# 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.

(excerpt from) Franceschi P. An Introduction to Analytic Philosophy: Paradoxes, Arguments and Contemporary Problems, 2nd edition, March, 2010

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 Courage 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 Doctrine of the mean Doomsday argument Dualities Dual poles Extreme opposition General cognitive distortions Instance of one-sidedness bias Liar paradox 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 Sorites paradox Specific cognitive distortions Surprise examination paradox System of taxa Two-sided viewpoint Viewpoint of a duality Viewpoint of a pole

# 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:

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 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 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

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)):

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

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.

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.

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:

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.

References

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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 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:

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:

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).

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:

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):

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.

# 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.

# A Solution to the Doomsday Argument

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

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.

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.

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.

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).

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.

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

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.