The classical version of the surprise examination paradox 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.
Franceschi, P., “Éléments d’un contextualisme dialectique” (in english), in Liber Amicorum Pascal Engel, J. Dutant, G. Fassio & A. Meylan (éd.), Université de Genève, 2014, pp. 581-608.
The classical version of the surprise examination paradox 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)):
day
non-surprise
surprise
1
1
1
2
1
1
3
1
1
4
1
1
5
1
1
6
1
1
7
1
1
Matrix structure of the version of the paradox corresponding to Quine’s solution for n = 7 (one week)
day
non-surprise
surprise
1
1
1
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.
day
non-surprise
surprise
1
0
1
…
…
…
365
1
0
Matrix structure of the version of the paradox corresponding to Williamson’s variation for n = 365 (one year)
The foregoing allows now to identify precisely what is at fault in the student’s reasoning, when applied to this particular version of the paradox. Under these circumstances, the student would then have reasoned as follows. The surprise cannot occur on the last day but it can occur on day 1, and the non-surprise can occur on the last day, but cannot occur on the first day. These are proper instances of non-surprise and of surprise, which prove to be disjoint. However, the notion of surprise is not captured exhaustively by the extension and the anti-extension of the surprise. But such a definition is consistent with the definition of a vague predicate, which is characterized by an extension and an anti-extension which are mutually exclusive and non-exhaustive. Thus, the notion of surprise associated with a disjoint structure is that of a vague notion. Thus, the student’s error of reasoning at the origin of the fallacy lies in not having taken into account the fact that the surprise is in the case of a disjoint structure, a vague concept and includes therefore the presence of a penumbra corresponding to borderline cases between non-surprise and surprise. Hence, the mere consideration of the fact that the surprise notion is here a vague notion would have prohibited the student to conclude that S[k, 1] = 0, for all values of k, that is to say that the examination can not occur surprisingly on any day of the period.
Finally, it turns out that the analysis leads to distinguish between two independent variations with regard to the surprise examination paradox. The matrix definition of the cases of non-surprise and of surprise leads to two variations of the paradox, according to the joint/disjoint duality. In the first case, the paradox is based on a joint definition of the cases of non-surprise and of surprise. In the second case, the paradox is grounded on a disjoint definition. Both of these variations lead to a structurally different variation of the paradox and to an independent solution. When the variation of the paradox is based on a joint definition, the solution put forth by Quine applies. However, when the variation of the paradox is based on a disjoint definition, the solution is based on the prior recognition of the vague nature of the concept of surprise associated with this variation of the paradox.
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.
Franceschi, P., “Éléments d’un contextualisme dialectique” (in english), in Liber Amicorum Pascal Engel, J. Dutant, G. Fassio & A. Meylan (éd.), Université de Genève, 2014, pp. 581-608.
Posprint in English (with additional illustrations) of an article appeared in French in the collective book (pages 581-608) written on the occasion of the 60th birthday of Pascal Engel.
Abstract In what follows, I strive to present the elements of a philosophical doctrine, which can be defined as dialectical contextualism. I proceed first to define the elements of this doctrine: dualities and polar contraries, the principle of dialectical indifference and the one-sidedness bias. I emphasize then the special importance of this doctrine in one specific field of meta-philosophy: the methodology for solving philosophical paradoxes. Finally, I describe several applications of this methodology on the following paradoxes: Hempel’s paradox, the surprise examination paradox and the Doomsday Argument.
In what follows, I will endeavour to present the elements of a specific philosophical doctrine, which can be defined as dialectical contextualism. I will try first to clarify the elements that characterise this doctrine, especially the dualities and dual poles, the principle of dialectical indifference and the one-sidedness bias. I will proceed then to describe its interest at a meta-philosophical level, especially as a methodology to assist in the resolution of philosophical paradoxes. Finally, I will describe an application of this methodology to the analysis of the following philosophical paradoxes: Hempel’s paradox , the surprise examination paradox and the Doomday Argument.
The dialectical contextualism described here is based on a number of constitutive elements which have a specific nature. Among these are: the dualities and dual poles, the principle of dialectical indifference and the one-sidedness bias. It is worth analysing in turn each of these elements.
1. Dualities and dual poles
To begin with, we shall focus on defining the concept of dual poles (polar opposites)1. Although intuitive, this concept needs to be clarified. Examples of dual poles are static/dynamic, internal/external, qualitative/quantitative, etc.. We can define the dual poles as concepts (which we shall denote by A and Ā), which come in pairs, and are such that each of them is defined as the opposite of the other. For example, internal can be defined as the opposite of external and symmetrically, external can be defined as the contrary of internal. In a sense, there is no primitive notion here and neither A nor Ā of the dual poles can be regarded as the primitive notion. Consider first a given duality, that we can denote by A/Ā, where A and Ā are dual concepts. This duality is shown in the figure below:
The dual poles A and Ā
At this point, we can also provide a list (which proves to be necessarily partial) of dualities:
In order to characterize more accurately the dual poles, it is worth distinguishing them from other concepts. We shall stress then several properties of the dual poles, which allow to differentiate them from other related concepts. The dual poles are neutral concepts, as well as simple qualities; in addition, they differ from vague notions. To begin with, two dual poles A and Ā constitute neutral concepts. They can thus be denoted by A0 and Ā0. This leads to represent both concepts A0 and Ā0 as follows:
The dual neutral poles A0 and Ā0
The dual poles are neutral concepts, i.e. concepts that present no ameliorative or pejorative nuance. In this sense, external, internal, concrete, abstract, etc.., are dual poles, unlike concepts such as beautiful, ugly, brave, which present either a ameliorative or pejorative shade, and are therefore non-neutral. The fact that the dual poles are neutral has its importance because it allows to distinguish them from concepts that have a positive or negative connotation. Thus, the pair of concepts beautiful/ugly is not a duality and therefore beautiful and ugly do not constitute dual poles in the sense of the present construction. Indeed, beautiful has a positive connotation and ugly has a pejorative connotation. In this context, we can denote them by beautiful+ and ugly–.
It should be emphasised, second, that the two poles of a given dual duality correspond to simple qualities, as opposed to composite qualities. The distinction between single and composite qualities can be made in the following manner. Let A1 and A2 be simple qualities. In this case, A1 ∧ A2, and A1 ∨ A2 are composite qualities. To take an example, static, qualitative, external are simple qualities, while static and qualitative, static and external, qualitative and external are composite qualities. A more general definition is as follows: let B1 and B2 be single or composite qualities, then B1 ∧ B2 and B1 ∨ B2 are composite qualities. Incidentally, this also highlights why the pairs of concepts red/non-red, blue/non-blue concepts can not be considered as dual poles. Indeed, non-red can thus be defined as follows as a composite quality: violet ∨ indigo ∨ blue ∨ green ∨ yellow ∨ orange ∨ white ∨ black. In this context, one can assimilate non-blue to the negation-complement of blue, such complement negation being defined with the help of composite qualities.
Given the above definition, we are also in a position to distinguish the dual poles from vague objects. We can first note that dual poles and vague objects have certain properties in common. Indeed, vague objects come in pairs in the same way as dual poles. Moreover, vague concepts are classically considered as having an extension and an anti-extension, which are mutually exclusive. Such a feature is also shared by the dual poles. For example, qualitative and quantitative can be assimilated respectively to an extension and an anti-extension, which also have the property of being mutually exclusive, and the same goes for static and dynamic, etc.. However, it is worth noting the differences between the two types of concepts. A first difference (i) lies in the fact that the union of the extension and the anti-extension of vague concepts is not exhaustive in the sense that they admit of borderline cases (and also borderline cases of borderline cases, etc., giving rise to a hierarchy of higher-order vagueness of order n), which is a penumbra zone. Conversely, the dual poles do not necessarily have such a characteristic. Indeed, the union of the dual poles can be either exhaustive or non-exhaustive. For example, the abstract/concrete duality is then intuitively exhaustive, since there does not seem to exist any objects that are neither abstract nor concrete. The same goes for the vague/precise duality: intuitively, there does no exist indeed objects that are neither vague nor precise, and that would belong to an intermediate category. Hence, there are dual poles whose extension and anti-extension turns out to be exhaustive, unlike vague concepts, such as the two poles of the abstract/concrete duality. It is worth mentioning, second, another difference (ii) between dual poles and vague objects. In effect, dual poles are simple qualities, while vague objects may consist of simple or compound qualities. There exist indeed some vague concepts which are termed multi-dimensional vague objects, such as the notion of vehicle, of machine, etc.. A final difference between the two categories of objects (iii) lies in the fact that some dual poles have an inherently precise nature. This is particularly the case of the individual/collective duality, which is susceptible to give rise to a very accurate definition.
2. The principle of dialectical indifference
From the notions of duality and of dual poles which have been just mentioned, we are in a position to define the notion of a viewpoint related to a given duality or dual pole. Thus, we have first the notion of viewpoint corresponding to a given A/Ā duality: it consists for example in the standpoint of the extension/restriction duality, or of the qualitative/quantitative duality or of the diachronic/synchronic duality, etc.. It also follows the concept of point of view related to a given pole of an A/Ā duality: we get then, for example (at the level of the extension/restriction duality) the standpoint by extension, as well as the viewpoint by restriction. Similarly, the qualitative viewpoint or perspective results from it, as well as the quantitative point of view, etc.. (at the level of the qualitative/quantitative duality). Thus, when considering a given object o (either a concrete or an abstract object such as a proposition or a reasoning), we may consider it in relation to various dualities, and at the level of the latter, relative to each of its two dual poles.
The underlying idea inherent to the viewpoints relative to a given duality, or to a given pole of a duality, is that each of the two poles of the same duality, all things being equal, deserve an equal legitimacy. In this sense, if we consider an object o in terms of a duality A/Ā, one should not favour one of the poles with respect to the other. To obtain an objective point of view with respect to a given duality A/Ā, one should place oneself in turn from the perspective of the pole A, and then from that of the pole Ā. For an approach that would only address the viewpoint of one of the two poles would prove to be partial and truncated. The fact of considering in turn the perspective of the two poles, in the study of an object o and of its associated reference class allows to avoid a subjective approach and to meet as much as possible the needs of objectivity.
As we can see it, the idea underlying the concept of point of view can be formalized in a principle of dialectical indifference, in the following way:
(PRINCIPLE OF DIALECTICAL INDIFFERENCE) When considering a given object o and the reference class E associated with it, from the angle of duality A/Ā, all things being equal, it should be given equal weight to the viewpoint of the A pole and the viewpoint of the Ā pole.
This principle is formulated in terms of a principle of indifference: if we consider an object o under the angle of an A/Ā duality, there is no reason to favour the viewpoint from A with regard to the viewpoint from Ā, and unless otherwise resulting from the context, we must weigh equally the viewpoints A and Ā. A direct consequence of this principle is that if one considers the perspective of the A pole, one also needs to take into consideration the standpoint of the opposite pole Ā (and vice versa). The need to consider both points of view, the one resulting from the A pole and the other associated with the Ā pole, meets the need of analysing the object o and the reference class associated with it from an objective point of view. This goal is achieved, as far as possible, by taking into account the complementary points of view which are those of the poles A and Ā. Each of these viewpoints has indeed, with regard to a given duality A/Ā, an equal relevance. Under such circumstances, when only the A pole or (exclusively) the pole Ā is considered, it consists then of a one-sided perspective. Conversely, the viewpoint which results from the synthesis of the standpoints corresponding to both poles A and Ā is of a two-sided type. Basically, this approach proves to be dialectical in essence. In effect, the step consisting of successively analysing the complementary views relative to a given reference class, is intended to allow, in a subsequent step, a final synthesis, which results from the joint consideration of the viewpoints corresponding to both poles A and Ā. In the present construction, the process of confronting the different perspectives relevant to an A/Ā duality is intended to build cumulatively, a more objective and comprehensive standpoint than the one, necessarily partial, resulting from taking into account those data that stem from only one of the two poles.
The definition of the dialectical principle of indifference proposed here refers to a reference class E, which is associated with the object o. The reference class2 is constituted by a number of phenomena or objects. Several examples can be given: the class of human beings who ever lived, the class of future events in the life of a person, the class of body parts of a given person, the class of ravens, etc.. We shall consider in what follows, a number of examples. Mention of such a reference class has its importance because its very definition is associated with the above-mentioned duality A/Ā. In effect, the reference class can be defined either from the viewpoint of A or from the viewpoint of Ā. Such a feature needs to be emphasized and will be useful in defining the bias which is associated with the very definition of the principle of dialectical indifference: the one-sidedness bias.
3. Characterisation of the one-sidedness bias
The previous formulation of the principle of dialectical indifference suggests straightforwardly an error of reasoning of a certain type. Informally, such a fallacy consists in focusing on a given standpoint when considering a given object, and of neglecting the opposite view. More formally, in the context described above, such a fallacy consists, when considering an object o and the reference class associated with it, in taking into account the viewpoint of the A pole (respectively Ā), while completely ignoring the viewpoint corresponding to its dual pole Ā (respectively A) to define the reference class. We shall term one-sidedness bias such type of fallacy. The conditions of this type of bias, in violation of the principle of dialectical indifference, needs however to be clarified. Indeed, in this context, we can consider that there are some cases where the two-sidedness with respect to a given duality A/Ā is not required. Such is the case when the elements of the context do not presuppose conditions of objectivity and exhaustiveness of views. Thus, a lawyer who would only emphasise the evidence in defence of his/her client, while completely ignoring the evidence against him/her does not commit the above-mentioned type of error of reasoning. In such a circumstance, in fact, the lawyer would not commit a faulty one-sidedness bias, since it is his/her inherent role. The same would go in a trial for the prosecutor, who conversely, would only focus on the evidence against the same person, by completely ignoring the exculpatory elements. In such a situation also the resulting one-sidedeness bias would not be inappropriate, because it follows well from the context that it consists well of the limited role assigned to the prosecutor. By contrast, a judge who would only take into account the evidence against the accused, or who would commit the opposite error, namely of only considering the exculpatory against the latter, would well commit an inappropriate one-sidedness bias because the mere role of the judge implies that he/she takes into account the two types of elements, and that his/her judgement is the result of the synthesis which is made.
In addition, as hinted at above, the mention of a reference class associated with the object o proves to be important. In effect, as we will have the opportunity to see it with the analysis of the following examples, the definition itself is associated with an A/Ā duality. And the reference class can be defined either from the viewpoint of A, or from the viewpoint of Ā. Such feature has the consequence that all objects are not likely to give rise to a one-sidedness bias. In particular, the objects that are not associated with a reference class that is itself likely to be envisaged in terms of an A/Ā duality, do not give rise to any such one-sidedness bias.
Before illustrating the present construction with the help of several practical examples, it is worth considering, at this stage, the one-sidedness bias which has been just defined, and which results from the very definition of the principle of dialectical indifference, in the light of several similar concepts. In a preliminary way, we can observe that a general description of this type of error of reasoning had already been made, in similar terms, by John Stuart Mill (On Liberty, II):
He who knows only his own side of the case, knows little of that. His reasons may be good, and no one may have been able to refute them. But if he is equally unable to refute the reasons on the opposite side; if he does not so much know what they are, he has no ground for preferring either opinion.
In the recent literature, some very similar concepts have also been described. It consists in particular of the dialectic bias notably described by Douglas Walton (1999). Walton (999, pp. 76-77) places then himself in the framework of the dialectical theory of bias, which opposes one-sided to two-sided arguments:
The dialectical theory of bias is based on the idea […] that an argument has two sides. […] A one-sided argument continually engages in pro-argumentation for the position supported and continually rejects the arguments of the opposed side in a dialogue. A two-sided (balanced) argument considers all arguments on both sides of a dialogue. A balanced argument weights each argument against the arguments that have been opposed to it.
Walton describes thus the dialectical bias as a one-sided perspective that occurs during the course of the argument. Walton emphasizes, though, that dialectic bias, which is universally common in human reasoning, does not necessarily constitute an error of reasoning. In line with the distinction between “good” and “bad” bias due to Antony Blair (1988), Walton considers that the dialectic bias is incorrect only under certain conditions, especially if it occurs in a context that is supposed to be balanced, that is to say where the two sides of the corresponding reasoning are supposed to be mentioned (p. 81):
Bad bias can be defined as “pure (one-sided) advocacy” in a situation where such unbalanced advocacy is normatively inappropriate in argumentation.
A very similar notion of one-sidedness bias is also described by Peter Suber (1998). Suber describes indeed a fallacy that he terms one-sidedness fallacy. He describes it as a fallacy which consists in presenting one aspect of the elements supporting a judgement or a viewpoint, by completely ignoring the other aspect of the relevant elements relating to the same judgement:
The fallacy consists in persuading readers, and perhaps ourselves, that we have said enough to tilt the scale of evidence and therefore enough to justify a judgment. If we have been one-sided, though, then we haven’t yet said enough to justify a judgment. The arguments on the other side may be stronger than our own. We won’t know until we examine them.
The error of reasoning consists then in taking only into account one viewpoint relating to the judgement in question, whereas the other viewpoint could as well prove to be decisive with regard to the conclusion to be drawn. Suber also undertakes to provide a characterization of the one-sidedness fallacy and notes in particular that the fallacy of one-sidedness constitutes a valid argument. For its conclusion is true if its premises are true. Moreover, Suber notes, it appears that the argument is not only valid but sound. For when the premises are true, the conclusion of the argument can be validly inferred. However, as hinted at by Suber, the argument is defective due to the fact that a number of premises are lacking. This is essential because if the missing premises are restored within the argument, the resulting conclusion can be radically different.
4. An instance of the one-sidedness bias
To illustrate the above concepts, it is worth at this stage providing an example of the one-sidedness bias. To this end, consider the following instance, which is a form of reasoning, mentioned by Philippe Boulanger (2000, p. 3)3, who attributes it to the mathematician Stanislaw Ulam. The one-sidedness bias shows up in a deductive form. Ulam estimates that if a company were to achieve a level of workforce large enough, its performance would be paralysed by the many internal conflicts that would result. Ulam estimates that the number of conflicts between people would increase according to the square of the number n of employees, while the impact on the work that would result would only grow as a function of n. Thus, according to this argument, it is not desirable that the number of employees within a company becomes important. However, it turns out that Ulam’s reasoning is fallacious, as Boulanger points it out, for it focuses exclusively on the conflictual relations between employees. But the n2 relationships among the company employees can well be confrontational, but may include as well collaborative relationships that are quite beneficial for the company. And so there is no reason to favour conflictual relationships with respect to collaborative ones. And when among n2 relationships established between the company employees, some are genuine collaborative relationships, the effect is, instead, of improving business performance. Therefore, we can not legitimately conclude that it is not desirable that the workforce of a company reaches a large size.
For the sake of clarity, it is worth formalizing the above reasoning. It turns out thus that Ulam’s reasoning can be described as follows:
(D1Ā ) if <a company has a large workforce>
(D2Ā ) then <n2 conflictual relationships will result>
(D3Ā ) then negative effects will result
(D4Ā ) ∴ the fact that <a company has a large workforce> is bad
This type of reasoning has the structure of a one-sidedness bias, since it focuses only on conflicting relationships (the dissociation pole of the association/dissociation duality), by ignoring a parallel argument with the same structure that could legitimately be raised, focusing on collaborative relationships (the association pole), which is the other aspect relevant to this particular topic. This parallel argument goes as follows:
(D1A) if <a company has a large workforce>
(D2A) then <n2 collaborative relationships will result>
(D3A) then positive effects will result
(D4A) ∴ the fact that <a company has a large workforce> is good
This finally casts light on how the two formulations of the argument lead to conflicting conclusions, i.e. (D4Ā) and (D4A). At this point, it is worth noting the very structure of the conclusion of the above reasoning, which is as follows:
(D5Ā ) the situation s is bad from the viewpoint of Ā (dissociation)
while the conclusion of the parallel reasoning is as follows:
(D5A) the situation s is good from the viewpoint of A (association)
But if the reasoning had been complete, by taking into account the two points of view, a different conclusion would have ensued:
(D5Ā ) the situation s is bad from the viewpoint of Ā (dissociation)
(D5A) the situation s is good from the viewpoint of A (association)
(D6A/Ā) the situation s is bad from the viewpoint of Ā (dissociation) and good from the viewpoint of A (association)
(D7A/Ā) the situation s is neutral from the viewpoint of the duality A/Ā (association/dissociation)
And such a conclusion turns out to be quite different from that resulting from (D5Ā ) and (D5A).
Finally, we are in a position to replace the one-sidedness bias which has just been described in the context of the present model: the object o is the above reasoning, the reference class is that of the relationships between the employees of a business, and the corresponding duality – allowing to define the reference class – is the dissociation/association duality.
5. Dichotomic analysis and meta-philosophy
The aforementioned principle of dialectic indifference and its corollary – one-sidedness bias – is likely to find applications in several domains4. We shall focus, in what follows, on its applications at a meta-philosophical level, through the analysis of several contemporary philosophical paradoxes. Meta-philosophy is that branch of philosophy whose scope is the study of the nature of philosophy, its purpose and its inherent methods. In this context, a specific area within meta-philosophy is the method to use to attach oneself to resolve, or make progress towards the resolution of philosophical paradoxes or problems. It is within this specific area that falls the present construction, in that it offers dichotomous analysis as a tool that may be useful to assist in the resolution of paradoxes or philosophical problems.
The dichotomous analysis as a methodology that can be used to search for solutions to some paradoxes and philosophical problems, results directly from the statement of the principle of dialectical indifference itself. The general idea underlying the dichotomous approach to paradox analysis is that two versions, corresponding to one and the other pole of a given duality, can be untangled within a philosophical paradox. The corresponding approach then is to find a reference class which is associated with the given paradox and the corresponding duality A/Ā, as well as the two resulting variations of the paradox that apply to each pole of this duality. Nevertheless, every duality is not suitable for this, as for many dualities, the corresponding version of the paradox remains unchanged, regardless of the pole that is being considered. In the dichotomous method, one focuses on finding a reference class and a relevant associated duality, such that the viewpoint of each of its poles actually lead to two structurally different versions of the paradox , or the disappearance of paradox from the point of view of one of the poles. Thus, when considering the paradox in terms of two poles A and Ā, and if it has no effect on the paradox itself, the corresponding duality A/Ā reveals itself therefore, from this point of view, irrelevant.
The dichotomous analysis is not by far a tool that claims to solve all philosophical problems, but only constitutes a methodology that is susceptible of shedding light on some of them. In what follows, we shall try to illustrate through several works of the author, how dichotomous analysis can be applied to progress towards the resolution of three contemporary philosophical paradoxes: Hempel’s paradox, the surprise examination paradox and the Doomsday argument.
In a preliminary way, we can observe here that in the literature, there is also an example of dichotomous analysis of a paradox in David Chalmers (2002). Chalmers attempts then to show how the two-envelope paradox leads to two fundamentally distinct versions, one of which corresponds to a finite version of the paradox and the other to an infinite version. Such an analysis, although conceived of independently of the present construction can thus be characterized as a dichotomous analysis based on the finite/infinite duality.
The dual poles in David Chalmers’ analysis of the two-envelope paradox
6. Application to the analysis of the philosophical paradoxes
Karl Hempel
At this point, it is worth applying the foregoing to the analysis of concrete problems. We shall illustrate this through the analysis of several contemporary philosophical paradoxes: Hempel’s paradox, the surprise examination paradox and the Doomsday argument. We will endeavour to show how a problem of one-sidednessn bias associated with a problem of definition of a reference class can be found in the analysis of the aforementioned philosophical paradoxes. In addition, we will show how the very definition of the reference class associated with each paradox is susceptible of being qualified with the help of the dual poles A and Ā of a given duality A/Ā as they have just been defined.
6.1. Application to the analysis of Hempel‘s paradox
Hempel’s paradox is based on the fact that the two following assertions:
(H) All ravens are black
(H*) All non-black things are non-ravens
are logically equivalent. By its structure (H*) presents itself indeed as the contrapositive form of (H). It follows that the discovery of a black raven confirms (H) and also (H*), but also that the discovery of a non-black thing that is not a raven such as a red flame or even a grey umbrella, confirms (H*) and therefore (H). However, this latter conclusion appears paradoxical.
We shall endeavour now to detail the dichotomous analysis on which is based the solution proposed in Franceschi (1999). The corresponding approach is based on finding a reference class associated with the statement of the paradox, which may be defined with the help of an A/Ā duality. If we scrutinise the concepts and categories that underlie propositions (H) and (H*), we first note that there are four categories: ravens, black objects, non-black objects and non- ravens. To begin with, a raven is precisely defined within the taxonomy in which it inserts itself. A category such as that of the ravens can be considered well-defined, since it is based on a precise set of criteria defining the species corvus corax and allowing the identification of its instances. Similarly, the class of black objects can be accurately described, from a taxonomy of colours determined with respect to the wave lengths of light. Finally, we can see that the class of non-black objects can also be a definition that does not suffer from ambiguity, in particular from the specific taxonomy of colours which has been just mentioned.
However, what about the class of non-ravens? What does constitute then an instance of a non-raven? Intuitively, a blue blackbird, a red flamingo, a grey umbrella and even a natural number, are non-ravens. But should we consider a reference class that goes up to include abstract objects? Should we thus consider a notion of non-raven that includes abstract entities such as integers and complex numbers? Or should we limit ourselves to a reference class that only embraces the animals? Or should we consider a reference class that encompasses all living beings, or even all concrete things, also including this time the artefacts? Finally, it follows that the initial proposition (H*) is susceptible of giving rise to several variations, which are the following:
(H1*) All that is non-black among the corvids is a non-raven
(H2*) All that is non-black among the birds is a non-raven
(H3*) All that is non-black among the animals is a non-raven
(H4*) All that is non-black among the living beings is a non-raven
(H5*) All that is non-black among the concrete things is a non-raven
(H6*) All that is non-black among the concrete and abstract objects is a non-raven
Thus, it turns out that the statement of Hempel’s paradox and in particular of proposition (H*) is associated with a reference class, which allow to define the non-ravens. Such a reference class can be assimilated to corvids, birds, animals, living beings, concrete things, or to concrete and abstract things, etc.. However, in the statement of Hempel’s paradox, there is no objective criterion for making such a choice. At this point, it turns out that one can choose such a reference class restrictively, by assimilating it for example to corvids. But in an equally legitimate manner, we can choose a reference class more extensively, by identifying it for example to the set of concrete things, thus notably including umbrellas. Why then choose such or such reference class defined in a restrictive way rather than another one extensively defined? Indeed, we are lacking a criterion allowing to justify the choice of the reference class, whether we proceed by restriction or by extension. Therefore, it turns out that the latter can only be defined arbitrarily. But the choice of such a reference class proves crucial because depending on whether you choose such or such class reference, a given object such as a grey umbrella will confirm or not (H*) and therefore (H). Hence, if we choose the reference class by extension, thus including all concrete objects, a grey umbrella will confirm (H). On the other hand, if we choose such a reference class by restriction, by assimilating it only to corvids, a grey umbrella will not confirm (H). Such a difference proves to be essential. In effect, if we choose a definition by extension of the reference class, the paradoxical effect inherent to Hempel’s paradox ensues. By contrast, if we choose a reference class restrictively defined, the paradoxical effect disappears.
The dual poles in the reference class of the non-ravens within Hempel’s paradox
The foregoing permits to describe accurately the elements of the preceding analysis of Hempel’s paradox in terms of one-sidedness bias such as it has been defined above: to the paradox and in particular to proposition (H*) are associated the reference class of non-ravens, which itself is susceptible of being defined with regard to the extension/restriction duality. However, for a given object such as a grey umbrella, the definition of the reference class by extension leads to a paradoxical effect, whereas the choice of the latter by restriction does not lead to such an effect.
6.2. Application to the analysis of the surprise examination paradox
The classical version of the surprise examination paradox (Quine 1953, Sorensen 1988) goes as follows: a teacher tells his students that an examination will take place on the next week, but they will not know in advance the precise date on which the examination will occur. The examination will thus occur surprisingly. The students reason then as follows. The examination cannot take place on Saturday, they think, otherwise they would know in advance that the examination would take place on Saturday and therefore it could not occur surprisingly. Thus, Saturday is eliminated. In addition, the examination can not take place on Friday, otherwise the students would know in advance that the examination would take place on Friday and so it could not occur surprisingly. Thus, Friday is also ruled out. By a similar reasoning, the students eliminate successively Thursday, Wednesday, Tuesday and Monday. Finally, every day of the week is eliminated. However, this does not preclude the examination of finally occurring by surprise, say on Wednesday. Thus, the reasoning of the students proved to be fallacious. However, such reasoning seems intuitively valid. The paradox lies here in the fact the students’ reasoning is apparently valid, whereas it finally proves inconsistent with the facts, i.e. that the examination can truly occur by surprise, as initially announced by the professor.
In order to introduce the dichotomous analysis (Franceschi 2005) that can be applied to the surprise examination paradox, it is worth considering first two variations of the paradox that turn out to be structurally different. The first variation is associated with the solution to the paradox proposed by Quine (1953). Quine considers then the student’s final conclusion that the examination can not take place surprisingly on any day of the week. According to Quine, the student’s error lies in the fact of not having envisaged from the beginning that the examination could take place on the last day. Because the fact of considering precisely that the examination will not take place on the last day finally allows the examination to occur by surprise on the last day. If the student had also considered this possibility from the beginning, he would not have been committed to the false conclusion that the examination can not occur surprisingly.
The second variation of the paradox that proves interesting in this context is the one associated with the remark made by several authors (Hall 1999, p. 661, Williamson 2000), according to which the paradox emerges clearly when the number n of units is large. Such a number is usually associated with a number n of days, but we may as well use hours, minutes, seconds, etc.. An interesting feature of the paradox is indeed that it emerges intuitively more significantly when large values of n are involved. A striking illustration of this phenomenon is thus provided by the variation of the paradox that corresponds to the following situation, described by Timothy Williamson (2000, p 139).
Advance knowledge that there will be a test, fire drill, or the like of which one will not know the time in advance is an everyday fact of social life, but one denied by a surprising proportion of early work on the Surprise Examination. Who has not waited for the telephone to ring, knowing that it will do so within a week and that one will not know a second before it rings that it will ring a second later?
The variation described by Williamson corresponds to the announcement made to someone that he/she will receive a phone call during the week, but without being able to determine in advance at what exact second the latter event will occur. This variation highlights how surprise may occur, in a quite plausible way, when the value of n is high. The unit of time considered here by Williamson is the second, in relation with a time duration that corresponds to one week. The corresponding value of n here is very high and equal to 604800 (60 x 60 x 24 x 7) seconds. However, it is not necessary to take into account a value as large of n, and a value of n equal to 365, for example, should also be well-suited.
The fact that two versions of the paradox that seem a priori quite different coexist suggests that two structurally different versions of the paradox could be inextricably intertwined within the surprise examination paradox. In fact, if we analyse the version of the paradox that leads to Quine’s solution, we find that it has a peculiarity: it is likely to occur for a value of n equal to 1. The corresponding version of the professor’s announcement is then as follows: “An examination will take place tomorrow, but you will not know in advance that this will happen and therefore it will occur surprisingly.” Quine’s analysis applies directly to this version of the paradox for which n = 1. In this case, the student’s error resides, according to Quine, in the fact of having only considered the hypothesis: (i) “the examination will take place tomorrow and I predict that it will take place.” In fact, the student should also have considered three cases: (ii) “the examination will not take place tomorrow, and I predict that it will take place” (iii) “the examination will not take place tomorrow and I do not predict that it will take place” (iv) “the examination will take place tomorrow and I do not predict that it will take place.” And the fact of having envisaged hypothesis (i), but also hypothesis (iv) which is compatible with the professor’s announcement would have prevented the student to conclude that the examination would not finally take place. Therefore, as Quine stresses, it is the fact of having only taken into account the hypothesis (i) that can be identified as the cause of the fallacious reasoning.
As we can see it, the very structure of the version of the paradox on which Quine’s solution is based has the following features: first, the non-surprise may actually occur on the last day, and second, the examination may also occur surprisingly on the last day. The same goes for the version of the paradox where n = 1: the non-surprise and the surprise may occur on day n. This allows to represent such structure of the paradox with the following matrix S[k, s] (where k denotes the day on which the examination takes place and S[k, s] denotes whether the corresponding case of non-surprise (s = 0) or surprise (s = 1) is possible (in this case, S[k, i] = 1) or not (in this case, S[k, i] = 0)):
day
non-surprise
surprise
1
1
1
2
1
1
3
1
1
4
1
1
5
1
1
6
1
1
7
1
1
Matrix structure of the version of the paradox corresponding to Quine’s solution for n = 7 (one week)
day
non-surprise
surprise
1
1
1
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.
day
non-surprise
surprise
1
0
1
…
…
…
365
1
0
Matrix structure of the version of the paradox corresponding to Williamson’s variation for n = 365 (one year)
The foregoing allows now to identify precisely what is at fault in the student’s reasoning, when applied to this particular version of the paradox. Under these circumstances, the student would then have reasoned as follows. The surprise cannot occur on the last day but it can occur on day 1, and the non-surprise can occur on the last day, but cannot occur on the first day. These are proper instances of non-surprise and of surprise, which prove to be disjoint. However, the notion of surprise is not captured exhaustively by the extension and the anti-extension of the surprise. But such a definition is consistent with the definition of a vague predicate, which is characterized by an extension and an anti-extension which are mutually exclusive and non-exhaustive. Thus, the notion of surprise associated with a disjoint structure is that of a vague notion. Thus, the student’s error of reasoning at the origin of the fallacy lies in not having taken into account the fact that the surprise is in the case of a disjoint structure, a vague concept and includes therefore the presence of a penumbra corresponding to borderline cases between non-surprise and surprise. Hence, the mere consideration of the fact that the surprise notion is here a vague notion would have prohibited the student to conclude that S[k, 1] = 0, for all values of k, that is to say that the examination can not occur surprisingly on any day of the period.
Finally, it turns out that the analysis leads to distinguish between two independent variations with regard to the surprise examination paradox. The matrix definition of the cases of non-surprise and of surprise leads to two variations of the paradox, according to the joint/disjoint duality. In the first case, the paradox is based on a joint definition of the cases of non-surprise and of surprise. In the second case, the paradox is grounded on a disjoint definition. Both of these variations lead to a structurally different variation of the paradox and to an independent solution. When the variation of the paradox is based on a joint definition, the solution put forth by Quine applies. However, when the variation of the paradox is based on a disjoint definition, the solution is based on the prior recognition of the vague nature of the concept of surprise associated with this variation of the paradox.
The dual poles in the class of the matrices associated with the surprise examination paradox
As we finally see it, the dichotomous analysis of the surprise examination paradox leads to consider the class of the matrices associated with the very definition of the paradox and to distinguish whether their structure is joint or disjoint. Therefore, it follows an independent solution for each of the resulting two structurally different versions of the paradox.
6.3. Application to the analysis of the Doomsday Argument
The Doomsday argument, attributed to Brandon Carter, was described by John Leslie (1993, 1996). It is worth recalling preliminarily its statement. Consider then proposition (A):
(A) The human species will disappear before the end of the XXIst century
We can estimate, to fix ideas, to 1 on 100 the probability that this extinction will occur: P(A) = 0.01. Let us consider also the following proposition:
(Ā) The human species will not disappear at the end of the XXIst century
Let also E be the event: I live during the 2010s. We can also estimate today to 60 billion the number of humans that ever have existed since the birth of humanity. Similarly, the current population can be estimated at 6 billion. One calculates then that one human out of ten, if event A occurs, will have known of the 2010s. We can then estimate accordingly the probability that humanity will be extinct before the end of the twenty-first century, if I have known of the 2010s: P(E, A) = 6×109/6×1010 = 0.1. By contrast, if humanity passes the course of the twenty-first century, it is likely that it will be subject to a much greater expansion, and that the number of human will be able to amount, for example to 6×1012. In this case, the probability that humanity will not be not extinct at the end of the twenty-first century, if I have known of the 2010s, can be evaluated as follows: P(E, Ā) = 6×109/6×1012 = 0,001. At this point, we can assimilate to two distinct urns – one containing 60 billion balls and the other containing 6,000,000,000,000 – the total human populations that will result. This leads to calculate the posterior probability of the human species’ extinction before the end of the XXIst century, with the help of Bayes’ formula: P'(A) = [P(A) x P(E, A)] / [P(A) x P(E, A) + P(Ā) x P(E, Ā )] = (0.01 x 0.1) / (0.01 x 0.1 + 0.99 x 0.001) = 0.5025. Thus, taking into account the fact that I am currently living makes pass the probability of the human species’ extinction before 2150 from 1% to 50.25 %. Such a conclusion appears counter-intuitive and is in this sense, paradoxical.
It is worth now describing how a dichotomous analysis (Franceschi, 1999, 2009) can be applied to the Doomsday Argument. We will endeavour, first, to point out how the Doomsday Argument has an inherent reference class5 problem definition linked to a duality A/Ā. Consider then the following statement:
(A) The human race will disappear before the end of the XXIst century
Such a proposition presents a dramatic, apocalyptic and tragic connotation, linked to the imminent extinction of the human species. It consists here of a prediction the nature of which is catastrophic and quite alarming. However, if we scrutinise such a proposition, we are led to notice that it conceals an inaccuracy. If the time reference itself – the end of the twenty-first century – proves to be quite accurate, the term “human species” itself appears to be ambiguous. Indeed, it turns out that there are several ways to define it. The most accurate notion in order to define the“’human race” is our present scientific taxonomy, based on the concepts of genus, species, subspecies, etc.. Adapting the latter taxonomy to the assertion (A), it follows that the ambiguous concept of “human species” is likely to be defined in relation to the genus, the species, the subspecies, etc.. and in particular with regard to the homo genus, the homo sapiens species, the homo sapiens sapiens subspecies, etc.. Finally, it follows that assertion (A) is likely to take the following forms:
(Ah) The homo genus will disappear before the end of the XXIst century
(Ahs) The homo sapiens species will disappear before the end of the XXIst century
(Ahss) The homo sapiens sapiens subspecies will disappear before the end of the XXIst century
At this stage, reading these different propositions leads to a different impact, given the original proposition (A). For if (Ah) presents well in the same way as (A) a quite dramatic and tragic connotation, it is not the case for (Ahss). Indeed, such a proposition that predicts the extinction of our current subspecies homo sapiens sapiens before the end of the twenty-first century, could be accompanied by the replacement of our present human race with a new and more advanced subspecies than we could call homo sapiens supersapiens. In this case, the proposition (Ahss) would not contain any tragic connotation, but would be associated with a positive connotation, since the replacement of an ancient race with a more evolved species results from the natural process of evolution. Furthermore, by choosing a reference class even more limited as that of the humans having not known of the computer (homo sapiens sapiens antecomputeris), we get the following proposition:
(Ahsss) The infra-subspecies homo sapiens sapiensantecomputeris will disappear before the end of the XXIst century
which is no longer associated at all with the dramatic connotation inherent to (A) and proves even quite normal and reassuring, being devoid of any paradoxical or counterintuitive nature. In this case, in effect, the disappearance of the infra-subspecies homo sapiens sapiens antecomputeris is associated with the survival of the much-evolved infra-subspecies homo sapiens sapiens postcomputeris. It turns out then that a restricted class of reference coinciding with an infra-subspecies goes extinct, but a larger class corresponding to a subspecies (homo sapiens sapiens) survives. In this case, we observe well the Bayesian shift described by Leslie, but the effect of this shift proves this time to be quite innocuous.
Thus, the choice of the reference class for proposition (A) proves to be essential for the paradoxical nature of the conclusion associated with the Doomsday Argument. If one chooses then an extended reference class for the very definition of humans, associated with e.g. the homo genus, one gets the dramatic and disturbing nature associated with proposition (A). By contrast, if one chooses such a reference class restrictively, by associating it for example with the infra-subspecies homo sapiens sapiens antecomputeris, a reassuring and normal nature is now associated with the proposition (A) underlying the Doomsday Argument.
Finally, we are in a position to replace the foregoing analysis in the present context. The very definition of the reference class of the “humans” associated with the proposition (A) inherent to the Doomsday Argument is susceptible of being made according to the two poles of the extension/restriction duality. An analysis based on a two-sided perspective leads to the conclusion that the choice by extension leads to a paradoxical effect, whereas the choice by restriction of the reference class makes this paradoxical effect disappear.
The dual poles within the reference class of “humans” in the Doomsday Argument
The dichotomous analysis, however, as regards the Doomsday argument, is not limited to this. Indeed, if one examines the argument carefully, it turns out that it contains another reference class which is associated with another duality. This can be demonstrated by analysing the argument raised by William Eckhardt (1993, 1997) against the Doomsday argument. According to Eckhardt, the human situation corresponding to DA is not analogous to the two-urn case described by Leslie, but rather to an alternative model, which can be termed the consecutive token dispenser. The consecutive token dispenser is a device that ejects consecutively numbered balls at regular intervals: “(…) suppose on each trial the consecutive token dispenser expels either 50 (early doom) or 100 (late doom) consecutively numbered tokens at the rate of one per minute.” Based on this model, Eckhardt (1997, p. 256) emphasizes that it is impossible to make a random selection, where there are many individuals who are not yet born within the corresponding reference class: “How is it possible in the selection of a random rank to give the appropriate weight to unborn members of the population?”. The strong idea of Eckhardt underlying this diachronic objection is that it is impossible to make a random selection when there are many members in the reference class who are not yet born. In such a situation, it would be quite wrong to conclude that a Bayesian shift in favour of the hypothesis (A) ensues. However, what can be inferred rationally in such a case is that the initial probability remains unchanged.
At this point, it turns out that two alternative models for modelling the analogy with the human situation corresponding to the Doomsday argument are competing: first, the synchronic model (where all the balls are present in the urn when the draw takes place) recommended by Leslie and second, Eckhardt’s diachronic model, where the balls can be added in the urn after the draw. The question that arises is the following: is the human situation corresponding to the Doomsday argument in analogy with (i) the synchronic urn model, or with (ii) the diachronic urn model? In order to answer, the following question arises: does there exist an objective criterion for choosing, preferably, between the two competing models? It appears not. Neither Leslie nor Eckhardt has an objective motivation allowing to justify the choice of their own favourite model, and to reject the alternative model. Under these circumstances, the choice of one or the other of the two models – whether synchronic or diachronic – proves to be arbitrary. Therefore, it turns out that the choice within the class of the models associated with the Doomsday argument is susceptible of being made according to the two poles of the synchronic/diachronic duality. Hence, an analysis based on a two-sided viewpoint leads to the conclusion that the choice of the synchronic model leads to a paradoxical effect, whereas the choice of the diachronic model makes this latter paradoxical effect disappear.
The dual poles within the models’ class of the Doomsday Argument
Finally, given the fact that the above problem related to the reference class of the humans and its associated choice within the extension/restriction duality only concerns the synchronic model, the structure of the dichotomous analysis at two levels concerning the Doomsday Argument can be represented as follows:
Structure of embedded dual poles Diachronic/Synchronic and Extension/Restriction for the Doomsday Argument
As we can see it, the foregoing developments implement the form of dialectical contextualism that has been described above by applying it to the analysis of three contemporary philosophical paradoxes. In Hempel’s paradox, the reference class of the non-ravens is associated with proposition (H*), which itself is susceptible of being defined with regard to the extension/restriction duality. However, for a given object x such as a grey umbrella, the definition of the reference class by extension leads to a paradoxical effect , whereas the choice of the latter reference class by restriction eliminates this specific effect. Secondly, the matrix structures associated with the surprise examination paradox are analysed from the angle of the joint/disjoint duality, thus highlighting two structurally distinct versions of the paradox , which themselves admit of two independent resolutions. Finally, at the level of the Doomsday argument, a double dichotomic analysis shows that the class of humans is related to the extension/restriction duality, and that the paradoxical effect that is evident when the reference class is defined by extension, dissolves when the latter is defined by restriction. It turns out, second, that the class of models can be defined according to the synchronic/diachronic duality; a paradoxical effect is associated with the synchronic view, whereas the same effect disappears if we place ourselves from the diachronic perspective.
Acknowledgements
This text is written starting from some entirely revised elements of my habilitation to direct research work report, presented in 2006. The changes introduced in the text, comprising in particular the correction of a conceptual error, follow notably from the comments and recommendations that Pascal Engel had made to me at that time.
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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.
English translation of a paper appeared in French in Philosophiques 2005, vol. 32, pages 399-421 (with minor changes with regard to the published version).
This paper proposes a new framework to solve the surprise examination paradox. I survey preliminary the main contributions to the literature related to the paradox. I introduce then a distinction between a monist and a dichotomic analysis of the paradox. With the help of a matrix notation, I also present a dichotomy that leads to distinguish two basically and structurally different notions of surprise, which are respectively based on a conjoint and a disjoint structure. I describe then how Quine’s solution and Hall’s reduction apply to the version of the paradox corresponding to the conjoint structure. Lastly, I expose a solution to the version of the paradox based on the disjoint structure.
A Dichotomic Analysis of the Surprise Examination Paradox
I shall present in what follows a new conceptual framework to solve the surprise examination paradox (henceforth, SEP), in the sense that it reorganizes, by adapting them, several elements of solution described in the literature. The solution suggested here rests primarily on the following elements: (i) a distinction between a monist and a dichotomic analysis of the paradox; (ii) the introduction of a matrix definition, which is used as support with several variations of the paradox; (iii) the distinction between a conjoint and a disjoint definition of the cases of surprise and of non-surprise, leading to two structurally different notions of surprise.
In section 1, I proceed to describe the paradox and the main solutions found in the literature. I describe then in section 2, in a simplified way, the solution to the paradox which results from the present approach. I also introduce the distinction between a monist and a dichotomic analysis of the paradox. I present then a dichotomy which makes it possible to distinguish between two basically and structurally different versions of the paradox: on the one hand, a version based on a conjoint structure of the cases of non-surprise and of surprise; in the other hand, a version based on a disjoint structure. In section 3, I describe how Quine’s solution and Hall’s reduction apply to the version of SEP corresponding to the conjoint structure of the cases of non-surprise and of surprise. In section 4, I expose the solution to SEP corresponding to the disjoint structure. Lastly, I describe in section 5, within the framework of the present solution, what should have been the student’s reasoning.
1. The paradox
The surprise examination paradox finds its origin in an actual fact. In 1943-1944, the Swedish authorities planned to carry out a civil defence exercise. They diffused then by the radio an announcement according to which a civil defence exercise would take place during the following week. However, in order to perform the latter exercise under optimal conditions, the announcement also specified that nobody could know in advance the date of the exercise. Mathematician Lennart Ekbom understood the subtle problem arising from this announcement of a civil defence exercise and exposed it to his students. A broad diffusion of the paradox throughout the world then ensued.
SEP first appeared in the literature with an article of D. O’ Connor (1948). O’ Connor presents the paradox under the form of the announcement of a military training exercise. Later on, SEP appeared in the literature under other forms, such as the announcement of the appearance of an ace in a set of cards (Scriven 1951) or else of a hanging (Quine 1953). However, the version of the paradox related to the professor’s announcement of a surprise examination has remained the most current form. The traditional version of the paradox is as follows: a professor announces to his/her students that an examination will take place during the next week, but that they will not be able to know in advance the precise day where the examination will occur. The examination will thus occur surprisingly. The students reason as follows. The examination cannot take place on Saturday, they think, for otherwise they would know in advance that the examination would take place on Saturday and thus it could not occur surprisingly. Thus, Saturday is ruled out. Moreover, the examination cannot take place on Friday, for otherwise the students would know in advance that the examination would take place on Friday and thus it could not occur surprisingly. Thus, Friday is also ruled out. By a similar reasoning, the students eliminate successively Thursday, Wednesday, Tuesday and Monday. Finally, all days of the week are then ruled out. However, this does not prevent the examination from finally occurring surprisingly, say, on Wednesday. Thus, the students’ reasoning proved to be fallacious. However, such a reasoning appears intuitively valid. The paradox lies here in the fact that the students’ reasoning seems valid, whereas it finally proves to be in contradiction with the facts, namely that the examination can truly occur surprisingly, in accordance with the announcement made by the professor.
In the literature, several solutions to SEP have been proposed. There does not exist however, at present time, a consensual solution. I will briefly mention the principal solutions which were proposed, as well as the fundamental objections that they raised.
A first attempt at solution appeared with O’ Connor (1948). This author pointed out that the paradox was due to the contradiction which resulted from the professor’s announcement and the implementation of the latter. According to O’ Connor, the professor’s announcement according to which the examination was to occur by surprise was in contradiction with the fact that the details of the implementation of the examination were known. Thus, the statement of SEP was, according to O’ Connor, self-refuting. However, such an analysis proved to be inadequate, because it finally appeared that the examination could truly take place under some conditions where it occurred surprisingly, for example on Wednesday. Thus, the examination could finally occur by surprise, confirming thus and not refuting, the professor’s announcement. This last observation had the effect of making the paradox re-appear.
Quine (1953) also proposed a solution to SEP. Quine considers thus the student’s final conclusion according to which the examination can occur surprisingly on no day of the week. According to Quine, the student’s error lies in the fact of having not considered from the beginning the hypothesis that the examination could not take place on the last day. For the fact of considering precisely that the examination will not take place on the last day makes it finally possible for the examination to occur surprisingly, on the last day. If the student had also taken into account this possibility from the beginning, he would not concluded fallaciously that the examination cannot occur by surprise. However, Quine’s solution has led to criticisms, emanating notably from commentators (Ayer 1973, Janaway 1989 and also Hall 1999) who stressed the fact that Quine’s solution did not make it possible to handle several variations of the paradox. Ayer imagines thus a version of SEP where a given person is informed that the cards of a set will be turned over one by one, but where that person will not know in advance when the ace of Spades will be issued. Nevertheless, the person is authorized to check the presence of the ace of Spades before the set of cards is mixed. The purpose of the objection to Quine’s solution based on such a variation is to highlight a situation where the paradox is quite present but where Quine’s solution does not find to apply any more, because the student knows with certainty, given the initial data of the problem, that the examination will take place as well.
According to another approach, defended in particular by R. Shaw (1958), the structure of the paradox is inherently self-referential. According to Shaw, the fact that the examination must occur by surprise is tantamount to the fact that the date of the examination cannot be deduced in advance. But the fact that the students cannot know in advance, by deduction, the date of the examination constitutes precisely one of the premises. The paradox thus finds its origin, according to Shaw, in the fact that the structure of the professor’s announcement is self-referential. According to the author, the self-reference which results from it constitutes thus the cause of the paradox. However, such an analysis did not prove to be convincing, for it did not make it possible to do justice to the fact that in spite of its self-referential structure, the professor’s announcement was finally confirmed by the fact that the examination could finally occur surprisingly, say on Wednesday.
Another approach, put forth by Richard Montague and David Kaplan (1960) is based on the analysis of the structure of SEP which proves, according to these authors, to be that of the paradox of the Knower. The latter paradox constitutes a variation of the Liar paradox. What thus ultimately proposes Montague and Kaplan, is a reduction of SEP to the Liar paradox. However, this last approach did not prove to be convincing. Indeed, it was criticized because it did not take account, on the one hand, the fact that the professor’s announcement can be finally confirmed and on the other hand, the fact that one can formulate the paradox in a non-self-referential way.
It is also worth mentioning the analysis developed by Robert Binkley (1968). In his article, Binkley exposes a reduction of SEP to Moore’s paradox. The author makes the point that on the last day, SEP reduces to a variation of the proposition ‘P and I don’t know that P’ which constitutes Moore’s paradox. Binkley extends then his analysis concerning the last day to the other days of the week. However, this approach has led to strong objections, resulting in particular from the analysis of Wright and Sudbury (1977).
Another approach also deserves to be mentioned. It is the one developed by Paul Dietl (1973) and Joseph Smith (1984). According to these authors, the structure of SEP is that of the sorites paradox. What then propose Dietl and Smith, is a reduction of SEP to the sorites paradox. However, such an analysis met serious objections, raised in particular by Roy Sorensen (1988).
It is worth lastly mentioning the approach presented by Crispin Wright and Aidan Sudbury (1977). The analysis developed by these authors1 results in distinguishing two cases: on the one hand, on the last day, the student is in a situation which is that which results from Moore’s paradox; in addition, on the first day, the student is in a basically different situation where he can validly believe in the professor’s announcement. Thus, the description of these two types of situations leads to the rejection of the principle of temporal retention. According to this last principle, what is known at a temporal position T0 is also known at a later temporal position T1 (with T0 < T1). However, the analysis of Wright and Sudbury appeared vulnerable to an argument developed by Sorensen (1982). The latter author presented indeed a version of SEP (the Designated Student Paradox) which did not rely on the principle of temporal retention, on which the approach of Wright and Sudbury rested. According to Sorensen’s variation, the paradox was quite present, but without the conditions of its statement requiring to rely on the principle of temporal retention. Sorensen describes thus the following variation of the paradox. Five students, A, B, C, D and E are placed, in this order, one behind the other. The professor then shows to the students four silver stars and one gold star. Then he places a star on the back of each student. Lastly, he announces to them that the one of them who has a gold star in the back has been designated to pass an examination. But, the professor adds, this examination will constitute a surprise, because the students will only know that who was designated when they break their alignment. Under these conditions, it appears that the students can implement a similar reasoning to that which prevails in the original version of SEP. But this last version is diachronic, whereas the variation described by Sorensen appears, by contrast, synchronic. And as such, it is thus not based on whatever principle of temporal retention.
Given the above elements, it appears that the stake and the philosophical implications of SEP are of importance. They are located at several levels and thus relate2 to the theory of knowledge, deduction, justification, the semantic paradoxes, self-reference, modal logic, and vague concepts.
2. Monist or dichotomic analysis of the paradox
Most analyses classically proposed to solve SEP are based on an overall solution which applies, in a general way, to the situation which is that of SEP. In this type of analysis, a single solution is presented, which is supposed to apply to all variations of SEP. Such type of solution has a unitary nature and appears based on what can be termed a monist theory of SEP. Most solutions to SEP proposed in the literature are monist analyses. Characteristic examples of this type of analysis of SEP are the solutions suggested by Quine (1953) or Binkley (1968). In a similar way, the solution envisaged by Dietl (1973) which is based on a reduction of SEP to the sorite paradox also constitutes a monist solution to SEP.
Conversely, a dichotomic analysis of SEP is based on a distinction between two different scenarios of SEP and on the formulation of an independent solution for each of the two scenarios. In the literature, the only analysis which has a dichotomic nature, as far as I know, is that of Wright and Sudbury mentioned above. In what follows, I will present a dichotomic solution to SEP. This solution is based on the distinction of two variations of SEP, associated with concepts of surprise that correspond to different structures of the cases of non-surprise and of surprise.
At this step, it proves to be useful to introduce the matrix notation. With the help of this latter, the various cases of non-surprise and of surprise be modelled with the following S[k, s] table, where k denotes the day where the examination takes place and S[k, s] denotes if the corresponding case of non-surprise (s = 0) or of surprise (s = 1) is made possible (S[k, s] = 1) or not (S[k, s] = 0) by the conditions of the announcement (with 1 k n).3 If one considers for example 7-SEP 4, S[7, 1] = 0 denotes the fact that the surprise is not possible on the 7th day, and conversely, S[7, 1] = 1 denotes the fact that the surprise is possible on the 7th day; in the same way, S[1, 0] = 0 denotes the fact that the non-surprise is not possible on the 1st day by the conditions of the announcement, and conversely, S[1, 0] = 1 denotes the fact that the non-surprise is possible on the 1st day.
The dichotomy on which rests the present solution results directly from the analysis of the structure which makes it possible to describe the concept of surprise corresponding to the statement of SEP. Let us consider first the following matrix, which corresponds to a maximal definition, where all cases of non-surprise and of surprise are made possible by the professor’s announcement (with ■ = 1 and □ = 0):
(D1)
S[k, 0]
S[k, 1]
S[7,s]
■
■
S[6,s]
■
■
S[5,s]
■
■
S[4,s]
■
■
S[3,s]
■
■
S[2,s]
■
■
S[1,s]
■
■
At the level of (D1), as we can see it, all values of the S[k, s] matrix are equal to 1, which corresponds to the fact that all the cases of non-surprise and of surprise are made possible by the corresponding version of SEP. The associated matrix can be thus defined as a rectangular matrix.
At this stage, it appears that one can conceive of some variations of SEP associated with more restrictive matrix structures, where certain cases of non-surprise and of surprise are not authorized by the announcement. In such cases, certain values of the matrix are equal to 0. It is now worth considering the structure of these more restrictive definitions. The latter are such that it exists at least one case of non-surprise or of surprise which is made impossible by the announcement, and where the corresponding value of the matrix S[k, s] is thus equal to 0. Such a condition leaves place [***room] with a certain number of variations, of which it is now worth studying the characteristics more thoroughly.
One can notice preliminarily that certain types of structures can be discarded from the beginning. It appears indeed that any definition associated with a restriction of (D1) is not adequate. Thus, there are minimal conditions for the emergence of SEP. In this sense, a first condition is that the base step be present. This base step is such that the non-surprise must be able to occur on the last day, that is to say S[n, 0] = 1. With the previously defined notation, it presents the general form n*n* and corresponds to 7*7* for 7-SEP. In the lack of this base step, there is no paradoxical effect of SEP. Consequently, a structure of matrix such as S[n, 0] = 0 can be discarded from the beginning.
One second condition so that the statement leads to a genuine version of SEP is that the examination can finally occur surprisingly. This renders indeed possible the fact that the professor’s announcement can be finally satisfied. Such a condition – let us call it the vindication step – is classically mentioned as a condition for the emergence of the paradox. Thus, a definition which would be such that all the cases of surprise are made impossible by the corresponding statement would also not be appropriate. Thus, the structure corresponding to the following matrix would not correspond either to a licit statement of SEP:
(D2)
S[k, 0]
S[k, 1]
S[7,s]
■
□
S[6,s]
■
□
S[5,s]
■
□
S[4,s]
■
□
S[3,s]
■
□
S[2,s]
■
□
S[1,s]
■
□
because the surprise is possible here on no day of the week (S[k, 1 ] = 0) and the validation step is thus lacking in the corresponding statement.
Taking into account what precedes, one is now in a position to describe accurately the minimal conditions which are those of SEP:
(C3) S[n, 0] = 1 (base step)
(C4) k (1 k n) such that S[k, 1] = 1 (validation step)
At this step, it is worth considering the structure of the versions of SEP based on the definitions which satisfy the minimal conditions for the emergence of the paradox which have just been described, i.e. which contain at the same time the basic step and the validation step. It appears here that the structure associated with the cases of non-surprise and of surprise corresponding to a variation with SEP can present two forms of a basically different nature. A first form of SEP is associated with a structure where the possible cases of non-surprise and of surprise are such that it exists during the n-period at least one day where the non-surprise and the surprise are simultaneously possible. Such a definition can be called conjoint. The following matrix constitutes an example of this type of structure:
(D5)
S[k, 0]
S[k, 1]
S[7,s]
■
■
S[6,s]
■
■
S[5,s]
■
■
S[4,s]
■
■
S[3,s]
□
■
S[2,s]
□
■
S[1,s]
□
■
because the non-surprise and the surprise are simultaneously possible here on the 7th, 6th, 5th and 4th days. However, it proves that one can also encounter a second form of SEP the structure of which is basically different, in the sense that for each day of the n-period, it is impossible to have simultaneously the surprise and the non-surprise.5 A definition of this nature can be called disjoint. The following matrix thus constitutes an example of this type of structure:
(D6)
S[k, 0]
S[k, 1]
S[7,s]
■
□
S[6,s]
■
□
S[5,s]
■
□
S[4,s]
□
■
S[3,s]
□
■
S[2,s]
□
■
S[1,s]
□
■
Consequently, it is worth distinguishing in what follows two structurally distinct versions of SEP: (a) a version based on a conjoint structure of the cases of non-surprise and of surprise made possible by the announcement; (b) a version based on a disjoint structure of these same cases. The need for making such a dichotomy finds its legitimacy in the fact that in the original version of SEP, the professor does not specify if one must take into account a concept of surprise corresponding to a disjoint or a conjoint structure of the cases of non-surprise and of surprise. With regard to this particular point, the professor’s announcement of SEP appears ambiguous. Consequently, it is necessary to consider successively two different concepts of surprise, respectively based on a disjoint or conjoint structure of the cases of non-surprise and of surprise, as well as the reasoning which must be associated with them.
3. The surprise notion corresponding to the conjoint structure
Let us consider first the case where SEP is based on a concept of surprise corresponding to a conjoint structure of the cases of non-surprise and of surprise. Let SEP(I) be the version associated with such a concept of surprise. Intuitively, this version corresponds to a situation where there exists in the n-period at least one day where the non-surprise and the surprise can occur at the same time. Several types of definitions are likely to satisfy this criterion. It is worth considering them in turn.
4.1 The definition associated with the rectangular matrix and Quine’s solution
To begin with, it is worth considering the structures which are such that all cases of non-surprise and of surprise are made possible by the statement. The corresponding matrix is a rectangular matrix. Let thus SEP(I□) be such a version. The definition associated with such a structure is maximal since all cases of non-surprise and of surprise are authorized. The following matrix corresponds thus to such a general structure:
(D7)
S[k, 0]
S[k, 1]
S[7,s]
■
■
S[6,s]
■
■
S[5,s]
■
■
S[4,s]
■
■
S[3,s]
■
■
S[2,s]
■
■
S[1,s]
■
■
and the associated professor’s announcement is the following:
(S7)
An examination will occur in the next week but the date of the examination will constitute a surprise.
At this step, it appears that we also get a version of SEP for n = 1 which satisfies this definition. The structure associated with 1-SEP(I□) is as follows:
(D8)
S[1, 0]
S[1, 1]
S[1,s]
■
■
which corresponds to the following professor’s announcement:
(S8)
An examination will occur on tomorrow but the date of the examination will constitute a surprise.
Thus, 1-SEP(I□) is the minimal version of SEP which satisfies not only the above condition, but also the base step (C3) according to which the non-surprise must possibly occur on the last day, as well as the validation step (C4) in virtue of which the examination can finally occur by surprise. Moreover, it is a variation which excludes, by its intrinsic structure, the emergence of the version of SEP based on a concept of surprise corresponding to a disjoint structure. For this reason, (D8) can be regarded as the canonical form of SEP(I□). Thus, it is the genuine core of SEP(I□) and in what follows, we will thus endeavour to reason on 1-SEP(I□).
At this stage, it is worth attempting to provide a solution to SEP(I□). For that purpose, let us recall first Quine’s solution. The solution to SEP proposed by Quine (1953) is well-known. Quine highlights the fact that the student eliminates successively the days n, n -1…, 1, by a reasoning based on backward-induction and concludes then that the examination will not take place during the week. The student reasons as follows. On day n, I will predict that the examination will take place on day n, and consequently the examination cannot take place on day n; on day n -1, I will predict that the examination will take place on day n-1, and consequently the examination cannot take place on day n -1; …; on day 1, I will predict that the examination will take place on day 1, and consequently the examination cannot take place on day 1. Finally, the student concludes that the examination will take place on no day of the week. But this last conclusion finally makes it possible to the examination to occur surprisingly, including on day n. According to Quine, the error in the student’s reasoning lies precisely in the fact of not having taken into account this possibility since the beginning, which would then have prevented the fallacious reasoning.6
Quine, in addition, directly applies his analysis to the canonical form 1-SEP(I□), where the corresponding statement is that of (S8). In this case, the error of the student lies, according to Quine, in the fact of having considered only the single following assumption: (a) “the examination will take place tomorrow and I will predict that it will take place”. In fact, the student should have also considered three other cases: (b) “the examination will not take place tomorrow and I will predict that it will take place”; (c) “the examination will not take place tomorrow and I will not predict that it will take place”; (d) “the examination will take place tomorrow and I will not predict that it will take place”. And the fact of considering the assumption (a) but also the assumption (d) which is compatible with the professor’s announcement would have prevented the student from concluding that the examination would not finally take place.7 Consequently, it is the fact of having taken into account only the hypothesis (a) which can be identified as the cause of the fallacious reasoning. Thus, the student did only take partially into account the whole set of hypotheses resulting from the professor’s announcement. If he had apprehended the totality of the relevant hypotheses compatible with the professor’s announcement, he would not have concluded fallaciously that the examination would not take place during the week.
At this stage, it proves to be useful to describe the student’s reasoning in terms of reconstitution of a matrix. For one can consider that the student’s reasoning classically based on backward-induction leads to reconstitute the matrix corresponding to the concept of surprise in the following way:
(D9)
S[1, 0]
S[1, 1]
S[1,s]
■
□
In reality, he should have considered that the correct way to reconstitute this latter matrix is the following :
(D8)
S[1, 0]
S[1, 1]
S[1,s]
■
■
4.2 The definition associated with the triangular matrix and Hall’s reduction
As we have seen, Quine’s solution applies directly to SEP(I□), i.e. to a version of SEP based on a conjoint definition of the surprise and a rectangular matrix. It is now worth being interested in some variations of SEP based on a conjoint definition where the structure of the corresponding matrix is not rectangular, but which satisfies however the conditions for the emergence of the paradox mentioned above, namely the presence of the base step (C3) and the validation step (C4). Such matrices have a structure that can be described as triangular. Let thus SEP(I∆) be the corresponding version.
Let us consider first 7-SEP, where the structure of the possible cases of non-surprise and of surprise corresponds to the matrix below:
(D10)
S[k, 0]
S[k, 1]
S[7,s]
■
□
S[6,s]
■
■
S[5,s]
■
■
S[4,s]
■
■
S[3,s]
■
■
S[2,s]
■
■
S[1,s]
■
■
and to the following announcement of the professor
(S10)
An examination will occur in the next week but the date of the examination will constitute a surprise. Moreover, the fact that the examination will take place constitutes an absolute certainty.
Such an announcement appears identical to the preceding statement to which the Quine’s solution applies, with however an important difference: the student has from now on the certainty that the examination will occur. And this has the effect of preventing him/her from questioning the fact that the examination can take place, and of making thus impossible the surprise to occur on the last day. For this reason, we note S[7, 1] = 0 in the corresponding matrix. The general structure corresponding to this type of definition is:
(D11)
S[k, 0]
S[k, 1]
S[n,s]
■
□
S[n-1,s]
■
■
…………
…………
…………
And similarly, one can consider the following canonical structure (from where the denomination of triangular structure finds its justification), which is that of SEP(I∆) and which corresponds thus to 2-SEP(I∆):
(D12)
S[k, 0]
S[k, 1]
S[2,s]
■
□
S[1,s]
■
■
Such a structure corresponds to the following announcement of the professor:
(S12)
An examination will occur on the next two days, but the date of the examination will constitute a surprise. Moreover, the fact that the examination will take place constitutes an absolute certainty.
As we see it, the additional clause of the statement according to which it is absolutely certain that the examination will occur prevents here the surprise of occurring on the last day. Such a version corresponds in particular to the variation of SEP described by A. J. Ayer. The latter version corresponds to a player, who is authorized to check, before a set of playing cards is mixed, that it contains the ace, the 2, 3…, 7 of Spades. And it is announced that the player that he will not be able to envisage in advance justifiably, when the ace of Spades will be uncovered. Finally the cards, initially hidden, are uncovered one by one. The purpose of such a version is to render impossible, before the 7th card being uncovered, the belief according to which the ace of Spades will not be uncovered. And this has the effect of forbidding to Quine’ solution to apply on the last day.
It is now worth presenting a solution to the versions of SEP associated with the structures corresponding to (D11). Such a solution is based on a reduction recently exposed by Ned Hall, of which it is worth beforehand highlighting the context. In the version of SEP under consideration by Quine (1953), it appears clearly that the fact that the student doubts that the examination will well take place during the week, at a certain stage of the reasoning, is authorized. Quine thus places himself deliberately in a situation where the student has the faculty of doubting that the examination will truly occur during the week. The versions described by Ayer (1973), Janaway (1989) but also Scriven (1951) reveal the intention to prevent this particular step in the student’s reasoning. Such scenarios correspond, in spirit, to SEP(I∆). One can also attach to it the variation of the Designated Student Paradox described by Sorensen (1982, 357)8, where five stars – a gold star and four silver stars – are attributed to five students, given that it is indubitable that the gold star is placed on the back of the student who was designated.
However, Ned Hall (1999, 659-660) recently exposed a reduction, which tends to refute the objections classically raised against Quine’s solution. The argumentation developed by Hall is as follows:
We should pause, briefly, to dispense with a bad – though oft-cited – reason for rejecting Quine’s diagnosis. (See for example Ayer 1973 and Janaway 1989). Begin with the perfectly sound observation that the story can be told in such a way that the student is justified in believing that, come Friday, he will justifiably believe that an exam is scheduled for the week. Just add a second Iron Law of the School : that there must be at least one exam each week. (…) Then the first step of the student’s argument goes through just fine. So Quine’s diagnosis is, evidently, inapplicable.
Perhaps – but in letter only, not in spirit. With the second Iron Law in place, the last disjunct of the professor’s announcement – that E5 & J(E5) – is, from the student’s perspective, a contradiction. So, from his perspective, the content of her announcement is given not by SE5 but by SE4 : (E1 & J1(E1)) … (E4 & J4(E4)). And now Quine’s diagnosis applies straightforwardly : he should simply insist that the student is not justified in believing the announcement and so, come Thursday morning, not justified in believing that crucial part of it which asserts that if the exam is on Friday then it will come as a surprise – which, from the student’s perspective, is tantamount to asserting that the exam is scheduled for one of Monday through Thursday. That is, Quine should insist that the crucial premise that J4(E1 E2 E3 E4) is false – which is exactly the diagnosis he gives to an ordinary 4-day surprise exam scenario. Oddly, it seems to have gone entirely unnoticed by those who press this variant of the story against Quine that its only real effect is to convert an n-day scenario into an n-1 day scenario.
Hall puts then in parallel two types of situations. The first corresponds to the situation in which Quine’s analysis finds classically to apply. The second corresponds to the type of situation under consideration by the opponents to Quine’s solution and in particular by Ayer (1973) and Janaway (1989). On this last hypothesis, a stronger version of SEP is taken into account, where one second Iron Law of the School is considered and it is given that the examination will necessarily take place during the week. The argumentation developed by Hall leads to the reduction of a version of n-SEP of the second type to a version of (n-1)-SEP of the quinean type. This equivalence has the effect of annihilating the objections of the opponents to Quine’s solution.9 For the effect of this reduction is to make it finally possible to Quine’s solution to apply in the situations described by Ayer and Janaway. In spirit, the scenario under consideration by Ayer and Janaway corresponds thus to a situation where the surprise is not possible on day n (i.e. S[n, 1] = 0). This has indeed the effect of neutralizing Quine’s solution based on n-SEP(I□). But Hall’s reduction then makes it possible to Quine’s solution to apply to (n-1)-SEP(I□). The effect of Hall’s reduction is thus of reducing a scenario corresponding to (D11) to a situation based on (D8). Consequently, Hall’s reduction makes it possible to reduce n-SEP(I∆) to (n-1)-SEP(I□). It results from it that any version of SEP(I∆) for one n-period reduces to a version of SEP(I□) for one (n-1)-period (formally n-SEP(I∆) (n-1)-SEP(I□) for n > 1). Thus, Hall’s reduction makes it finally possible to apply Quine’s solution to SEP(I∆).10
4. The surprise notion corresponding to the disjoint structure
It is worth considering, second, the case where the notion of surprise is based on a disjoint structure of the possible cases of non-surprise and of surprise. Let SEP(II) be the corresponding version. Intuitively, such a variation corresponds to a situation where for a given day of the n-period, it is not possible to have at the same time the non-surprise and the surprise. The structure of the associated matrix is such that one has exclusively on each day, either the non-surprise or the surprise.
At this step, it appears that a preliminary question can be raised: can’t Quine’s solution apply all the same to SEP(II)? However, the preceding analysis of SEP(I) shows that a necessary condition in order to Quine’s solution to apply is that there exists during the n-period at least one day when the non-surprise and the surprise are at the same time possible. However such a property is that of a conjoint structure and corresponds to the situation which is that of SEP(I). But in the context of a disjoint structure, the associated matrix, in contrast, verifies k S[k, 0] + S[k, 1] = 1. Consequently, this forbids Quine’s solution to apply to SEP(II).
In the same way, one could wonder whether Hall’s reduction wouldn’t also apply to SEP(II). Thus, isn’t there a reduction of SEP(II) for a n-period to SEP(I) for a (n – 1)-period? It also appears that not. Indeed, as we did see it, Quine’s solution cannot apply to SEP(II). However, the effect of Hall’s reduction is to reduce a given scenario to a situation where Quine’s solution finally finds to apply. But, since Quine’s solution cannot apply in the context of SEP(II), Hall’s reduction is also unable to produce its effect.
Given that Quine’s solution does not apply to SEP(II), it is now worth attempting to provide an adequate solution to the version of SEP corresponding to a concept of surprise associated with a disjoint structure of the cases of non-surprise and of surprise. To this end, it proves to be necessary to describe a version of SEP corresponding to a disjoint structure, as well as the structure corresponding to the canonical version of SEP(II).
In a preliminary way, one can observe that the minimal version corresponding to a disjoint version of SEP is that which is associated with the following structure, i.e. 2-SEP(II):
(D13)
S[1, 0]
S[1, 1]
S[2,s]
■
□
S[1,s]
□
■
However, for reasons that will become clearer later, the corresponding version of SEP(II) does not have a sufficient degree of realism and of plausibility to constitute a genuine version of SEP, i.e. such that it is susceptible of inducing in error our reasoning.
In order to highlight the canonical version of SEP(II) and the corresponding statement, it is first of all worth mentioning the remark, made by several authors11, according to which the paradox emerges clearly, in the case of SEP(II), when n is large. An interesting characteristic of SEP(II) is indeed that the paradox emerges intuitively in a clearer way when great values of n are taken into account. A striking illustration of this phenomenon is thus provided to us by the variation of the paradox which corresponds to the following situation, described by Timothy Williamson (2000, 139):
Advance knowledge that there will be a test, fire drill, or the like of which one will not know the time in advance is an everyday fact of social life, but one denied by a surprising proportion of early work on the Surprise Examination. Who has not waited for the telephone to ring, knowing that it will do so within a week and that one will not know a second before it rings that it will ring a second later?
The variation suggested by Williamson corresponds to the announcement made to somebody that he will receive a phone call during the week, without being able however to determine in advance at which precise second the phone call will occur. This variation underlines how the surprise can appear, in a completely plausible way, when the value of n is high. The unit of time considered by Williamson is here the second, associated with a period which corresponds to one week. The corresponding value of n is here very high and equals 604800 (60 x 60 x 24 x 7) seconds. This illustrates how a great value of n makes it possible to the corresponding variation of SEP(II) to take place in both a plausible and realistic way. However, taking into account such large value of n is not indeed essential. In effect, a value of n which equals, for example, 365, seems appropriate as well. In this context, the professor’s announcement which corresponds to a disjoint structure is then the following:
(S14)
An examination will occur during this year but the date of the examination will constitute a surprise.
The corresponding definition presents then the following structure :
(D14)
S[1, 0]
S[1, 1]
S[365,s]
■
□
…………
…………
…………
S[1,s]
□
■
which is an instance of the following general form :
(D15)
S[1, 0]
S[1, 1]
S[n,s]
■
□
…………
…………
…………
S[1,s]
□
■
This last structure can be considered as corresponding to the canonical version of SEP(II), with n large. In the specific situation associated with this version of SEP, the student predicts each day – in a false way but justified by a reasoning based on backward-induction – that the examination will take place on no day of the week. But it appears that at least one case of surprise (for example if the examination occurs on the first day) makes it possible to validate, in a completely realistic way, the professor’s announcement..
The form of SEP(II) which applies to the standard version of SEP is 7-SEP(II), which corresponds to the classical announcement:
(S7)
An examination will occur on the next week but the date of the examination will constitute a surprise.
but with this difference with the standard version that the context is here exclusively that of a concept of surprised associated with a disjoint structure.
At this stage, we are in a position to determine the fallacious step in the student’s reasoning. For that, it is useful to describe the student’s reasoning in terms of matrix reconstitution. The student’s reasoning indeed leads him/her to attribute a value for S[k, 0] and S[k, 1]. And when he is informed of the professor’s announcement, the student’s reasoning indeed leads him/her to rebuild the corresponding matrix such that all S[k, 0] = 1 and all S[k, 1] = 0, in the following way (for n = 7):
(D16)
S[k, 0]
S[k, 1]
S[7,s]
■
□
S[6,s]
■
□
S[5,s]
■
□
S[4,s]
■
□
S[3,s]
■
□
S[2,s]
■
□
S[1,s]
■
□
One can notice here that the order of reconstitution proves to be indifferent. At this stage, we are in a position to identify the flaw which is at the origin of the erroneous conclusion of the student. It appears indeed that the student did not take into account the fact that the surprise corresponds here to a disjoint structure. Indeed, he should have considered here that the last day corresponds to a proper instance of non-surprise and thus that S[n, 0] = 1. In the same way, he should have considered that the 1st day12 corresponds to a proper instance of surprise and should have thus posed S[1, 1] = 1. The context being that of a disjoint structure, he could have legitimately added, in a second step, that S[n, 1] = 0 and S[1, 0] = 0. At this stage, the partially reconstituted matrix would then have been as follows:
(D17)
S[k, 0]
S[k, 1]
S[7,s]
■
□
S[6,s]
S[5,s]
S[4,s]
S[3,s]
S[2,s]
S[1,s]
□
■
The student should then have continued his reasoning as follows. The proper instances of non-surprise and of surprise which are disjoint here do not capture entirely the concept of surprise. In such context, the concept of surprise is not captured exhaustively by the extension and the anti-extension of the surprise. However, such a definition is in conformity with the definition of a vague predicate, which characterizes itself by an extension and an anti-extension which are mutually exclusive and non-exhaustive13. Thus, the surprise notion associated with a disjoint structure is a vague one.
What precedes now makes it possible to identify accurately the flaw in the student’s reasoning, when the surprise notion is a vague notion associated with a disjoint structure. For the error which is at the origin of the student’s fallacious reasoning lies in lack of taking into account the fact that the surprise corresponds in the case of a disjoint structure, to a vague concept, and thus comprises the presence of a penumbral zone corresponding to borderline cases between the non-surprise and the surprise. There is no need however to have here at our disposal a solution to the sorites paradox. Indeed, whether these borderline cases result from a succession of intermediate degrees, from a precise cut-off between the non-surprise and the surprise whose exact location is impossible for us to know, etc. is of little importance here. For in all cases, the mere fact of taking into account the fact that the concept of surprise is here a concept vague forbids to conclude that S[k, 1] = 0, for all values of k.
Several ways thus exist to reconstitute the matrix in accordance with what precedes. In fact, there exists as many ways of reconstituting the latter than there are conceptions of vagueness. One in these ways (based on a conception of vagueness based on fuzzy logic) consists in considering that there exists a continuous and gradual succession from the non-surprise to the surprise. The corresponding algorithm to reconstitute the matrix is then the one where the step is given by the formula 1/(n–p) when p corresponds to a proper instance of surprise. For p = 3, we have here 1/(7-3) = 0,25, with S[3, 1] = 1. And the corresponding matrix is thus the following one:
(D18)
S[k, 0]
S[k, 1]
S[7,s]
1
0
S[6,s]
0,75
0,25
S[5,s]
0,5
0,5
S[4,s]
0,25
0,75
S[3,s]
0
1
S[2,s]
0
1
S[1,s]
0
1
where the sum of the values of the matrix associated with a day given is equal to 1. The intuition which governs SEP (II) is here that the non-surprise is total on day n, but that there exists intermediate degrees of surprise si (0 < si < 1), such as the more one approaches the last day, the higher the effect of non-surprise. Conversely, the effect of surprise is total on the first days, for example on days 1, 2 and 3.
One can notice here that the definitions corresponding to SEP (II) which have just been described, are such that they present a property of linearity (formally, k (for 1 < k n), S[k, 0] S[k-1, 0]). It appears indeed that a structure corresponding to the possible cases of non-surprise and of surprise which would not present such a property of linearity, would not capture the intuition corresponding to the concept of surprise. For this reason, it appears sufficient to limit the present study to the structures of definitions that satisfy this property of linearity.
An alternative way to reconstitute the corresponding matrix, based on the epistemological conception of vagueness, could also have been used. It consists of the case where the vague nature of the surprise is determined by the existence of a precise cut-off between the cases of non-surprise and of surprise, of which it is however not possible for us to know the exact location. In this case, the matrix could have been reconstituted, for example, as follows:
(D19)
S[k, 0]
S[k, 1]
S[7,s]
■
□
S[6,s]
■
□
S[5,s]
■
□
S[4,s]
□
■
S[3,s]
□
■
S[2,s]
□
■
S[1,s]
□
■
At this stage, one can wonder whether the version of the paradox associated with SEP(II) cannot be assimilated with the sorites paradox. The reduction of SEP to the sorites paradox is indeed the solution which has been proposed by some authors, notably Dietl (1973) and Smith (1984). The latter solutions, based on the assimilation of SEP to the sorites paradox, constitute monist analyses, which do not lead, to the difference of the present solution, to two independent solutions based on two structurally different versions of SEP. In addition, with regard to the analyses suggested by Dietl and Smith, it does not clearly appear whether each step of SEP is fully comparable to the corresponding step of the sorites paradox, as underlined by Sorensen.14 But in the context of a conception of surprise corresponding to a disjoint structure, the fact that the last day corresponds to a proper instance of non-surprise can be assimilated here to the base step of the sorites paradox.
Nevertheless, it appears that such a reduction of SEP to the sorites paradox, limited to the notion of surprise corresponding to a disjoint structure, does not prevail here. On the one hand, it does not appear clearly if the statement of SEP can be translated into a variation of the sorites paradox, in particular for what concerns 7-SEP(II). Because the corresponding variation of the sorites paradox would run too fast, as already noted by Sorensen (1988).15 It is also noticeable, moreover, as pointed out by Scott Soames (1999), than certain vague predicates are not likely to give rise to a corresponding version of the sorites paradox. Such appears well to be the case for the concept of surprise associated with 7-SEP(II). Because as Soames16 points out, the continuum which is semantically associated with the predicates giving rise to the sorites paradox, can be fragmented in units so small that if one of these units is intuitively F, then the following unit is also F. But such is not the case with the variation consisting in 7-SEP(II), where the corresponding units (1 day) are not fine enough with regard to the considered period (7 days).
Lastly and overall, as mentioned above, the preceding solution to SEP(II) applies, whatever the nature of the solution which will be adopted for the sorites paradox. For it is the ignorance of the semantic structure of the vague notion of surprise which is at the origin of the student’s fallacious reasoning in the case of SEP(II). And this fact is independent of the solution which should be provided, in a near or far future, to the sorites paradox – whether this approach be of epistemological inspiration, supervaluationnist, based on fuzzy logic…, or of a very different nature.
5. The solution to the paradox
The above developments make it possible now to formulate an accurate solution to the surprise examination paradox. The latter solution can be stated by considering what should have been the student’s reasoning. Let us consider indeed, in the light of the present analysis, how the student should have reasoned, after having heard the professor’s announcement:
– The student: Professor, I think that two semantically distinct conceptions of surprise, which are likely to influence the reasoning to hold, can be taken into account. I also observe that you did not specify, at the time of your announcement, to which of these two conceptions you referred. Isn’t it?
– The professor: Yes, it is exact. Continue.
– The student: Since you refer indifferently to one or the other of these conceptions of surprise, it is necessary to consider each one of them successively, as well as the reasoning to be held in each case.
– The professor: Thus let us see that.
– The student: Let us consider, on the one hand, the case where the surprise corresponds to a conjoint definition of the cases of non-surprise and of surprise. Such a definition is such that the non-surprise and the surprise are possible at the same time, for example on the last day. Such a situation is likely to arise on the last day, in particular when a student concludes that the examination cannot take place on this same last day, since that would contradict the professor’s announcement. However, this precisely causes to make it possible for the surprise to occur, because this same student then expects that the examination will not take place. And in a completely plausible way, as put forth by Quine, such a situation corresponds then to a case of surprise. In this case, the fact of taking into account the possibility that the examination can occur surprisingly on the last day, prohibits eliminating successively the days n, n-1, n-2, …, 2, and 1. In addition, the concept of surprise associated with a conjoint structure is a concept of total surprise. For one faces on the last day either the non-surprise or the total surprise, without there existing in this case some intermediate situations.
– The professor: I see that. You did mention a second case of surprise…
– The student: Indeed. It is also necessary to consider the case where the surprise corresponds to a disjoint definition of the cases of non-surprise and of surprise. Such a definition corresponds to the case where the non-surprise and the surprise are not possible on the same day. The intuition on which such a conception of the surprise rests corresponds to the announcement made to students that they will undergo an examination in the year, while being moreover unaware of the precise day where it will be held. In such a case, it results well from our experience that the examination can truly occur surprisingly, on many days of the year, for example on whatever day of the first three months. It is an actual situation that can be experienced by any student. Of course, in the announcement that you have just made to us, the period is not as long as one year, but corresponds to one week. However, your announcement also leaves place to such a conception of surprise associated with a disjoint structure of the cases of non-surprise and of surprise. Indeed, the examination can indeed occur surprisingly, for example on the 1st day of the week. Thus, the 1st day constitutes a proper instance of surprise. In parallel, the last day constitutes a proper instance of non-surprise, since it results from the announcement that the examination cannot take place surprisingly on this day. At this stage, it also appears that the status of the other days of the corresponding period is not determined. Thus, such a disjoint structure of the cases of non-surprise and of surprise is at the same time disjoint and non-exhaustive. Consequently, the concept of corresponding surprise presents here the criteria of a vague notion. And this casts light on the fact that the concept of surprise associated with a conjoint structure is a vague one, and that there is thus a zone of penumbra between the proper instances of non-surprise and of surprise, which corresponds to the existence of borderline cases. And the mere existence of these borderline cases prohibits to eliminate successively, by a reasoning based on backward-induction, the days n, n-1, n-2, …, 2, and then 1. And I finally notice, to the difference of the preceding concept of surprise, that the concept of surprise associated with a conjoint structure leads to the existence of intermediate cases between the non-surprise and the surprise.
– The professor: I see. Conclude now.
– The student: Finally, the fact of considering successively two different concepts of surprise being able to correspond to the announcement which you have just made, resulted in both cases in rejecting the classical reasoning which results in eliminating successively all days of the week. Here, the motivation to reject the traditional reasoning appears different for each of these two concepts of surprise. But in both cases, a convergent conclusion ensues which leads to the rejection of the classical reasoning based on backward-induction.
6. Conclusion
I shall mention finally that the solution which has been just proposed also applies to the variations of SEP mentioned by Sorensen (1982). Indeed, the structure of the canonical forms of SEP(I□), SEP(I∆) or SEP(II) indicates that whatever the version taken into account, the solution which applies does not require to make use of whatever principle of temporal retention. It is also independent of the order of elimination and can finally apply when the duration of the n-period is unknown at the time of the professor’s announcement.
Lastly, it is worth mentioning that the strategy adopted in the present study appears structurally similar to the one used in Franceschi (1999): first, establish a dichotomy which makes it possible to divide the given problem into two distinct classes; second, show that each resulting version admits of a specific resolution.17 In a similar way, in the present analysis of SEP, a dichotomy is made and the two resulting categories of problems lead then to an independent resolution. This suggests that the fact that two structurally independent versions are inextricably entangled in philosophical paradoxes could be a more widespread characteristic than one could think at first glance and could also partly explain their intrinsic difficulty.18
REFERENCES
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CHOW, T. Y. 1998, “The Surprise Examination or Unexpected Hanging Paradox”, The American Mathematical Monthly 105, pp. 41-51.
DIETL, P. 1973, “The Surprise Examination”, Educational Theory 23, pp. 153-158.
FRANCESCHI, P. 1999, “Comment l’urne de Carter et Leslie se déverse dans celle de Hempel”, Canadian Journal of Philosophy 29, pp. 139-156. English translation.
HALL, N. 1999, “How to Set a Surprise Exam”, Mind 108, pp. 647-703.
HYDE, D. 2002 “Sorites Paradox”, The Stanford Encyclopedia of Philosophy (Fall 2002 Edition), E. N. Zalta (ed.), http ://plato.stanford.edu/archives/fall2002/entries/sorites-paradox.
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MONTAGUE, R. & KAPLAN, D. 1960, “A Paradox Regained”, Notre Dame Journal of Formal Logic 3, pp. 79-90.
O’ CONNOR, D. 1948, “Pragmatic paradoxes”, Mind 57, pp. 358-359.
QUINE, W. 1953, “On a So-called Paradox”, Mind 62, pp. 65-66.
SAINSBURY, R. M. 1995, Paradoxes, 2ème édition, Cambridge : Cambridge University Press.
SCRIVEN, M. 1951, “Paradoxical announcements”, Mind 60, pp. 403-407.
SHAW, R. 1958, “The Paradox of the Unexpected Examination”, Mind 67, pp. 382-384.
SMITH, J. W. 1984, “The surprise examination on the paradox of the heap”, Philosophical Papers 13, pp. 43-56.
SOAMES, S. 1999, Understanding Truth, New York, Oxford : Oxford University Press.
SORENSEN, R. A. 1982, “Recalcitrant versions of the prediction paradox”, Australasian Journal of Philosophy 69, pp. 355-362.
SORENSEN, R. A. 1988, Blindspots, Oxford : Clarendon Press.
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WRIGHT, C. & SUDBURY, A. 1977, “The Paradox of the Unexpected Examination”, Australasian Journal of Philosophy 55, pp. 41-58.
3 In what follows, n denotes the last day of the term corresponding to the professor’s announcement.
4 Let 1-SEP, 2-SEP,…, n-SEP be the problem for respectively 1 day, 2 days,…, n days.
5 The cases where neither the non-surprise nor the surprise are made possible on the same day (i.e. such that S[k, 0] + S[k, 1] = 0) can be purely and simply ignored.
6 Cf. (1953, 65) : ‘It is notable that K acquiesces in the conclusion (wrong, according to the fable of the Thursday hanging) that the decree will not be fulfilled. If this is a conclusion which he is prepared to accept (though wrongly) in the end as a certainty, it is an alternative which he should have been prepared to take into consideration from the beginning as a possibility.’
7 Cf. (1953, 66) : ‘If K had reasoned correctly, Sunday afternoon, he would have reasoned as follows : “We must distinguish four cases : first, that I shall be hanged tomorrow noon and I know it now (but I do not) ; second, that I shall be unhanged tomorrow noon and do not know it now (but I do not) ; third, that I shall be unhanged tomorrow noon and know it now ; and fourth, that I shall be hanged tomorrow noon and do not know it now. The latter two alternatives are the open possibilities, and the last of all would fulfill the decree. Rather than charging the judge with self-contradiction, let me suspend judgment and hope for the best.”‘
8 ‘The students are then shown four silver stars and one gold star. One star is put on the back of each student.’.
9 Hall refutes otherwise, but on different grounds, the solution proposed by Quine.
10 Hall’s reduction can be easily generalised. It is then associated with a version of n-SEP(I∆) such that the surprise will not possibly occur on the m last days of the week. Such a version is associated with a matrix such that (a) m (1 m < n) and S[n–m, 0] = S[n–m, 1] = 1 ; (b) p > n–m S[p, 0] = 1 and S[p, 1] = 0 ; (c) q < n–m S[q, 0] = S[q, 1] = 1. In this new situation, a generalised Hall’s reduction applies to the corresponding version of SEP. In this case, the extended Hall’s reduction leads to : n-SEP(I∆) (n–m)-SEP(I□).
11 Cf. notably Hall (1999, 661), Williamson (2000).
12 It is just an example. Alternatively, one could have chosen here the 2nd or the 3rd day.
13 This definition of a vague predicate is borrowed from Soames. Considering the extension and the anti-extension of a vague predicate, Soames (1999, 210) points out thus: “These two classes are mutually exclusive, though not jointly exhaustive”.
14 Cf. Sorensen (1988, 292-293) : ‘Indeed, no one has simply asserted that the following is just another instance of the sorites.
i. Base step : The audience can know that the exercise will not occur on the last day.
ii. Induction step : If the audience can know that the exercise will not occur on day n, then they can also know that the exercise will not occur on day n – 1
iii. The audience can know that there is no day on which the exercise will occur.
Why not blame the whole puzzle on the vagueness of ‘can know’? (…) Despite its attractiveness, I have not found any clear examples of this strategy.’
15 Cf. (1988, 324): ‘One immediate qualm about assimilating the prediction paradox to the sorites is that the prediction paradox would be a very ‘fast’ sorites. (…) Yet standard sorites arguments involve a great many borderline cases.’
16 Cf. Soames (1999, 218): ‘A further fact about Sorites predicates is that the continuum semantically associated with such a predicate can be broken down into units fine enough so that once one has characterized one item as F (or not F), it is virtually irresistible to characterize the same item in the same way’.
17 One characteristic example of this type of analysis is also exemplified by the solution to the two-envelope paradox described by David Chalmers (2002, 157) : ‘The upshot is a disjunctive diagnosis of the two-envelope paradox. The expected value of the amount in the envelopes is either finite or infinite. If it is finite, then (1) and (2) are false (…). If it is infinite, then the step from (2) to (3) is invalid (…)’.
18 I am grateful toward Timothy Chow, Ned Hall, Claude Panaccio and the anonymous referees for very useful comments concerning previous versions of this paper.