate, A0 is connex to Love+ and anti-connex to Hate, Ā0 is connex to Hate and anti-connex to Love+. Given these elements, we are now in a position to reconstitute the corresponding canonical matrix: {Love+, Attraction0, A, Defiance+, Repulsion0, Hate}.14

Let us examine now the case of the concept Indifference. Such a concept inserts itself into a matrix the structure of which is: {E+, E0, E, Ē+, Ē0, Indifference}. Just as before, it is now necessary to reconstitute the complete matrix. This can be done with the help of the corresponding definitions: Ē+ is corollary to Indifference, E is complementary to Indifference, E+ is contrary to Indifference, Ē0 is connex to Indifference and to the corollary of Indifference, E0 is anti-connex to Indifference and to the corollary of Indifference. The associated matrix is then: {E+, Interest0, E, Phlegm+, Detachment0, Indifference}.15

Figure 4

It should be observed now that Interest0 = Attraction0 Repulsion0 i.e. that Interest0 is an includer for Attraction0 and Repulsion0. At this step, given that {Love+, Attraction0, A, Repulsion+, Repulsion0, Hate} {E+, Interest0, E, Phlegm+, Detachment0, Indifference}, the relationship of Love+/Indifference and Hate/Indifference now apply straightforwardly. In effect, it ensues from the above definitions that, on the one hand, Love+ and Indifferenceare trichotomic contraries and on the other hand, Hate and Indifferenceare trichotomic complementaries. At this point, one is finally in a position to formulate a solution to the LHI problem:

(i) love is contrary to hate

(ii) love is 2-contrary to indifference

(iii) hate is 2-complementary to indifference

Hence, R, S, T identify respectively themselves with contrary, trichotomic contrary, trichotomic complementarity.

4. Concluding remarks

At this point, it is tempting not to consider the above analysis as a solution to the LHI problem per se. In effect, the concepts love, hate and indifference seem to be instances of a wider class of concepts whose relationships are of the same nature. This suggests that the same type of solution should be provided to the general problem of the definition of the relations of three given concepts , , . At first sight, certain concepts such as true, false and indeterminate, fall under the scope of the current analysis. Nevertheless, such a claim should be envisaged with caution. To what extent does the present analysis apply to other concepts? This is another problem that needs to be addressed, but whose resolution goes beyond the scope of the present account.16


References

Benzaken, Claude (1991). “Systèmes formels”. Paris, Masson

Franceschi, Paul (2002). “Une Classe de Concepts”. Semiotica, 139, pp. 211-26, English translation

Garlikov, Rick (1998). “Understanding, Shallow Thinking, and School”. At http://www.garlikov.com/writings.htm

1 My translation. The original text is as follows: ‘La difficulté cependant peut provenir de paires de mots dont l’un exprime le contraire (négation) de l’autre; “haïr” peut être pris comme la négation forte de “aimer” tandis que “être indifférent” en serait la négation faible. (p. 63).

2 With the latter notation, the matrix of the canonical poles is rendered as follows: {(A/Ā, -1, 1), (A/Ā, -1, 0), (A/Ā, -1, -1), (A/Ā, 1, 1), (A/Ā, 1, 0), (A/Ā, 1, -1)}.

3 Formally 1 and 2 are dual if and only if c[1] = – c[2] and p[1] = p[2] = 0.

4 Formally 1 and 2 are antinomical if and only if c[1] = – c[2] and p[1] = – p[2] with p[1], p[2] 0.

5 Formally 1 and 2 are complementary if and only if c[1] = – c[2] and p[1] = p[2] with p[1], p[2] 0.

6 Formally 1 and 2 are corollary if and only if c[1] = c[2] and p[1] = – p[2] with p[1], p[2] 0.

7 Formally 1 and 2 are connex if and only if c[1] = c[2] and │p[1] – p[2]│ = 1.

8 Formally 1 and 2 are anti-connex if and only if c[1] = – c[2] and │p[1] – p[2]│ = 1.

9 It should be observed that one of the three conditions is sufficient. In effect, E+ = A+ Ā+ entails E0 = A0 Ā0 and E = A Ā; E0 = A0 Ā0 implies E+ = A+ Ā+ and E = A Ā; E = A Ā entails E0 = A0 Ā0 and E+ = A+ Ā+.

10 The generalisation to n matrices (n > 1) of the present construction ensues, with the relations of n-duality, n-antinomy, n-complementarity, n-anti-connexity.

11 Or 2-antinomical.

12 In this context, E+, E0 and E can be omitted without loss of content, given their nature of includers.

13 The fact of considering alternatively p[indifference] > -0.5 and thus p[Indifference] = 0 also leads to a solution in the present framework. In this last case, the relations S and T both identify themselves with trichotomic anti-connexity.

14 In the process of reconstitution of the complete matrix, some concepts may be missing. The reason is that they are not lexicalized in the corresponding language. This is notably the case for A. This last concept semantically corresponds to inappropriate, excessive attraction.

15 As far as I can see, the concepts associated with E+ and E are not lexicalized. They respectively correspond to appropriate interest and inappropriate, excessive interest.

16 I thank Professor Claude Panaccio and Rick Garlikov for useful comments on an earlier draft.

The Dialectical Plan: an Alternative to the Paradigm

Posprint in English (with additional illustrations from wikimedia commons) of a paper published in French in Semiotica, vol. 146(1-4), 2003, 353-367, under the title “Le plan dialectique: pour une alternative au paradigme”. I apply the theory developed in On a Class of Concepts (2002) to the methodology for conceiving a plan. Regarding the dialectical plan, the paradigm is a plan whose structure is thesis-antithesis-synthesis. I describe a new type of matricial dialectical plan, which presents several advantages in regard to the classical dialectical plan and proposes to constitute an alternative to this latter.


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 The Dialectical Plan: an Alternative to the Paradigm

In Franceschi (2002), I exposed a theory which aims to constitute an alternative to the classification proposed by Greimas in the field of paradigmatic analysis. In the present article, I proceed to draw the consequences of this latter theory by applying it to the technique of conception of a plan. Regarding the dialectic plan, the current paradigm is in effect a plan of the type thesis-antithesis-synthesis. This form of plan is very widespread and its use proves to be consensual. In what follows, I shall propose a novel type of dialectic plan as an alternative to the classical one. It consists of a type of plan which can be qualified as matrix-based, and which presents several advantages with regard to the classical dialectic plan.

The classical dialectic plan

Hegel: Portrait by Jakob Schlesinger

The current paradigm regarding the dialectic plan is a plan of the thesis-antithesis-synthesis type1. This plan finds its origin in the dialectical approach2 developed by Hegel. The association of the three concepts thesis-antithesis-synthesis, which is now associated with the dialectical line of reasoning, was elaborated by Hegel and Marx3. The dialectical approach constitutes thus a process of reasoning that proceeds by the statement of two contradictory theses and by their reconciliation at the stage of the synthesis. According to Hegel4, every thesis presents then an inherently incomplete and partial nature, which gives then birth to its contrary, the antithesis. From Hegel’s standpoint, the contraries present, beyond the contradiction underlying them, an indissociable nature. This last property allows thus to make their final union, at a thought level which places itself beyond the one where the contradiction manifests itself. The contraries present thus by essence a genuine unity, from which it is worth grasping the fecund principle, allowing thus to reach, at a higher level, a genuine knowledge. This latter phase constitutes the synthesis, which can thus be considered as the step of reasoning which reconciliates veritably, at a greater level, the contradiction observed between the thesis and the antithesis. The synthesis allows thus to go beyond the conflict raised between the thesis and the antithesis, by further unifying the part of truth simultaneously contained in both of them. However, the process is not limited to that. For the synthesis thus obtained constitutes in turn a novel thesis, which itself yields a novel antithesis and then a novel synthesis, and so on… Within the current language, the dialectical approach designates now the general methodology which allows to go beyond and to solve the contradictions. It is in this dialectical approach that the classical plan of the thesis-antithesis-synthesis type finds its origin.

At this step, it is worth considering in turn each component of the thesis-antithesis-synthesis plan. Consider, to begin with, the thesis. This latter constitutes a standpoint expressed by a given author. It consists of the viewpoint on which the discussion is based, and toward which the structure of the plan is oriented. For simplicity, let us assimilate here the thesis to a given proposition. On the other hand, the antithesis is a standpoint which proves to be contrary to that of the thesis. Like the thesis, it is useful to reduce the antithesis, for the sake of simplicity, to a given proposition. At this step, the viewpoints expressed by the thesis and the antithesis are of an antinomical nature. Lastly, the synthesis constitutes the part of the discourse where the antagonist viewpoints developed in the thesis and the antithesis are overcome. The synthesis aims thus classically to go beyond the antinomy existing between the thesis and the antithesis and to encompass it.

In a general way, the advantage of the dialectic plan of the type thesis-antithesis-synthesis is to allow to apprehend the double aspect of a given problem or reality. By placing oneself alternatively from one side and from the other, by considering successively the thesis and then the antithesis, this type of plan allows to avoid a partial or truncated vision of the particular problem raised by the thesis. The aim of the classical dialectic plan is thus to apprehend the two-faceted nature of a given reality and to go beyond the contradiction which results from a preliminary study.

Matrices of concepts

mtcg-image1
A matrix of concepts

In Franceschi (2002), I described the structure of a matrix of concepts, the scope of which extends to many concepts. For the sake of the present discussion, it is not necessary to describe in a detailed way the structure of concepts put forth in this article. Nevertheless, the type of dialectic plan which will be proposed later derives directly from the notion of a matrix of concepts. It proves then necessary to present the main lines of the basic structure of a matrix of concepts.

Consider first a given duality. Let us denote it by A/Ā. At this step, A and Ā constitute dual concepts. One can then consider that A and Ā are concepts which characterize themselves by a contrary component c  {-1, 1} at the level of a given duality A/Ā, such that c[A] = -1 and c[Ā] = 1. One can also consider that A and Ā are neutral concepts which can thus be denoted by A0 and Ā0.

At this step, we are in a position to define the class of the canonical poles. It suffices to consider an extension of the preceding class {A0, Ā0}, such that A0 and Ā0 respectively admit of both a positive and a negative concept which are correlative. Such concepts possess a certain intuitive support. Let us denote them respectively by {A+, A} and {Ā+, Ā}. At this step, for a given duality A/Ā, we get the following concepts: {A+, A0, A, Ā+, Ā0, Ā}, which constitute the canonical poles. It is worth mentioning here that the notation α(A/Ā, c, p) could be used alternatively, for a given canonical pole5. In all cases, the components of a canonical pole are: a duality A/Ā, a contrary component c  {-1, 1} and a canonical polarity p  {-1, 0, 1}. This definition of the canonical poles leads to distinguish between the positive (A+, Ā+), neutral (A0, Ā0) and negative (A, Ā) canonicalpoles. Lastly, the class made up of the six canonical poles of a same matrix can be dubbed the canonical matrix: {A+, A0, A, Ā+, Ā0, Ā}.

Let us focus now on the nature of the relationships existing between the canonical poles of a given matrix. Among the combinations of relationships existing between the six canonical poles (A+, A0, A, Ā+, Ā0, Ā) of a same duality A/Ā, one will retain the following relations: duality, antinomy, complementarity, corollarity, connexity, anti-connexity. Thus, two canonical poles α1(A/Ā, c1, p1) and α2(A/Ā, c2, p2) of a same matrix are:

(a) dual if their contrary components are opposite and their polarities are neutral6

(b) contrary (or antinomical) if their components are opposite and their polarities are non-neutral and opposite7

(c) complementary if their contrary components are opposite and their polarities are non-neutral and equal8

(d) corollary if their contrary components are equal and their polarities are non-neutral and opposite9

(e) connex if their contrary components are equal and the absolute value of the difference of their polarities equals 110

(f) anti-connex if their contrary components are opposite and the absolute value of the difference of their polarities equals 111

To sum up: {A0, Ā0} are dual; {A+, Ā} and {A, Ā+} are contraries; {A+, Ā+} and {A, Ā} are complementary; {A+, A} and {Ā+, Ā} are corollary; {A0, A+}, {A0, A}, {Ā0, Ā+} and {Ā0, Ā} are connex; {A0, Ā+}, {A0, Ā}, {Ā0, A+} and {Ā0, A} are anti-connex.

To fix ideas, let us take the example of the matrix12 {eclecticism+, multi-disciplinarity0, dispersion, expertise+, monodisciplinarity0, compartmentalization}. One has then the following relationships:

(a’) {multi-disciplinarity0, monodisciplinarity0} are dual

(b’) {eclecticism+, compartmentalization}, {dispersion, expertise+} are antinomical

(c’) {eclecticism+, expertise+}, {dispersion, compartmentalization} are complementary

(d’) {eclecticism+, dispersion}, {expertise+, compartmentalization} are corollary

(e’) {multi-disciplinarity0, eclecticism+}, {multi-disciplinarity0, dispersion}, {monodisciplinarity0, expertise+}, {monodisciplinarity0, compartmentalization} are connex

(f’) {multi-disciplinarity0, expertise+}, {multi-disciplinarity0, compartmentalization}, {monodisciplinarity0, eclecticism+}, {monodisciplinarity0, dispersion} are anti-connex

Structure of a thesis

At this step, it is worth delving more deeply into the internal structure of the thesis to which the plan dialectical applies. I shall draw a distinction here between simple and complex theses.

Simples theses

In general, a simple thesis presents a structure which is that of an appreciation – negative, neutral or positive – relative to a given concept. Let α be such a concept; one denotes then by p(α) such structure of thesis, where p denotes a negative polarity, neutral or positive such that respectively p  {-1, 0, 1}. The negative appreciation can be assimilated to a blame and the positive appreciation to a praise. The blame of a given concept α is thus denoted by (α), the neutral appreciation by 0(α) and the praise by +(α). In a general way, the propositions corresponding to the simple theses present the following structure: p(α), with p  {-1, 0, 1} and α  {A+, A0, A, Ā+, Ā0, Ā}. By referring to the matrix notion, one notes that the different theoretical cases are the following, with regard to the six concepts of a given matrix: {(A+), (A0), (A), +), 0), ), 0(A+), 0(A0), 0(A), 0+), 00), 0), +(A+), +(A0), +(A), ++), +0), +)}. At this step, it appears that the neutral appreciation is somewhat rarely found. Thus, for the sake of simplicity, we shall be mainly concerned here with describing more accurately the theses which present the structure of a blame or of a praise.

Let us begin with the blame. A number of theses are thus composed of a depreciative appreciation, related to a behavior, a way of doing or apprehending things, a given situation. Such statements correspond to propositions that present the structure of a blame. Such propositions can be denoted by (s) where s designates a way of apprehending or of doing things.

Let us take, to fix ideas, a few examples. Consider the following thesis:

In the contempt of ambition is to be found one of the essential principles of happiness on earth. (Edgar Poe, The Domain of Arnheim)

(1) In the contempt of ambition is to be found one of the essential principles of happiness on earth.(Edgar Poe, The Domain of Arnheim)

The author considers here the “contempt of ambition” as a fundamental principle allowing to reach happiness. Such a viewpoint can be analyzed as a negative, depreciative judgment toward ambition. This latter concept can be considered as a neutral notion13. Hence, such a simple thesis presents the structure which is that of the blame of ambition0 and can be thus denoted by (ambition0).

Consider also this other thesis:

Love, the scourge of the world, atrocious folly. (Alfred of Musset, Premières poésies)

(2) Love, the scourge of the world, atrocious folly. (Alfred of Musset, Premières poésies)

The content of this latter thesis can be analyzed as a very pejorative appreciation formulated with regard to love+. Here also, such thesis presents a structure that can be analyzed as a blame of love+, that one can thus denote by (love+).

Conversely, one also frequently encounters some theses which are composed of a flattering appreciation with regard to a given behavior, a propensity to act, a situation or a way of apprehending things. The structure of the corresponding proposition is then that of a praise. One denotes such propositions by +(s) where s designates a way of considering things or a given behavior.

Consider then a few examples. To begin with, the following viewpoint illustrates this type of structure:

(3) Nothing of great importance came true in the world without passion. (Hegel, Introduction to the Philosophy of History)

The author formulates here a praise related to the passion, considering thus that “nothing of great importance” ever came true without this latter. One can consider here the passion as a neutral notion14. Such a viewpoint presents thus the structure of a praise of passion0, i.e. formally +(passion0).

One also encounters an identical type of structure, regarding the following affirmation:

Kant: “Passion is an illness that abhors all medication.”

(4) Passion is an illness that abhors all medication. (Kant)

which can be analyzed as a blame of passion0, i.e. formally ( passion0).

Lastly, the following simple thesis:

Oscar Wilde: “The worst vice of the fanatic is his sincerity.”

(5) The worst vice of the fanatic is his sincerity. (Oscar Wilde)

constitutes an example of praise of the negative concept of fanaticism, i.e. formally +(fanaticism).

At this step, we are in a position to determine the truth value of the simple theses. The truth value of each type of praise, of neutral appreciation or of blame indicates if the considered affirmation is plausible and coherent or not, given that the praise of a positive concept is true, in the same way as the neutral appreciation of a neutral concept and the blame of a negative concept. Conversely, the praise of a non-positive concept15, the neutral appreciation of a non-neutral concept or well the blame of a non-negative concept16 are false. Formally, the truth value [v] of propositions of the type P = pq), with p, q  {-1, 0, 1} and α  {A+, A0, A, Ā+, Ā0, Ā} can be calculated as follows: [v] = 1 (true) if p = q and [v] = -1 (false) if pq.17 Hence, among the different cases which have just been enumerated, those whose truth value is true are: {(A), ), 0(A0), 00), +(A+), ++)}. And those whose truth value is false are: {(A+), (A0), +), 0), 0(A+), 0(A), 0+), 0), +(A0), +(A), +0), +)}.

Complex theses

Whereas simple theses contain a judgment related to one single concept belonging to a given matrix, complex theses are composed of appreciations relative to several concepts of a same matrix. A complex thesis can thus be defined in a general way as the conjunction of several simple theses. A complex thesis can thus be composed of appreciations relative to two, three, …, n different concepts. One will use accordingly the term of n-complex thesis. Under these circumstances, the combinations prove to be numerous, without it being nevertheless necessary to enumerate them exhaustively. A given proposition P constituting a complex thesis presents thus the following structure: P = Q1  Q2  …  Qn, for n > 1, and Qi = piqi), with pi, qi  {-1, 0, 1} and α  {A+, A0, A, Ā+, Ā0, Ā}. We have then the 2-complex, 3-complex, …, n-complex theses.

At this step, it appears necessary to consider first the 2-complex theses, which constitute, among the complex theses, the most common case. The 2-complex theses are composed of some appreciations relative to two concepts of a same matrix. They present the structure: p1(A/Ā, c1, q))  r2(A/Ā, c2, s)). The following appreciation constitutes thus an example of 2-complex thesis:

(6) All theory is gray, but the golden tree of life is green. (Goethe)

This 2-complex thesis is in effect composed of both the blame of theory (“all theory is gray”) and the praise of pragmatism (“the golden tree of life is green”). It proves here that the concepts of interest for theory and of pragmatism belong to the following matrix: {capacity of abstraction+, interest for theory0, dogmatism, pragmatism+, interest for practise0, prosaicness}. The structure of the thesis is thus (interest for theory0)  +(pragmatism+) i.e. (A0)  ++).

In the same way, the following appreciation constitutes a case of 2-complex thesis:

Napoleon Bonaparte: “The art of being sometimes very audacious, sometimes very cautious is the art of success.”

(7) The art of being sometimes very audacious, sometimes very cautious is the art of success. (Napoleon Bonaparte)

This 2-complex thesis is composed of both the praise of boldness (“The art of being (…) very audacious (…) is the art of success”) and the praise of the cautiousness (“the art of being (…) very cautious is the art of success”). It appears that these latter concepts belong to the following matrix: {boldness+, propensity to take risk0, temerity, cautiousness+, propensity to avoid the risk0, cowardice}. The thesis is thus composed here of the praise of two complementary positive concepts of a same matrix. The particular structure of this type of complex thesis is thus composed of the praise of A+ and the praise of Ā+, i.e. formally +(boldness+)  +(cautiousness+).

Consider lastly the following thesis, which also constitutes a case of 2-complex thesis:

(8) Two excesses: to exclude reason, and to admit nothing else than reason.(Pascal, Thoughts)

This last thesis is in effect composed of both the blame of irrationality (“exclude the reason”) and the blame of hyper-rationalism (“to admit nothing else than reason”). The corresponding reconstituted matrix is the following: {imagination+, inspiration0, irrationality, rationality+, reason0, hyper-rationalism}. As we see it, we face here a 2-complex thesis whose structure is (irrationality)  (hyper-rationalism) i.e. (A)  ).

Lastly, the following 2-complex thesis:

(9) How can we tolerate that passion be placed on the same level than reason? (Sénèque, De Ira)

can be analyzed as a blame of passion0 and a praise of reason0, i.e. formally (passion0)  +(reason0), i.e. (A0)  +0) at the level of the matrix {motivation+, passion0, fanaticism, level-headedness+, reason0, lukewarmness}.

It is worth noting here that this last type of 2-complex thesis corresponds to a common case, for motives of internal coherence. It is in effect logical when one criticizes or depreciates such or such value or concept, of flattering its contrary. To blame such or such thing amounts naturally to praising its opposite, and conversely. For that reason, the 2-complex theses whose particular structure is (A)  ++) or well +(A+)  ) also constitute, among all possible combinations of 2-complex theses, a common case.

For what concerns the truth value of the 2-complex theses, it can be determined in the same way as for the simple theses. Let thus P  Q be a 2-complex thesis, such that P = pq) and Q = rs), with p, q, r, s  {-1, 0, 1} and α, β  {A+, A0, A, Ā+, Ā0, Ā}. Formally, the truth value [v] of a 2-complex thesis P  Q is true if v[P] = v[Q] = true, and false in other cases18. It is worth noting that the most common types of 2-complex theses are those whose truth value are true. Such is the case when the truth-value of each of the two propositions included within the complex thesis is true. Under this hypothesis, the two propositions reinforce themselves. It consists thus of the cases corresponding to: {+(A+)  (A), +(A+)  ++), +(A+)  ), (A)  ++), (A)  ), ++)  )}.

Dual theses

At this step, it is worth focusing on the notion of a dual thesis of a given thesis. This last notion applies both to the simple theses and to the complex ones. The dual thesis constitutes here an element of the dialectical discussion, which proves to be important since it is the basis of the discussion related to the thesis under consideration.

Let us focus, to begin with, on dual theses of simple theses. Let us begin by giving a general definition. Formally, a simple thesis p1(A/Ā, c, q)) admits of a dual thesis that corresponds to the following definition: p2(A/Ā, –c, q)). Thus, a dual thesis of a simple thesis presents the following characteristics: (i) the polarities of the appreciation of the dual thesis and of the simple thesis are identical; (ii) the contrary components of the concepts on which bear the appreciations of the dual thesis and of the simple thesis are opposite; (iii) the polarities of the concepts on which bear the appreciations of the dual thesis and of the simple thesis are identical.

Let us consider first the dual theses of the true simple theses. The types of true simple theses can be thus enumerated as follows: {+(A+), 0(A0), (A), ++), 00), )}. Formally, a true simple thesis p1(A/Ā, c, p)) presents a dual thesis which responds to the following definition: p2(A/Ā, –c, p)). Thus, the dual theses of the true simple theses are respectively: {++), 00), ), +(A+), 0(A0), (A)}.

To take an example, consider the following true simple thesis:

(10) What you can do, or dream you can do, begin it. Boldness has genius, power and magic in it. (Goethe)

which presents the structure +(boldness+) i.e. +(A+) at the level of the matrix {boldness+, propensity to take risk0, temerity, cautiousness+, propensity to avoid risk0, cowardice}. The thesis below whose structure is +(cautiousness+) i.e. ++) constitutes thus its dual thesis:

(11) Cautiousness is as much superior to the other virtues as sight is to the other senses. (Bion of Phlossa)

Consider also the dual theses of the false simple theses. The types of false simple theses are: {(A+), (A0), +), 0), 0(A+), 0(A), 0+), 0), +(A0), +(A), +0), +)}. And the dual theses of the false simple theses are respectively: {+), 0), (A+), (A0), 0+), 0), 0(A+), 0(A), +0), +), +(A0), +(A)}.

To take an example, the following false simple thesis:

(4) Passion is an illness that abhors all medication. (Kant)

presents the structure (passion0) i.e. (A0) at the level of the matrix {motivation+, passion0, fanaticism, level-headedness+, reason0, lukewarmness}. The following thesis whose structure is (reason0) i.e. 0) constitutes thus its dual thesis:

(12) If reason dominated on the earth, nothing would happen there. (Bernard Fontenelle)

It is worth considering now, on the other hand, the dual theses of the complex theses. These latter are such that the contrary components of the concepts on which bear the appreciations of the two simple theses, which are part of the dual thesis and of the considered thesis, are opposite19. Consider then the true 2-complex theses. Thus, the dual thesis of +(A+)  ) is ++)  (A). And also, the dual thesis of 0(A0)  +(A+) is 00)  ++). It is worth noting here in particular that the dual thesis of 0(A0)  00) is 00)  0(A0), that the dual thesis of +(A+)  ++) is + +)  +(A+) and that the dual thesis (A)  ) is (A)  ).

Let us also give a few examples. Thus, the true 2-complex thesis corresponding to the following proposition:

Goethe: “All theory is gray, but the golden tree of life is green.”

(6) All theory is gray, but the golden tree of life is green. (Goethe)

presents the structure (A0)  ++) i.e. (interest for theory0)  +(pragmatism+) at the level of the matrix {capacity of abstraction+, interest for theory0, dogmatism, pragmatism+, interest for practice0, prosaicness}. The following thesis whose structure is 0)  +(A+) i.e. (interest for practice0)  +(capacity of abstraction+) constitutes thus its dual thesis:

(13) All practice is vile, but fecund and elevated is the quest of the genuine abstraction.

Similarly, the following proposition:

Pascal: “Two excesses: to exclude reason, and to admit nothing else than reason. “

(8) Two excesses: to exclude reason, and to admit nothing else than reason.(Pascal, Thoughts)

constitutes a true 2-complex thesis whose structure is (irrationality)  (hyper-rationalism) i.e. (A)  ) at the level of the matrix: {imagination+, inspiration0, irrationality, rationality+, reason0, hyper-rationalism}. The thesis below whose structure is +(imagination+)  +(rationality+) i.e. +(A+)  ++) constitutes thus its dual thesis:

(14) The art of being sometimes very imaginative, sometimes very rational is the art of success.

Lastly, it is worth noting that one has also analogous definitions for 3-complex, 4-complex, etc. theses. To take then an example, the dual thesis of the 3-complex thesis +(A+)  0(A0)  00) is ++)  00)  0(A0). In the same way, the dual thesis of the 3-complex thesis +(A+)  0(A0)  (A) is ++)  00)  ).

The matrix-based dialectic plan

The preceding developments allow now to describe the steps of the dialectical reasoning applicable to the analysis of a given particular thesis, from the above-mentioned principles. The first step consists thus in the accurate determination of the structure of the thesis under consideration. The second step, which results directly from it, is the attribution of a truth-value to this latter thesis. The following step consists then in the reconstitution of the whole matrix applicable to the concept(s) which are the object of the thesis. One is then in a position to determine the dual thesis of the considered thesis in the same way as the true simple theses other than the considered thesis and its dual thesis. Lastly, the final step is the synthesis which consists in the conjunction of the true simple theses relative to each of the 6 concepts of the considered matrix: +(A+)  0(A0)  (A)  ++)  00)  ). Such a synthesis allows to encompass a threefold antinomy: the one existing between A+ and Ā, A0 and Ā0, and A and Ā+. It should be observed here that one can eventually retain from the synthesis but a simplified form consisting of the conjunction of the true simple theses constituting a praise or a blame: +(A+)  (A)  ++)  ). In the same way, one may sometimes limit oneself to a truncated form of synthesis consisting in +(A+)  ++), which emphasizes the complementarity between A+ and Ā+.20

At this step, we are in a position to present the matrix-based dialectic plan. Such a plan results directly from the structure of matrix of concepts which has been just described. The corresponding matrix-based dialectic plan presents thus the following structure:21

(15) 1. From the viewpoint of A0

1.1 Praise of A+

1.2 Blame of A

2. From the viewpoint of Ā0

2.1 Praise of Ā+

2.2 Blame of Ā

3. Complementarity between A+ and Ā+ 22

Consider then, to take an example the following true simple thesis:

(16) Success was always a child of audacity. (Prosper Crebillon, Catilina)

whose structure is +(boldness+) i.e. +(A+) at the level of the matrix {boldness+, propensity to take risk0, temerity, cautiousness+, propensity to avoid risk0, cowardice}. It results then the following matrix-based plan:

(17) 1. From the viewpoint of risk taking0

1.1 The necessity of boldness+

1.2 The dangers of temerity

2. From the viewpoint of risk avoidance0

2.1 The advantages of the cautiousness+

2.2 The risk of cowardice

3. The necessary complementarity between boldness+ and cautiousness+

Consider also the following false simple thesis:

(12) If reason dominated on the earth, nothing would happen there. (Bernard Fontenelle)

whose structure is (reason0). The corresponding matrix is: {level-headedness+, reason0, lukewarmness, motivation+, passion0, fanaticism}. And the following matrix-based plan then ensues:

(18) Introduction: (i) structure of the thesis; (ii) truth value; (iii) matrix

1. From the viewpoint of reason0

1.1 The pitfall of lukewarmness

1.2 The necessity of level-headedness+

2. From the viewpoint of passion0

2.1 The dangers of fanaticism

2.2 The necessity of motivation+

3. The necessary complementarity between level-headedness+ and motivation+

Lastly, such a type of plan also proves to be adapted to a true 2-complex thesis such as the following:

(19) In the first place comes your profession, because doing just one thing well will procure a higher development for you than doing one hundred by halves. (Goethe)

This latter thesis can be analyzed as a 2-complex thesis whose structure is +(expertise+)  (superficiality) i.e. +(A+)  ) at the level of the matrix: {expertise+, monodisciplinarity0, compartmentalization, eclecticism+, multi-disciplinarity0, superficiality}. And the following matrix-based plan23 then ensues:

(20) 1. From the viewpoint of monodisciplinarity0

1.1 The advantages of expertise+

1.2 The risk of compartmentalization

2. From the viewpoint of multi-disciplinarity0

2.1 The necessity of eclecticism+

2.2 The dangers of superficiality

3. The necessary complementarity between expertise+ and eclecticism+

Conclusion

From the above developments, it should be noted that the matrix-based dialectic plan presents a number of advantages with regard to the classical dialectic plan. First, the dialectical approach which has just been described performs first an analysis of the structure of the thesis under consideration, which leads then to assign a truth value to it, on objective grounds.

Second, it appears that the matrix-based dialectic plan replaces the thesis or the main proposition in a context that comprises a greater number of concepts than the classical dialectic plan. In effect, the classical dialectic plan usually places the thesis in an environment comprising in general two, or even three concepts. By contrast, the matrix-based dialectic plan replaces the thesis in a context comprising six concepts which are related to this latter.

Third, one of the advantages of the matrix-based dialectic plan is that it also allows to take into account some concepts which are not lexicalized. In effect, a matrix of concepts describes six canonical concepts. But it is rare that the totality of these latter concepts are lexicalized. In effect, the most common situation is that only some concepts – in general two or three – among the six described by the corresponding matrix, are lexicalized. Here also, the advantage of the matrix-based dialectic plan is to allow to take into account exhaustively the six concepts of a same matrix and to incorporate them in the corresponding discussion.

It should also be noted that the step of the antithesis at the level of the classical dialectic plan is replaced here by the determination of the dual thesis, which presents an identical structure to that of the initial thesis. The dual thesis, which serves here as a basis for dialectical reasoning, presents by its simple or well n-complex structure a more elaborated nature than the traditional antithesis.

Lastly, it proves that the classical dialectic plan allows to overcome an antinomy existing between two concepts, which serve respectively as a support to the thesis and to the antithesis. It consists most often of A+ and Ā, of A0 and Ā0, or well of A and Ā+. Most of the time, it consists of a dual or antinomical pair of concepts which present the property of being lexicalized. Conversely, the matrix-based dialectic plan constitutes the expression of a dialectical move of the thought which allows to go beyond a threefold antinomy: the one existing at the same time between A+ and Ā, A0 and Ā0, and finally A and Ā+, whether these concepts are lexicalized or not.


References

Franceschi, Paul (2002). Une classe de concepts. Semiotica 139 (1-4), 211-226. English translation.

Hegel, Georg Wilhelm Friedrich (1812-1816). Wissenschaft der Logik. Science de la logique, trad. Bourgeois, Paris, Aubier Montaigne, 1972.

Hegel, Georg Wilhelm Friedrich (1817). Die Encyclopädie der philosophischen Wissenschaften im Grundrisse. Précis de l’encyclopédie des sciences philosophiques, trad. J. Gibelin. Vrin, Paris, 1978


1 One also finds the antithesis-thesis-synthesis variant.

2 Platon envisaged dialectic under the form of a dialogue between two persons, based on alternate questions and responses. One also finds a dialectical approach in Kant, but also in Fichte and Schelling.

3 In the context of dialectical materialism, the dialectic finds its expression on the social terrain, through the conflict or the struggle, which are viewed as the manifestation, at a material level, of the contradiction. Historical progress and social advances ensue once this conflict has been overcome. For Marx also, the dialectical objective situates itself veritably at the level of the reality, finding thus its expression in the facts and the phenomena. Conversely, the dialectical move observed at the level of human thought only constitutes the subjective reflect of the essential dialectic, a simple transposition of the latter at the level of the humain brain.

4 Cf. Hegel (1812-1816) and (1817).

5 With this last notation, the matrix of the canonical poles is rendered as follows: {α(A/Ā, -1, 1), α(A/Ā, -1, 0), α(A/Ā, -1, -1), α(A/Ā, 1, 1), α(A/Ā, 1, 0), α(A/Ā, 1, -1)}.

6 Formally α1 and α2 are dual if and only if c1] = – c2] and p1] = p2] = 0.

7 Formally α1 and α2 are antinomical if and only if c1] = – c2] and p1] = – p2] with p1], p2]  0.

8 Formally α1 and α2 are complementary if and only if c1] = – c2] and p1] = p2] with p1], p2]  0.

9 Formally α1 and α2 are corollary if and only if c1] = c2] and p1] = – p2] with p1], p2]  0.

10 Formally α1 and α2 are connex if and only if c1] = c2] and │p1] – p2]│ = 1.

11 Formally α1 and α2 are anti-connex if and only if c1] = – c2] and │p1] – p2]│ = 1.

12 For a more comprehensive list of matrices of concepts, see Franceschi (2002).

13 Personal ambition could be fruitful (ambition+) or well excessive, or even immoderate (ambition).

14 A passion could be positive (passion+) or well excessive, destructive (passion).

15 Negative or neutral.

16 Positive or neutral.

17 One could as well distinguish here degrees of truth value, by making use of degrees of appreciation, with p  [-1, 1]. An approach by degree of the truth value ensues, by calculating thus this latter with regard to the absolute value of the difference between p and q: [v] = 1- |(pq)/2|.

18 Such a definition generalizes to the determination of the truth values of the 3-composed theses, …, n-composed.

19 Formally, let thus P  Q be a 2-composed thesis, such that P = p11(A/Ā, c1, q1)) and Q = P22(A/Ā, c2, q2), with p1, p2, q1, q2  {-1, 0, 1}, c1, c2  {-1, 1} and α, β  {A+, A0, A, Ā+, Ā0, Ā}; then the dual thesis of P  Q is of the form: p11 (A/Ā, –c1, q1))  P22 (A/Ā, –c2, q2). Such definition generalizes easily to the dual theses of the n-composed theses.

20 The description of the different steps of the dialectical process thus defined also suggests other types of plans than the one which has been emphasized here. Alternative plans can notably highlight a part related to the step of determination of the truth value of the considered thesis, or well to the dual thesis of this latter.

21 Alternatively, one could also consider the following variation:

1. From an analytic point of view

1.1 From the viewpoint of A0

1.1.1 Praise of A+

1.1.2 Blame of A

1.2 From the viewpoint of Ā0

1.2.1 Praise of Ā+

1.2.2 Blame of Ā

2. From a synthetic point of view: the complementarity between A+ and Ā+ and between A and Ā

22 A variation of this type of plan consists evidently in assimilating the part 3 with the conclusion.

23 For this last type of thesis whose structure is +(A+)  ), it is also possible to recur to another type of plan which emphasizes more the dual thesis ++)  (A). Such a type of plan proves to be close to the classical dialectic plan and stresses on the dual thesis of the considered thesis, e.g. +(eclecticism+)  (compartmentalization). Such a type of plan presents then the following structure:

1. Thesis

1.1 The advantages of expertise+

1.2 The dangers of superficiality

2. Dual thesis

2.1 The necessity of eclecticism+

2.2 The risk of compartmentalization

3. The necessary synthesis between eclecticism+ and expertise+, and superficiality and compartmentalization

The Simulation Argument and the Reference Class Problem: the Dialectical Contextualist’s Standpoint

Postprint. I present in this paper an analysis of the Simulation Argument from a dialectical contextualist’s standpoint. This analysis is grounded on the reference class problem. I begin with describing in detail Bostrom’s Simulation Argument. I identify then the reference class within the Simulation Argument. I also point out a reference class problem, by applying the argument successively to three different reference classes: aware-simulations, imperfect simulations and immersion-simulations. Finally, I point out that there are three levels of conclusion within the Simulation Argument, depending on the chosen reference class, that yield each final conclusions of a fundamentally different nature.

This article supersedes my preceding work on the Simulation argument. Please do not cite previous work.


The Simulation Argument and the Reference Class Problem : a Dialectical Contextualism Analysis

Paul FRANCESCHI

www.paulfranceschi.com

English postprint of a paper initially published in French in Philosophiques, 2016, 43-2, pp. 371-389, under the title L’argument de la Simulation et le problème de la classe de référence : le point de vue du contextualisme dialectique

ABSTRACT. I present in this paper an analysis of the Simulation Argument from a dialectical contextualist’s standpoint. This analysis is grounded on the reference class problem. I begin with describing in detail Bostrom’s Simulation Argument. I identify then the reference class within the Simulation Argument. I also point out a reference class problem, by applying the argument successively to three different reference classes: aware-simulations, imperfect simulations and immersion-simulations. Finally, I point out that there are three levels of conclusion within the Simulation Argument, depending on the chosen reference class, that yield each final conclusions of a fundamentally different nature.

1. The Simulation Argument

I shall propose in what follows an analysis of the Simulation Argument, recently described by Nick Bostrom (2003). I will first describe in detail the Simulation Argument (SA for short), focusing in particular on the resulting counter-intuitive consequence. I will then show how such a consequence can be avoided, based on the analysis of the reference class underlying SA, without having to give up one’s pre-theoretical intuitions.

The general idea behind SA can be stated as follows. It is very likely that post-human civilizations will possess a computing power that will be completely out of proportion with that of ours today. Such extraordinary computing power should give them the ability to carry out completely realistic human simulations, such as ensuring that the inhabitants of these simulations are aware of their own existence, in all respects similar to ours. In such a context, it is likely that post-human civilizations will devote part of their computer resources to carrying out simulations of the human civilizations that preceded them. In this case, the number of simulated humans should greatly exceed the number of authentic humans. Under such conditions, taking into account the simple fact that we exist leads to the conclusion that it is more likely that we are part of the simulated humans, rather than of the authentic humans.

Bostrom thus points out that the Simulation Argument is based on the following three hypotheses:

(1)it is very likely that humanity will not reach a post-human stage
(2)it is very unlikely that post-human civilizations will carry out simulations of the human races that preceded them
(3)it is very likely that we are currently living in a simulation carried out by a post-human civilization

and it follows that at least one of these three assumptions is true.

For the purposes of the present analysis, it is also useful at this stage to emphasize the underlying dichotomous structure of SA. The first step in the reasoning consists then in considering, by dichotomy, that either (i) humanity will not reach a post-human stage, or (ii) it will actually reach such a post-human stage. The first of these two hypotheses corresponds to the disjunct (1) of the argument. We consider then the hypothesis that humanity will reach a post-human stage and thus continue its existence for many millennia. In such a case, it can also be considered likely that post-human civilizations will possess both the technology and the skills necessary to perform human simulations. A new dichotomy then arises: either (i) these post-human civilizations will not perform such simulations — this is the disjunct (2) of the argument; or (ii) these post-human civilizations will actually perform such simulations. In the latter case, it will follow that the number of simulated humans will greatly exceed the number of humans. The probability of living in a simulation will therefore be much greater than that of living in the shoes of an ordinary human. The conclusion then follows that we, the inhabitants of the Earth, are probably living in a simulation carried out by a post-human civilization. This last conclusion constitutes the disjunct(3) of the argument. An additional step leads then to the conclusion that at least one of the hypotheses (1), (2) and (3) is true. The dichotomous structure underlying SA can thus be described step by step as follows:

(4)humanity will either not reach a post-human stage or reach a post-human stagedichotomy 1
(1)humanity will not reach a post-human stagehypothesis 1.1
(5)humanity will reach a post-human stagehypothesis 1.2
(6)post-human civilizations will be able to perform human simulationsfrom (5)
(7)post-human civilizations will either not perform human simulations or will perform themdichotomy 2
(2)post-human civilizations will not perform human simulationshypothesis 2.1
(8)post-human civilizations will perform human simulationshypothesis 2.2
(9)the proportion of simulated humans will far exceed that of humansfrom (8)
(3)it is very likely that we are currently living in a simulation carried out by a post-human civilizationfrom (9)
(10)at least one of the hypotheses (1), (2) and (3) is truefrom (1), (2), (3)

It is also worth mentioning an element that results from the very interpretation of the argument. For as Bostrom (2005) points out, the Simulation Argument must not be misinterpreted. This is not an argument that leads to the conclusion that (3) is true, namely that we are currently living in a simulation carried out by a post-human civilization. The core of SA is thus that one of the hypotheses (1), (2) or (3) at least is true.

This nuance of interpretation being mentioned, the Simulation Argument is not without its problems. Because SA leads to the conclusion that at least one of the assumptions (1), (2) or (3) is true, and that in the situation of ignorance in which we find ourselves, we can consider the latter as equiprobable. As Bostrom himself notes (Bostrom, 2003): “In the dark forest of our current ignorance, it seems sensible to apportion one’s credence roughly evenly between (1), (2) and (3)”. However, according to our pre-theoretical intuition, the probability of (3) is nil or at best extremely close to 0, so the conclusion of the argument has the consequence of increasing the probability that (3) is true from zero to a probability of about 1/3. Thus, the problem with the Simulation Argument is precisely that it shifts — via its disjunctive conclusion — from a zero or almost zero probability concerning (3) to a much higher probability of about 1/3. Because a probability of 1/3 for the hypotheses (1) and (2) is not a priori shocking, but is completely counter-intuitive as far as hypothesis (3) is concerned. It is in this sense that we can talk about the problem posed by the Simulation Argument and the need to find a solution to it.

As a preliminary point, it is worth considering what constitutes the paradoxical aspect of SA. What indeed gives SA a paradoxical nature? For SA differs from the class of paradoxes that lead to a contradiction. In paradoxes such as the Liar or the sorites paradox, the corresponding reasoning leads to a contradiction1. However, nothing of the sort can be seen at the level of SA, which belongs, from this point of view, to a different class of paradoxes, including the Doomsday Argument and Hempel’s problem. It is indeed a class of paradoxes whose conclusion is contrary to intuition, and which comes into conflict with the set of all our beliefs. In the Doomsday Argument then, the conclusion that taking into account our rank within the class of humans who have ever existed has the effect that an apocalypse is much more likely than one might have initially thought, offends the set of all our beliefs. Similarly, in Hempel’s problem, the fact that a blue umbrella confirms the hypothesis that all crows are black comes in conflict with the body of our knowledge. Similarly within SA, what finally appears paradoxical at first analysis is that SA leads to a probability of the hypothesis that we are currently living in a simulation created by post-humans, which is higher than that resulting from our pre-theoretical intuition.

2. The reference class problem and the Simulation Argument

The conclusion of the reasoning underlying SA, based on the calculation of the future ratio between real and simulated humans, albeit counter-intuitive, nevertheless results from a reasoning that appears a priori valid. However, such reasoning raises a question, which is related to the reference class that is inherent to the argument itself2. Indeed, it appears that SA has, indirectly, a particular reference class, which is that of human simulations. But what constitutes a simulation? The original argument implicitly refers to a reference class which is that of virtual simulations of humans, of a very high quality and by nature indistinguishable from authentic humans. However, there is some ambiguity about the very notion of a simulation and the question arises as to the applicability of SA to other types of human simulations3. Indeed, we are in a position to conceive of somewhat different types of simulations which also fall intuitively within the scope of the argument.

As a preliminary point, it is worth specifying here the nature of the simulations carried out by computer means referred to in the original argument. Implicitly, SA refers to computer simulations carried out by means of conventional computers composed of silicon chips. But it can also be envisaged that simulations are carried out using computers built from components using DNA properties and molecular biology. Recent research has shown that it is possible to implement high-performance algorithms (Adleman 1994, 1998) and to produce computer components (Benenson & al. 2001, MacDonald & al. 2006) based on bio-calculation techniques that exploit in particular the combinations of the four components (adenine, cytosine, guanine, thymine) of the DNA molecule. If such a field of research were to expand significantly and make it possible to produce computers at least as powerful as conventional computers, this type of bio-computers could legitimately fall within the scope of SA as well. Because the fact that the simulations are carried out using conventional or biological computers4 does not alter the scope of the argument. In any case, the result is that the proportion of simulated humans will be much higher than that of real humans, due to the properties of simulated reality using digital means, because the computer does not know the physical limits that are those of matter.

It can also be observed preliminarily that Bostrom explicitly refers to simulations carried out using computer means. However, the question arises as to whether simulated humans could not consist of perfectly successful physical copies of real humans. In such a case, simulations5 could be extremely difficult to discern. A priori, such a variation also constitutes an acceptable version of SA. However, there is a difference with the original argument, which also highlights Bostrom’s preferential choice of computer simulations. Indeed, in the original argument there is a very significant disproportion between humans simulated by computer means on the one hand and real humans on the other. This is the premise (9) of the argument: “the proportion of simulated humans will far exceed that of humans”. As Bostrom points out, the former would then be much more numerous than the latter, due to the very nature of computer simulations. It is this disproportion that then allows us to conclude (3) “we most probably live in a simulation carried out by a post-human civilization”. With simulations of a physical nature, one would not a priori have such a disproportion, and the scope of the conclusion would be somewhat different. Suppose, for example, that post-humans manage to perform simulations of a physical nature, the number of which would be equal to that of real humans. In this case, the proportion of simulated humans would be 1/2 (whereas it is close to 1 in the original argument). Premise (9) would then become: “the proportion of simulated humans and actual humans will be 1/2”. And this would only allow us to conclude (3) “the probability that we are simulations performed by a post-human civilization is equal to 1/2”. As can be seen, this would result in a significantly attenuated version of SA. The difference with the original version of SA is that the simulation argument for physical simulations applies with less force than the original argument. However, if the conditions were to change and this would result in the future in a disproportion of the same nature as with computer simulations for physical simulations, SA would then apply with all its force. In any event, the following analysis would then apply in the same way to this last category of simulations.

With these preliminary considerations in mind, we shall focus in turn on different types of human simulations, which are likely to be part of the SA reference class, and the ensuing conclusions at the argument level. Because the very question of defining the reference class for SA leads to questions about whether or not several types of simulations should be included within the SA scope. However, the question of the definition of the reference class for SA thus appears closely related to the nature of the future taxonomy of the beings and entities that will populate the Earth in the near or distant future. There is no question here of claiming exhaustiveness, given the speculative nature of such an area. However, it is possible to determine to what extent SA can also be applied to simulations of a different nature from those mentioned in the original argument, but which have equal legitimacy. We shall examine then in turn: conscious simulations, imperfect simulations, and immersion simulations.

3. The reference class problem : the caseof conscious simulations

At this step, it is not yet possible to really talk about the problem of the reference class within SA. To do so, it must be shown that the choice of one or the other reference class has completely different consequences at the level of the argument, and in particular that the nature of its conclusion is affected, i.e. fundamentally modified. In what follows, we will now focus on showing that depending on which reference class is chosen, radically different conclusions ensue at the level of the argument itself and that, consequently, there is a reference class problem within SA. For this purpose, we will consider several reference classes in turn, focusing on how conclusions of a fundamentally different nature result from them at the level of the argument itself.

The original version of SA implicitly depicts simulations of humans of a certain type. These are virtual simulations, almost indistinguishable from real humans and that present thus a very high degree of sophistication. Moreover, these are a type of simulations that are not aware that they are themselves simulated and are therefore convinced that they are genuine humans. This is implicit in the terms of the argument itself and in particular, the inference from (9) to (3) which leads to the conclusion that ʻweʼ are currently living in an indistinguishable simulation carried out by post-humans. In fact, these are simulations that are somehow abused and misled by post-humans regarding their true identity. For the purposes of this discussion, we shall term quasi-humans the simulated humans who are not aware that they are human.

At this stage, it appears that it is also possible to conceive of indistinguishable simulations that have an identical degree of sophistication but that, on the other hand, would be aware that they are being simulated. We shall then call quasi-humans+ the simulated humans who are aware that they are themselves simulations. Such simulations are in all respects identical to the quasi-humans to which SA implicitly refers, with the only difference that they are this time clearly aware of their intrinsic nature of simulation. Intuitively, SA also applies to this type of simulation. A priori, there is no justification for excluding such a type of simulation. Moreover, there are several reasons to believe that quasi-humans+ may be more numerous than quasi-humans. For ethical reasons (i) first of all, it may be thought that post-humans might be inclined to prefer quasi-humans+ to quasi-humans. For the fact of conferring an existence on quasi-humans constitutes a deception as to their true identity, whereas such an inconvenient is absent in the case of quasi-humans+. Such deception could reasonably be considered unethical and lead to some form of prohibition of quasi-humans. Another reason (ii) is that simulations of humans who are aware of their own simulation nature should not be dismissed a priori. Indeed, we can think that the level of intelligence acquired by some quasi-humans in the near future could be extremely high and in this case, the simulations would very quickly become aware that they are themselves simulations. It may be thought that from a certain degree of intelligence, and in particular that which may be obtained by humanity in the not too distant future (Kurtzweil, 2000, 2005; Bostrom, 2006), quasi-humans should be able — at least much more easily than at present — to collect evidence that they are the subject of a simulation. Furthermore (iii), the very concept of “unconscious simulation that it is a simulation” could be inherently contradictory, because it would then be necessary to limit one’s intelligence and therefore, it would no longer constitute an indistinguishable and sufficiently realistic simulation6. These three reasons suggest that quasi-humans+ may well exist in greater numbers than quasi-humans — or even that they may even be the only type of simulation implemented by post-humans.

At this stage, it is worth considering the consequences of taking into account the quasi-humans+ within the simulation reference class inherent to SA. For this purpose, let us first consider the variation of SA (let us term it SA*) that applies, exclusively, to the class of quasi-humans+. Such a choice, first of all, has no consequence on the disjunct (1) of SA, which refers to a possible disappearance of our humanity before it has reached the post-human stage. Nor does this has any effect on the disjunct (2), according to which post-humans will not perform quasi-humans+, i.e. conscious simulations of human beings. On the other hand, the choice of such a reference class has a direct consequence on the disjunct (3) of SA. Certainly, it follows, in the same way as for the original argument, the first level conclusion that the number of quasi-humans+ will far exceed the number of authentic humans (the disproportion). However, the second level conclusion that “we” are currently quasi-humans no longer follows. Indeed, such a conclusion (let us call it self-applicability) no longer applies to us, since we are not aware that we are being simulated and are completely convinced that we are authentic humans. Thus, in this particular context, the inference from (9) to (3) no longer prevails. Indeed, what constitutes SA’s worrying conclusion no longer results from step (9), since we cannot identify with the quasi-humans+, the latter being clearly aware that they are evolving in a simulation. Thus, unlike the original version of SA based on the reference class that associates humans with quasi-humains, this new version associating humans with quasi-humans+ is not associated with such a disturbing conclusion. The conclusion that now follows, as we can see, is quite reassuring, and in any case very different from the deeply worrying7 conclusion that results from the original argument.

At this stage, it appears that a question arises: should we identify, in the context of SA, the reference class to the quasi-humans or the quasi-humans+?8 It appears that no objective element in SA’s statement supports the a priori choice of the quasi-humans or the quasi-humans+. Thus, any version of the argument that includes the preferential choice of the quasi-humans or the quasi-humans+ appears to be biased. This is the case for the original version of SA, which thus contains a bias in favor of the quasi-humans, which results from Bostrom’s choice of a class of simulations that is exclusively assimilated to quasi-humans, i.e. simulations that are not aware of their simulation nature and are therefore abused and misled by post-humans about the very nature of their identity. And this is also the case for SA*, the alternative version of SA that has just been described, which includes a particular bias in favor of quasi-humans+, simulations that are aware of their own simulation nature. However, the choice of the reference class is fundamental here, because it has an essential consequence: if we choose a reference class that associates simulations with quasi-humans, the result is the worrying conclusion that we are most likely currently experiencing in a simulation. On the other hand, if a reference class is chosen that identifies simulations with quasi-humans+, the result is a scenario that reassuringly does not include such a conclusion. At this stage, it is clear that the choice of the quasi-humans i.e., non-conscious simulations — in the original version of SA, to the detriment of conscious simulations, constitutes an arbitrary choice. Indeed, what makes it possible to prefer the choice of quasi-humans, compared to quasi-humans+? Such justification is lacking in the context of the argument. At this stage, it appears that SA’s original argument contains a bias that leads to the preferential choice of quasi-humans, and to the alarming conclusion associated with it9.

4. The reference class problem : the case of imperfect simulations

The problem of the reference class within SA relates, as mentioned above, to the very nature and to the type of simulations referred to in the argument. Is this problem limited to the preferential choice, at the level of the original argument, of unconscious simulations, to the detriment of the alternative choice of conscious simulations, which correspond to very sophisticated simulations of humans, capable of creating illusion, but endowed with the awareness that they themselves are simulations? It appears not. Indeed, as mentioned above, other types of simulations can also be envisaged for which the argument also works, but which are of a somewhat different nature. In particular, it is conceivable that post-humans may design and implement simulations that are identical to those of the original argument, but that are not as perfect in essence. Such a situation is quite likely and does not have the ethical disadvantages that could accompany the indistinguishable simulations staged in the original argument. The choice to carry out such simulations could be the result of the necessary technological level, or of deliberate and pragmatic choices, designed to save time and resources. These could be, for example, simulations of excellent quality such that the scientific inhabitants of the simulations could only discover their artificial nature after, for example, ten years of research. Such simulations could be carried out in very large numbers and, given their less resource-intensive nature, could occur in even greater numbers than quasi-humans. For the purposes of this discussion, we will call imperfect simulations this category of simulations.

At this stage, one can ask oneself what are the consequences on SA of taking into account a reference class that identifies itself with imperfect simulations? In this case, it follows, in the same way as the original argument, that the first level conclusion that the number of imperfect simulations will far exceed the number of authentic humans (the disproportion). But here too, however, the second level conclusion that “we” are currently imperfect simulations (self-applicability) no longer follows. The latter no longer applies to us and a reassuring conclusion replaces it, since we are clearly aware that we are not such imperfect simulations. Finally, it turns out that the conclusion that results from taking into account the class of imperfect simulations is of the same nature as that which follows when considering the class of the quasi-humans+.

5. The reference class problem : the caseof immersion simulations

As we have seen, extending the SA reference class to conscious simulations leads to a conclusion of a different nature from the one that results from the original argument. The same applies to another category of simulations — imperfect simulations — which lead to a conclusion of the same nature as conscious simulations, and which in any case turns out to be different from that resulting from taking into account the simulations mentioned in the original argument. At this stage, the question arises as to whether the reference class can not be assimilated to other types of simulations relevant from the point of view of SA and whose consideration would lead to a conclusion that is inherently different from that which follows when considering the simulations of the original argument, or conscious or imperfect simulations.

In particular, the question arises as to whether human simulations, which would be such as to apply to ourselves — in a sense that may differ from the original argument — and which would include the conclusion of self-applicability inherent in SA, could not exist in a more or less near future. Some answers can be provided by considering an evolution of the concepts of virtual reality that are already being implemented in different fields such as psychiatry, surgery, industry, military training, entertainment, etc. In psychiatry in particular, virtual universes are used to implement techniques related to behavioral therapies, and offer advantages over traditional in vivo scenarios (Powers & Emmelkamp, 2008). In this type of treatment, the patient himself is simulated using an avatar and the universe in which he evolves is also simulated in the most realistic way possible. Convincing results have been obtained in the treatment of some phobias (Choy & al., 2007, Parsons & Rizzo, 2008), as well as post-traumatic stress disorder (Cukor & al., 2009, Baños & al., 2011).

In this context, it is conceivable that developments in this concept of virtual reality could lead to the realization of simulated humans, which would require a high degree of realism. This would require, in particular, the completion of current research, particularly on the simulation of the human brain. It is possible that significant progress may be made in the near future (Moravec, 1998; Kurzweil, 2005; Sandberg and Bostrom, 2008; De Garis et al. al., 2010). It is also conceivable that we will then have the ability to immerse ourselves in simulated universes by borrowing the personalities of humans thus simulated, while really having — the time of immersion — the impression that this is our real existence10. In addition, the same human simulation could take the form of multiple variations that would correspond to the purpose — therapeutic, scientific, playful, utilitarian, historical, etc. — sought during the immersion. For example, it is conceivable that some variations may only include important elements of the simulated personality’s life, neglecting uninteresting details. For the purposes of this discussion, we can term this type of simulation: immersion simulations. In this context, humans could thus frequently resort to immersion in a simulated anterior human personality. It is also possible that individuals may use simulations of themselves: they could be simulations of themselves at earlier times in their lives, with eventual slight variations, however, depending on the purpose sought for the immersion in question. In such circumstances, it is conceivable that very large quantities of this type of simulation could be carried out by computer means. In any case, it appears that the number of simulations at our disposal would be much greater than the inhabitants of our planet. In this context, it appears that SA functions in the same way as the original argument if we reason in relation to a reference class that identifies itself with this type of immersion simulations.

At this point, it is worth considering the effect on SA of assimilating the reference class to immersion simulations. In such a context, it appears that the first-level consequence based on the humans/simulations disproportion would apply here, in the same way as the original argument. Secondly, and this is an important consequence, the second level conclusion based on self-applicability would now apply, since we can conclude that “we” are also, in this extended sense, simulations. On the other hand, it would no longer follow the alarming conclusion, which is that of the original argument and which manifests itself at a third level, that we are unconscious simulations, since the fact that we are in this sense simulations does not imply here that we are mistaken about our first identity. Thus, unlike the original argument, the result is a reassuring conclusion: humans are occasionally immersion simulations, while being aware that they use them.

Could we not object here that we have not yet reached the state where we can identify, even if only temporarily, with such immersion simulations and that this does not make the above developments relevant to SA? Strictly speaking, the virtual reality implemented in our time can indeed be considered too coarse in nature to be assimilated to the very realistic simulations hinted at by Bostrom. However, it can be assumed that only high-quality immersion simulations, which would give the illusion at least the time of their use that they are a real existence, could be carried out, for such simulations to become relevant for the SA reference class. The hypothesis that such a technological level, based on an explosion of artificial intelligence, could be achieved within a few decades has thus been put forward (Kurzweil, 2005; Eden et al. al., 2013). If such a technological evolution were to occur within, for example, a few decades, could we not then legitimately consider that such simulations also fall within the reference class of SA? Given this possible temporal proximity, it seems appropriate to take into account the case of immersion simulations and to evaluate their consequences for SA11.

6. The different levels of conclusion according to the chosen reference class

Finally, the preceding discussion emphasizes that if SA is considered in light of its inherent reference class problem, there are actually several levels in the conclusion of SA: (C1) disproportion; (C2) self-applicability; (C3) unconsciousness (the worrying fact that we are fooled, deceived about our primary identity). In fact, the previous discussion shows that (C1) is true regardless of the chosen (by restriction or extension) reference class: quasi-humans, quasi-humans+, imperfect simulations and immersion simulations. In addition, (C2) is also true for the original reference class of quasi-humans — and for immersion simulations, but is false for the class of quasi-humans+ and imperfect simulations. Finally, (C3) is true for the original reference class of quasi-humans, but it proves to be false for quasi-humans+, imperfect simulations and immersion simulations. These three levels of conclusion are represented in the table below:

levelconclusioncasequasi-humansquasi-humans+imperfect simulationsimmersion simulations
C1the proportion of simulated humans will far exceed that of humans (disproportion)C1Atruetruetruetrue
the proportion of simulated humans will not significantly exceed that of humansC1Āfalsefalsefalsefalse
C2we are most likely simulations (self-applicability)C2Atruefalsefalsetrue
we are most likely not simulationsC2Āfalsetruetruefalse
C3we are unconscious simulations of their simulation nature (unconsciousness)C3Atruefalsefalsefalse
we are not unconscious simulations of their simulation natureC3Āfalsetruetruetrue

Figure 1. The different levels of conclusion within SA

as well as in the following tree structure:

Figure 2. Treeof the different levels of conclusion of SA

While SA’s original conclusion suggests that there is only one level of conclusion, it turns out, however, as just pointed out, that there are in fact several levels of conclusion in SA, when the argument is examined from a broader perspective, in the light of the reference class problem. The conclusion of the original argument (C3A) is itself worrying and alarming, in that it concludes that there is a much higher probability than we had imagined a priori that we are humans simulated without being aware of it. However, the above analysis shows that, depending on the chosen reference class, some conclusions of a very different nature can be inferred by the simulation argument. Thus, a completely different conclusion is associated with the choice of the reference class of the quasi-humans+ or imperfect simulations. The resulting conclusion is that we are not such simulations (C2Ā). Finally, another possible conclusion, itself associated with the choice of the immersion simulation class, is that we are eventually part of such a simulation class, but we are aware of it and therefore it is not a cause for concern (C3Ā).

The above analysis finally highlights what is wrong with the original version of SA, which is at a twofold level. First, the original argument focuses on the class of simulations that are not aware of their own simulation nature. This leads to a succession of conclusions that there will be a greater proportion of simulated humans than authentic humans (C1A), that we are part of simulated humans (C2A) and finally that we are, more likely than we might have imagined a priori, simulated humans unaware of being (C3A). However, as mentioned above, the very notion of human simulation is ambiguous, and such a class can in fact be defined in different ways, given that there is no objective criterion in SA for choosing such a class in a way that is not arbitrary. We can indeed choose the reference class by identifying the simulations with unconscious simulations, i.e. quasi-humans simulations. But the alternative choice of a reference class that identifies itself with simulations that are conscious of being simulations themselves, i.e. quasi-humans+, has equal legitimacy. In the original argument, there is no objective criterion for choosing the reference class in a non-arbitrary way. Thus, the fact of favoring, in the original argument, the choice of quasi-humans — with the alarming conclusion associated with them — over quasi-humans+, constitutes a bias, as well as the choice of a reference class that identifies itself with quasi-humans+, leads this time to a reassuring conclusion.

Secondly, it appears that the reference class of SA can be defined at a certain level of restriction or extension. The choice in the original argument of the quasi-humans — occurs at a certain level of restriction. But if we now move to a certain level of extension, the reference class now includes imperfect simulations. And if we place ourselves at an even greater level of extension, simulations include not only imperfect simulations, but also immersion simulations. But depending on whether the class is chosen at a particular level of restriction or extension, a completely different conclusion will follow. Thus, the choice, at a higher level of extension, of imperfect simulations leads to a reassuring conclusion. Similarly, at an even greater level of extension, which this time includes immersion simulations, there also follows a new reassuring conclusion. Thus, the above analysis shows that in the original version of SA, the choice is made preferentially, by restriction, on the reference class of quasi-humans, to which is associated a worrying conclusion, as well as a choice by extension, also taking into account imperfect simulations or immersion simulations, leads to a reassuring conclusion.

Can we not object, at this stage, that the above analysis leads to a change in the original scenario of SA and that it is no longer the same problem12? To this, it can be replied that the previous analysis is based on variations in SA that preserve the very structure of the original argument. What this analysis shows is that this same structure is likely to produce conclusions of a very different nature, as long as the reference class is varied within reasonable limits that correspond to the context of SA, and even though the original SA statement suggests a single type of conclusion. Bostrom himself emphasizes that it is the structure of the argument that constitutes its real core: “The structure of the Simulation Argument does not depend on the nature of the hypothetical beings that would be created by the technologically mature civilizations. If instead of computer simulations they created enormous numbers of brains in vats connected to a suitable virtual reality simulation, the same effect could in principle be achieved.” (Bostrom, 2005). In addition, the different levels of extension used here to highlight variations in the SA reference class are intended to illustrate how different levels of conclusion can result. But if we wish to preserve the very form of the original argument, we can then limit the variation of the reference class to what really constitutes the core of this analysis, by considering only a reference class that identifies itself with the quasi-humans. The reference class is then made up of both quasi-humans and quasi-humans+. This is sufficient to generate a reassuring conclusion — which is not taken into account in the original argument — and thus modify the general conclusion resulting from the argument. In this case, it is the same reference class as the one underlying the original argument, with the only difference that simulations knowing that they are simulated are now part of it. Because the latter, whose possible existence is not mentioned in the original argument, nevertheless have an equal right to legitimacy in the context of SA.

Finally, the preferential choice in the original argument of the quasi-humans class, appears to be an arbitrary choice that no objective criterion justifies, while other choices deserve equal legitimacy. For the SA statement does not contain any objective element allowing the choice of the reference class to be made in a non-arbitrary manner. In this context, the worrying conclusion associated with the original argument also turns out to be an arbitrary conclusion, since there are several other reference classes that have an equal degree of relevance to the argument itself, and from which a quite reassuring conclusion follows.13 14

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1 The Liar is thus both true and false. In the sorites paradox, an object with a certain number of grains of sand is both a heap and a non-heap. Similarly, in Goodman’s paradox, an emerald is both green and grue, and therefore both green and blue after a certain date. Finally, in the Sleeping Beauty paradox, the probability that the piece fell on heads before the awakening of the Sleeping Beauty is 1/2 by virtue of one reasoning mode, and only 1/3 by virtue of an alternative reasoning.

2 William Eckhardt (2013, p. 15) considers that — in the same way as the Doomsday Argument (Eckhardt 1993, 1997, Franceschi, 2009) — the problem inherent in SA comes from the use of retrocausality and the problem related to the definition of the reference class: “if simulated, are you random among human sims? hominid sims? conscious sims?”.

3 We will leave aside here the question of whether an infinite number of simulated humans should be taken into account. This could be the case if the ultimate level of reality were abstract. In this case, the reference class could include simulated humans who identify themselves, for example, with matrices of very large integers. But Bostrom answers such an objection in his FAQ (www.simulation-argument.com/faq.html) and points out that in this case, the calculations are no longer valid (the denominator is infinite) and the ratio is not defined. We will therefore leave this hypothesis aside, focusing our argument on what constitutes the core of SA, i.e. the case where the number of human simulations is finite.

4 The same would be true if simulations were carried out using quantum computers.

5 I thank an anonymous referee for highlighting this point, as well as the point about computers built from components using DNA properties and molecular biology.

6 It seems difficult to rule out here the case where quasi-humans discover, at least fortuitously, that they are simulated humans, thus becoming quasi-humans+ from that moment on. However, in order to advantage the paradox, we will consider here that the very notion of an indistinguishable simulation is not plagued with contradiction.

7 Bostrom (2003) considers that the fact that we live in a simulation would only moderately affect our daily lives: “Supposing we live in a simulation, what are the implications for us humans? The foregoing remarks notwithstanding, the implications are not all that radical”. However, it may be thought that the effect should be much more profound, given that the fundamental level of reality is not where the simulation subjects believe it to be and that, as a result, many of their beliefs are completely erroneous. As David Chalmers (2005) points it out: “The brain is massively deluded, it seems. It has all sorts of false beliefs about the world. It believes that it has a body, but it has no body. It believes that it is walking outside in the sunlight, but in fact it is inside a dark lab. It believes it is one place, when in fact it may be somewhere quite different”.

8 For the purposes of this discussion, we present things as an alternative between quasi-humans and quasi-humans+. However, one could conceive that post-humans – perhaps different post-human civilizations – create both quasi-humans and quasi-humans+. We would then have a tripartite situation involving humans, quasi-humans and quasi-humans+. For the sake of simplicity, we can assimilate here such a situation to the one that prevails when post-humans only create quasi-humans since it is sufficient that the latter are present in very large numbers to create the worrying effect inherent to SA.

9 This type of bias can be analyzed in one instance of the one-sidedness bias (Walton, 1999, p. 76-81, Franceschi, 2014, p. 587-592) where the reference class is that of the simulations and the associated duality is consciousness/unconsciousness.

10 A complete simulation of a human brain is also called an upload. One definition (Sandberg & Bostrom, 2008, p. 7) is as follows:  : “The basic idea is to take a particular brain, scan its structure in detail, and construct a software model of it that is so faithful to the original that, when run on appropriate hardware, it will behave in essentially the same way as the original brain.”

11 The above also shows that when examining SA carefully, it can be seen that the argument contains a second reference class. This second reference class is that of post-humans. What is a post-human? Should we assimilate this class to civilizations far superior to ours, to those that will evolve in the 25th century or the 43th century? Should the descendants of our current human race who will live in the 22nd century be counted among the post-humans if they were to make considerable technological progress in the field of simulations? In any case, the definition of the post-human class appears to be closely linked to that of simulations. Because if we are interested, in a broad sense, in immersion simulations, then post-humans can be assimilated to a generation of humans not very far from us. If we consider imperfect simulations, then they should be associated with a more distant time. On the other hand, if we consider, in a more restrictive sense, simulations of humans that are completely indistinguishable from our current humanity, then we should be interested in post-humans from a much more distant era. Thus, the class of post-humans appears to be closely correlated with that of simulations, because the degree of evolution of simulations is related to the level reached by the post-human civilizations that implement them. For this reason, we shall limit the present discussion to the reference class of the simulations.

12 I thank an anonymous referee for raising this objection.

13 The resulting double weakening of SA finally makes it possible to reconcile SA with our pre-theoretical intuitions, because the worrying scenario of the original argument now coexists with several scenarios of a quite reassuring nature.

14 The present analysis is a direct application to the Simulation Argument of the form of dialectical contextualism described in Franceschi (2014).

I thank two anonymous referees for Philosophiques, for very useful comments on an earlier version of this article.

An Introduction to Analytic Philosophy

In this book, Paul Franceschi provides us with an introduction to analytic philosophy. In a concrete way, he chooses to describe forty paradoxes, arguments or philosophical issues that represent so many challenges for contemporary philosophy and human intelligence, for some paradoxes of millennial origin—such as the Liar or the sorites paradox—are still unresolved in the present day. Some other philosophical puzzles, however—such as the Doomsday argument—appeared only recently in the literature. The author strives to introduce us clearly to each of these problems as well as to major attempts that have been formulated to solve them.

In 2021, “An Introduction to Analytic Philosophy” entered the “64 Best Analytic Philosophy eBooks of All Time” list established by the bookauthority.org site.

“I’m really impressed by this very neat and stimulating book. I highly recommend it both to students for pedagogy and general culture (prisoner’s dilemma, twin-earth, etc.), and to professionals as well for the reference tool and even more generally to those who like to think.”

Julien Dutant, Philotropes, Philosophical blog


The Kindle version is also available.

Complements to a Theory of Cognitive Distortions

English preprint of a paper published in French under the title “Compléments pour une théorie des distorsions cognitives”, in the Journal de Thérapie Comportementale et Cognitive, 2007, 17-2, pp. 84-88.

The purpose of this study is to describe a conceptual framework for cognitive distortions, which notably allows to specify more accurately their intrinsic relationships. This conceptual framework aims at inserting itself within the apparatus of cognitive therapy and of critical thinking. The present analysis is based on the following fundamental concepts: the reference class, the duality and the system of taxa. With the help of these three notions, each cognitive distortion can be defined. A distinction is also made between, on the one hand, general cognitive distortions and on the other hand, specific cognitive distortions. The present model allows then to define within the same conceptual framework the general cognitive distortions such as dichotomous reasoning, disqualifying a given pole, minimisation and maximisation. It also allows to describe as specific cognitive distortions: disqualifying the positive, selective abstraction and catastrophism. Furthermore, the present model predicts the existence of two other general cognitive distortions: the omission of the neutral and requalifying in the other pole.


This paper is cited in:

  • Paul Franceschi, Théorie des distorsions cognitives : application à l’anxiété généralisée, Journal de Thérapie Comportementale et Cognitive, Volume 18, Issue 4, December 2008, Pages 127-131, English translation
  • Paul Franceschi, Théorie des distorsions cognitives : la sur-généralisation et l’étiquetage, Journal de Thérapie Comportementale et Cognitive, Volume 19, Issue 4, December 2009, Pages 136-140, English translation
  • Pramod Pandey, On the Nature of Reason in the present-day research, in Hasnain, Imtiaz, S. and Chaudhary, S. C. (eds.), Problematizing Language Studies: Cultural, Theoretical and Applied Perspectives- Essays in Honour of of Rama Kant Agnihotri. New Delhi: Aakar Books. Pp. 387-97, 2010.
  • Paul Franceschi, Traitement cognitif différentiel des délires oolythématiques et du trouble anxieux généralisé, Journal de Thérapie Comportementale et Cognitive, Volume 21, Issue 4, November 2011, Pages 121-125, English translation
  • Lisa Wake, Karl Nielsen, Nandana Nielsen & Catalin Zaharia, Depression symptom clusters, in “The Clinical Effectiveness of Neurolinguistic Programming”, Routledge 2013, edited by Lisa Wake, Richard M. Gray and Frank S. Bourke
  • Masoumeh Nozari, Ali Razipour Jouybari, Azam Nozari, Roya Raoufi Ahmad (2013) The Relationship between Moral Intelligence and Cognitive Distortions among Employees, Journal of Basic and Applied Scientific Research, 3(9), pages 345-348.
  • Lizet Fernandez-Jammet, Evaluation longitudinale de l’efficacité d’une prise en charge cognitivo-comportementale de groupe destinée à des patients atteints de fibromyalgie, dissertation doctorale, 2016
  • Hélène Richard-Lepouriel, Trouble bipolaire, auto-stigmatisation et restructuration cognitive : une première tentative de prise en charge, Journal de Thérapie Comportementale et Cognitive, Volume 27-4, November 2017, Pages 177-183
  • Valérie Pennequin & Nicolas Combalbert, L’influence des biais cognitifs sur l’anxiété chez des adultes non cliniques, Annales Médico-psychologiques, Volume 175, Issue 2, February 2017, Pages 103-107
  • Anita Robert, Nicolas Combalbert, Valérie Pennequin, Etude des profils de distorsion cognitive en fonction des états anxieux et dépressifs chez des adultes tout-venant, Annales Médico-Psychologiques 176 (2018) 225–230
  • Nawal Ouhmad, Nicolas Combalbert, Wissam El-Hage, Cognitive distortions and emotion regulation among post traumatic stress disorder victims, in Psychological Applications and Trends, Ed. by C. Pracana & M. Wang, InScience Press, 2019
  • Paul Franceschi, For a Typology of Auditory Verbal Hallucinations Based on their Content, Activitas Nervosa Superior, volume 62, pages 104–109, 2020
  • A.Robert, N.Combalbert, V.Pennequin, R.Deperrois, N.Ouhmad, Création de l’Échelle de Distorsions Cognitives pour adultes (EDC-A) : étude des propriétés psychométriques en population générale et association avec l’anxiété et la dépression, Psychologie Française, 2021
  • Deperrois Romain & Nicolas Combalbert, Links between cognitive distortions and cognitive emotion regulation strategies in non-clinical young adulthood, in Psychological Applications and Trends, Ed. by C. Pracana & M. Wang, InScience Press, 2021
  • Ouhmad, Nawal & El-Hage, Wissam & Combalbert, Nicolas. (2022), Maladaptive cognitions and emotional regulation in PTSD, 3-7, Conference: International Psychological Applications Conference and Trend, doi:10.36315/2022.

Complements to a Theory of Cognitive Distortions

Paul FRANCESCHI

The cognitive distortions, introduced by Aaron Beck (1963, 1964) and Albert Ellis (1962) are traditionally defined as fallacious reasoning that plays a crucial role in the emergence of certain mental disorders. The cognitive therapy in particular is based on the identification of these cognitive distortions within the everyday way of thinking of the patient, and their replacement by alternative reasoning. Traditionally, the cognitive distortions are represented as one of the twelve following irrational modes of reasoning: 1. Emotional reasoning 2. Overgeneralization 3. Jumping to conclusions (or arbitrary inference) 4. Dichotomous reasoning 5. Should statements (Ellis 1962) 6. Fortune telling or mind reading 7. Selective abstraction 8. Disqualifying the positive 9. Maximisation and minimisation 10. Catastrophism 11. Personalisation 12. Labelling.

Under their classical form which is that of an enumeration, the cognitive distortions plays a central role within the field of cognitive therapy. Considering also their widespread nature in normal reasoning, one can think however that an accurate understanding of the cognitive distortions proves also to be useful outside the field of psychopathology. In particular, the cognitive distortions can also be seen as part of the apparatus which constitutes critical thinking. For these reasons, it appears that a conceptual framework, notably allowing to define the relationships between the different cognitive distortions, could also turn out to be useful. In what follows, we shall set out to present a general theory of the cognitive distortions, which brings a certain number of supplementary elements in comparison with classical theory.

1. Main notions

The present framework allows to represent several classical cognitive distortions: dichotomous reasoning, disqualification of one of the poles, selective abstraction, minimisation and maximisation. To these, one can add two other cognitive distortions of which the present model allows to predict the existence and which are closely related to the classical cognitive distortions, although they do not appear, to the knowledge of the author, to the number of these last. It consists of the omission of the neutral and the requalification in the other pole.

The cognitive distortions can be constructed, in the present model, from three main notions: the reference class,the duality and the system of taxa. It is necessary, in a preliminary way, to set out to describe these three notions. The reference class, above all, is constituted by a group of phenomena or objects. Several examples can be given here: the class composed of the events and facts of the patient’s life; the class of the future events of the patient’s life; the class constituted by all the parts of the patient’s body; the class which is made up of the patient’s character’s traits.

The notion of duality, second, corresponds to a pair of concepts such as Positive/Negative, Internal/External, Collective/Individual, Nice/Ugly, etc. A duality corresponds then to a criterion under the angle of which the elements of the reference class can be considered or evaluated. Let us denote by A/Ā a given duality, where A and Ā constitute then dual concepts. An enumeration (necessarily partial) of the dualities is as follows: Positive/Negative, Internal/External, Quantitative/Qualitative, Visible/Invisible, Analytical/synthetic, Absolute/Relative, Abstract/Concrete, Static/Dynamic, Unique/Multiple, Aesthetics/Practice, Definite/Vague, Finite/Infinite, Simple/Composite, Individual/Collective, Implicit/Explicit, Intentional/Unintentional.

Finally, thepatient’s system of taxa consists of a taxonomy which allows the patient to evaluate and to classify the elements of the reference class, according to the criterion corresponding to a given duality A/Ā. These taxa can be considered as “what can see” the patient. It consists of a system of values that is inherent to the patient or of a filter through which the patient “sees” the elements of the reference class, i.e. the phenomena or the objects of reality. The figure below represents an optimal system of taxa.

Fig.1. The optimal system of taxa

This last is composed of 11 spheres which represent each a given taxon. The system of taxa is optimal, because all taxa are present. On the other hand, if the patient does not have some taxa, he cannot see nor count the corresponding elements. For example, if he/she lacks the taxa of the duality A/Ā corresponding to pole A, he cannot see the corresponding elements. In the same way, if the patient has no neutral taxon in his/her system of taxa, he cannot see the neutral elements of the reference class. More formally, let us consider then a series of n elements E1, E2, …, En such that each of them has, in a objective way, a degree d[Ei] in a duality A/Ā comprised between -1 and 1 (d Î [-1, +1]). We can consider then a series including 11 elements, E1, E2, …, E11, which present an objective increasing degree (the choice of 11 elements is here arbitrary, and any other number would also do the job). Let us pose then: d[E1] = -1, d[E2] = -4/5, d[E3] = -3/5, d[E4] = -2/5, d[E5] = -1/5, d[E6] = 0, d[E7] = 1/5, d[E8] = 2/5, d[E9] = 3/5, d[E10] = 4/5, d[E11] = 1. Let us also define a subjective degree D[Ei] such that it is attributed by the patient to each of the Ei. So, E1-E5 corresponds to the pole A of duality A/Ā, E6 to the neutral taxon and E7-E11 corresponds to the pole Ā. Moreover, this optimal system of taxa can be assimilated with one Likert scale with 11 degrees.

At this stage, we are in a position to define the main cognitive distortions, and it is worth considering them in turn. The cognitive distortions can be defined as a type of reasoning which leads to favour, without objective grounds, a subset of the taxa applicable to a given duality A/Ā, in order to qualify a given reference class. It also proves to be useful to draw a distinction, in a preliminary way, between the general cognitive distortions and the specific cognitive distortions. The general cognitive distortions relate to all reference classes and all dualities. By contrast, the specific cognitive distortions are mere instances of the general cognitive distortions which are inherent to a given reference class and to a given duality.

2. The cognitive distortions

2.1 Dichotomous reasoning

In the present context, dichotomous reasoning (or all-or-nothing thinking) can be defined as a general cognitive distortion which leads the patient to consider a given reference class only according to the two extreme taxa which relate to every pole of a given duality. With this type of reasoning, the patient ignores completely the presence of degrees or of intermediate steps. In his/her taxa system, the patient has as well the two extreme taxa corresponding to poles A and Ā. The defect in that way of considering things is that facts or objects corresponding to intermediary taxa are not taken into account. So it results from it a reasoning without nuances nor gradation, which proves to be maladapted to properly apprehend the diversity of human situations. Formally, dichotomous reasoning consists in taking into account only the elements of the reference class such as|d[Ei]| = 1, ord[E1] = 1 ord[E11] = -1, by ignoring all the others.

Fig. 2. Dichotomous reasoning

2.2 The disqualification of one of the poles

In the present model, the disqualification of one of the poles is the general cognitive distortion which leads to grant an arbitrary priority in one of the poles of a given duality, in order to qualify the elements of a reference class. It consists then in the fact of attributing more importance to one of the poles rather than to the other one, in the lack of objective motivation. The taxa corresponding to one of the poles of a given duality are lacking in the patient’s system of taxa. So, the patient sees things only through the prism of pole A (respectively Ā), by ignoring completely the viewpoint of the opposed pole Ā (respectively A). Formally, the disqualification of one of the poles leads to consider only the Ei such that d[Ei] ≤ (respectivelyd[Ei] ≥ 0), by ignoring any events such thatd[Ei] > 0 (respectivelyd[Ei] < 0).

Fig. 3. The disqualification of one of the poles

An instance of the disqualification of one of the poles consists in the disqualification of the positive. This last can be analysed, in the present context, as a specific instance of the disqualification of one of the poles, which applies to the Positive/Negative duality and to the reference class including the facts and events of the patient’s life. The patient tends then to ignore positive events, by considering that they do not count, for any reason. Such instance finds to apply in the cognitive therapy of depression.

Another instance of the disqualification of one of the poles also applies to the Positive/Negative duality and to the reference class which comprises the character’s traits of the patient. This one completely ignores his/her positive character’s traits (qualities) and only directs his/her attention to his/her negative character’s traits (defects). This encourages then him/her to conclude that he/she “is worth nothing”, that he/she is “a failure”. Such instance also applies in the cognitive therapy of depression.

2.3 Arbitrary focusing on a given modality

Another type of cognitive general distortion consists in arbitrary focusing on a modality of a given duality. In the present context, this type of general cognitive distortion leads to favour one taxon in the patient’s system of taxa, by ignoring all the others. In arbitrary focusing, the taxon being discussed is present in the patient’s system of taxa, and is affected to an unique element of the reference class. There is eclipsing (in general temporary) of others taxa and other elements of the reference class, so that the patient is haunted by this specific element.

Fig. 4. Arbitrary focusing

A particular instance of this type of general cognitive distortion, relates to the reference class of the facts of the patient’s life, and to the Positive/Negative duality. It is a specific cognitive distortion, which consists in focusing on a negative event of the patient’s life. It is then one of the classical cognitive distortions, defined as selective abstraction (Mental filter), which consists in the fact of choosing one detail with a negative connotation and to focalise on it. Suchlike, the patient sees only this detail, and his/her vision{view} of reality is darkened because it is entirely tinted with this particular event. Such instance applies in the cognitive therapy of depression.

One can also mention another instance of arbitrary focusing, which also applies to the Positive/Negative duality, but relates to the class of reference composed of the hypothetical future events of the patient’s life. In that case, the patient focalises on the possible happening of a very negative event. Such instance finds to apply in the cognitive therapy of generalised anxiety disorder.

Another specific instance of arbitrary focusing applies to the Nice/Ugly duality and to a reference class which identifies itself with all the parts of the patient’s body. The patient focalises then on a detail of his/her anatomy which he considers to be ugliness. The patient has well, in his/her system of taxa the Ugly taxon in question. Moreover, he/she affects this taxon to an unique part of his/her body, while all the others taxa are temporarily eclipsed. Such specific cognitive distortion finds to apply in the cognitive therapy of body dysmorphic disorder (Neziroglu and Yaryura-Tobias 1993, Veale and Riley 2001, Veale 2004).

2.4 The omission of the neutral

The present model also leads to predict the existence of another type of general cognitive distortion, which consists in the omission of the neutral. This latter cognitive distortion results from the absence, in the patient’s system of taxa, of the neutral taxon. It follows that the elements of the reference class which can objectively be defined as neutral with regard to a given duality A/Ā, are not taken into account by the patient. Formally, the patient omits to consider the Ei such that d[Ei] = 0. The omission of the neutral sometimes plays an important role, notably when there is a gaussian distribution of the elements of the reference class, where the elements objectively corresponding to the neutral taxon are precisely those which are the most numerous.

Fig. 5. The omission of the neutral

2.5 The requalification in the other pole

The present model also leads to predict the existence of another type of general cognitive distortion. That is the reasoning which consists in re-qualifying an event belonging to a given duality A, in the other duality Ā. Formally, the subjective degree attributed by the patient to a given event E is the opposite of its objective degree, so that: D[E] = (-1) x d[E].

Fig. 6. The requalification into the other pole

A characteristic instance of requalification in the other pole consists in the specific cognitive distortion which applies to the class of the events of the patient’s life and to the Positive/Negative duality. This consists typically in re-describing as negative an event which should be objectively considered as positive. By requalifying positive events in a negative way, the patient can reach the conclusion that all events of his/her life are of a negative nature. For instance, by considering the past events of his/her life, the patient notes that he/she made no act of violence. He/she considers this to be “suspect”. This type of instance also finds to apply within the cognitive therapy of depression.

Another instance of requalification in the other pole consists in the specific cognitive distortion which applies to the class of the parts of the patient’s body and to the Nice/Ugly duality. Typically, the patient re-qualifies as “ugly” a part of his/her body which is objectively “nice”. Such specific cognitive distortion relates to the cognitive therapy of body dysmorphic disorder.

2.6 Minimisation and maximisation

This general cognitive distortion consists in attributing to an element of the reference class, a taxon according to the criterion of a duality A/Ā which proves to be lower (minimisation) or greater (maximisation) than its objective value. It consists here of a classical cognitive distortion. The subjective degree D[E] which is attributed by the patient to an event E differs significantly from its objective degree d[E]. In minimisation, this subjective degree is distinctly less than, so that |D[E]| < |d[E]|. In maximisation, by contrast, the subjective degree is distinctly greater, such that |D[E]| > |d[E]|.

Fig. 7. Maximisation and minimisation

A specific instance of minimisation relates to the class of the facts of the patient’s life and to the Positive/Negative duality. The patient tends to consider certain facts of his/her existence as less positive than they in reality are. In maximisation, he/she considers certain facts of his/her life as more negative than they really are. In the present context, the classical cognitive distortion of catastrophism (or dramatisation) can be considered as a specific cognitive distortion, which consists of a maximisation applied to the negative pole of the Positive/Negative duality. The patient attributes then a subjective degree D[E] in the Positive/Negative duality to an event, while the absolute value of its objective degree d[E] is very distinctly lesser. Such instance applies to the cognitive therapy of depression.

3. Conclusion

As we see it, the present theory provides several elements, in comparison with classical theory, that allow to define and to classify the classical cognitive distortions, within the same conceptual framework. These last are considered, either as general cognitive distortions, or as specific cognitive distortions, i.e. as instances of the general cognitive distortions which relate to a given reference class and duality. Thus, dichotomous reasoning, maximisation and minimisation constitute general cognitive distortions. In addition, disqualification of the positive, selective abstraction, selective negative focus and catastrophism constitute then specific cognitive distortions. Besides, the present analysis allowed to describe two additional general cognitive distortions: the omission of the neutral and the requalification in the other pole.

References

Beck AT.: 1963, Thinking and depression: Idiosyncratic content and cognitive distortions. Archives of General Psychiatry 9, 324-333.

Beck AT.: 1964, Thinking and depression: Theory and therapy, Archives of General Psychiatry 10, 561-571.

Ellis A.: 1962, Reason and Emotion in Psychotherapy, Lyle Stuart, New York.

Neziroglu FA et JA. Yaryura-Tobias: 1993, Exposure, response prevention, and cognitive therapy in the treatment of body dysmorphic disorder. Behav Ther 24, 431-438.

Veale D et S. Riley: 2001, Mirror, mirror on the wall, who is the ugliest of them all? The psychopathology of mirror gazing in previous termbody dysmorphic disorder.next term. Behaviour Research and Therapy 39, 1381-1393.

Veale D.: 2004, Advances in a cognitive behavioural model of body dysmorphic disorder. Body Image 1, 113-125.

Viewpoint of a duality

Viewpoints relative to a given duality are viewpoints that concerne a given duality A/Ā.

For example, one can place oneself under the angle of the Static/Dynamic duality. Or, one can place oneself according to the point of view of the Absolute/Relative duality.

Internal/External
Quantitative/Qualitative
Visible/Invisible
Absolute/Relative
Abstract/Concrete
Static/Dynamic
Diachronic/Synchronic
Single/Multiple
Extension/Restriction
Aesthetic/Practical
Precise/Vague
Finite/Infinite
Single/compound, Individual/Collective
Analytical/Synthetic
Implicit/Explicit
Voluntary/Involuntary


Ambiguous images Arbitrary focus Bistable perception Complementarity relationship Conflict resolution Conflict resolution with matrices of concepts Conflict types relating to matrices of concepts Contrary relationship Dialectical contextualism Dialectical monism Dialectical monism in Aztec philosophy Dialectical monism in Heraclitus Dichotomic analysis Dichotomic analysis applied to paradox resolution Dichotomous reasoning Disqualification of one pole Disqualification of the positive Dualities Dual poles Extreme opposition General cognitive distortions Instance of one-sidedness bias Matrix of concepts Maximization Mental filter Minimization Bistable cognition Omission of the neutral One-sidedness bias One-sided viewpoint Opposition relationship Principle of dialectical indifference Requalification into the other pole Reference class Reference class problem Reference class problem in philosophical paradoxes Reference class problem in the Doomsday argument Reference class problem in Hempel’s paradox Reference class problem in the surprise examination paradox Selective abstraction Specific cognitive distortions System of taxa Two-sided viewpoint Viewpoint of a duality Viewpoint of a pole

A Solution to the Doomsday Argument

doomsday

Un article publié en français dans the Canadian Journal of Philosophy Volume 28, Juillet 1998, pages 227-46.

Cet article présente une solution pour l’Argument de l’Apocalypse (DA). Je montre tout d’abord qu’il n’existe pas de critère objectif pour le choix en général d’une classe de référence: dans ce cas, le calcul inhérent à DA ne peut pas prendre place. En second lieu, j’envisage le choix particulier d’une classe de référence donnée, ainsi que Leslie le recommande. Mais le caractère arbitraire de la sélection rend légitime de multiples possibilités de choix, soit par extension, soit par restriction: DA peut alors être établi en particulier pour le genre Homo, pour l’espèce Homo sapiens, pour la sous-espèce Homo sapiens sapiens, … , pour une classe définie de manière restreinte correspondant aux humains n’ayant pas connu l’ordinateur, etc. Finalement, il apparaît que DA “fonctionne”, mais sa conclusion se révèle inoffensive.

The Doomsday Argument and Hempel’s Problem

Posprint in English (with additional illustrations from wikimedia commons) of a paper published in French in the Canadian Journal of Philosophy Vol.29, July 1999, pp. 139-56 under the title “Comment l’Urne de Carter et Leslie se Déverse dans celle de Hempel”.
I begin by describing a solution to Hempel’s Problem. I recall, second, the solution to the Doomsday Argument described in my previous Une Solution pour l’Argument de l’Apocalypse (Canadian Journal of Philosophy 1998-2) and remark that both solutions are based on a similar line of reasoning. I show thirdly that the Doomsday Argument can be reduced to the core of Hempel’s Problem.


This paper is cited in:

Koji Sawa, Junki Yokokawa and Tatsuji Takahashi (2013) Logical Equivalence: Symmetric and Asymmetric Features, Symmetry: Culture and Science, Vol. 24, No. x.

Milan M. Cirkovic, A Resource Letter on Physical eschatology, Am.J.Phys. 71 (2003) 122-133

Nick Bostrom, Anthropic Bias: Observation Selection Effects in Science and Philosophy, Routledge (2002)

Alasdair Richmond, The Doomsday Argument, Philosophical Books Vol. 47 No. 2 April 2006, pp. 129–142


The Doomsday Argument and Hempel’s Problem

Postprint – with additional illustrations from Wikimedia commons) – of a paper originally pubslihed in French in the Canadian Journal of Philosophy under the title « Comment l’urne de Carter et Leslie se déverse dans celle de Carter », vol. 29, March 1999, pages 139-156.

Paul Franceschi

I Hempel’s Problem

Hempel’s Problem (thereafter, HP) is based on the fact that the two following assertions:

(H) All ravens are black

(H’) Everything that is non-black is a non-raven

are logically equivalent. The logical structure of (H) is:

(H1) All X are Y

that is to say x (Xx  Yx), whereas that of (H’) has the form:

(H1′) All non-Y are non-X

Carl Gustav Hempel.jpg
Carl Hempel

that is to say x (~Yx  ~Xx). In fact, the structure of the contrapositive form (H1′) is clearly equivalent to that of (H1). It follows that the discovery of a black raven confirms (H) and also (H’), but also that the discovery of a non-black thing which is not a raven such as a pink flame or even a grey umbrella, confirms (H’) and thus (H). This last conclusion appears paradoxical. The propositions (H1) and (H1′) are based on four properties X, ~X, Y and ~Y, respectively corresponding to raven, non-raven, black, and non-black in the original version of HP. These four properties determine four categories of objects: XY, X~Y, ~XY and ~X~Y, which correspond respectively to black ravens, non-black ravens, black non-ravens and non-black non-ravens. One can observe here that a raven is defined with precision in the taxonomy within which it fits. A category as that of the ravens can be regarded as well defined, because it is based on a set of precise criteria defining unambiguously the species corvus corax and allowing the identification of its instances. It also appears that one can build without difficulty a version of HP where a variation with regard to the X class is operated. If one replace the X class with that of the tulips or that of the dolphins, etc. by adapting correlatively the Y property, one still obtains a valid version of HP. It appears thus that changes can be operated at the level of the X class without loosing the problem inherent to HP.

Corvus corax

Similarly, the black property can be specified with precision, on the basis of a taxonomy of colours established with regard to the wavelengths of the light.1 Moreover, one can consider variations with regard to the Y property. One will thus be able to choose properties such as whose length is smaller than 50 cm, living less than 10 years, etc. Such variations also lead to acceptable versions of HP. Lastly, it should be noted that the non-black property can be the subject of a definition which does not suffer from ambiguity, in particular with the help of the precise taxonomy of colours which has been just mentioned. Similarly, if one takes into account variations of the Y property such as smaller than 40 cm, or whose diameter is larger than 25 cm, etc, one arrives to definitions of the non-Y property which just as non-black are established with precision and lead in addition to versions of HP presenting the same problem as the original version. Thus, the X class, just as the properties Y and non-Y can be the subject of a precise and nonambiguous definition. Moreover, variations operated at the level of these classes lead to acceptable versions of HP. In contrast, the situation is not the same for the non-X class.

II The reference class Z

The concept of non-raven present in the original version of HP leads to highlight an important problem. What constitutes an instance of a non-raven? Intuitively a blue jay, a pink flame, a grey umbrella and even a natural integer constitute non-ravens. One is thus confronted with the definition of a new reference class – call it Z – including X and non-X. The Z class allows defining complementarily the class of non-X, and in the original version of Hempel, the class of non-ravens. Thus Z is the implicit reference class with regard to which the definition of the X class allows that of non-X. Does one have then to consider a Z class that goes until including abstract objects? Is it necessary to consider a concept of non-raven including abstract entities such as natural integers and complex numbers? Or is it necessary to limit oneself to a Z class, which only embraces concrete things? Such a discussion has its importance, because there are infinitely many abstract objects, whereas there are only finitely many individualised concrete objects. This fact is likely to influence later importantly the possible application of a bayesian reasoning. One could thus have a reference class Z including at the same time abstract objects (natural integers, real and complex numbers, etc.) and concrete objects such as artefacts but also natural entities such as humans, animals, plants, meteorites, stars, etc. Such a reference class is defined very extensively. And the consequence of such a choice is that the discovery of any object confirms (H’) and thus (H). At this stage, anything2 confirms (H). It should be noted that one can also have a definition of Z including all concrete objects that have been just mentioned, but excluding this time the abstract objects.

Larus audouinii

The instances of this class are now finitely denumerable, just as the cardinal of the corresponding set: the reference class Z then includes animals, plants, stars, etc. But alternatively, one could still consider a Z class associating the ravens (corvus corax) and the Audouin’s gulls3 (larus audouinii). In this case, the instances of the X class (corvus corax) are in a number larger than those of the non-X class (larus audouinii). And we always face the corresponding version of HP.4

Lastly, nothing seems to prohibit, at a very restrictive level, to choose a Z class made up of the X class, only added with one single element such as a red tulip. With this definition of Z, we still face a minimal version of HP. Of course, any object, added to the class of X and constituting the non-X class will be appropriate and then confirm at the same time (H’) and (H). Thus, any object ~X~Y will lead to confirm (H). The remarks which have just been made call however an immediate objection. With various degrees, it is allowed to think that the choice of each reference class Z that has been just mentioned is arbitrary. Because it is allowed to reject on those grounds extreme definitions of Z such as the one defined above and including all abstract objects. Similarly, a Z class including the natural integers or the complex numbers can also be eliminated. The X class is defined with regard to the concrete objects that are the ravens and there is not particular reason to choose a Z class including abstract entities.

A red tulip

Similarly, one will be able to reject a definition of Z based on a purely artificial restriction, simply associating with X a determinate object such as a red tulip. Because I can choose arbitrarily, the object that constitutes the complement of X, i.e. I can define Z as I wish. Such an extreme conception appears as without relationship with the initial definition of X. A Z class thus defined is not homogeneous. And there is no justification to legitimate the association of a red tulip to the class of the ravens to build that of Z. The association within a same Z class of the ravens and the Audouin’s gulls, appears analogously as an illegitimate choice. Why not then the association of the ravens and the goldfinches? Such associations are symptomatic of a purely artificial selection. Thus, the choices of reference classes Z mentioned above reveal an arbitrary and artificial nature. Indeed, shouldn’t one make one’s possible to find a Z class which is the most natural and the most homogeneous possible, taking into account the given definition of X? One can think that one must attempt to operate a determination of the Z class, which is the most objective possible. In the original version of HP, doesn’t the choice of the ravens for the X class implicitly determine a Z class which is directly in connection with that of the ravens? A Z class naturally including that of the ravens such as that of the corvidae, or that of the birds, seems a good candidate. Because such a class is at least implicitly determined by the contents of the X class. But before analysing versions of HP built accordingly, it is worth considering before some nonparadoxical versions of HP.

III The analogy with the urn

It is notoriously admitted that certain versions5 of HP are not paradoxical. Such is in particular the case if one considers a reference class Z associated with boxes, or a set of playing cards. One can also consider a version of HP associated with an urn. An X class is thus considered where the objects are finitely denumerable and which only includes balls and tetrahedrons. The Y class itself is reduced to two colours: red and green. One has thus four types of objects: red balls, green balls, red tetrahedrons and green tetrahedrons. In this context, we have the following version of HP:

(H2) All balls are red

(H2′) All non-red objects are non-balls

It appears here that the case of red tetrahedrons can be ignored. Indeed, their role is indifferent and one can thus ignore their presence in the urn. They can be regarded as parasitic objects, whose eventual presence in the urn does not have importance. One is thus brought to take into account an urn containing the significant objects consisting in red balls, green balls and green tetrahedrons. And the fact that non-red objects can only be green, and that non-balls can only be tetrahedrons leads to consider equivalently:

(H3) All balls are red

(H3′) All green objects are tetrahedrons

that clearly constitutes a nonparadoxical version of HP. Indeed, the draw of a red ball confirms (H3) and (H3′) whereas the draw of a green tetrahedron confirms (H3′) and (H3).

Consider now the case where the urn contains six significant objects.6 One has just drawn three red balls and one green tetrahedron (the draw is 3-0-17) and one makes then the hypothesis (H3). At this stage, the probability that all balls are red corresponds to three draws (3-0-3, 4-0-2 and 5-0-1) among six possible draws (3-0-3, 3-1-2, 3-2-1, 4-0-2, 4-1-1, 5-0-1). Similarly, the probability that all green objects are tetrahedrons is identical. Thus, P(H3) = P(H3′) = 1/2 and also P(~H3) = P(~H3′) = 1/2. These initial probabilities being stated, consider now the case where one has just carried out a new draw in the urn. Another red ball is drawn (the draw is 4-0-1). This corresponds to three possible compositions of the urn (4-0-2, 4-1-1, 5-0-1). Let E be the event consisting in the draw of a red ball in the urn. We have then the probability of drawing a red ball if all the balls of the urn are red, i.e. P(E, H3) such as P(E, H3) = 2/3, since two cases (4-0-2, 5-0-1) correspond to the fact that all balls are red. In the same way, P(E, ~H3) = 1/3. The situation is identical if one considers P(E, H3′) and P(E, ~H3′). One is then in a position to calculate the posterior probability that all balls are red using Bayes formula: P'(H3) = [P(H3) X P(E, H3)] / [P(H3) X P(E, H3) + P(~H3) X P(E, ~H3)] = (0,5 X 2/3) / (0,5 X 2/3 + 0,5 X 1/3) = 2/3. And P'(~H3) = 1/3. There are identical results concerning P'(H3′) and P'(~H3′). Thus, P'(H3) > P(H3) and P'(H3′) > P(H3′), so that the hypothesis (H3) just as the equivalent hypothesis (H3′) are confirmed by the draw of a new red ball.

Let us examine finally the situation where, instead of a red ball, one draws a green tetrahedron (the draw is 3-0-2) in the urn. Let thus F be the event consisting in the draw of a green tetrahedron. In this case, we have three possible combinations (3-0-3, 3-1-2, 4-0-2). But among these, two (3-0-3, 4-0-2) correspond to a situation where hypotheses (H3) and (H3′) are confirmed. Thus, P(F, H3) = P(F, H3′) = 2/3 and P(F, ~H3) = P(F, ~H3′) = 1/3. A bayesian calculation provides the same results as on the preceding hypothesis of the draw of a red ball. Thus, on the hypothesis of the draw of a green tetrahedron, one calculates the posterior probabilities P'(H3) = P'(H3′) = 2/3 and P'(~H3) = P'(~H3′) = 1/3. Thus, the draw of a green tetrahedron confirms at the same time (H3′) and (H3). It should be noted that one can easily build versions of HP allowing to establish nonparadoxically the preceding reasoning. Consider a cubic mineral block of 1m on side. Such an object of 1m3 is divided into 1000 cubic blocks of 1 dm3, consisting either of quartz, or of white feldspar. One examines fifty of these blocks, and one notes that several of them consist of white feldspar of gemmeous quality. One is brought to make the hypothesis that all blocks of white feldspar are of gemmeous quality. We have then the following version of HP:

(H4) All blocks of white feldspar are of gemmeous quality

(H4′) All blocks of non-gemmeous quality are not white feldspar

that is equivalent to:

(H5) All blocks of white feldspar are of gemmeous quality

(H5′) All blocks of non-gemmeous quality are quartz

where we have in effect the equivalence between (H5) and (H5′) and where a correct bayesian reasoning can be established. Such an example (call it the mineral urn) can also be transposed to other properties X and Y, since identical conditions are preserved.

IV A solution to the problem

One must, taking into account the above developments,8 attempt to highlight a definition of the Z class that does not present an arbitrary and artificial nature, but proves on the contrary the most natural and the most homogeneous possible, with regard to the given definition of X. Consider accordingly the following9 version of HP:

(H6) All Corsican-Sardinian goshawks have a wingspan smaller than 3,50 m

(H6′) All birds having a wingspan larger than 3,50 m are not Corsican-Sardinian goshawks

In this particular version of (H’), the X class is that of the Corsican-Sardinian goshawks,10 and the reference class Z is that of the birds. This last class presents an obvious relationship with that of the Corsican-Sardinian goshawks. It is allowed to think that such a way of defining Z with regard to X is a natural one. Indeed such a definition does not present an arbitrary nature as obviously as that was the case with the examples of Z classes mentioned above. Of course, one can observe that it is possible to choose, in a more restricted but so natural way, a Z class corresponding to the accipiter genus. Such a class presents a homogeneous nature. It includes in particular the species accipiter gentilis (northern goshawk) but also accipiter nisus (European sparrowhawk), accipiter novaehollandiae (grey goshawk), accipiter melanoleucus (black and white goshawk).

Accipiter gentilis

However, alternatively and according to the same viewpoint, one could also extend the Z class to the instances of the – wider – family of accipitridae11 including at the same time the accipiter genus which have been just mentioned, but also the milvus (kite), buteo (buzzard), aquila (eagle), etc. genus. Such a class includes in particular the species milvus migrans (black kite), milvus milvus (red kite), buteo buteo (common buzzard), aquila chrysaetos (golden eagle), etc. These various acceptable definitions of the Z class find their justification in the taxonomy within which the Corsican-Sardinian goshawk inserts itself. More systematically, the latter belongs to the subspecies accipiter gentilis arrigonii, to the species accipiter gentilis, to the accipiter genus, to the family of accipitridae, to the order of falconiformes, to the class of birds, to the subphylum of vertebrates, to the phylum of chordates,12 to the animal reign, etc. It ensues that the following variations of (H’) are acceptable, in the context which has just been defined:

(H7′) All northern goshawks having a wingspan larger than 3,50 m are not Corsican-Sardinian goshawks

(H8′) All goshawks having a wingspan larger than 3,50 m are not Corsican-Sardinian goshawks

(H9′) All accipitridae having a wingspan larger than 3,50 m are not Corsican-Sardinian goshawks

(H10′) All falconiformes having a wingspan larger than 3,50 m are not Corsican-Sardinian goshawks

(H11′) All birds having a wingspan larger than 3,50 m are not Corsican-Sardinian goshawks

(H12′) All vertebrates having a wingspan larger than 3,50 m are not Corsican-Sardinian goshawks

(H13′) All chordates having a wingspan larger than 3,50 m are not Corsican-Sardinian goshawks

(H14′) All animals having a wingspan larger than 3,50 m are not Corsican-Sardinian goshawks

There are thus several versions of (H’) corresponding to variations of the Z class which themselves are made possible by the fact that the Corsican-Sardinian goshawk belongs to n categories, determined by the taxonomy to which it belongs. And in fact, when I meet one northern goshawk belonging to the nominal form (accipiter gentilis gentilis), it is at the same time a northern goshawk (accipiter gentilis) non- Corsican-Sardinian (non-accipiter gentilis arrigonii), a goshawk (accipiter) non-Corsican-Sardinian goshawk, an accipitridae non-Corsican-Sardinian goshawk, a falconiformes non-Corsican-Sardinian goshawk, a bird (aves) non-Corsican-Sardinian goshawk, but also a vertebrate non-Corsican-Sardinian goshawk, a chordate non-Corsican-Sardinian goshawk, an animal non-Corsican-Sardinian goshawk. Thus, the instance of accipiter gentilis gentilis that I have just observed, belongs at the same time to all these categories. And when I meet a grey whale, it is not a bird non-Corsican-Sardinian goshawk, but it is indeed a vertebrate non-Corsican-Sardinian goshawk, as well as a chordate non-Corsican-Sardinian goshawk, and also an animal non-Corsican-Sardinian goshawk.

In general, a given object x which has just been discovered belongs to n levels in the taxonomy within which it fits. It belongs thus to a subspecies, 13 a species, a sub-genus, a genus, a super-genus, a subfamily, a family, a super-family, a subphylum, a junction, a reign… One can assign to the subspecies the level14 1 in the taxonomy, to the species the level 2…, to the super-family the level 8, etc. And if within (H), the class X is at a level p, it is clear that Z must be placed at a level q such as q > p. But how to fix Z at a level q which is not arbitrary? Because the reference class Z corresponds to a level of integration. But where must one stop? Does one have to attach Z to the level of the species, the sub-genus, the genus…, the reign? One does not have an objective criterion allowing the choice of a level q among the possibilities that are offered. I can choose q close to p by proceeding by restriction; but in a so conclusive way, I am authorised to choose q distant from p, by applying a principle of extension. Then why choose such class of reference restrictively defined rather than such other extensively defined? One does not have actually a criterion to legitimate the choice, according to whether one proceeds by restriction or by extension, of the Z class. Consequently, it appears that the latter can only be defined arbitrarily. And it follows clearly here that the determination of the Z class and thus of the non-X class is arbitrary. But the choice of the reference class Z appears fundamental. Because according to whether I choose such or such reference class Z, it will result from it that a given object x will confirm or not (H). For any object x, I can build a Z class such as x belongs to non-X, as I can choose a Z class such as x does not belong to non-X. Thus, this choice is left to my arbitrary.

For a given object x, I can build a Z class such as this object confirms (H) and another Z class such as this object does not confirm (H). Of course, if Z is selected arbitrarily, the bayesian reasoning inherent to HP “works”, but corresponds to an arbitrary and artificial point of view: having found an object x, (H) is confirmed. But one can as well choose, in a so artificial and more restrictive way, a Z class where x misses and where x does not confirm (H). Thus, one is not enabled to conclude objectively that the discovery of the object x confirms (H). Because to reason thus would amount to conferring a universal and general value to a viewpoint which is only the expression of an arbitrary choice.

How this result can be reconciled with the facts mentioned above,15 concerning the existence of nonparadoxical versions of HP? It is worth noting here that the bayesian reasoning can be established in each case where the Z class is finite, and where this fact is known before the experiment.16 One can then show a bayesian shift. But at this stage, it is worth distinguishing the cases where the Z class is determined before the experiment by an objective criterion and the cases where it is not the case. In the first case, the contents of the Z class are given before the experiment and the Z class is thus not selected arbitrarily, but according to an objective criterion. Consequently, the bayesian reasoning is correct and provides relevant information. Such is in particular the case when one considers a version of HP applied to an urn, or a version such as the mineral urn. On this last hypothesis, the composition of the Z class is fixed in advance. There is then a significant difference with Nicod’s criterion:17 an object ~X~Y confirms (H) and an object XY confirms (H’).

Conversely, when the Z class is not fixed and is not determined before the experiment by an objective criterion, one can subjectively choose Z at any level of extension or restriction, but the conclusions resulting from the bayesian reasoning must be regarded as purely arbitrary and do not present thus an objective value. Because one then does not have a base and a justification to choose such or such level of restriction or extension. Thus, in this case, Nicod’s criterion according to which any object ~X~Y is neutral with respect to (H) and any object XY is neutral with respect to (H’), can apply itself. It should be observed that the present solution has the effect of preserving the equivalence of a proposition and its contraposition. And similarly, the principle of the confirmation of a generalisation by each of its instances is also preserved.

V A common solution to Hempel’s Problem and the Doomsday Argument

The Doomsday Argument (thereafter, DA) attributed to Brandon Carter, has been described by John Leslie (1992).18 DA can be described as follows. Consider an event A: the final extinction of the human race will occur before year 2150. One can estimate at 1 chance from 100 the probability that this extinction occurs: P(A) = 0,01. Let also ~A be the event: the final extinction of the human race will not occur before 2150. Consider also the event E: I live during the 1990s. In addition one can estimate today at 50 billions the number of humans having existed since the birth of humanity: let H1997 be such a number. In the same way, the current population can be evaluated to 5 billions: P1997 = 5×109. One calculates thus that one human from ten, if event A occurs, will have known the 1990s. The probability that humanity is extinct before 2150 if I have known the 1990s, is thus evaluated: P(E, A) = 5×109/5×1010 = 0,1. On the other hand, if the human race passes the course of the 2150s, one can think that it will be destined to a much more significant expansion, and that the number of humans will be able to rise for example to 5×1012. In this case, the probability that the human race is not extinct after 2150 if I have known the 1990s, can be evaluated as follows: P(E, ~A) = 5×109/5×1012 = 0,001. This now makes it possible to calculate the posterior probability of the human race extinction before 2150, using Bayes formula: P'(A) = [P(A) x P(E, A)] / [P(A) x P(E, A) + P(~A) X P(E, ~A)] = (0,01 x 0,1) / (0,01 x 0,1 + 0,99 x 0,001)  0,5025. Thus, the fact of taking into account the fact that I live currently has made the probability of the human race extinction before 2150 shift from 0,01 to 50,25.

640px-Dmanisi-D3844._Homo_erectus_or_Homo_georgicus
Homo erectus’ skull (photo by Ryan Somma)

I have presented in my paper ‘Une Solution pour l’Argument de l’Apocalypse’19 a solution to DA, whose main lines can be described as follows. The DA reasoning is based on a single reference class, which is that of the humans.20 But how this reference class has to be defined? Should it be limited to the only representatives of our current subspecies Homo sapiens sapiens? Or does one have to extend it to all the representatives of the species Homo sapiens, by including this time, in addition to Homo sapiens sapiens, Homo sapiens neandertalensis…? Or is it necessary to include in the reference class the entire Homo genus, including then all the successive representatives of Homo erectus, Homo habilis, Homo sapiens, etc? And isn’t it still necessary to go until envisaging a wider class, including all the representatives of a super-genus S, made up not only of the Homo genus, but also of the new genus Surhomo, Hyperhomo, etc. which will result from the foreseeable evolutions from our current species? It appears thus that one can consider a reduced reference class by proceeding by restriction, or apprehend a larger class by making the choice of a reference class by extension. One can thus operate for the choice of the reference class by applying either a principle of restriction or a principle of extension. And according to whether one applies one or the other principle, various levels of choice are each time possible.

But it appears that one does not have an objective criterion, which makes it possible to legitimate the choice of such or such a reference class. And even our current subspecies Homo sapiens sapiens cannot be regarded as a natural and an adequate choice for the reference class. Because isn’t it allowed to think that our paradigmatic concept of human has to undergo evolutions? And in addition, the fact of excluding from the reference class a subspecies such as Homo sapiens neandertalensis or the future evolutions of our species, doesn’t it reveal an anthropocentric viewpoint? Since one does not have an objective selection criterion, one can choose arbitrarily one or the other of the classes that have been just described. One can for example identify the reference class to the species Homo sapiens, and observe a bayesian shift. There is indeed then an increase in the posterior probability of the extinction of Homo sapiens. But this bayesian shift is worth as well for a still more restricted reference class, such as our subspecies Homo sapiens sapiens. There too, the application of Bayes formula leads to an appreciable increase in the posterior probability of the nearest end of Homo sapiens sapiens. However identically, the bayesian shift also applies to a still more reduced reference class, which is that of the representatives of Homo sapiens sapiens having not known the computer. Such a reference class will certainly face a nearest extinction. There however, such a conclusion is not likely to frighten us, because the evolutionary potentialities of our species are such that the succession of a new species to those which preceded them, constitutes one of the characteristics of our evolution mode.

It should be mentioned that this solution leads here to accept the conclusion (the bayesian shift) of Carter and Leslie for a given reference class, while placing it in comparison with conclusions of comparable nature relating to other reference classes, completely inoffensive. The fact of taking into account various levels of restriction, made legitimate by the lack of an objective criterion of choice, leads finally to the harmlessness of the argument. Thus, it appears that the argument based on the reference class and its arbitrary choice by restriction or extension constitutes a common solution to HP and DA. HP and DA are ultimately underlain by the same problem inherent to the definition of the Z class of HP and the single reference class of DA. One thus has a solution of comparable nature for the two paradoxes. It is worth here concluding by presenting an element that tends to confirm the common source of the two problems. One will observe first that one is not able to highlight a version of DA corresponding veritably to the original version of HP, a reference class such as that of the ravens being not transposable in DA. The inherent argument in DA is indeed based on the use of the anthropic principle and requires obviously a reference class made up of intelligent beings. When Leslie21 considers the extension of the reference class, he specifies expressly that the condition for the membership of the reference class is the aptitude to produce an anthropic reasoning. On the other hand it is possible to describe a version of HP made up from the elements of DA. If one takes X for our current subspecies Homo sapiens sapiens and Y for are alive only before 2150, one obtains the following version of HP:

(H15) All Homo sapiens sapiens will be alive only before the year 2150

(H15′) All those which will live after 2150 will be non-Homo sapiens sapiens

In this context, an alive human being in 1997 constitutes an instance confirming (H15). In parallel, the discovery of an Homo sapiens sapiens after 2150 leads to refute (H15). Lastly, the discovery of an alive non-Homo sapiens sapiens after 2150 constitutes a confirmation of (H15′) and thus of (H15). Taking into account this particular formulation, it is clear that one currently only observes instances confirming (H15). On the other hand, after 2150, one will be able to have instances refuting (H15) or instances confirming (H15′).

It is worth noting here that (H15) does not allow veritably to be used as support of a version of DA. Indeed, the reference class identifies itself here precisely as Homo sapiens sapiens, whereas in the original version of DA, the reference class consists in the human race. Consequently, one has not, strictly speaking, an identity between the event underlie by (H15) and A, so that (H15)-(H15′) does not constitute a joint version22 of DA and HP.

But this version of HP being made up with the elements of DA, one must be able, at this stage, to verify the common origin of the two problem, by showing how the argument raised in defence of DA with regard to the reference class, can also be used in support of HP. One knows the response made by Leslie to the objection that the reference class for DA is ambiguous or, due to the evolutions of Homo sapiens sapiens, leads to a heterogeneous reference class, of composite nature. It is exposed in the response made to Eckhardt:

How far should the reference class extend? (…) One can place the boundary more or less where one pleases, provided that one adjusts one’s prior probability accordingly. Exclude, if you really want to, all future beings with intelligence quotients above five thousand, calling them demi-gods and not humans23.

and developed in The End of the World24:

The moral could seem to be that one’s reference class might be made more or less what one liked. (…) What if we wanted to count our much-modified descendants, perhaps with three arms or with godlike intelligence, as ‘genuinely human’? There would be nothing wrong in this. Yet if we were instead interested in the future only of two-armed humans, or of humans with intelligence much like that of humans today, then there would be nothing wrong in refusing to count any others25.

For Leslie, one can go until including in the reference class, the descendants of humanity become very distant from our current species due to the fact of evolution. But Leslie also accepts liberally that one limits the reference class to the only individuals close to our current humanity. One is thus free to choose the reference class that one wishes, while operating either by extension, or by restriction. It will be enough in each case to adjust the initial probability accordingly. It appears here that this type of answer can be transposed, literally, to an objection to HP of comparable nature, based on the reference class of (H15)-(H15′). One can fix, so the objection goes, the Z class as one wishes, and assign to “all those” the desired content. One can for example limit Z to the species Homo sapiens, or well associate it to the whole of the Homo genus, including then the evolutions of our species such as Homo spatialis, Homo computeris, etc. What is important – could continue this defender – is to determine preliminarily the reference class and to conserve this definition when the various instances are then met. Thus, it proves that the arguments advanced in support of the reference class of DA can be transposed in defence of HP. This constitutes an additional element, going in the direction of the common origin of the two problems, dependent on the definition of a reference class. DA and HP need consequently a same type of answer. Thus, the urn of Carter and Leslie flows in that of Hempel.26

References

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FRANCESCHI, P. 1998, “Une Solution pour l’Argument de l’Apocalypse.” Canadian Journal of Philosophy, 28 (1998): 227-46
GOODMAN, N. 1955. Fact, Fiction and Forecast. Cambridge: Harvard University Press.
HEMPEL, C. 1945. “Studies in the logic of confirmation.” Mind, 54 (1945): 1-26 et 97-121
LESLIE, J. 1992. “Time and the Anthropic Principle.” Mind, 101 (1992): 521-40
—. 1993. “Doom and probabilities.” Mind, 102 (1993): 489-91
—. 1996. The End of the World: the science and ethics of human extinction. London and New York: Routledge.
PAPINEAU, D. 1995. “Methodology: the Elements of the Philosophy of Science.” In Philosophy A Guide Through the Subject, ed. A.C. Grayling. Oxford: Oxford University Press.
SAINSBURY, M. 1988. Paradoxes. New York: Cambridge University Press.
THIBAULT, J-C. 1983. Les oiseaux de Corse. Paris: De Gerfau.

1 It is known that a monochromatic light, of single wavelength, meets practically only in laboratory. But the natural colours can be modelled in terms of subtraction of lights of certain wavelengths, starting from the white light of the Sun.

2 Any object ~X~Y in the Z class thus extensively defined.

3 The total population of Audouin’s gulls is evaluated with approximately 3000 couples (cf. Thibault 1983, 132).

4 This incidentally makes it possible to verify that HP does not find its origin in a disproportion of the X class compared to that of the non-X. The fact that the instances of the X class are in a number larger than those of the non-X does not prevent the emergence of a version of HP.

5 Properly speaking, these are not thus versions of HP, since they are nonparadoxical. But the corresponding propositions have the logical structure of (H) and (H’).

6 The red tetrahedrons possibly found in the urn are regarded as nonsignificant objects.

7 With the notation: npq (red balls – green balls – green tetrahedrons).

8 Cf. § II.

9 This particular version of HP is chosen here because it is based on an X class corresponding to the subspecies accipiter gentilis arrigonii. Conversely, the original version of HP is grounded on the species corvus corax. The choice of a subspecies for the X class allows simply here a supplementary level of integration.

10 The Corsican-Sardinian goshawks (accipiter gentilis arrigonii) constitute a subspecies of the northern goshawk, specific to Corsica and Sardinia. This endemic subspecies differs from the nominal form of the northern goshawk by the following characteristics (cf. Thibault 1983): the colouring of the head is blackish instead of brown blackish; the back is brown; the lower part is darker.

11 The ornithologists still distinguish the class of the accipitriformes, corresponding to all accipitridae, to which are added the pandlionidae, such as pandlion haliaetus (osprey), etc.

12 The phylum of chordata includes all vertebrates and some invertebrates, which present the property of having a dorsal chord, at least at a given period of their life.

13 It is possible to consider alternatively, if one wishes, another taxonomy that our current scientific taxonomy. That does not affect the current reasoning, since the conclusions are identical, since the principles of classification are respected.

14 It is obviously possible to take into account finer taxonomies and including additional subdivisions starting from the various subspecies. Obviously, that does not affect the current line of reasoning.

15 Cf. § III.

16 As we have seen, the bayesian reasoning cannot take place when one considers a Z class including infinite sets such as natural integers, real numbers, etc.

17 Nicod’s criterion is defined as follows (Hempel 1945, 11), with S1 = (H) and S2 = (H’): ‘(…) let has, B, C, D Be furnace objects such that has is has raven and black, B is has raven goal not black, C not has raven goal black and D neither has raven NOR black. Then, according to Nicod’ S criterion, has would confirm S1, goal Be neutral with respect to S2; B would disconfirm both S1 and S2; C would Be neutral with respect to both S1 and S2, and D would confirm S1, goal Be neutral with respect to S2.’

18 John Leslie, ‘Time and the Anthropic Principle.’ Mind, 101 (1992): 521-40.

19 Canadian Journal of Philosophy 28 (1998) 227-46.

20 Leslie uses the terms of human race.

21 ‘How much widening of the reference class is appropriate when we look towards the future? There are strong grounds for widening it to include our evolutionarily much-altered descendants, three-armed or otherwise, as ‘humans’ for doomsday argument purposes – granted, that’s to say, that their intelligence would remain well above the chimpanzee level.’ (1996, 262)

22 I.e. comprising simultaneously the two problems.

23 W. Eckhardt, ‘Probability Theory and the Doomsday Argument.’ Mind, 102 (1993): 483-8; cf. John Leslie, ‘Doom and probabilities.’ Mind, 102 (1993): 489-91

24 This point of view is detailed by Leslie, in the part entitled ‘Just who should count have being human?’ (The End of the World, 256-63).

25 Cf. Leslie (1996, 260).

26 I thank two anonymous referees for the Canadian Journal of Philosophy for their comments, concerning an earlier draft of this paper.