Dichotomous analysis is a technique of analysis that tends to consider an object o from the point of view of a given duality, and to distinguish two different situations, each of which corresponds to one of the two poles of the duality considered. Examples of dualities are: Quantitative/Qualitative, Absolute/Relative, Static/Dynamic, Diachronic/Synchronic, Extension/Restriction.

The dichotomous analysis is directly linked to the very statement of the principle of dialectical indifference. Its use responds to the concern not to be vulnerable to one-sidedness bias and to demonstrate, on the contrary, a two-sided viewpoint.

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, p. 581-608.

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 authors^{1} 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 T_{0} is also known at a later temporal position T_{1} (with T_{0} < T_{1}). 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 relate^{2} 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 E_{5} & J(E_{5}) – is, from the student’s perspective, a contradiction. So, from his perspective, the content of her announcement is given not by SE_{5} but by SE_{4} : (E_{1} & J_{1}(E_{1})) … (E_{4} & J_{4}(E_{4})). 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 J_{4}(E_{1} E_{2} E_{3} E_{4}) 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 authors^{11}, 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 day^{12} 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-exhaustive^{13}. 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 s_{i} (0 < s_{i} < 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 Soames^{16} points out, the continuum which is semantically associated with the predicates giving rise to the sorites paradox, can be fragmented in units so small that if one of these units is intuitively F, then the following unit is also F. But such is not the case with the variation consisting in 7-SEP(II), where the corresponding units (1 day) are not fine enough with regard to the considered period (7 days).

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

5. The solution to the paradox

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

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

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

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

– The professor: Thus let us see that.

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

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

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

– The professor: I see. Conclude now.

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

6. Conclusion

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

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

REFERENCES

AYER, A. J. 1973, “On a Supposed Antinomy”, Mind 82, pp. 125-126.

BINKLEY, R. 1968, “The Surprise Examination in Modal Logic”, Journal of Philosophy 65, pp. 127-136.

CHALMERS, D. 2002, “The St. Petersburg two-envelope paradox”, Analysis 62, pp. 155-157.

CHOW, T. Y. 1998, “The Surprise Examination or Unexpected Hanging Paradox”, The American Mathematical Monthly 105, pp. 41-51.

DIETL, P. 1973, “The Surprise Examination”, Educational Theory 23, pp. 153-158.

FRANCESCHI, P. 1999, “Comment l’urne de Carter et Leslie se déverse dans celle de Hempel”, Canadian Journal of Philosophy 29, pp. 139-156. English translation.

HALL, N. 1999, “How to Set a Surprise Exam”, Mind 108, pp. 647-703.

HYDE, D. 2002 “Sorites Paradox”, The Stanford Encyclopedia of Philosophy (Fall 2002 Edition), E. N. Zalta (ed.), http ://plato.stanford.edu/archives/fall2002/entries/sorites-paradox.

JANAWAY, C. 1989, “Knowing About Surprises : A Supposed Antinomy Revisited”, Mind 98, pp. 391-410.

MONTAGUE, R. & KAPLAN, D. 1960, “A Paradox Regained”, Notre Dame Journal of Formal Logic 3, pp. 79-90.

O’ CONNOR, D. 1948, “Pragmatic paradoxes”, Mind 57, pp. 358-359.

QUINE, W. 1953, “On a So-called Paradox”, Mind 62, pp. 65-66.

SAINSBURY, R. M. 1995, Paradoxes, 2ème édition, Cambridge : Cambridge University Press.

SCRIVEN, M. 1951, “Paradoxical announcements”, Mind 60, pp. 403-407.

SHAW, R. 1958, “The Paradox of the Unexpected Examination”, Mind 67, pp. 382-384.

SMITH, J. W. 1984, “The surprise examination on the paradox of the heap”, Philosophical Papers 13, pp. 43-56.

SOAMES, S. 1999, Understanding Truth, New York, Oxford : Oxford University Press.

SORENSEN, R. A. 1982, “Recalcitrant versions of the prediction paradox”, Australasian Journal of Philosophy 69, pp. 355-362.

SORENSEN, R. A. 1988, Blindspots, Oxford : Clarendon Press.

WILLIAMSON, T. 2000, Knowledge and its Limits, London & New York : Routledge.

WRIGHT, C. & SUDBURY, A. 1977, “The Paradox of the Unexpected Examination”, Australasian Journal of Philosophy 55, pp. 41-58.

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.

Postprint. I present in this paper an analysis of the Simulation Argument from a dialectical contextualist’s standpoint. This analysis is grounded on the reference class problem. I begin with describing in detail Bostrom’s Simulation Argument. I identify then the reference class within the Simulation Argument. I also point out a reference class problem, by applying the argument successively to three different reference classes: aware-simulations, imperfect simulations and immersion-simulations. Finally, I point out that there are three levels of conclusion within the Simulation Argument, depending on the chosen reference class, that yield each final conclusions of a fundamentally different nature.

This article supersedes my preceding work on the Simulation argument. Please do not cite previous work.

The Simulation Argument and the Reference Class Problem : a Dialectical Contextualism Analysis

ABSTRACT. I present in this paper an analysis of the Simulation Argument from a dialectical contextualist’s standpoint. This analysis is grounded on the reference class problem. I begin with describing in detail Bostrom’s Simulation Argument. I identify then the reference class within the Simulation Argument. I also point out a reference class problem, by applying the argument successively to three different reference classes: aware-simulations, imperfect simulations and immersion-simulations. Finally, I point out that there are three levels of conclusion within the Simulation Argument, depending on the chosen reference class, that yield each final conclusions of a fundamentally different nature.

1. The Simulation Argument

I shall propose in what follows an analysis of the Simulation Argument, recently described by Nick Bostrom (2003). I will first describe in detail the Simulation Argument (SA for short), focusing in particular on the resulting counter-intuitive consequence. I will then show how such a consequence can be avoided, based on the analysis of the reference class underlying SA, without having to give up one’s pre-theoretical intuitions.

The general idea behind SA can be stated as follows. It is very likely that post-human civilizations will possess a computing power that will be completely out of proportion with that of ours today. Such extraordinary computing power should give them the ability to carry out completely realistic human simulations, such as ensuring that the inhabitants of these simulations are aware of their own existence, in all respects similar to ours. In such a context, it is likely that post-human civilizations will devote part of their computer resources to carrying out simulations of the human civilizations that preceded them. In this case, the number of simulated humans should greatly exceed the number of authentic humans. Under such conditions, taking into account the simple fact that we exist leads to the conclusion that it is more likely that we are part of the simulated humans, rather than of the authentic humans.

Bostrom thus points out that the Simulation Argument is based on the following three hypotheses:

(1)

it is very likely that humanity will not reach a post-human stage

(2)

it is very unlikely that post-human civilizations will carry out simulations of the human races that preceded them

(3)

it is very likely that we are currently living in a simulation carried out by a post-human civilization

and it follows that at least one of these three assumptions is true.

For the purposes of the present analysis, it is also useful at this stage to emphasize the underlying dichotomous structure of SA. The first step in the reasoning consists then in considering, by dichotomy, that either (i) humanity will not reach a post-human stage, or (ii) it will actually reach such a post-human stage. The first of these two hypotheses corresponds to the disjunct (1) of the argument. We consider then the hypothesis that humanity will reach a post-human stage and thus continue its existence for many millennia. In such a case, it can also be considered likely that post-human civilizations will possess both the technology and the skills necessary to perform human simulations. A new dichotomy then arises: either (i) these post-human civilizations will not perform such simulations — this is the disjunct (2) of the argument; or (ii) these post-human civilizations will actually perform such simulations. In the latter case, it will follow that the number of simulated humans will greatly exceed the number of humans. The probability of living in a simulation will therefore be much greater than that of living in the shoes of an ordinary human. The conclusion then follows that we, the inhabitants of the Earth, are probably living in a simulation carried out by a post-human civilization. This last conclusion constitutes the disjunct(3) of the argument. An additional step leads then to the conclusion that at least one of the hypotheses (1), (2) and (3) is true. The dichotomous structure underlying SA can thus be described step by step as follows:

(4)

humanity will either not reach a post-human stage or reach a post-human stage

dichotomy 1

(1)

humanity will not reach a post-human stage

hypothesis 1.1

(5)

humanity will reach a post-human stage

hypothesis 1.2

(6)

post-human civilizations will be able to perform human simulations

from (5)

(7)

post-human civilizations will either not perform human simulations or will perform them

dichotomy 2

(2)

post-human civilizations will not perform human simulations

hypothesis 2.1

(8)

post-human civilizations will perform human simulations

hypothesis 2.2

(9)

the proportion of simulated humans will far exceed that of humans

from (8)

(3)

it is very likely that we are currently living in a simulation carried out by a post-human civilization

from (9)

(10)

at least one of the hypotheses (1), (2) and (3) is true

from (1), (2), (3)

It is also worth mentioning an element that results from the very interpretation of the argument. For as Bostrom (2005) points out, the Simulation Argument must not be misinterpreted. This is not an argument that leads to the conclusion that (3) is true, namely that we are currently living in a simulation carried out by a post-human civilization. The core of SA is thus that one of the hypotheses (1), (2) or (3) at least is true.

This nuance of interpretation being mentioned, the Simulation Argument is not without its problems. Because SA leads to the conclusion that at least one of the assumptions (1), (2) or (3) is true, and that in the situation of ignorance in which we find ourselves, we can consider the latter as equiprobable. As Bostrom himself notes (Bostrom, 2003): “In the dark forest of our current ignorance, it seems sensible to apportion one’s credence roughly evenly between (1), (2) and (3)”. However, according to our pre-theoretical intuition, the probability of (3) is nil or at best extremely close to 0, so the conclusion of the argument has the consequence of increasing the probability that (3) is true from zero to a probability of about 1/3. Thus, the problem with the Simulation Argument is precisely that it shifts — via its disjunctive conclusion — from a zero or almost zero probability concerning (3) to a much higher probability of about 1/3. Because a probability of 1/3 for the hypotheses (1) and (2) is not a priori shocking, but is completely counter-intuitive as far as hypothesis (3) is concerned. It is in this sense that we can talk about the problem posed by the Simulation Argument and the need to find a solution to it.

As a preliminary point, it is worth considering what constitutes the paradoxical aspect of SA. What indeed gives SA a paradoxical nature? For SA differs from the class of paradoxes that lead to a contradiction. In paradoxes such as the Liar or the sorites paradox, the corresponding reasoning leads to a contradiction^{1}. However, nothing of the sort can be seen at the level of SA, which belongs, from this point of view, to a different class of paradoxes, including the Doomsday Argument and Hempel’s problem. It is indeed a class of paradoxes whose conclusion is contrary to intuition, and which comes into conflict with the set of all our beliefs. In the Doomsday Argument then, the conclusion that taking into account our rank within the class of humans who have ever existed has the effect that an apocalypse is much more likely than one might have initially thought, offends the set of all our beliefs. Similarly, in Hempel’s problem, the fact that a blue umbrella confirms the hypothesis that all crows are black comes in conflict with the body of our knowledge. Similarly within SA, what finally appears paradoxical at first analysis is that SA leads to a probability of the hypothesis that we are currently living in a simulation created by post-humans, which is higher than that resulting from our pre-theoretical intuition.

2. The reference class problem and the Simulation Argument

The conclusion of the reasoning underlying SA, based on the calculation of the future ratio between real and simulated humans, albeit counter-intuitive, nevertheless results from a reasoning that appears a priori valid. However, such reasoning raises a question, which is related to the reference class that is inherent to the argument itself^{2}. Indeed, it appears that SA has, indirectly, a particular reference class, which is that of human simulations. But what constitutes a simulation? The original argument implicitly refers to a reference class which is that of virtual simulations of humans, of a very high quality and by nature indistinguishable from authentic humans. However, there is some ambiguity about the very notion of a simulation and the question arises as to the applicability of SA to other types of human simulations^{3}. Indeed, we are in a position to conceive of somewhat different types of simulations which also fall intuitively within the scope of the argument.

As a preliminary point, it is worth specifying here the nature of the simulations carried out by computer means referred to in the original argument. Implicitly, SA refers to computer simulations carried out by means of conventional computers composed of silicon chips. But it can also be envisaged that simulations are carried out using computers built from components using DNA properties and molecular biology. Recent research has shown that it is possible to implement high-performance algorithms (Adleman 1994, 1998) and to produce computer components (Benenson & al. 2001, MacDonald & al. 2006) based on bio-calculation techniques that exploit in particular the combinations of the four components (adenine, cytosine, guanine, thymine) of the DNA molecule. If such a field of research were to expand significantly and make it possible to produce computers at least as powerful as conventional computers, this type of bio-computers could legitimately fall within the scope of SA as well. Because the fact that the simulations are carried out using conventional or biological computers^{4} does not alter the scope of the argument. In any case, the result is that the proportion of simulated humans will be much higher than that of real humans, due to the properties of simulated reality using digital means, because the computer does not know the physical limits that are those of matter.

It can also be observed preliminarily that Bostrom explicitly refers to simulations carried out using computer means. However, the question arises as to whether simulated humans could not consist of perfectly successful physical copies of real humans. In such a case, simulations^{5} could be extremely difficult to discern. A priori, such a variation also constitutes an acceptable version of SA. However, there is a difference with the original argument, which also highlights Bostrom’s preferential choice of computer simulations. Indeed, in the original argument there is a very significant disproportion between humans simulated by computer means on the one hand and real humans on the other. This is the premise (9) of the argument: “the proportion of simulated humans will far exceed that of humans”. As Bostrom points out, the former would then be much more numerous than the latter, due to the very nature of computer simulations. It is this disproportion that then allows us to conclude (3) “we most probably live in a simulation carried out by a post-human civilization”. With simulations of a physical nature, one would not a priori have such a disproportion, and the scope of the conclusion would be somewhat different. Suppose, for example, that post-humans manage to perform simulations of a physical nature, the number of which would be equal to that of real humans. In this case, the proportion of simulated humans would be 1/2 (whereas it is close to 1 in the original argument). Premise (9) would then become: “the proportion of simulated humans and actual humans will be 1/2”. And this would only allow us to conclude (3) “the probability that we are simulations performed by a post-human civilization is equal to 1/2”. As can be seen, this would result in a significantly attenuated version of SA. The difference with the original version of SA is that the simulation argument for physical simulations applies with less force than the original argument. However, if the conditions were to change and this would result in the future in a disproportion of the same nature as with computer simulations for physical simulations, SA would then apply with all its force. In any event, the following analysis would then apply in the same way to this last category of simulations.

With these preliminary considerations in mind, we shall focus in turn on different types of human simulations, which are likely to be part of the SA reference class, and the ensuing conclusions at the argument level. Because the very question of defining the reference class for SA leads to questions about whether or not several types of simulations should be included within the SA scope. However, the question of the definition of the reference class for SA thus appears closely related to the nature of the future taxonomy of the beings and entities that will populate the Earth in the near or distant future. There is no question here of claiming exhaustiveness, given the speculative nature of such an area. However, it is possible to determine to what extent SA can also be applied to simulations of a different nature from those mentioned in the original argument, but which have equal legitimacy. We shall examine then in turn: conscious simulations, imperfect simulations, and immersion simulations.

3. The reference class problem : the caseof conscious simulations

At this step, it is not yet possible to really talk about the problem of the reference class within SA. To do so, it must be shown that the choice of one or the other reference class has completely different consequences at the level of the argument, and in particular that the nature of its conclusion is affected, i.e. fundamentally modified. In what follows, we will now focus on showing that depending on which reference class is chosen, radically different conclusions ensue at the level of the argument itself and that, consequently, there is a reference class problem within SA. For this purpose, we will consider several reference classes in turn, focusing on how conclusions of a fundamentally different nature result from them at the level of the argument itself.

The original version of SA implicitly depicts simulations of humans of a certain type. These are virtual simulations, almost indistinguishable from real humans and that present thus a very high degree of sophistication. Moreover, these are a type of simulations that are not aware that they are themselves simulated and are therefore convinced that they are genuine humans. This is implicit in the terms of the argument itself and in particular, the inference from (9) to (3) which leads to the conclusion that ʻweʼ are currently living in an indistinguishable simulation carried out by post-humans. In fact, these are simulations that are somehow abused and misled by post-humans regarding their true identity. For the purposes of this discussion, we shall term quasi-humans^{–} the simulated humans who are not aware that they are human.

At this stage, it appears that it is also possible to conceive of indistinguishable simulations that have an identical degree of sophistication but that, on the other hand, would be aware that they are being simulated. We shall then call quasi-humans^{+} the simulated humans who are aware that they are themselves simulations. Such simulations are in all respects identical to the quasi-humans^{–} to which SA implicitly refers, with the only difference that they are this time clearly aware of their intrinsic nature of simulation. Intuitively, SA also applies to this type of simulation. A priori, there is no justification for excluding such a type of simulation. Moreover, there are several reasons to believe that quasi-humans^{+} may be more numerous than quasi-humans^{–}. For ethical reasons (i) first of all, it may be thought that post-humans might be inclined to prefer quasi-humans^{+} to quasi-humans^{–}. For the fact of conferring an existence on quasi-humans constitutes a deception as to their true identity, whereas such an inconvenient is absent in the case of quasi-humans^{+}. Such deception could reasonably be considered unethical and lead to some form of prohibition of quasi-humans^{–}. Another reason (ii) is that simulations of humans who are aware of their own simulation nature should not be dismissed a priori. Indeed, we can think that the level of intelligence acquired by some quasi-humans in the near future could be extremely high and in this case, the simulations would very quickly become aware that they are themselves simulations. It may be thought that from a certain degree of intelligence, and in particular that which may be obtained by humanity in the not too distant future (Kurtzweil, 2000, 2005; Bostrom, 2006), quasi-humans should be able — at least much more easily than at present — to collect evidence that they are the subject of a simulation. Furthermore (iii), the very concept of “unconscious simulation that it is a simulation” could be inherently contradictory, because it would then be necessary to limit one’s intelligence and therefore, it would no longer constitute an indistinguishable and sufficiently realistic simulation^{6}. These three reasons suggest that quasi-humans^{+} may well exist in greater numbers than quasi-humans^{–} — or even that they may even be the only type of simulation implemented by post-humans.

At this stage, it is worth considering the consequences of taking into account the quasi-humans^{+} within the simulation reference class inherent to SA. For this purpose, let us first consider the variation of SA (let us term it SA*) that applies, exclusively, to the class of quasi-humans^{+}. Such a choice, first of all, has no consequence on the disjunct (1) of SA, which refers to a possible disappearance of our humanity before it has reached the post-human stage. Nor does this has any effect on the disjunct (2), according to which post-humans will not perform quasi-humans^{+}, i.e. conscious simulations of human beings. On the other hand, the choice of such a reference class has a direct consequence on the disjunct (3) of SA. Certainly, it follows, in the same way as for the original argument, the first level conclusion that the number of quasi-humans^{+} will far exceed the number of authentic humans (the disproportion). However, the second level conclusion that “we” are currently quasi-humans no longer follows. Indeed, such a conclusion (let us call it self-applicability) no longer applies to us, since we are not aware that we are being simulated and are completely convinced that we are authentic humans. Thus, in this particular context, the inference from (9) to (3) no longer prevails. Indeed, what constitutes SA’s worrying conclusion no longer results from step (9), since we cannot identify with the quasi-humans^{+}, the latter being clearly aware that they are evolving in a simulation. Thus, unlike the original version of SA based on the reference class that associates humans with quasi-humains^{–}, this new version associating humans with quasi-humans^{+} is not associated with such a disturbing conclusion. The conclusion that now follows, as we can see, is quite reassuring, and in any case very different from the deeply worrying^{7} conclusion that results from the original argument.

At this stage, it appears that a question arises: should we identify, in the context of SA, the reference class to the quasi-humans^{–} or the quasi-humans^{+}?^{8} It appears that no objective element in SA’s statement supports the a priori choice of the quasi-humans^{–} or the quasi-humans^{+}. Thus, any version of the argument that includes the preferential choice of the quasi-humans^{–} or the quasi-humans^{+} appears to be biased. This is the case for the original version of SA, which thus contains a bias in favor of the quasi-humans^{–}, which results from Bostrom’s choice of a class of simulations that is exclusively assimilated to quasi-humans^{–}, i.e. simulations that are not aware of their simulation nature and are therefore abused and misled by post-humans about the very nature of their identity. And this is also the case for SA*, the alternative version of SA that has just been described, which includes a particular bias in favor of quasi-humans^{+}, simulations that are aware of their own simulation nature. However, the choice of the reference class is fundamental here, because it has an essential consequence: if we choose a reference class that associates simulations with quasi-humans, the result is the worrying conclusion that we are most likely currently experiencing in a simulation. On the other hand, if a reference class is chosen that identifies simulations with quasi-humans^{+}, the result is a scenario that reassuringly does not include such a conclusion. At this stage, it is clear that the choice of the quasi-humans^{–} i.e., non-conscious simulations — in the original version of SA, to the detriment of conscious simulations, constitutes an arbitrary choice. Indeed, what makes it possible to prefer the choice of quasi-humans^{–}, compared to quasi-humans^{+}? Such justification is lacking in the context of the argument. At this stage, it appears that SA’s original argument contains a bias that leads to the preferential choice of quasi-humans^{–}, and to the alarming conclusion associated with it^{9}.

4. The reference class problem : the case of imperfect simulations

The problem of the reference class within SA relates, as mentioned above, to the very nature and to the type of simulations referred to in the argument. Is this problem limited to the preferential choice, at the level of the original argument, of unconscious simulations, to the detriment of the alternative choice of conscious simulations, which correspond to very sophisticated simulations of humans, capable of creating illusion, but endowed with the awareness that they themselves are simulations? It appears not. Indeed, as mentioned above, other types of simulations can also be envisaged for which the argument also works, but which are of a somewhat different nature. In particular, it is conceivable that post-humans may design and implement simulations that are identical to those of the original argument, but that are not as perfect in essence. Such a situation is quite likely and does not have the ethical disadvantages that could accompany the indistinguishable simulations staged in the original argument. The choice to carry out such simulations could be the result of the necessary technological level, or of deliberate and pragmatic choices, designed to save time and resources. These could be, for example, simulations of excellent quality such that the scientific inhabitants of the simulations could only discover their artificial nature after, for example, ten years of research. Such simulations could be carried out in very large numbers and, given their less resource-intensive nature, could occur in even greater numbers than quasi-humans^{–}. For the purposes of this discussion, we will call imperfect simulations this category of simulations.

At this stage, one can ask oneself what are the consequences on SA of taking into account a reference class that identifies itself with imperfect simulations? In this case, it follows, in the same way as the original argument, that the first level conclusion that the number of imperfect simulations will far exceed the number of authentic humans (the disproportion). But here too, however, the second level conclusion that “we” are currently imperfect simulations (self-applicability) no longer follows. The latter no longer applies to us and a reassuring conclusion replaces it, since we are clearly aware that we are not such imperfect simulations. Finally, it turns out that the conclusion that results from taking into account the class of imperfect simulations is of the same nature as that which follows when considering the class of the quasi-humans^{+}.

5. The reference class problem : the caseof immersion simulations

As we have seen, extending the SA reference class to conscious simulations leads to a conclusion of a different nature from the one that results from the original argument. The same applies to another category of simulations — imperfect simulations — which lead to a conclusion of the same nature as conscious simulations, and which in any case turns out to be different from that resulting from taking into account the simulations mentioned in the original argument. At this stage, the question arises as to whether the reference class can not be assimilated to other types of simulations relevant from the point of view of SA and whose consideration would lead to a conclusion that is inherently different from that which follows when considering the simulations of the original argument, or conscious or imperfect simulations.

In particular, the question arises as to whether human simulations, which would be such as to apply to ourselves — in a sense that may differ from the original argument — and which would include the conclusion of self-applicability inherent in SA, could not exist in a more or less near future. Some answers can be provided by considering an evolution of the concepts of virtual reality that are already being implemented in different fields such as psychiatry, surgery, industry, military training, entertainment, etc. In psychiatry in particular, virtual universes are used to implement techniques related to behavioral therapies, and offer advantages over traditional in vivo scenarios (Powers & Emmelkamp, 2008). In this type of treatment, the patient himself is simulated using an avatar and the universe in which he evolves is also simulated in the most realistic way possible. Convincing results have been obtained in the treatment of some phobias (Choy & al., 2007, Parsons & Rizzo, 2008), as well as post-traumatic stress disorder (Cukor & al., 2009, Baños & al., 2011).

In this context, it is conceivable that developments in this concept of virtual reality could lead to the realization of simulated humans, which would require a high degree of realism. This would require, in particular, the completion of current research, particularly on the simulation of the human brain. It is possible that significant progress may be made in the near future (Moravec, 1998; Kurzweil, 2005; Sandberg and Bostrom, 2008; De Garis et al. al., 2010). It is also conceivable that we will then have the ability to immerse ourselves in simulated universes by borrowing the personalities of humans thus simulated, while really having — the time of immersion — the impression that this is our real existence^{10}. In addition, the same human simulation could take the form of multiple variations that would correspond to the purpose — therapeutic, scientific, playful, utilitarian, historical, etc. — sought during the immersion. For example, it is conceivable that some variations may only include important elements of the simulated personality’s life, neglecting uninteresting details. For the purposes of this discussion, we can term this type of simulation: immersion simulations. In this context, humans could thus frequently resort to immersion in a simulated anterior human personality. It is also possible that individuals may use simulations of themselves: they could be simulations of themselves at earlier times in their lives, with eventual slight variations, however, depending on the purpose sought for the immersion in question. In such circumstances, it is conceivable that very large quantities of this type of simulation could be carried out by computer means. In any case, it appears that the number of simulations at our disposal would be much greater than the inhabitants of our planet. In this context, it appears that SA functions in the same way as the original argument if we reason in relation to a reference class that identifies itself with this type of immersion simulations.

At this point, it is worth considering the effect on SA of assimilating the reference class to immersion simulations. In such a context, it appears that the first-level consequence based on the humans/simulations disproportion would apply here, in the same way as the original argument. Secondly, and this is an important consequence, the second level conclusion based on self-applicability would now apply, since we can conclude that “we” are also, in this extended sense, simulations. On the other hand, it would no longer follow the alarming conclusion, which is that of the original argument and which manifests itself at a third level, that we are unconscious simulations, since the fact that we are in this sense simulations does not imply here that we are mistaken about our first identity. Thus, unlike the original argument, the result is a reassuring conclusion: humans are occasionally immersion simulations, while being aware that they use them.

Could we not object here that we have not yet reached the state where we can identify, even if only temporarily, with such immersion simulations and that this does not make the above developments relevant to SA? Strictly speaking, the virtual reality implemented in our time can indeed be considered too coarse in nature to be assimilated to the very realistic simulations hinted at by Bostrom. However, it can be assumed that only high-quality immersion simulations, which would give the illusion at least the time of their use that they are a real existence, could be carried out, for such simulations to become relevant for the SA reference class. The hypothesis that such a technological level, based on an explosion of artificial intelligence, could be achieved within a few decades has thus been put forward (Kurzweil, 2005; Eden et al. al., 2013). If such a technological evolution were to occur within, for example, a few decades, could we not then legitimately consider that such simulations also fall within the reference class of SA? Given this possible temporal proximity, it seems appropriate to take into account the case of immersion simulations and to evaluate their consequences for SA^{11}.

6. The different levels of conclusion according to the chosen reference class

Finally, the preceding discussion emphasizes that if SA is considered in light of its inherent reference class problem, there are actually several levels in the conclusion of SA: (C1) disproportion; (C2) self-applicability; (C3) unconsciousness (the worrying fact that we are fooled, deceived about our primary identity). In fact, the previous discussion shows that (C1) is true regardless of the chosen (by restriction or extension) reference class: quasi-humans^{–}, quasi-humans^{+}, imperfect simulations and immersion simulations. In addition, (C2) is also true for the original reference class of quasi-humans^{–} — and for immersion simulations, but is false for the class of quasi-humans^{+} and imperfect simulations. Finally, (C3) is true for the original reference class of quasi-humans^{–}, but it proves to be false for quasi-humans^{+}, imperfect simulations and immersion simulations. These three levels of conclusion are represented in the table below:

level

conclusion

case

quasi-humans^{–}

quasi-humans^{+}

imperfect simulations

immersion simulations

C1

the proportion of simulated humans will far exceed that of humans (disproportion)

C1A

true

true

true

true

the proportion of simulated humans will not significantly exceed that of humans

C1Ā

false

false

false

false

C2

we are most likely simulations (self-applicability)

C2A

true

false

false

true

we are most likely not simulations

C2Ā

false

true

true

false

C3

we are unconscious simulations of their simulation nature (unconsciousness)

C3A

true

false

false

false

we are not unconscious simulations of their simulation nature

C3Ā

false

true

true

true

Figure 1. The different levels of conclusion within SA

as well as in the following tree structure:

Figure 2. Treeof the different levels of conclusion of SA

While SA’s original conclusion suggests that there is only one level of conclusion, it turns out, however, as just pointed out, that there are in fact several levels of conclusion in SA, when the argument is examined from a broader perspective, in the light of the reference class problem. The conclusion of the original argument (C3A) is itself worrying and alarming, in that it concludes that there is a much higher probability than we had imagined a priori that we are humans simulated without being aware of it. However, the above analysis shows that, depending on the chosen reference class, some conclusions of a very different nature can be inferred by the simulation argument. Thus, a completely different conclusion is associated with the choice of the reference class of the quasi-humans^{+} or imperfect simulations. The resulting conclusion is that we are not such simulations (C2Ā). Finally, another possible conclusion, itself associated with the choice of the immersion simulation class, is that we are eventually part of such a simulation class, but we are aware of it and therefore it is not a cause for concern (C3Ā).

The above analysis finally highlights what is wrong with the original version of SA, which is at a twofold level. First, the original argument focuses on the class of simulations that are not aware of their own simulation nature. This leads to a succession of conclusions that there will be a greater proportion of simulated humans than authentic humans (C1A), that we are part of simulated humans (C2A) and finally that we are, more likely than we might have imagined a priori, simulated humans unaware of being (C3A). However, as mentioned above, the very notion of human simulation is ambiguous, and such a class can in fact be defined in different ways, given that there is no objective criterion in SA for choosing such a class in a way that is not arbitrary. We can indeed choose the reference class by identifying the simulations with unconscious simulations, i.e. quasi-humans^{–} simulations. But the alternative choice of a reference class that identifies itself with simulations that are conscious of being simulations themselves, i.e. quasi-humans^{+}, has equal legitimacy. In the original argument, there is no objective criterion for choosing the reference class in a non-arbitrary way. Thus, the fact of favoring, in the original argument, the choice of quasi-humans^{–} — with the alarming conclusion associated with them — over quasi-humans^{+}, constitutes a bias, as well as the choice of a reference class that identifies itself with quasi-humans^{+}, leads this time to a reassuring conclusion.

Secondly, it appears that the reference class of SA can be defined at a certain level of restriction or extension. The choice in the original argument of the quasi-humans^{–} — occurs at a certain level of restriction. But if we now move to a certain level of extension, the reference class now includes imperfect simulations. And if we place ourselves at an even greater level of extension, simulations include not only imperfect simulations, but also immersion simulations. But depending on whether the class is chosen at a particular level of restriction or extension, a completely different conclusion will follow. Thus, the choice, at a higher level of extension, of imperfect simulations leads to a reassuring conclusion. Similarly, at an even greater level of extension, which this time includes immersion simulations, there also follows a new reassuring conclusion. Thus, the above analysis shows that in the original version of SA, the choice is made preferentially, by restriction, on the reference class of quasi-humans^{–}, to which is associated a worrying conclusion, as well as a choice by extension, also taking into account imperfect simulations or immersion simulations, leads to a reassuring conclusion.

Can we not object, at this stage, that the above analysis leads to a change in the original scenario of SA and that it is no longer the same problem^{12}? To this, it can be replied that the previous analysis is based on variations in SA that preserve the very structure of the original argument. What this analysis shows is that this same structure is likely to produce conclusions of a very different nature, as long as the reference class is varied within reasonable limits that correspond to the context of SA, and even though the original SA statement suggests a single type of conclusion. Bostrom himself emphasizes that it is the structure of the argument that constitutes its real core: “The structure of the Simulation Argument does not depend on the nature of the hypothetical beings that would be created by the technologically mature civilizations. If instead of computer simulations they created enormous numbers of brains in vats connected to a suitable virtual reality simulation, the same effect could in principle be achieved.” (Bostrom, 2005). In addition, the different levels of extension used here to highlight variations in the SA reference class are intended to illustrate how different levels of conclusion can result. But if we wish to preserve the very form of the original argument, we can then limit the variation of the reference class to what really constitutes the core of this analysis, by considering only a reference class that identifies itself with the quasi-humans. The reference class is then made up of both quasi-humans^{–} and quasi-humans^{+}. This is sufficient to generate a reassuring conclusion — which is not taken into account in the original argument — and thus modify the general conclusion resulting from the argument. In this case, it is the same reference class as the one underlying the original argument, with the only difference that simulations knowing that they are simulated are now part of it. Because the latter, whose possible existence is not mentioned in the original argument, nevertheless have an equal right to legitimacy in the context of SA.

Finally, the preferential choice in the original argument of the quasi-humans^{–} class, appears to be an arbitrary choice that no objective criterion justifies, while other choices deserve equal legitimacy. For the SA statement does not contain any objective element allowing the choice of the reference class to be made in a non-arbitrary manner. In this context, the worrying conclusion associated with the original argument also turns out to be an arbitrary conclusion, since there are several other reference classes that have an equal degree of relevance to the argument itself, and from which a quite reassuring conclusion follows.^{13}^{14}

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1 The Liar is thus both true and false. In the sorites paradox, an object with a certain number of grains of sand is both a heap and a non-heap. Similarly, in Goodman’s paradox, an emerald is both green and grue, and therefore both green and blue after a certain date. Finally, in the Sleeping Beauty paradox, the probability that the piece fell on heads before the awakening of the Sleeping Beauty is 1/2 by virtue of one reasoning mode, and only 1/3 by virtue of an alternative reasoning.

2 William Eckhardt (2013, p. 15) considers that — in the same way as the Doomsday Argument (Eckhardt 1993, 1997, Franceschi, 2009) — the problem inherent in SA comes from the use of retrocausality and the problem related to the definition of the reference class: “if simulated, are you random among human sims? hominid sims? conscious sims?”.

3 We will leave aside here the question of whether an infinite number of simulated humans should be taken into account. This could be the case if the ultimate level of reality were abstract. In this case, the reference class could include simulated humans who identify themselves, for example, with matrices of very large integers. But Bostrom answers such an objection in his FAQ (www.simulation-argument.com/faq.html) and points out that in this case, the calculations are no longer valid (the denominator is infinite) and the ratio is not defined. We will therefore leave this hypothesis aside, focusing our argument on what constitutes the core of SA, i.e. the case where the number of human simulations is finite.

4 The same would be true if simulations were carried out using quantum computers.

5 I thank an anonymous referee for highlighting this point, as well as the point about computers built from components using DNA properties and molecular biology.

6 It seems difficult to rule out here the case where quasi-humans^{–} discover, at least fortuitously, that they are simulated humans, thus becoming quasi-humans^{+} from that moment on. However, in order to advantage the paradox, we will consider here that the very notion of an indistinguishable simulation is not plagued with contradiction.

7 Bostrom (2003) considers that the fact that we live in a simulation would only moderately affect our daily lives: “Supposing we live in a simulation, what are the implications for us humans? The foregoing remarks notwithstanding, the implications are not all that radical”. However, it may be thought that the effect should be much more profound, given that the fundamental level of reality is not where the simulation subjects believe it to be and that, as a result, many of their beliefs are completely erroneous. As David Chalmers (2005) points it out: “The brain is massively deluded, it seems. It has all sorts of false beliefs about the world. It believes that it has a body, but it has no body. It believes that it is walking outside in the sunlight, but in fact it is inside a dark lab. It believes it is one place, when in fact it may be somewhere quite different”.

8 For the purposes of this discussion, we present things as an alternative between quasi-humans^{–} and quasi-humans^{+}. However, one could conceive that post-humans – perhaps different post-human civilizations – create both quasi-humans^{–} and quasi-humans^{+}. We would then have a tripartite situation involving humans, quasi-humans^{–} and quasi-humans^{+}. For the sake of simplicity, we can assimilate here such a situation to the one that prevails when post-humans only create quasi-humans^{–} since it is sufficient that the latter are present in very large numbers to create the worrying effect inherent to SA.

9 This type of bias can be analyzed in one instance of the one-sidedness bias (Walton, 1999, p. 76-81, Franceschi, 2014, p. 587-592) where the reference class is that of the simulations and the associated duality is consciousness/unconsciousness.

10 A complete simulation of a human brain is also called an upload. One definition (Sandberg & Bostrom, 2008, p. 7) is as follows: : “The basic idea is to take a particular brain, scan its structure in detail, and construct a software model of it that is so faithful to the original that, when run on appropriate hardware, it will behave in essentially the same way as the original brain.”

11 The above also shows that when examining SA carefully, it can be seen that the argument contains a second reference class. This second reference class is that of post-humans. What is a post-human? Should we assimilate this class to civilizations far superior to ours, to those that will evolve in the 25th century or the 43th century? Should the descendants of our current human race who will live in the 22nd century be counted among the post-humans if they were to make considerable technological progress in the field of simulations? In any case, the definition of the post-human class appears to be closely linked to that of simulations. Because if we are interested, in a broad sense, in immersion simulations, then post-humans can be assimilated to a generation of humans not very far from us. If we consider imperfect simulations, then they should be associated with a more distant time. On the other hand, if we consider, in a more restrictive sense, simulations of humans that are completely indistinguishable from our current humanity, then we should be interested in post-humans from a much more distant era. Thus, the class of post-humans appears to be closely correlated with that of simulations, because the degree of evolution of simulations is related to the level reached by the post-human civilizations that implement them. For this reason, we shall limit the present discussion to the reference class of the simulations.

12 I thank an anonymous referee for raising this objection.

13 The resulting double weakening of SA finally makes it possible to reconcile SA with our pre-theoretical intuitions, because the worrying scenario of the original argument now coexists with several scenarios of a quite reassuring nature.

14 The present analysis is a direct application to the Simulation Argument of the form of dialectical contextualism described in Franceschi (2014).

I thank two anonymous referees for Philosophiques, for very useful comments on an earlier version of this article.

Cet article présente une solution pour l’Argument de l’Apocalypse (DA). Je montre tout d’abord qu’il n’existe pas de critère objectif pour le choix en général d’une classe de référence: dans ce cas, le calcul inhérent à DA ne peut pas prendre place. En second lieu, j’envisage le choix particulier d’une classe de référence donnée, ainsi que Leslie le recommande. Mais le caractère arbitraire de la sélection rend légitime de multiples possibilités de choix, soit par extension, soit par restriction: DA peut alors être établi en particulier pour le genre Homo, pour l’espèce Homo sapiens, pour la sous-espèce Homo sapiens sapiens, … , pour une classe définie de manière restreinte correspondant aux humains n’ayant pas connu l’ordinateur, etc. Finalement, il apparaît que DA “fonctionne”, mais sa conclusion se révèle inoffensive.

Posprint in English (with additional illustrations from wikimedia commons) of a paper published in French in the Canadian Journal of Philosophy Vol.29, July 1999, pp. 139-56 under the title “Comment l’Urne de Carter et Leslie se Déverse dans celle de Hempel”. I begin by describing a solution to Hempel’s Problem. I recall, second, the solution to the Doomsday Argument described in my previous Une Solution pour l’Argument de l’Apocalypse (Canadian Journal of Philosophy 1998-2) and remark that both solutions are based on a similar line of reasoning. I show thirdly that the Doomsday Argument can be reduced to the core of Hempel’s Problem.

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

The Doomsday Argument and Hempel’s Problem

Postprint – with additional illustrations from Wikimedia commons) – of a paper originally pubslihed in French in the Canadian Journal of Philosophy under the title « Comment l’urne de Carter et Leslie se déverse dans celle de Carter », vol. 29, March 1999, pages 139-156.

Paul Franceschi

I Hempel’s Problem

Hempel’s Problem (thereafter, HP) is based on the fact that the two following assertions:

(H) All ravens are black

(H’) Everything that is non-black is a non-raven

are logically equivalent. The logical structure of (H) is:

(H1) All X are Y

that is to say x (Xx Yx), whereas that of (H’) has the form:

(H1′) All non-Y are non-X

that is to say x (~Yx ~Xx). In fact, the structure of the contrapositive form (H1′) is clearly equivalent to that of (H1). It follows that the discovery of a black raven confirms (H) and also (H’), but also that the discovery of a non-black thing which is not a raven such as a pink flame or even a grey umbrella, confirms (H’) and thus (H). This last conclusion appears paradoxical. The propositions (H1) and (H1′) are based on four properties X, ~X, Y and ~Y, respectively corresponding to raven, non-raven, black, and non-black in the original version of HP. These four properties determine four categories of objects: XY, X~Y, ~XY and ~X~Y, which correspond respectively to black ravens, non-black ravens, black non-ravens and non-black non-ravens. One can observe here that a raven is defined with precision in the taxonomy within which it fits. A category as that of the ravens can be regarded as well defined, because it is based on a set of precise criteria defining unambiguously the species corvus corax and allowing the identification of its instances. It also appears that one can build without difficulty a version of HP where a variation with regard to the X class is operated. If one replace the X class with that of the tulips or that of the dolphins, etc. by adapting correlatively the Y property, one still obtains a valid version of HP. It appears thus that changes can be operated at the level of the X class without loosing the problem inherent to HP.

Similarly, the black property can be specified with precision, on the basis of a taxonomy of colours established with regard to the wavelengths of the light.^{1} Moreover, one can consider variations with regard to the Y property. One will thus be able to choose properties such as whoselength is smaller than 50 cm, living less than 10 years, etc. Such variations also lead to acceptable versions of HP. Lastly, it should be noted that the non-black property can be the subject of a definition which does not suffer from ambiguity, in particular with the help of the precise taxonomy of colours which has been just mentioned. Similarly, if one takes into account variations of the Y property such as smaller than 40 cm, or whose diameter is larger than 25 cm, etc, one arrives to definitions of the non-Y property which just as non-black are established with precision and lead in addition to versions of HP presenting the same problem as the original version. Thus, the X class, just as the properties Y and non-Y can be the subject of a precise and nonambiguous definition. Moreover, variations operated at the level of these classes lead to acceptable versions of HP. In contrast, the situation is not the same for the non-X class.

II The reference class Z

The concept of non-raven present in the original version of HP leads to highlight an important problem. What constitutes an instance of a non-raven? Intuitively a blue jay, a pink flame, a grey umbrella and even a natural integer constitute non-ravens. One is thus confronted with the definition of a new reference class – call it Z – including X and non-X. The Z class allows defining complementarily the class of non-X, and in the original version of Hempel, the class of non-ravens. Thus Z is the implicit reference class with regard to which the definition of the X class allows that of non-X. Does one have then to consider a Z class that goes until including abstract objects? Is it necessary to consider a concept of non-raven including abstract entities such as natural integers and complex numbers? Or is it necessary to limit oneself to a Z class, which only embraces concrete things? Such a discussion has its importance, because there are infinitely many abstract objects, whereas there are only finitely many individualised concrete objects. This fact is likely to influence later importantly the possible application of a bayesian reasoning. One could thus have a reference class Z including at the same time abstract objects (natural integers, real and complex numbers, etc.) and concrete objects such as artefacts but also natural entities such as humans, animals, plants, meteorites, stars, etc. Such a reference class is defined very extensively. And the consequence of such a choice is that the discovery of any object confirms (H’) and thus (H). At this stage, anything^{2} confirms (H). It should be noted that one can also have a definition of Z including all concrete objects that have been just mentioned, but excluding this time the abstract objects.

The instances of this class are now finitely denumerable, just as the cardinal of the corresponding set: the reference class Z then includes animals, plants, stars, etc. But alternatively, one could still consider a Z class associating the ravens (corvus corax) and the Audouin’s gulls^{3} (larus audouinii). In this case, the instances of the X class (corvus corax) are in a number larger than those of the non-X class (larus audouinii). And we always face the corresponding version of HP.^{4}

Lastly, nothing seems to prohibit, at a very restrictive level, to choose a Z class made up of the X class, only added with one single element such as a red tulip. With this definition of Z, we still face a minimal version of HP. Of course, any object, added to the class of X and constituting the non-X class will be appropriate and then confirm at the same time (H’) and (H). Thus, any object ~X~Y will lead to confirm (H). The remarks which have just been made call however an immediate objection. With various degrees, it is allowed to think that the choice of each reference class Z that has been just mentioned is arbitrary. Because it is allowed to reject on those grounds extreme definitions of Z such as the one defined above and including all abstract objects. Similarly, a Z class including the natural integers or the complex numbers can also be eliminated. The X class is defined with regard to the concrete objects that are the ravens and there is not particular reason to choose a Z class including abstract entities.

Similarly, one will be able to reject a definition of Z based on a purely artificial restriction, simply associating with X a determinate object such as a red tulip. Because I can choose arbitrarily, the object that constitutes the complement of X, i.e. I can define Z as I wish. Such an extreme conception appears as without relationship with the initial definition of X. A Z class thus defined is not homogeneous. And there is no justification to legitimate the association of a red tulip to the class of the ravens to build that of Z. The association within a same Z class of the ravens and the Audouin’s gulls, appears analogously as an illegitimate choice. Why not then the association of the ravens and the goldfinches? Such associations are symptomatic of a purely artificial selection. Thus, the choices of reference classes Z mentioned above reveal an arbitrary and artificial nature. Indeed, shouldn’t one make one’s possible to find a Z class which is the most natural and the most homogeneous possible, taking into account the given definition of X? One can think that one must attempt to operate a determination of the Z class, which is the most objective possible. In the original version of HP, doesn’t the choice of the ravens for the X class implicitly determine a Z class which is directly in connection with that of the ravens? A Z class naturally including that of the ravens such as that of the corvidae, or that of the birds, seems a good candidate. Because such a class is at least implicitly determined by the contents of the X class. But before analysing versions of HP built accordingly, it is worth considering before some nonparadoxical versions of HP.

III The analogy with the urn

It is notoriously admitted that certain versions^{5} of HP are not paradoxical. Such is in particular the case if one considers a reference class Z associated with boxes, or a set of playing cards. One can also consider a version of HP associated with an urn. An X class is thus considered where the objects are finitely denumerable and which only includes balls and tetrahedrons. The Y class itself is reduced to two colours: red and green. One has thus four types of objects: red balls, green balls, red tetrahedrons and green tetrahedrons. In this context, we have the following version of HP:

(H2) All balls are red

(H2′) All non-red objects are non-balls

It appears here that the case of red tetrahedrons can be ignored. Indeed, their role is indifferent and one can thus ignore their presence in the urn. They can be regarded as parasitic objects, whose eventual presence in the urn does not have importance. One is thus brought to take into account an urn containing the significant objects consisting in red balls, green balls and green tetrahedrons. And the fact that non-red objects can only be green, and that non-balls can only be tetrahedrons leads to consider equivalently:

(H3) All balls are red

(H3′) All green objects are tetrahedrons

that clearly constitutes a nonparadoxical version of HP. Indeed, the draw of a red ball confirms (H3) and (H3′) whereas the draw of a green tetrahedron confirms (H3′) and (H3).

Consider now the case where the urn contains six significant objects.^{6} One has just drawn three red balls and one green tetrahedron (the draw is 3-0-1^{7}) and one makes then the hypothesis (H3). At this stage, the probability that all balls are red corresponds to three draws (3-0-3, 4-0-2 and 5-0-1) among six possible draws (3-0-3, 3-1-2, 3-2-1, 4-0-2, 4-1-1, 5-0-1). Similarly, the probability that all green objects are tetrahedrons is identical. Thus, P(H3) = P(H3′) = 1/2 and also P(~H3) = P(~H3′) = 1/2. These initial probabilities being stated, consider now the case where one has just carried out a new draw in the urn. Another red ball is drawn (the draw is 4-0-1). This corresponds to three possible compositions of the urn (4-0-2, 4-1-1, 5-0-1). Let E be the event consisting in the draw of a red ball in the urn. We have then the probability of drawing a red ball if all the balls of the urn are red, i.e. P(E, H3) such as P(E, H3) = 2/3, since two cases (4-0-2, 5-0-1) correspond to the fact that all balls are red. In the same way, P(E, ~H3) = 1/3. The situation is identical if one considers P(E, H3′) and P(E, ~H3′). One is then in a position to calculate the posterior probability that all balls are red using Bayes formula: P'(H3) = [P(H3) X P(E, H3)] / [P(H3) X P(E, H3) + P(~H3) X P(E, ~H3)] = (0,5 X 2/3) / (0,5 X 2/3 + 0,5 X 1/3) = 2/3. And P'(~H3) = 1/3. There are identical results concerning P'(H3′) and P'(~H3′). Thus, P'(H3) > P(H3) and P'(H3′) > P(H3′), so that the hypothesis (H3) just as the equivalent hypothesis (H3′) are confirmed by the draw of a new red ball.

Let us examine finally the situation where, instead of a red ball, one draws a green tetrahedron (the draw is 3-0-2) in the urn. Let thus F be the event consisting in the draw of a green tetrahedron. In this case, we have three possible combinations (3-0-3, 3-1-2, 4-0-2). But among these, two (3-0-3, 4-0-2) correspond to a situation where hypotheses (H3) and (H3′) are confirmed. Thus, P(F, H3) = P(F, H3′) = 2/3 and P(F, ~H3) = P(F, ~H3′) = 1/3. A bayesian calculation provides the same results as on the preceding hypothesis of the draw of a red ball. Thus, on the hypothesis of the draw of a green tetrahedron, one calculates the posterior probabilities P'(H3) = P'(H3′) = 2/3 and P'(~H3) = P'(~H3′) = 1/3. Thus, the draw of a green tetrahedron confirms at the same time (H3′) and (H3). It should be noted that one can easily build versions of HP allowing to establish nonparadoxically the preceding reasoning. Consider a cubic mineral block of 1m on side. Such an object of 1m^{3} is divided into 1000 cubic blocks of 1 dm^{3}, consisting either of quartz, or of white feldspar. One examines fifty of these blocks, and one notes that several of them consist of white feldspar of gemmeous quality. One is brought to make the hypothesis that all blocks of white feldspar are of gemmeous quality. We have then the following version of HP:

(H4) All blocks of white feldspar are of gemmeous quality

(H4′) All blocks of non-gemmeous quality are not white feldspar

that is equivalent to:

(H5) All blocks of white feldspar are of gemmeous quality

(H5′) All blocks of non-gemmeous quality are quartz

where we have in effect the equivalence between (H5) and (H5′) and where a correct bayesian reasoning can be established. Such an example (call it the mineral urn) can also be transposed to other properties X and Y, since identical conditions are preserved.

IV A solution to the problem

One must, taking into account the above developments,^{8} attempt to highlight a definition of the Z class that does not present an arbitrary and artificial nature, but proves on the contrary the most natural and the most homogeneous possible, with regard to the given definition of X. Consider accordingly the following^{9} version of HP:

(H6) All Corsican-Sardinian goshawks have a wingspan smaller than 3,50 m

(H6′) All birds having a wingspan larger than 3,50 m are not Corsican-Sardinian goshawks

In this particular version of (H’), the X class is that of the Corsican-Sardinian goshawks,^{10} and the reference class Z is that of the birds. This last class presents an obvious relationship with that of the Corsican-Sardinian goshawks. It is allowed to think that such a way of defining Z with regard to X is a natural one. Indeed such a definition does not present an arbitrary nature as obviously as that was the case with the examples of Z classes mentioned above. Of course, one can observe that it is possible to choose, in a more restricted but so natural way, a Z class corresponding to the accipiter genus. Such a class presents a homogeneous nature. It includes in particular the species accipiter gentilis (northern goshawk) but also accipiter nisus (European sparrowhawk), accipiter novaehollandiae (grey goshawk), accipiter melanoleucus (black and white goshawk).

However, alternatively and according to the same viewpoint, one could also extend the Z class to the instances of the – wider – family of accipitridae^{11} including at the same time the accipiter genus which have been just mentioned, but also the milvus (kite), buteo (buzzard), aquila (eagle), etc. genus. Such a class includes in particular the species milvus migrans (black kite), milvus milvus (red kite), buteo buteo (common buzzard), aquila chrysaetos (golden eagle), etc. These various acceptable definitions of the Z class find their justification in the taxonomy within which the Corsican-Sardinian goshawk inserts itself. More systematically, the latter belongs to the subspecies accipiter gentilis arrigonii, to the species accipiter gentilis, to the accipiter genus, to the family of accipitridae, to the order of falconiformes, to the class of birds, to the subphylum of vertebrates, to the phylum of chordates,^{12} to the animal reign, etc. It ensues that the following variations of (H’) are acceptable, in the context which has just been defined:

(H7′) All northern goshawks having a wingspan larger than 3,50 m are not Corsican-Sardinian goshawks

(H8′) All goshawks having a wingspan larger than 3,50 m are not Corsican-Sardinian goshawks

(H9′) All accipitridae having a wingspan larger than 3,50 m are not Corsican-Sardinian goshawks

(H10′) All falconiformes having a wingspan larger than 3,50 m are not Corsican-Sardinian goshawks

(H11′) All birds having a wingspan larger than 3,50 m are not Corsican-Sardinian goshawks

(H12′) All vertebrates having a wingspan larger than 3,50 m are not Corsican-Sardinian goshawks

(H13′) All chordates having a wingspan larger than 3,50 m are not Corsican-Sardinian goshawks

(H14′) All animals having a wingspan larger than 3,50 m are not Corsican-Sardinian goshawks

There are thus several versions of (H’) corresponding to variations of the Z class which themselves are made possible by the fact that the Corsican-Sardinian goshawk belongs to n categories, determined by the taxonomy to which it belongs. And in fact, when I meet one northern goshawk belonging to the nominal form (accipiter gentilis gentilis), it is at the same time a northern goshawk (accipiter gentilis) non- Corsican-Sardinian (non-accipiter gentilis arrigonii), a goshawk (accipiter) non-Corsican-Sardinian goshawk, an accipitridae non-Corsican-Sardinian goshawk, a falconiformes non-Corsican-Sardinian goshawk, a bird (aves) non-Corsican-Sardinian goshawk, but also a vertebrate non-Corsican-Sardinian goshawk, a chordate non-Corsican-Sardinian goshawk, an animal non-Corsican-Sardinian goshawk. Thus, the instance of accipiter gentilis gentilis that I have just observed, belongs at the same time to all these categories. And when I meet a grey whale, it is not a bird non-Corsican-Sardinian goshawk, but it is indeed a vertebrate non-Corsican-Sardinian goshawk, as well as a chordate non-Corsican-Sardinian goshawk, and also an animal non-Corsican-Sardinian goshawk.

In general, a given object x which has just been discovered belongs to n levels in the taxonomy within which it fits. It belongs thus to a subspecies,^{ 13} a species, a sub-genus, a genus, a super-genus, a subfamily, a family, a super-family, a subphylum, a junction, a reign… One can assign to the subspecies the level^{14} 1 in the taxonomy, to the species the level 2…, to the super-family the level 8, etc. And if within (H), the class X is at a level p, it is clear that Z must be placed at a level q such as q > p. But how to fix Z at a level q which is not arbitrary? Because the reference class Z corresponds to a level of integration. But where must one stop? Does one have to attach Z to the level of the species, the sub-genus, the genus…, the reign? One does not have an objective criterion allowing the choice of a level q among the possibilities that are offered. I can choose q close to p by proceeding by restriction; but in a so conclusive way, I am authorised to choose q distant from p, by applying a principle of extension. Then why choose such class of reference restrictively defined rather than such other extensively defined? One does not have actually a criterion to legitimate the choice, according to whether one proceeds by restriction or by extension, of the Z class. Consequently, it appears that the latter can only be defined arbitrarily. And it follows clearly here that the determination of the Z class and thus of the non-X class is arbitrary. But the choice of the reference class Z appears fundamental. Because according to whether I choose such or such reference class Z, it will result from it that a given object x will confirm or not (H). For any object x, I can build a Z class such as x belongs to non-X, as I can choose a Z class such as x does not belong to non-X. Thus, this choice is left to my arbitrary.

For a given object x, I can build a Z class such as this object confirms (H) and another Z class such as this object does not confirm (H). Of course, if Z is selected arbitrarily, the bayesian reasoning inherent to HP “works”, but corresponds to an arbitrary and artificial point of view: having found an object x, (H) is confirmed. But one can as well choose, in a so artificial and more restrictive way, a Z class where x misses and where x does not confirm (H). Thus, one is not enabled to conclude objectively that the discovery of the object x confirms (H). Because to reason thus would amount to conferring a universal and general value to a viewpoint which is only the expression of an arbitrary choice.

How this result can be reconciled with the facts mentioned above,^{15} concerning the existence of nonparadoxical versions of HP? It is worth noting here that the bayesian reasoning can be established in each case where the Z class is finite, and where this fact is known before the experiment.^{16} One can then show a bayesian shift. But at this stage, it is worth distinguishing the cases where the Z class is determined before the experiment by an objective criterion and the cases where it is not the case. In the first case, the contents of the Z class are given before the experiment and the Z class is thus not selected arbitrarily, but according to an objective criterion. Consequently, the bayesian reasoning is correct and provides relevant information. Such is in particular the case when one considers a version of HP applied to an urn, or a version such as the mineral urn. On this last hypothesis, the composition of the Z class is fixed in advance. There is then a significant difference with Nicod’s criterion:^{17} an object ~X~Y confirms (H) and an object XY confirms (H’).

Conversely, when the Z class is not fixed and is not determined before the experiment by an objective criterion, one can subjectively choose Z at any level of extension or restriction, but the conclusions resulting from the bayesian reasoning must be regarded as purely arbitrary and do not present thus an objective value. Because one then does not have a base and a justification to choose such or such level of restriction or extension. Thus, in this case, Nicod’s criterion according to which any object ~X~Y is neutral with respect to (H) and any object XY is neutral with respect to (H’), can apply itself. It should be observed that the present solution has the effect of preserving the equivalence of a proposition and its contraposition. And similarly, the principle of the confirmation of a generalisation by each of its instances is also preserved.

V A common solution to Hempel’s Problem and the Doomsday Argument

The Doomsday Argument (thereafter, DA) attributed to Brandon Carter, has been described by John Leslie (1992).^{18} DA can be described as follows. Consider an event A: the final extinction of the human race will occur before year 2150. One can estimate at 1 chance from 100 the probability that this extinction occurs: P(A) = 0,01. Let also ~A be the event: the final extinction of the human race will not occur before 2150. Consider also the event E: I live during the 1990s. In addition one can estimate today at 50 billions the number of humans having existed since the birth of humanity: let H_{1997} be such a number. In the same way, the current population can be evaluated to 5 billions: P_{1997} = 5×10^{9}. One calculates thus that one human from ten, if event A occurs, will have known the 1990s. The probability that humanity is extinct before 2150 if I have known the 1990s, is thus evaluated: P(E, A) = 5×10^{9}/5×10^{10} = 0,1. On the other hand, if the human race passes the course of the 2150s, one can think that it will be destined to a much more significant expansion, and that the number of humans will be able to rise for example to 5×10^{12}. In this case, the probability that the human race is not extinct after 2150 if I have known the 1990s, can be evaluated as follows: P(E, ~A) = 5×10^{9}/5×10^{12} = 0,001. This now makes it possible to calculate the posterior probability of the human race extinction before 2150, using Bayes formula: P'(A) = [P(A) x P(E, A)] / [P(A) x P(E, A) + P(~A) X P(E, ~A)] = (0,01 x 0,1) / (0,01 x 0,1 + 0,99 x 0,001) 0,5025. Thus, the fact of taking into account the fact that I live currently has made the probability of the human race extinction before 2150 shift from 0,01 to 50,25.

I have presented in my paper ‘Une Solution pour l’Argument de l’Apocalypse’^{19} a solution to DA, whose main lines can be described as follows. The DA reasoning is based on a single reference class, which is that of the humans.^{20} But how this reference class has to be defined? Should it be limited to the only representatives of our current subspecies Homo sapiens sapiens? Or does one have to extend it to all the representatives of the species Homo sapiens, by including this time, in addition to Homo sapiens sapiens, Homo sapiens neandertalensis…? Or is it necessary to include in the reference class the entire Homo genus, including then all the successive representatives of Homo erectus, Homo habilis, Homo sapiens, etc? And isn’t it still necessary to go until envisaging a wider class, including all the representatives of a super-genus S, made up not only of the Homo genus, but also of the new genus Surhomo, Hyperhomo, etc. which will result from the foreseeable evolutions from our current species? It appears thus that one can consider a reduced reference class by proceeding by restriction, or apprehend a larger class by making the choice of a reference class by extension. One can thus operate for the choice of the reference class by applying either a principle of restriction or a principle of extension. And according to whether one applies one or the other principle, various levels of choice are each time possible.

But it appears that one does not have an objective criterion, which makes it possible to legitimate the choice of such or such a reference class. And even our current subspecies Homo sapiens sapiens cannot be regarded as a natural and an adequate choice for the reference class. Because isn’t it allowed to think that our paradigmatic concept of human has to undergo evolutions? And in addition, the fact of excluding from the reference class a subspecies such as Homo sapiens neandertalensis or the future evolutions of our species, doesn’t it reveal an anthropocentric viewpoint? Since one does not have an objective selection criterion, one can choose arbitrarily one or the other of the classes that have been just described. One can for example identify the reference class to the species Homo sapiens, and observe a bayesian shift. There is indeed then an increase in the posterior probability of the extinction of Homo sapiens. But this bayesian shift is worth as well for a still more restricted reference class, such as our subspecies Homo sapiens sapiens. There too, the application of Bayes formula leads to an appreciable increase in the posterior probability of the nearest end of Homo sapiens sapiens. However identically, the bayesian shift also applies to a still more reduced reference class, which is that of the representatives of Homo sapiens sapienshaving not known the computer. Such a reference class will certainly face a nearest extinction. There however, such a conclusion is not likely to frighten us, because the evolutionary potentialities of our species are such that the succession of a new species to those which preceded them, constitutes one of the characteristics of our evolution mode.

It should be mentioned that this solution leads here to accept the conclusion (the bayesian shift) of Carter and Leslie for a given reference class, while placing it in comparison with conclusions of comparable nature relating to other reference classes, completely inoffensive. The fact of taking into account various levels of restriction, made legitimate by the lack of an objective criterion of choice, leads finally to the harmlessness of the argument. Thus, it appears that the argument based on the reference class and its arbitrary choice by restriction or extension constitutes a common solution to HP and DA. HP and DA are ultimately underlain by the same problem inherent to the definition of the Z class of HP and the single reference class of DA. One thus has a solution of comparable nature for the two paradoxes. It is worth here concluding by presenting an element that tends to confirm the common source of the two problems. One will observe first that one is not able to highlight a version of DA corresponding veritably to the original version of HP, a reference class such as that of the ravens being not transposable in DA. The inherent argument in DA is indeed based on the use of the anthropic principle and requires obviously a reference class made up of intelligent beings. When Leslie^{21} considers the extension of the reference class, he specifies expressly that the condition for the membership of the reference class is the aptitude to produce an anthropic reasoning. On the other hand it is possible to describe a version of HP made up from the elements of DA. If one takes X for our current subspecies Homo sapiens sapiens and Y for are alive only before 2150, one obtains the following version of HP:

(H15) All Homo sapiens sapiens will be alive only before the year 2150

(H15′) All those which will live after 2150 will be non-Homo sapiens sapiens

In this context, an alive human being in 1997 constitutes an instance confirming (H15). In parallel, the discovery of an Homo sapiens sapiens after 2150 leads to refute (H15). Lastly, the discovery of an alive non-Homo sapiens sapiens after 2150 constitutes a confirmation of (H15′) and thus of (H15). Taking into account this particular formulation, it is clear that one currently only observes instances confirming (H15). On the other hand, after 2150, one will be able to have instances refuting (H15) or instances confirming (H15′).

It is worth noting here that (H15) does not allow veritably to be used as support of a version of DA. Indeed, the reference class identifies itself here precisely as Homo sapiens sapiens, whereas in the original version of DA, the reference class consists in the human race. Consequently, one has not, strictly speaking, an identity between the event underlie by (H15) and A, so that (H15)-(H15′) does not constitute a joint version^{22} of DA and HP.

But this version of HP being made up with the elements of DA, one must be able, at this stage, to verify the common origin of the two problem, by showing how the argument raised in defence of DA with regard to the reference class, can also be used in support of HP. One knows the response made by Leslie to the objection that the reference class for DA is ambiguous or, due to the evolutions of Homo sapiens sapiens, leads to a heterogeneous reference class, of composite nature. It is exposed in the response made to Eckhardt:

How far should the reference class extend? (…) One can place the boundary more or less where one pleases, provided that one adjusts one’s prior probability accordingly. Exclude, if you really want to, all future beings with intelligence quotients above five thousand, calling them demi-gods and not humans^{23}.

and developed in The End of the World^{24}:

The moral could seem to be that one’s reference class might be made more or less what one liked. (…) What if we wanted to count our much-modified descendants, perhaps with three arms or with godlike intelligence, as ‘genuinely human’? There would be nothing wrong in this. Yet if we were instead interested in the future only of two-armed humans, or of humans with intelligence much like that of humans today, then there would be nothing wrong in refusing to count any others^{25}.

For Leslie, one can go until including in the reference class, the descendants of humanity become very distant from our current species due to the fact of evolution. But Leslie also accepts liberally that one limits the reference class to the only individuals close to our current humanity. One is thus free to choose the reference class that one wishes, while operating either by extension, or by restriction. It will be enough in each case to adjust the initial probability accordingly. It appears here that this type of answer can be transposed, literally, to an objection to HP of comparable nature, based on the reference class of (H15)-(H15′). One can fix, so the objection goes, the Z class as one wishes, and assign to “all those” the desired content. One can for example limit Z to the species Homo sapiens, or well associate it to the whole of the Homo genus, including then the evolutions of our species such as Homo spatialis, Homo computeris, etc. What is important – could continue this defender – is to determine preliminarily the reference class and to conserve this definition when the various instances are then met. Thus, it proves that the arguments advanced in support of the reference class of DA can be transposed in defence of HP. This constitutes an additional element, going in the direction of the common origin of the two problems, dependent on the definition of a reference class. DA and HP need consequently a same type of answer. Thus, the urn of Carter and Leslie flows in that of Hempel.^{26}

References

ECKHARDT, W. 1993. “Probability Theory and the Doomsday Argument.” Mind, 102 (1993): 483-8

FRANCESCHI, P. 1998, “Une Solution pour l’Argument de l’Apocalypse.” Canadian Journal of Philosophy, 28 (1998): 227-46

GOODMAN, N. 1955. Fact, Fiction and Forecast. Cambridge: Harvard University Press.

HEMPEL, C. 1945. “Studies in the logic of confirmation.” Mind, 54 (1945): 1-26 et 97-121

LESLIE, J. 1992. “Time and the Anthropic Principle.” Mind, 101 (1992): 521-40

—. 1993. “Doom and probabilities.” Mind, 102 (1993): 489-91

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

PAPINEAU, D. 1995. “Methodology: the Elements of the Philosophy of Science.” In Philosophy A Guide Through the Subject, ed. A.C. Grayling. Oxford: Oxford University Press.

SAINSBURY, M. 1988. Paradoxes. New York: Cambridge University Press.

THIBAULT, J-C. 1983. Les oiseaux de Corse. Paris: De Gerfau.

1 It is known that a monochromatic light, of single wavelength, meets practically only in laboratory. But the natural colours can be modelled in terms of subtraction of lights of certain wavelengths, starting from the white light of the Sun.

2 Any object ~X~Y in the Z class thus extensively defined.

3 The total population of Audouin’s gulls is evaluated with approximately 3000 couples (cf. Thibault 1983, 132).

4 This incidentally makes it possible to verify that HP does not find its origin in a disproportion of the X class compared to that of the non-X. The fact that the instances of the X class are in a number larger than those of the non-X does not prevent the emergence of a version of HP.

5 Properly speaking, these are not thus versions of HP, since they are nonparadoxical. But the corresponding propositions have the logical structure of (H) and (H’).

6 The red tetrahedrons possibly found in the urn are regarded as nonsignificant objects.

7 With the notation: n–p–q (red balls – green balls – green tetrahedrons).

9 This particular version of HP is chosen here because it is based on an X class corresponding to the subspecies accipiter gentilis arrigonii. Conversely, the original version of HP is grounded on the species corvus corax. The choice of a subspecies for the X class allows simply here a supplementary level of integration.

10 The Corsican-Sardinian goshawks (accipiter gentilis arrigonii) constitute a subspecies of the northern goshawk, specific to Corsica and Sardinia. This endemic subspecies differs from the nominal form of the northern goshawk by the following characteristics (cf. Thibault 1983): the colouring of the head is blackish instead of brown blackish; the back is brown; the lower part is darker.

11 The ornithologists still distinguish the class of the accipitriformes, corresponding to all accipitridae, to which are added the pandlionidae, such as pandlion haliaetus (osprey), etc.

12 The phylum of chordata includes all vertebrates and some invertebrates, which present the property of having a dorsal chord, at least at a given period of their life.

13 It is possible to consider alternatively, if one wishes, another taxonomy that our current scientific taxonomy. That does not affect the current reasoning, since the conclusions are identical, since the principles of classification are respected.

14 It is obviously possible to take into account finer taxonomies and including additional subdivisions starting from the various subspecies. Obviously, that does not affect the current line of reasoning.

16 As we have seen, the bayesian reasoning cannot take place when one considers a Z class including infinite sets such as natural integers, real numbers, etc.

17 Nicod’s criterion is defined as follows (Hempel 1945, 11), with S1 = (H) and S2 = (H’): ‘(…) let has, B, C, D Be furnace objects such that has is has raven and black, B is has raven goal not black, C not has raven goal black and D neither has raven NOR black. Then, according to Nicod’ S criterion, has would confirm S1, goal Be neutral with respect to S2; B would disconfirm both S1 and S2; C would Be neutral with respect to both S1 and S2, and D would confirm S1, goal Be neutral with respect to S2.’

18 John Leslie, ‘Time and the Anthropic Principle.’ Mind, 101 (1992): 521-40.

19Canadian Journal of Philosophy 28 (1998) 227-46.

21 ‘How much widening of the reference class is appropriate when we look towards the future? There are strong grounds for widening it to include our evolutionarily much-altered descendants, three-armed or otherwise, as ‘humans’ for doomsday argument purposes – granted, that’s to say, that their intelligence would remain well above the chimpanzee level.’ (1996, 262)

22 I.e. comprising simultaneously the two problems.

23 W. Eckhardt, ‘Probability Theory and the Doomsday Argument.’ Mind, 102 (1993): 483-8; cf. John Leslie, ‘Doom and probabilities.’ Mind, 102 (1993): 489-91

24 This point of view is detailed by Leslie, in the part entitled ‘Just who should count have being human?’ (The End of the World, 256-63).