The Simulation Argument and the Reference Class Problem: the Dialectical Contextualist’s Standpoint

chap31Preprint. I present in this paper an analysis of the Simulation argument from a dialectical contextualist standpoint. This analysis is grounded on the reference class problem. I begin with describing Bostrom’s Simulation Argument step-by-step. I identify then the reference class within the Simulation argument. I also point out a reference class problem, by applying the argument successively to several references classes: aware-simulations, rough simulations and cyborg-type 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 preprint supersedes my preceding work on the Simulation argument. Please do not cite previous work.

Comments are welcome.

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December 2016: An updated version of my analysis of the Simulation Argument has appeared in the canadian Philosophiques journal (in French) under the
title: L’argument de la Simulation et le problème de la classe de référence : le point de vue du contextualisme dialectique

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The Simulation Argument and the Reference Class Problem:

the Dialectical Contextualist’s Standpoint

1. The Simulation Argument

I shall propose in what follows a solution to solve the problem posed by the Simulation argument, recently described by Nick Bostrom (2003). I shall first attempt to describe in detail the Simulation argument, by exposing in particular its inherent problem. I will show then how a solution can be brought to such a problem, based on the analysis of the reference class which underlies the Simulation argument, and without it being necessary to give up one’s pretheoretical intuitions.

The general idea which underlies the Simulation argument (SA, for short) can be expressed as follows. It is very likely that post-human civilizations will possess a calculus computing power completely out of proportion with the one which is ours at present time. Such an extraordinary computing power should confer them the capacity to implement completely realistic human simulations, such in particular that the inhabitants of these simulations would be conscious of their own existence, in every respect similar to ours. In such a context, we can think that it is likely that post-human civilizations will actually dedicate a part of their computing resources to realize simulations of the human civilizations which preceded them. In this case, the number of the simulated human beings should very largely encompass that of the genuine human beings. In such conditions, the fact of taking into account the mere fact that we exist leads to the conclusion that it is more likely that we belong to the simulated human beings, rather than to the genuine ones.

Bostrom also manages to describe the Simulation argument accurately. He underlines that SA is based on the following three hypotheses:

(1)

humanity will face a nearest extinction

(2)

the post-human civilizations will not realize human beings’ simulations

(3)

we currently live in a simulation realized by a post-human civilization

The first step of the reasoning consists in considering, by dichotomy, that either (i) humankind will face a nearest extinction, or (ii) she will pursue its existence in the distant future. The first of these two hypotheses constitutes the disjunct (1) of the argument. We consider then the hypothesis according to which the humanity will not face a nearest extinction and will then pursue its existence through numerous millenniums. In such a case, we can also consider that it is likely that the post-human civilizations will possess at the same time the technology and the required capacities to realize human beings’ simulations. A new dichotomy then follows: either (i) these post-human civilizations will not realize such simulations—it consists of the disjunct (2) of the argument; or (ii) these post-human civilizations will actually realize such simulations. In this last case, it will follow that the number of simulated human beings will largely exceed that of the human beings. The probability to live in a simulation will thus be much greater than that to live as an ordinary human being. It follows then the conclusion that we, inhabitants of the Earth, probably live in a simulation realized by a post-human civilization. This last conclusion constitutes the disjunct (3) of the argument. An additional step leads then to consider that in the lack of any evidence in favour of the one or the other of them, we can consider the hypotheses (1), (2) and (3) as equiprobable.

The Simulation argument can thus be described step-by-step as follows:

(4)

either humankind will face a nearest extinction, or humankind will not face a nearest extinction

dichotomy 1

(1)

humankind will face a nearest extinction

hypothesis 1.1

(5)

humankind will not face a nearest extinction

hypothesis 1.2

(6)

the post-human civilizations will be capable of realizing human beings’ simulations

from (5)

(7)

either the post-human civilizations will not realize human beings’ simulations, or they will realize them

dichotomy 2

(2)

the post-human civilizations will not realize human beings’ simulations

hypothesis 2.1

(8)

the post-human civilizations will realize human beings’ simulations

hypothesis 2.2

(9)

the proportion of the simulated human beings will very largely exceed that of the human beings

from (8)

(3)

we currently live in a simulation realized by a post-human civilization

from (9)

(10)

in the lack of evidence in favour of one of them, the hypotheses (1), (2) and (3) are equiprobable

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

It is also worth mentioning an element which results from the interpretation of the argument. For as Bostrom himself (2005) notes, the Simulation argument must not be wrongly interpreted. It is not indeed an argument which leads to the conclusion that (3) is true, namely that we currently live in a simulation realized by a post-human civilization. The core of the Simulation argument lies then in the fact that the propositions (1), (2) or (3) are equiprobable.

This nuance of interpretation being mentioned, the Simulation argument does not however miss to raise a problem. For the argument leads to the conclusion that one of the propositions (1), (2) or (3) at least is true, and that in the situation of ignorance where we are, we can consider them as equiprobable. As Bostrom himself puts it: In the dark forest of our current ignorance, it seems sensible to apportion one’s credence roughly evenly between (1), (2) and (3). (Bostrom 2003). However, according to our pre-theoretical intuition, the probability of (3) is null or at best extremely close to 0. So, the conclusion of the argument has for consequence to make pass the probability that (3) is the true, from zero to a probability of about 1/3. So, the problem posed by SA is precisely that it makes shift—via its disjunctive conclusion—a probability of zero or of near zero concerning (3) to a much more considerable probability of about 1/3. For a probability of 1/3 concerning proposition (1) and (2) has nothing shocking a priori, but reveals itself on the other hand completely counter-intuitive as regards proposition (3). It is in this sense that we can speak of the problem posed by the Simulation argument and of the need of looking for a solution to the latter.

In a preliminary way, it is worth wondering about what constitutes the paradoxical nature of SA. What is it indeed that confers a paradoxical aspect to SA? For SA distinguishes itself from the class of paradoxes that lead to a contradiction. In the paradoxes such as the Liar or the sorites paradox, the corresponding reasoning leads to a contradiction: the Liar is both true and false. In the sorites paradox, an object containing a certain number of grains of sand is both a heap and a non-heap. No such thing manifests itself at the level of SA which belongs, from this point of view, to a different class of paradoxes of which is also part the Doomsday argument. It consists indeed of a class of paradoxes the conclusion of which presents a counter-intuitive nature, and comes in conflict with the whole set of our beliefs. In the Doomsday argument, the conclusion according to which the fact of taking into consideration our own birth rank within the class of the human beings having never existed results in the fact that a doomsday is much more likely than we would have possibly envisaged initially, comes to strike the set of all our beliefs. In a similar way, what appears finally here as paradoxical, in first analysis, is that SA leads to a probability of the hypothesis according to which we currently live in a simulation created by post-humans, which is greater than the one which results from our pre-theoretical intuition.

2. The reference class within the Simulation Argument

The conclusion of the reasoning which underlies SA, based on the calculation of the future ratio between the real human beings and the simulated ones, as it proves to be counter-intuitive, results nevertheless from a reasoning which seems a priori valid. However, such a reasoning leads to an interrogation, which is associated with the reference class which is inherent to the argument itself.1 Indeed, it turns out that SA contains, in an indirect way, a specific class of reference, which is that of the human beings’ simulations. But what is it then that constitutes a simulation? The original argument refers, in an implicit way, to a reference class which is that of virtual simulations of human beings, of a very high quality and by nature indiscernible from the genuine ones. However, a certain ambiguity lies in the mere notion of simulation and the question arises of the applicability of SA to other types of human beings’ simulations.2 We can indeed conceive of somewhat different types of simulations which, in an intuitive way, also enter the scope of the argument.

It is possible to imagine, first, a type of simulations in every respect identical to those described in the original argument, i.e. almost indiscernible from genuine human beings, but with the only difference that they would be aware of their own nature of simulation. The only difference with the type of simulation described in the original argument would thus be that these last simulations would clearly be conscious of being not authentic human beings. A priori, nothing excludes that the post-humans would choose to implement any such simulations and intuitively, SA is also susceptible of applying to this particular type of simulations.

In the same way, SA refers implicitly to sophisticated simulations, of very high quality, which are by nature indiscernible from authentic human beings. However, we can conceive of various degrees in the quality of the human simulations. So the question notably arises of whether we can include in the reference class of SA some virtual simulations of a very slightly lower quality? With any such simulations, the nature of simulation which constitutes their deep identity would be susceptible of being one day discovered by the very subject. If the argument has to apply to this class of simulations, the question then arises of its applicability to other types of simulations of this nature, because we can conceive of numerous intermediate degrees between on the one hand, the indiscernible simulations and on the other hand, the simulations which we are currently capable of realizing, notably by means of computer generated images. So, the question does arise of whether the reference class of SA can go as far as to include simulations of lower quality than those evoked in the original argument?

Finally, it appears that SA also works if we apply it to human beings whose brain is interfaced with uploads, simulations of the human mind including memorized events, knowledge, personality’s traits, ways of reasoning, etc. relative to a given individual. We can imagine indeed that in a not very distant future, the emulation of the human brain could be achieved (Moravec 1998, Sandberg & Bostrom 2008, Garis & al. 2010), so that the realization of uploads could become common and intensively used. A very large number of uploads could be so realized and used in different purposes: scientific, cultural, social, utilitarian, etc. If we assimilate then the uploads to the simulations of SA, the argument also works. In a sense, the human beings endowed with uploads can be considered as simulations of partial nature, which only concern the brain or a part of the brain, even though the rest of the human body remains authentic and not simulated. In such a case, the human beings of which only the brain is simulated by means of an upload, can be assimilated to a particular type of cyborgs. We can so raise the general question of to what extent the class of the simulations of SA can be widened to partial simulations and to the types of cyborgs who have just been described. We can indeed conceive of cyborgs of various types, depending on the parts of the body and organs of replacement or substitution which are theirs. So the question does arise of to what extent SA also applies to this type of cyborgs?

As we can see it, the very question of the definition of the reference class for SA leads to wonder about the inclusion or not within the scope of SA of several types of simulations. Without pretending to be exhaustive, we can mention at this stage, among the latter: aware-simulations, the more or less rough-simulations and the partial simulations of a cyborg-type. The question of the definition of the reference class for SA seems then closely related to the nature of the future taxonomy of the beings and creatures which will populate the Earth in a near or distant future.

At this step, it turns out in first approach that the types of human beings’ simulations present a somewhat varied nature, and that we can define the reference class of the simulations in several ways. We could then choose the reference class in a more or less restrictive or extensive way. In this context, it is worth delving more deeply into the consequences of the one or the other choice.

3. The reference class problem: the aware-simulations case

At this stage, we still cannot speak veritably of a reference class problem within SA. For it indeed, we need to show that the choice of the 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 finds itself modified in a fundamental way. In what follows, we shall from now on attempt to show that according to the choice of the one or the other reference class, some radically different conclusions follow at the level of the very argument and that consequently, there exists well a reference class problem within SA. For this purpose, we shall consider successively several reference classes, by attaching ourselves to show that some conclusions of a fundamentally different nature result from them at the level of the argument itself.

The original version of SA stages implicitly human beings’ simulations of a certain type. They consist of virtual type simulations, almost indiscernible for ourselves and which present then a very high degree of sophistication. More still, they consist of a type of simulations which are not aware that they are themselves simulated and which are thus persuaded to be genuine human beings. This results implicitly of the terms of the argument itself and in particular, of the inference from (9) to (3) which leads to conclude that we currently live in an indiscernible simulation realized by the post-humans. In fact, it consists of simulations which are somewhat abused and deceived by the post-humans as regards their true identity. For the needs of the present discussion, we shall term quasi-humans the simulated human beings who are not aware that they are themselves simulated.

At this step, it turns out that we can also conceive of indiscernible simulations which present a completely identical degree of sophistication but which, on the contrary, would be aware that they are simulated. We shall then term quasi-humans+ those simulated human beings who are aware that they are themselves simulations. Such simulations are in every respect identical to the quasi-humans to which SA refers implicitly, with the only difference that they are this time clearly aware of their intrinsic nature of simulation. In an intuitive way, SA also applies to this type of simulation. A priori, we lack the justification to exclude such a type of simulations. More still, several reasons lead to think that the quasi-humans+ could be more numerous than the quasi-humans. For ethical reasons (i) first, we can think that the post-humans could be inclined to prefer quasi-humans+ to quasi-humans. For the fact of conferring an existence to the quasi-humansconstitutes a deceit on their real identity, while such an inconvenient is absent with regard to quasi-humans+. Such a deceit could be reasonably considered as unethical and lead to one or the other form of quasi-humans interdiction. Another reason (ii) militates for the fact of not pushing aside a priori those human beings’ simulations that are aware of their own nature of simulation. We can think indeed that the level of intelligence acquired by certain quasi-human beings in a near future could be extremely high and make that in this case, the simulations would become very quickly aware that they are themselves simulations. We can think that starting from a certain degree of intelligence, and in particular the one susceptible of being obtained by humankind in a not very distant future (Kurtzweil 2000, 2005, Bostrom 2006), the quasi-humans should be able—at least much more easily than at present—to collect the evidence that they are the object of a simulation. Moreover, the concept of ‘simulation which is unaware that it is a simulation could be plagued with contradiction, because it would then be necessary to limit its intelligence and from then on, it would not consist any more of an indiscernible and enough realistic simulation. These two reasons suggest that the quasi-humans+ could well exist in greater number than the quasi-humans.

At this stage, it turns out to be necessary to envisage the consequences of the consideration of the quasi-humans+ within the reference class of the simulations inherent to SA. For that purpose, let us consider first the variation of SA (let us call it SA*) which applies, in an exclusive way, to the class of the quasi-humans+. Such a choice has no consequence, first, on the disjunct (1) of SA, which refers to a possible next disappearance of our humanity. It has no effect either on the disjunct (2), according to which the post-humans will not realize quasi-humans+, i.e. aware-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 with the original argument, the first-level conclusion according to which the number of quasi-humans+ will largely exceed the number of genuine human beings (the disproportion). However, from now, the second-level conclusion according to which we currently are quasi-humans+, does not follow any more. Indeed, such a conclusion (let us term it the self-applicability) does no longer apply to us from now on, since we are not conscious of being simulated and are fully convinced of being genuine human beings. In effect, what constitutes the disturbing conclusion of SA does not result any more from now on from step (9), for we cannot identify ourselves with the quasi-humans+, the latter being clearly aware that they live in a simulation. So, unlike the original version of SA based on the reference class which associates the human beings with the quasi-humans, this new version associating the human beings and the quasi-humans+, is not related to such a disturbing conclusion. The conclusion which follows from now on, as we can see it, turns out to be completely reassuring, and in any case very different from that, profoundly disturbing, which results from the original argument.

At this step, it turns out that a question arises: must we identify, in the context of SA, the reference class with the quasi-humans or with the quasi-humans+? It turns out that no objective element, in the statement of SA, comes to justify the a priori choice of the quasi-humans or of the quasi-humans+. So, any version of the argument which contains the preferential choice of either the quasi-humans or the quasi-humans+ can be considered as exemplifying a bias. Such is then the case for the original version of SA, which contains then a bias in favour of the quasi-humans, which results from the choice by Bostrom of a class of simulations which assimilates itself exclusively with the quasi-humans, i.e. simulations which are unaware of their nature of simulations and which are consequently abused and deceived by the post-humans on the true nature of their identity. And such is also the case for SA* the alternate version of SA which has been just described, which contains a specific bias in favour of the quasi-humans+, i.e. simulations that are aware of their own nature of simulation. However, the choice of the reference class proves here to be fundamental, for it contains an essential consequence: if we choose a reference class which associates the human beings with the quasi-humans, it results from it the disturbing conclusion that we currently very probably live in a simulation. On the other hand, if we choose a reference class which associates the human beings with the quasi-humans+, if follows a scenario which in a reassuring way, does not entail such a conclusion. At this stage, it appears that the choice of the quasi-humans, i.e. unaware-simulations, in the original version of SA, to the detriment of aware-simulations, constitutes an arbitrary choice. In effect, what allows to prefer the choice of the quasi-humans over the quasi-humans+? Such a justification is lacking in the context of the argument. At this stage, it turns out that the original argument of SA contains a bias which leads to the preferential choice of the quasi-humans, and to the alarming conclusion which is associated with it. This remark being made, it is worth considering now the problem under a still wider perspective, by taking into consideration other possible types of simulations.

4. The reference class problem: the rough-simulations case

Robotic hand by Richard Greenhill and Hugo Elias (wiki commons)

The reference class problem within SA bears, as mentioned above, on the very nature and the type of simulations implemented within the argument. Does this problem limit itself to the preferential choice, at the level of the original argument, of unaware-simulations, to the detriment of the alternate choice of aware-simulations, which correspond to human beings’ very sophisticated simulations, capable of creating the illusion, but endowed with the awareness that they are themselves simulations? It seems not. Indeed, as mentioned above, we can also conceive of other types of simulations for which the argument also works, but which are themselves of a slightly different nature. In particular, we can imagine that the post-humans will conceive of and implement simulations that are identical to those of the original argument, but who do not however present a so perfect character. Such a situation presents a completely likely nature and does not present the ethical drawbacks which could accompany the indiscernible simulations hinted at in the original argument. The choice of realizing this type of simulations could result either from the required technological level, or from deliberate and pragmatic choices, intended to save time and resources. We can so conceive of various degrees in the realization of such type of simulations. It could involve for example simulations of very good quality, the artificial nature of which our current scientists could only discover after, say, ten years of research. But in an alternative way, such simulations could be of average quality, even rather rough, with regard to the above-mentioned almost indiscernible simulations. For the sake of the present discussion, however, we shall call rough-simulations this whole category of simulations.

What are then the consequences on SA of the consideration of a reference class which assimilates itself with rough-simulations? In such circumstances, a large number of such simulations would be detectable by us human beings. In this case, the first-level consequence based on the human beings/simulations disproportion always applies, in the same way as with the original argument. On the other hand, the second level conclusion based on the self-applicability does not apply any more now. For we can no longer conclude from now on that we are simulations, since in the presence of any such simulations, we would quickly notice that they consist of simulated human beings and are not genuine human beings. So, in such a case, it turns out that the alarming conclusion inherent to the original version of SA and based on the self-applicability does no longer apply. A reassuring conclusion substitutes indeed itself to it, based on the fact that we human beings do not belong to this type of simulations.

Robotic leg by David Buckley (Wiki commons)

Robotic leg by David Buckley (Wiki commons)

At this stage, it appears that SA, in its original version, opts for the preferential choice of very sophisticated, undetectable simulations by us human beings and unaware of their nature of simulation. But as mentioned above, we can conceive of other types of simulations, of a more unrefined nature, to which the argument also applies. Until what level of detectable simulation can we go? Do we have to go as far as including in the reference class, at a higher level of extension, rather unrefined simulations, such as for example some improved versions of the simulations that which we are already capable of realizing by means of computer generated images? In this case, this leads to a slightly different formulation from the original argument, for we can then assimilate the class of the post-humans to the human beings who will live on Earth in ten years, or even in one year, or even—at a much greater level of extension—in one month. In this case, the disjunct (1) according to which humankind will not last until this time does no longer prevail, since such a technological level has already been reached. Also, the disjunct (2) has no longer any raison d’être, since we already realize such unrefined simulations. Thus, there only remains in this case the disjunct (3), which constitutes then the unique proposition which underlies the argument and constitutes the first-level conclusion of SA, according to which the number of the simulated human beings will largely encompass that of the genuine human beings. In this case, it follows well, in an identical way as with the original argument, the first-level conclusion according to which the number of quasi-humans+ will largely exceed the number of genuine human beings (the disproportion). But there also, the second-level conclusion according to which we currently are quasi-humans+ (the self-applicability) does no longer follow. The latter does no longer apply to us from now on and a conclusion of reassuring nature substitutes itself to it, since we are clearly aware of being not such rough-simulations.

5. The reference class problem: the cyborg-case

As evoked above, another question that arises is whether the reference class can be widened to the cyborgs and in particular to this category of cyborgs who are indiscernible from human beings. We can indeed conceive of various types of cyborgs, going from those for which some parts of the body have been replaced by synthetic organs of substitution or more powerful, to those for which almost all organs—including the brain—have been replaced. A priori, such a class also enters the field of the argument. Here, the argument applies naturally to those elaborate, indiscernible from human beings cyborgs, for which a large part of the original organs have been replaced or transformed. In particular, the cyborgs for which a part of the brain was replaced by a—partial or not—upload enters naturally the field of the argument. Partial uploads are the ones for which only one part of the brain has been replaced by an upload. We can also imagine numerous types of uploads of this kind: uploads which reconstitute the memory by restoring the forgotten events can be so envisaged. They can prove themselves useful not only for healthy people, but also for those who suffer from diseases in which the memory functions are altered. We can conceive of that such types of partial uploads could be implemented in a more or less close future (Moravec 1998, Kurzweil 2005, Garis & al. 2010). And in the same way as with the original argument, we can conceive of that very large quantities of these uploads could be realized by computing means. In a general way, it turns out that the discussion about the inclusion of the cyborgs within the reference class of SA has its importance, because if we consider the class of the cyborgs in a wide sense, we are nearly already all cyborgs. If we indeed consider that organs or parts of the human body that have been replaced or improved so that they work correctly makes us cyborgs, such is today already the case, given the generalization of synthetic teeth, pacemakers, prostheses, etc. So the question arises of up to which degree we can include certain types of cyborgs within the scope of the argument.

What would then be the effect on SA of taking into account the class of the partial cyborgs, if we place ourselves at such a degree of extension? As well as for the rough-simulations, it turns out that the disjunct (1) according to which the human beings will not reach until this time does no longer prevail, since such a technological stage is already levelled off. In an identical way, the disjunct (2) does not justify itself any more either, because in such a context, we are already almost all any such partial cyborgs. So, the disjunct (3) only remains in this case as a unique proposition, but which emerges however under a different form from that of the original argument. In effect, the first-level consequence based on the human beings/simulations disproportion also applies here, in the same way as with the original argument. In addition, and it is an important difference here, the second-level conclusion based on the self-applicability also applies, since we can conclude from it that we are also, in this wide sense, simulations. On the other hand, the alarming conclusion of the original argument that we are unaware-simulations, which manifests itself at a third level, does no longer follow, since the fact that we are simulations in this sense does not involve here that we are deceived on our true identity. So there finally ensues, unlike the original argument, a reassuring conclusion: we are simulations, who are fully aware of their own nature of partial cyborgs.

What precedes also shows that by examining SA with attention, we can notice that the argument holds a second reference class. This second reference class is that of the post-humans. What is then a post-human being? Must we assimilate this class to those civilizations that are very widely superior to ours, to those who will evolve either in the XXVth century or in the XLIIIth century? Must the descendants of our current human race who will live in the XXIIth century be counted among the post-humans? The fact that important evolutions associated with the increase of human intelligence (Moravec 1998, Kurzweil 2005) can arise in a more or less close future, constitutes in particular an argument which supports this assertion. But must we go as far as including the descendants of the current human beings who will live on Earth in 5 years? Such questions arise and require an answer. The question of how we have to define the post-humans, also constitutes then an element of the reference class problem of SA. In any case, the definition of the post-humans‘ class seems closely related to that of the simulations. For if one considers, in a wide sense, those cyborgs hardly more evolved than we currently are in a certain sense, then the post-humans can be assimilated with the human beings’ next generation. The same goes if we consider rough-simulations slightly improved with regard to those that we are currently capable of producing. On the other hand, if we consider, in a more restrictive sense, simulations that are completely indiscernible from our current humanity, it is then worth considering post-humans of a clearly more distant time. In any case, it appears here that the reference class of the post-humans, as well as the class of the simulations which is associated with it, can be chosen at different levels of restriction or of extension.

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

Finally, the foregoing discussion emphasizes the fact that if we consider SA in the light of its inherent reference class problem, there are in reality several levels in the conclusion of SA: (C1) the disproportion; (C2) the self-applicability; (C3) the unawareness (the worrying fact that we are abused, deceived on our true identity). In fact, the previous discussion shows that (C1) is true whatever the chosen reference class (by restriction or by extension): the quasi-humans, the quasi-humans+, rough-simulations and cyborg-type simulations. In addition, (C2) is also true for the original reference class of the quasi-humans and for that of cyborg-type simulations, but proves however to be false for the class of the quasi-humans+ and also for that of rough-simulations. Finally, (C3) is true for the original reference class of the quasi-humans, but proves to be false for the quasi-humans+, rough-simulations and cyborg-type simulations. These three levels of conclusion are represented on the table below:

level

conclusion

case

quasi-humans

quasi-humans+

rough simulations

cyborg-type simulations

C1

the proportion of the simulated human beings will largely exceed that of the human beings (disproportion)

C1A

true

true

true

true

the proportion of the simulated human beings will not largely exceed that of the human beings

C1Ā

false

false

false

false

C2

very probably, we are simulations (self-applicability)

C2A

true

false

false

true

very probably, we are not simulations

C2Ā

false

true

true

false

C3

we are simulations which are unaware of their nature of simulation (unawareness)

C3A

true

false

false

false

we are not simulations which are unaware of their nature of simulation

C3Ā

false

true

true

true

Figure 1. The different levels of conclusion within SA

and on the following tree structure:

1 William Eckhardt (2013, p. 15) considers thatin the same way as with the Doomsday argument (Eckhardt 1993, 1997, Franceschi 2009)the problem inherent to SA results from the use of reverse-causality and from the problem associated with the definition of the reference class: ‘if simulated, are you random among human sims? hominid sims? conscious sims?’.

2 We shall set aside here the question of whether or not we need to take into account an infinite number of simulated human beings. Such could be the case if the ultimate level of reality was abstract. In this case, the reference class could include simulated human beings who would identify themselves, for example, with matrices of very large integer numbers. But Bostrom answers such an objection in his FAQ (www.simulation-argument.com/faq.html) and indicates that in this case, the calculations do not apply any more (the denominator is infinite) and the ratio is not defined. We shall thus leave aside this hypothesis, by concentrating our argumentation on what constitutes the core of SA, i.e. the case where the number of human beings’ simulations is finite.

figure

Figure 2. Tree structure of the different levels of conclusion within SA

Even though the original conclusion of SA suggests that there is only one single level of conclusion, it turns out however, as it has been revealed, that there are in reality several levels of conclusion within SA, inasmuch as we examine the argument from a wider perspective, in the light of the reference class problem. The conclusion of the original argument is itself disturbing and alarming, in the sense that it concludes to a much stronger probability that we had imagined it a priori, that we are simulated human beings who are not aware of it. Such a conclusion results from the path C1-C1A-C2-C2A-C3-C3A of the above tree. However, the preceding analysis shows that according to the chosen reference class, some conclusions of a very different nature can be inferred from the simulation argument. Hence, a conclusion of a completely different nature is associated with the choice of the reference class of the quasi-humans+, but also of that of rough-simulations. The resulting conclusion is that we are not such simulations (C2Ā). This last conclusion is associated with the path C1-C1A-C2-C2Ā of the above tree. Finally, another possible conclusion, itself associated with the choice of the class of cyborg-type simulations, is that we belong to such a class of simulations, but that we are aware of it and that it presents thus nothing disturbing (C3Ā). This last conclusion is represented by the path C1-C1A-C2-C2A-C3-C3Ā.

Finally, the foregoing analysis casts light on the flaw in the original version of SA. The original argument focuses indeed on the class of simulations which are not conscious of their own nature of simulation. It follows then a succession of conclusions according to which there will be a larger proportion of simulated human beings than genuine human beings (C1A), that we belong to the simulated human beings (C2A) and finally that we are, more probably than we would have a priori imagined it, simulated human beings unaware of their being simulated (C3A). However, as evoked above, the very notion of human beings’ simulation—itself associated with the class of the post-humans—shows itself ambiguous, and such a class can be defined in reality by different ways, given that there does not exist, within SA, an objective criterion allowing to choose such a class non-arbitrarily. In effect, we can choose the reference class by restriction, by identifying the simulations with the quasi-humans, or with the quasi-humans+; in this case, the post-humans are the ones to which refers the original argument, of a much more distant time than ours. On the other hand, if we place ourselves at certain level of extension, the simulations assimilate themselves with less perfect simulations than those referred to in the original argument, as well as those of cyborg-type containing evolved uploads; in such a case, the associated post-humans are the ones of a less distant time. Finally, if we make the choice of the class of reference at a greater level of extension, the simulations are rough-simulations, hardly better than we are at present capable of realizing, or cyborg-type simulations with a degree of integration of simulated parts slightly much-evolved than the one that we know at present; in such a case, the class of associated post-humans is that of the human beings who will succeed us by a few years. As we can see it, we can make the choice of the reference class underlying SA at different levels of restriction or of extension. But according to whether the class will be chosen at such or such level of restriction or extension, a completely different conclusion will follow. So, the choice by restriction of perfect simulations that are unaware of their nature of simulation, as the argument original makes it, leads to a disturbing conclusion. On the other hand, the choice at slightly greater level of extension, of perfect simulations but aware of their nature of simulation, leads to a reassuring conclusion. And also, the choice, at a still greater level of extension, of rough-simulations or of cyborg-type simulations, also entails a reassuring conclusion. Hence, the preceding analysis shows that in the original version of SA, the choice concerns in a preferential way, by restriction, the reference class of the quasi-humans, with which a worrying conclusion is associated, whereas a choice by extension, taking into account the quasi-humans+, rough-simulations, cyborg-type simulations, etc., leads to a reassuring conclusion. Finally, the preferential choice in the original argument of the class of the quasi-humans, appears then as an arbitrary choice that nothing comes to justify, whereas there exist several other classes that deserve an equal legitimacy. For there is no objective element in the statement of SA allowing to make the choice of the reference class non-arbitrarily. In this context, the disturbing conclusion associated with the original argument also turns out to be an arbitrary conclusion, whereas there exists several other reference classes which possess an equal degree of relevance with regard to the argument itself, and from which ensue a completely reassuring conclusion.1

1 The present analysis constitutes a direct application to the Simulation argument of the form of dialectical contextualism described in Franceschi (2014).

Bostrom, N. (2003) Are You a Living in a Computer Simulation?, Philosophical Quarterly, 53, 243-55

Bostrom, N. (2005) Reply to Weatherson, Philosophical Quarterly, 55, 90-97

Bostrom, N. (2006) ‘How long before superintelligence?’, Linguistic and Philosophical Investigations, 5-1, 11-30

De Garis, H.D., Shuo, C., Goertzel, B., Ruiting, L. (2010) A world survey of artificial brain projects, part i: Large-scale brain simulations, Neurocomputing, 74(1-3), 3-29

Eckhardt, W. (1993) ‘Probability Theory and the Doomsday Argument’, Mind, 102, 483-88

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

Eckhardt, W. (2013) Paradoxes in probability Theory, Dordrecht, New York : Springer

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

Franceschi, P. (2014) Eléments d’un contextualisme dialectique, in Liber Amicorum Pascal Engel, edited by J. Dutant, D. Fassio & A. Meylan, 581-608, English translation under the title Elements of Dialectical Contextualism, cogprints.org/9225

Kurzweil, R. (2000) The Age of Spiritual Machines: When Computers Exceed Human Intelligence, New York & London: Penguin Books

Kurzweil, R. (2005) The Singularity is Near, New York : Viking Press

Moravec, H. (1998) When will computer hardware match the human brain?, Journal of Evolution and Technology, vol. 1

Sandberg, A & Bostrom, N. (2008) Whole Brain Emulation: a Roadmap, Technical Report #2008-3, Future of Humanity Institute, Oxford University

1 William Eckhardt (2013, p. 15) considers thatin the same way as with the Doomsday argument (Eckhardt 1993, 1997, Franceschi 2009)the problem inherent to SA results from the use of reverse-causality and from the problem associated with the definition of the reference class: ‘if simulated, are you random among human sims? hominid sims? conscious sims?’.

2 We shall set aside here the question of whether or not we need to take into account an infinite number of simulated human beings. Such could be the case if the ultimate level of reality was abstract. In this case, the reference class could include simulated human beings who would identify themselves, for example, with matrices of very large integer numbers. But Bostrom answers such an objection in his FAQ (www.simulation-argument.com/faq.html) and indicates that in this case, the calculations do not apply any more (the denominator is infinite) and the ratio is not defined. We shall thus leave aside this hypothesis, by concentrating our argumentation on what constitutes the core of SA, i.e. the case where the number of human beings’ simulations is finite.

3 The present analysis constitutes a direct application to the Simulation argument of the form of dialectical contextualism described in Franceschi (2014).

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Polythematic Delusions and Logico-Theoretical vs. Experimentalist Turn of Mind

George Grie-Panic-attackpaper published in the Journal for Neurocognitive Research,  Vol.  2013, 55, No. 1-2.

This article aims to contribute to cognitive therapy of polythematic delusions by proposing a preliminary step to the implementation of traditional cognitive therapy, based on the construction of alternative hypotheses to delusions and testing of the latter. This additional step resides in the construction in the patient of the necessary skills to use the general experimentalist method of knowledge acquisition. Such an approach is based on the contrast between the logico-theoretical and the experimentalist turn of mind. Some elements such as to allow any such construction in the patient are then described and analyzed.

This article is cited in:

Ondrej Pec, Petr Bob,and Jiri Raboch (2014) Splitting in Schizophrenia and Borderline Personality Disorder, PLoS One 9(3) e91228.

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Polythematic Delusions and Logico-Theoretical vs.  Experimentalist Turn of Mind

Classical cognitive therapy targeted at polythematic delusions associated with schizophrenia is based on the search for evidence related to delusional ideas and the construction of alternative hypotheses to the latter. This article aims to contribute to cognitive therapy for polythematic delusions by proposing a preliminary step to this classical cognitive therapy. Such a step aims to strengthen the patient’s ability to use the general approach of experimentalist type for knowledge acquisitionan approach which is based on the opposition between the theoretical-logical and the experimentalist turn of mind. Some elements such as to enable the reinforcement of such a capability in the patient are thus described and analyzed.

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On the Plausibility of Psychotic Hallucinations

anonyme flamand_cauchemarA paper published in the Journal for Neurocognitive Research,  Vol. 53, No 1-2 (2011).

In this paper, we describe several factors that can contribute, from the patient’s viewpoint, to the plausibility of psychotic hallucinations. We sketch then a Plausibility of Hallucinations Scale, consisting of a 50-item questionnaire, which aims at evaluating the degree of plausibility of hallucinations. We also emphasize the utility of pointing out to the patient the several factors that contribute to the plausibility of his/her hallucinations, in the context of cognitive therapy for schizophrenia.

 

This paper is cited in:

Mark Grimshaw, Tom Garner, Sonic Virtuality: Sound as Emergent Perception, New York: Oxford University Press, 2015

I. de Chazeron, B. Pereirae, I. Chereau-Boudete, G. Broussee, D. Misdrahie, G. Fénelone, A.-M. Tronchee, R. Schwane, C. Lançone, A. Marquese, B. Debillye, F. Durife, P.M. Llorca, Validation of a Psycho-Sensory Hallucinations Scale (PSAS) in schizophrenia and Parkinson’s disease, Volume 161, Issues 2-3, Pages 269–276

 

 

 

 

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On the plausibility of psychotic hallucinations

Edvard_Munch_-_The_Scream_-_Google_Art_Project

Edvard Munch: The Scream.

Cognitive therapy of hallucinations is part of cognitive therapy for schizophrenia. Several accounts of cognitive therapy of hallucinations have been described in the literature (Chadwick et al.,1996; Rector & Beck, 2002; Kingdon & Turkington, 2005). On the one hand, Chadwick et al. (1996) stress the importance of the ABC model for cognitive therapy of hallucinations: the hallucinations are the activating events, which engender cognitions, which in turn yield emotional distress and anger. By working on beliefs about the voices, they primary aim at reducing the negative emotions which are the consequences of automatic thoughts following the occurrence of hallucinations. Chadwick et al. also have a special emphasis on the omnipotence and omniscience of the voices. On the other hand, Kingdon & Turkington (2005) propose the cognitive model of hallucinations as an alternative explanation for the voices: auditory hallucinations are the patient’s automatic thoughts that are perceived as originating from outside the patient’s mind. Kingdon & Turkington weigh the available evidence for both competing explanations and finally work on reattribution of auditory hallucinations. Rector & Beck (2002) take a similar stance, and stress that the final aim of the therapy “is to help patients recognize that the voices simply reflect either their own attitudes about themselves or those they imagine others to have about them”.

The purpose of the present paper is to contribute to cognitive therapy for schizophrenia by focusing on the plausibility of psychotic hallucinations. Our concern will be with providing an account of complex hallucinations encountered in schizophrenia that stresses multiple factors which reinforce, from the patient’s viewpoint, the intrinsic plausibility of the hallucinations. The purpose of this paper is then to expose how hallucinations can seem plausible and credible to the patient. In section 1, we describe several factors that contribute to the plausibility of hallucinations occurring in schizophrenia. We sketch then in section 2 a scale which is designed to measure accurately the plausibility of hallucinations. In section 3, we point out what could be the impact on cognitive therapy for schizophrenia of the present account. Finally, we point out the limitations of the present study and some directions for further research.

1. Factors of plausibility of hallucinations

We shall enumerate in what follows several factors that can contribute, from the patient’s viewpoint, to the plausibility of the hallucinations that he/she experiences. Hallucinations are one major symptom of schizophrenia. According to DSM-IV, a hallucination is defined as “A sensory perception that has the compelling sense of reality of a true perception but that occurs without external stimulation of the relevant sensory organ”. (DSM-IV, p. 767). By plausibility, we mean the fact that the patient’s abnormal perceptions are seemingly attributable to an external source (usually, other people). The plausibility that results from certain phenomenological features of auditory hallucinations has notably been hinted at by Stephane et al. (2003):

(…) hearing “multiple voices” is associated with attribution of the “voices” to others, which is plausible intuitively as well. This indicates that the patients’ experiences of hallucinations could be understood, intuitively, based on common sense experiences of the world.

In this paper, we shall expand this idea, by pointing out that multiple factors are susceptible of congruently strengthening the patient’s conviction that his/her abnormal perceptions come from the external world.

Hallucinations come in a variety of modalities. In order to shed light on the factors that can lead to the plausibility of hallucinations occurring in schizophrenia, it is worth drawing first some useful distinctions.

1.1 Unimodal and multi-modal hallucinations

Let us consider, to begin with, the distinction between unimodal and multi-modal hallucinations. Unimodal hallucinations can be classified into five types, corresponding to our five sensory pathways: auditory, visual, olfactory, tactile and gustatory. Multi-modal hallucinations are made up of unimodal hallucinations of different types which occur simultaneously (or quasi-simultaneously). There are accordingly 26 combinations of multi-modal hallucinations (plus 5 unimodal ones). The latter can be enumerated exhaustively as follows (we also describe an instance of some common multi-modal cases, since it can be useful for explanatory purposes):

(i) 1-modal: auditory, visual, olfactory, tactile, gustatory

The_temptation_of_St_Antony._Wellcome_L0034926

The temptation of St Antony. Pastel drawing by W.S.

(ii) 2-modal: auditory-visual (“I saw x sitting on my bed and I heard him saying ‘Bastard!’”), auditory-olfactory (“I heard x saying ‘I will smoke a cigar! ’ and at this very moment I smell a taste of tobacco”), auditory-tactile (“I heard x saying ‘You will be stung by a scorpion!’ and at this very moment I felt a sharp sting of pain on my left arm”), auditory-gustatory, visual-olfactory (“I saw x on my bed smoking a cigar and I also smell the taste of tobacco”), visual-tactile (“I saw a scorpion on my left arm and at this very moment I felt a sharp sting of pain there”), visual-gustatory (“I saw blood dripping from my finger and it had the taste of blood when I put it on my tongue”), olfactory-tactile, olfactory-gustatory, tactile-gustatory

(iii) 3-modal: auditory-visual-olfactory, auditory-visual-tactile (“I heard x saying ‘You will be stung by a scorpion!’ and at this very moment I saw a scorpion on my left arm while feeling a sharp sting of pain there”), auditory-visual-gustatory (“I heard ‘I will harm you’ and at this very moment I saw blood dripping from my finger and it had the taste of blood when I put it on my tongue”), auditory-olfactory-tactile, auditory-olfactory-gustatory, auditory-tactile-gustatory, visual-olfactory-tactile, visual-olfactory-gustatory, visual-tactile-gustatory, olfactory-tactile-gustatory

(iv) 4-modal: auditory-visual-olfactory-tactile, auditory-visual-olfactory-gustatory, auditory-visual-tactile-gustatory (“I heard ‘I will harm you’ and then I saw blood dripping from my finger, while I felt a sharp pain there. It had the taste of blood when I put it on my tongue”), auditory-olfactory-tactile-gustatory, visual-olfactory-tactile-gustatory

(v) 5-modal: auditory-visual-olfactory-tactile-gustatory

At this step, it should be noted that multi-modal hallucinations retain their force from the plausibility that results from the simultaneous (or quasi-simultaneous) occurrence of two or more unimodal hallucinations of different types. For this reason, multi-modal hallucinations retain considerable power with regard to unimodal ones. The sense of reality that results from multi-modal hallucinations is due to the fact that several sensory pathways are congruently involved in the hallucinatory process. If we consider, for example, bimodal hallucinations of the auditory-visual type, it proves to be that the simultaneous occurrence of an additional visual hallucination strongly reinforces the sense of reality that results from the auditory hallucination. As the above examples illustrate, multi-modal hallucinations are seemingly highly more plausible and realistic than unimodal ones, and result in a much greater sense of realism. More generally, it illustrates how (n + 1)-modal hallucinations are seemingly much more realistic than n-modal ones, a supplementary sensory pathway being involved in the hallucinatory process.

1.2 Factors of plausibility of auditory hallucinations

It is worth mentioning, second, several factors that can contribute to the intrinsic plausibility of auditory hallucinations:

(i) structured versus unstructured auditory hallucinations: structured sounds notably consist of comments on the patient’s thoughts or activities, conversations of several persons, or commands ordering the patient to do things, etc., while on the other hand, unstructured sounds consist of ringing, buzzing, whistling, etc.

(ii) auditory hallucinations having an external versus an internal origin: auditory hallucinations seemingly coming out from outer space could reinforce the idea that the voices have an external origin, e.g. are attributable to other people.

(iii) the locus (Chadwick et al., 1996, p. 103) – i.e. the space location – of auditory hallucinations is also susceptible of reinforcing their intrinsic plausibility. We can consider, for example, a patient who hears the voice of the presenter of the show saying ‘Bastard!’. Now this sounds more realistic if the locus of the voice is the television device rather than the ashtray. Let us suppose now that the patient hears a voice saying ‘I can read your thoughts’. Now it sounds more likely to the patient if the voice comes out from the telephone than from the halogen lamp.

It is worth noting here that this criterion is susceptible to vary from culture to culture.1 In effect, depending on the individuals, a speaking tree or a speaking animal could be, in certain cases, consistent with the patient’s cultural background.

(iv) bilateral versus unilateral auditory hallucinations: auditory hallucinations coming indifferently from the patient’s right or from his/her left are more plausible than unilateral ones.2

(v) time location related versus unrelated to the patient’s thoughts, emotions or actions (Stephane et al. 2003 make mention of the “relation to the moment”). In this regard, auditory hallucinations that are simultaneous with the patients internal phenomena gain more plausibility.

(vi) phrases versus single words: in this context, phrases, conversations, elaborate sentences gain are more plausible than single words.

(vii) multiple voices versus single voice (Stephane et al., 2003).

(viii) auditory hallucinations fitting versus not fitting with the patient’s desires or fears: this factor consists of whether the hallucinations experienced by a patient fit adequately or not with his/her individual fears or desires. For in the affirmative, it would greatly increase the plausibility of the corresponding hallucinations. Let us take an example. The patient is very anxious about the evolution of his/her illness. He/she hears a voice that says: ‘You will relapse next month’. Now the content of this auditory hallucination fits adequately with the patient’s own fears. The reason why auditory hallucinations fitting with desires or fears are more plausible, is that they are coherent with the patient’s belief system. By contrast, had the content of auditory hallucinations been unrelated or contradictory with the patient’s desires and fears, the corresponding information would have then resulted in a lack of coherence with the patient’s belief system (this is in line with the approach to hallucinations exposed in Rector & Beck, 2002, which is concerned with: “(…) how the specific voice content and beliefs about the voices reflect the person’s prehallucinatory fears, concerns, interests, preoccupations and fantasies”).

(ix) interactive versus non-interactive voices: whether the patient can interact or not with voices, i.e. discuss or engage in dialog with them.

1.3 Factors of plausibility of visual hallucinations

Several factors can contribute, third, to the intrinsic plausibility of visual hallucinations:

(i) formed versus unformed visual hallucinations: formed hallucinations are made up of figures, faces, morphing objects or scenes. By contrast, unformed hallucinations consist of dots, lines, geometrical figures, flashes, etc.

(ii) ordinary versus bizarre or extraordinary visual hallucinations: for obvious reasons, objects that look ordinary gain more likeliness than bizarre, unreal objects.

(iii) objects in color versus in black and white.

(iv) visual hallucinations fitting versus not fitting with surroundings: as noted by Teunisse et al. (1996), the relationship to surroundings could play an important role in the plausibility of complex hallucinations. Such or such unimodal hallucination could fit well (e.g. a person lying on a bed, a scorpion walking on the ground) or not (a figure on the ceiling) with surroundings. Now it should be apparent that fitting with surroundings visual hallucinations are consistent with the patient’s knowledge of the physical world. This renders, from the patient’s viewpoint, the hallucination very plausible. By analogy with the locus of auditory hallucinations, fitting with surroundings can be assimilated to the locus – i.e. space location – of visual hallucinations.

(v) bilateral versus unilateral visual hallucinations.

(vi) time location of visual hallucinations related versus unrelated to the patient’s thoughts, emotions or actions (e.g. the patient thinks to a scorpion and at this very moment he/she sees a scorpion on the ground).

(vii) animated versus static images.

1.4 Factors of plausibility of olfactory hallucinations

Several factors can contribute, fourth, to the plausibility of olfactory hallucinations:

(i) bilateral versus unilateral olfactory hallucinations.

(ii) olfactory hallucinations fitting versus not fitting with the patient’s desires or fears: the patient fears of being killed and smells a poisonous gas in his/her room.

(iii) transient versus permanent olfactory hallucinations: some patients experience olfactory hallucinations that occur any time of day and also last for hours (Tousi & Frankel 2004).

1.5 Factors of plausibility of tactile hallucinations

Certain factors can contribute, fifth, to the plausibility of tactile hallucinations:

(i) bilateral versus unilateral tactile hallucinations.

(ii) tactile hallucinations fitting versus not fitting with the patient’s desires or fears: the patient fears of being murdered and feels an electric-shock sensation.

1.6 Factors of plausibility of gustatory hallucinations

Some factors can contribute, sixth, to the plausibility of gustatory hallucinations:

(i) common versus strange gustatory hallucinations: in some cases, the patient may find that his/her food tastes strange. This could decrease the plausibility of the corresponding hallucination, in contrast with gustatory hallucinations where the patient experiences normal and common tastes.

(ii) gustatory hallucinations fitting versus not fitting with the patient’s desires or fears: the patient fears of being poisoned and feels the taste of poison in his/her mouth.

2. Plausibility of hallucinations scale

From the above, it results that it could be useful to measure accurately the plausibility of the hallucinations occurring in schizophrenia. For this purpose, we shall now sketch a 50-item scale, which is targeted at evaluating the plausibility of hallucinations experienced by a patient. This binary scale consists of a questionnaire which allows for yes/no answers (each yes answer weighting 2 points):

item

questions (0-100)

Unimodal hallucinations

Auditory hallucinations

1

Does the patient hear auditory hallucinatory which consist of structured sounds?

2

Does the patient experience auditory hallucinations which come out from outer space?

3

Does the patient experience auditory hallucinations whose locus sounds realistic?

4

Does the patient experience bilateral auditory hallucinations?

5

Does the patient experience auditory hallucinations whose time location is related to the patient’s thoughts, emotions or actions?

6

Does the patient experience auditory hallucinations which consist of phrases, conversations?

7

Does the patient experience auditory hallucinations with multiple voices?

8

Does the patient experience auditory hallucinations whose content fits with his/her fears or desires?

10

Can the patient interact with auditory hallucinations, i.e. discuss or engage in dialog with them?

Visual hallucinations

11

Does the patient experience formed visual hallucinations?

12

Does the patient experience visual hallucinations with ordinary objects?

13

Does the patient experience visual hallucinations in color?

14

Does the patient experience visual hallucinations whose locus fits with surroundings?

15

Does the patient experience bilateral visual hallucinations?

16

Does the patient experience visual hallucinations whose time location is related to his/her thoughts, emotions or actions?

17

Does the patient experience visual hallucinations consisting of scenes or sequences of animated images?

Olfactory hallucinations

18

Does the patient experience bilateral olfactory hallucinations?

19

Does the patient experience olfactory hallucinations whose content fits with his/her fears or desires?

20

Does the patient experience transient olfactory hallucinations?

Tactile hallucinations

21

Does the patient experience bilateral tactile hallucinations?

22

Does the patient experience tactile hallucinations whose content fits with his/her fears or desires?

Gustatory hallucinations

23

Does the patient experience gustatory hallucinations of a common type?

24

Does the patient experience gustatory hallucinations whose content fits with his/her fears or desires?

Bimodal hallucinations

25

Does the patient experience bimodal hallucinations of the auditory-visual type?

26

Does the patient experience bimodal hallucinations of the auditory-olfactory type?

27

Does the patient experience bimodal hallucinations of the auditory-tactile type?

28

Does the patient experience bimodal hallucinations of the auditory-gustatory type?

29

Does the patient experience bimodal hallucinations of the visual-olfactory type?

30

Does the patient experience bimodal hallucinations of the visual-tactile type?

31

Does the patient experience bimodal hallucinations of the visual-gustatory type?

32

Does the patient experience bimodal hallucinations of the olfactory-tactile type?

33

Does the patient experience bimodal hallucinations of the olfactory-gustatory type?

34

Does the patient experience bimodal hallucinations of the tactile-gustatory type?

Trimodal hallucinations

35

Does the patient experience trimodal hallucinations of the auditory-visual-olfactory type?

36

Does the patient experience trimodal hallucinations of the auditory-visual-tactile type?

37

Does the patient experience trimodal hallucinations of the auditory-visual-gustatory type?

38

Does the patient experience trimodal hallucinations of the auditory-olfactory-tactile type?

39

Does the patient experience trimodal hallucinations of the auditory-olfactory-gustatory type?

40

Does the patient experience trimodal hallucinations of the auditory-tactile-gustatory type?

41

Does the patient experience trimodal hallucinations of the visual-olfactory-tactile type?

42

Does the patient experience trimodal hallucinations of the visual-olfactory-gustatory type?

43

Does the patient experience trimodal hallucinations of the visual-tactile-gustatory type?

44

Does the patient experience trimodal hallucinations of the olfactory-tactile-gustatory type?

quadri-modal hallucinations

45

Does the patient experience quadri-modal hallucinations of the auditory-visual-olfactory-tactile type?

46

Does the patient experience quadri-modal hallucinations of the auditory-visual-olfactory-gustatory type?

47

Does the patient experience quadri-modal hallucinations of the auditory-visual-tactile-gustatory type?

48

Does the patient experience quadri-modal hallucinations of the auditory-olfactory-tactile-gustatory type?

49

Does the patient experience quadri-modal hallucinations of the visual-olfactory-tactile-gustatory type?

quinti-modal hallucinations

50

Does the patient experience quinti-modal hallucinations of the auditory-visual-tactile-olfactory-gustatory type?

It is worth noting that this 50-item scale can be regarded as non-specific to psychotic hallucinations. It is also suited to other disorders or illnesses involving hallucinations. Among these are other mental illnesses, but also Charles Bonnet syndrome (Teunisse et al., 1996; Menon et al., 2003), epilepsy (Sachdev, 1998; Schwartz & Marsh, 2000), etc. In particular, the Charles Bonnet syndrome occurs in the elderly and is usually associated with ocular pathology and severe visual impairment. The Charles Bonnet syndrome is characterized by the presence of complex and persistent visual hallucinations. The syndrome is usually associated with an absence of hallucinations in other sensory modalities. It is worth noting that the Charles Bonnet syndrome affects psychologically normal individuals with full or partial insight and the patients are accordingly non-delusional. On the other hand, auditory hallucinations are frequently associated with temporal lobe epilepsy, where hallucinations in other modalities can also occur.

It is worth mentioning that the Plausibility of Hallucinations Scale could also be used in association with other instruments for measuring insight, such as the Beck Cognitive Insight Scale (Beck et al., 2003) in order to gain more accurate knowledge of the patient’s state. For schizophrenia is usually associated with lack of insight into the internal origin of the hallucinations. By contrast, in other illnesses such as Charles Bonnet syndrome or in pseudo-hallucinations related to brain trauma or PTSD (Stephane et al., 2004), the patient usually retains insight into the internal origin of his/her hallucinations.

3. Impact on Cognitive Therapy

We suggest that the above emphasis on the plausibility of hallucinations could be usefully incorporated into the process of cognitive-behavior therapy of schizophrenia (Kingdon & Turkington, 1994, 2005; Chadwick et al., 1996; Rector & Beck, 2002). The general idea would be to point out to the patient who experiences highly plausible hallucinations those factors that confer to his/her hallucinations their intrinsic plausibility. Hopefully, this could insert itself well into the process of cognitive-behavior therapy, whose primary goal is to help the patient gaining more insight into the nature of his/her hallucinations and in particular to understand that they do not originate from an external source. In this context, stressing to the patient the plausibility of his/her hallucinations, could help him/her understand better how hallucinations can be self-deceiving.

It is worth focusing, to begin with, on multi-modal hallucinations. In this context, a first step would be to point out to the patient that multi-modal hallucinations are capable of seeming very plausible and realistic. It could then be argued and explained to the patient that multi-modal hallucinations are more plausible than unimodal ones. This could be illustrated through some examples. This latter strategy could make use of “what if statements” (Ellis & Dryden, 1997). Along these lines, it could be pointing out to the patient that if someone, instead of experiencing one single auditory hallucination, would experience simultaneously one supplementary visual hallucination, then the resulting multi-modal (of the auditory-visual type) hallucination would sound much more realistic. Along these lines, it could be pointed out to the patient that the particular case of multi-modal hallucination that he/she experiences is potentially very realistic and inherently capable of deceiving him/her.

Once the patient familiar with the concept of multi-modal hallucinations, another goal could be to learn the patient how to use by himself/herself the preceding taxonomy of multi-modal hallucinations and to apply it when he/she experiences these types of complex hallucinations. He/she would then be capable of identifying the corresponding case at hand. Hopefully, this could help the patient rationalize his/her abnormal perceptions and perhaps accept better the internal origin of his/her hallucinations as an alternative explanation.

The fact of classifying multi-modal hallucinations would be helpful to the patient, it seems, to help him/her rationalize and explain the phenomena he/she experiences. For we should bear in mind that the patient experiences abnormal phenomena, which are unfamiliar to psychologically normal people. In this context, helping the patient rationalize, classify and describe accurately the phenomena of his/her own internal world, proves then to be a valuable practical goal to attain. Accordingly, identifying, recognizing and labeling a given type of multi-modal hallucination could help lessen its associated omnipotence (Chadwick et al., 1996). This could be helpful to the patient, we suggest, who ordinarily faces an unexplained and upsetting phenomenon. More generally, the fact of identifying the various factors that render his/her complex hallucinations so plausible could help the patient gaining more insight into the internal origin of his/her hallucinations. At this step, it should be noted that the present account is notably in line – for what concerns the delusion that consists in attributing an external origin to the hallucinations – with the views advocated by Brendan Maher (1988, 1999), who sees delusions as a patient’s attempt to explain some perplexing and puzzling phenomena. According to Maher, delusions arise from normal (mainly rational but occasionally irrational) reasoning applied to abnormal phenomena. Among these abnormal phenomena which are very perplexing to the patient are the hallucinations.

4. Limitations and directions for further research

The main limitation of the present study is that the psychometric properties of the Plausibility of Hallucinations Scale have not been tested. However, given the number of items of the scale, one can expect a good sensibility. On the other hand, the reliability and validity of the scale remain to be tested.

Finally, the above developments suggest several questions, which could be usefully the subject of further study, based on the Plausibility of Hallucinations Scale. A first question that arises is the following: (i) Is the plausibility of hallucinations rate higher in schizophrenia than in other illnesses involving hallucinations, e.g. other mental illnesses, Charles Bonnet syndrome, temporal lobe epilepsy, etc.? A comparison of the plausibility of hallucinations rate occurring in schizophrenia and other illnesses involving hallucinations could be made accordingly. We suggest that such comparison could provide some useful information about the relationships of these illnesses (Sachdev, 1998). Although schizophrenia (paranoid subtype) should prima facie involve a higher rating, it seems that an accurate measure of the degree of plausibility of hallucinations could result in some interesting information. Along these lines, a comparison of the plausibility of hallucinations ratings occurring in different subtypes of schizophrenia could also be informative.

The above Plausibility of Hallucinations Scale is also designed to allow for comparisons between different chronological states in the same patient. This suggests a second type of question: (ii) Does the plausibility of hallucinations rating evolve during the course of schizophrenia? Along these lines, Nayani & David (1996) observed an increase in the complexity of auditory hallucinations over time, seemingly related to lesser distress and better coping. A similar question could be raised for other illnesses involving hallucinations. In this context, Menon et al. (2003) reported accordingly that “Elementary hallucinations may progressively evolve into complex visual hallucinations” in the Charles Bonnet syndrome.

Lastly, a third interesting question goes as follows: (iii) Is the plausibility rate of hallucinations occurring in schizophrenia correlated with the I.Q. of the patient, i.e. do patients with a high I.Q. more frequently experience complex hallucinations with a high level of plausibility? In other words, is the following hypothesis confirmed: The higher the I.Q., the higher the plausibility of hallucinations rating? Hopefully, the answer to these questions will provide some information that might well be useful to the understanding of the illness and to cognitive-behavior therapy of schizophrenia.

Acknowledgments

We thank Peter Brugger, Paul Gilbert and Hélène Verdoux for very useful comments on earlier drafts.

References

American Psychiatric Association (1994). Diagnostic and Statistical Manual of Mental Disorders (4th edition). American Psychiatric Association: Washington.

Beck, A. T., Rector, N. A. (2003). A Cognitive Model of Hallucinations. Cognitive Therapy and Research, 27(1), 19-52.

Beck, A. T., Baruch, E., Balter, J. M., Steer, R. A., Warman, D. M. (2004). A new instrument for measuring insight: the Beck Cognitive Insight Scale. Schizophrenia Research, 68(2-3), 319-329.

Chadwick, P., Birchwood, M. & Trower, P. (1996). Cognitive Therapy for Delusions, Voices, and Paranoia. Chichester: Wiley.

Ellis, A., Dryden, W. (1997). The practice of rational emotive behaviour therapy, London: Free Association Books.

Kingdon, D. & Turkington, D. (1994). Cognitive-behavioural Therapy of Schizophrenia. Guilford: New York.

Kingdon, D., & Turkington, D. (2005). Cognitive Therapy of Schizophrenia. New York, London: Guilford.

Maher, B. A. (1988). Anomalous experiences and delusional thinking: the logic of explanations. In: T.F. Oltmanns & B.A. Maher (Eds.), Delusional Beliefs, pp. 15-33. Wiley: New York.

Maher, B. A. (1999). Anomalous experience in everyday life: Its significance for psychopathology. The Monist, 82, 547-570.

Menon, G. J., Rahman, I., Menon, S. J., Dutton, G. N. (2003). Complex visual hallucinations in the visually impaired: the Charles Bonnet Syndrome. Survey of Ophthalmology, 48, 58-72.

Nayani, T. H., David, A. S. (1996). The auditory hallucination: a phenomenological survey. Psychological Medicine, 26(1), 177-189.

Rector, N. A., Beck A. T. (2002). Cognitive Therapy for Schizophrenia: From Conceptualization to Intervention. Canadian Journal of Psychiatry, 47(1), 41-50.

Sachdev, P. (1998). Schizophrenia-Like Psychosis and Epilepsy: The Status of the Association. American Journal of Psychiatry, 155(3), 325-336.

Schwartz, J. M., Marsh, L. (2000). The Psychiatric Perspectives of Epilepsy. Psychosomatics, 41(1), 31-38.

Siddle, R. (2002). Communications from my parents. In: D. Kingdon & D. Turkington (Eds.), The Case Study Guide to Cognitive Behaviour Therapy of Psychosis, pp. 109-121. Chichester: Wiley.

Stephane, M., Thuras, P., Nasrallah, H., Georgopoulos, A. P. (2003). The internal structure of the phenomenology of auditory verbal hallucinations. Schizophrenia Research, 61, 185-193.

Stephane, M., Hill, T., Matthew, E., & Folstein, M. (2004) New phenomenon of abnormal auditory perception associated with emotional and head trauma: Pathological confirmation by SPECT scan. Brain and Language, 89, 503-507.

Teunisse, R. J., Cruysberg, J. R., Hoefnagels, W. H. (1996). Visual hallucinations in psychologically normal people: Charles Bonnet’s syndrome. Lancet, 347, 794-797.

Toone, B. K. (2000). The psychoses of epilepsy. Journal of Neurology, Neurosurgery and Psychiatry, 69(1), 1-3.

Tousi, B. & Frankel, M. (2004). Olfactory and visual hallucinations in Parkinson’s disease, Parkinsonism and Related Disorders, 10, 253-254.

1 We thank Paul Gilbert for the suggestion of taking into account cultural beliefs with regard to this specific criterion.

2 We owe the suggestion to include the bilateral/unilateral distinction related to hallucinations in all modalities to Peter Brugger.

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Elements of Dialectical Contextualism

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

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

 

 

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Elements of Dialectical Contextualism

 

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

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

1. Dualities and dual poles

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

The dual poles A and Ā

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

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

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

fig2

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

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

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

2. The principle of dialectical indifference

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

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

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

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

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

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

3. Characterisation of the one-sidedness bias

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

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

John_Stuart_Mill_by_London_Stereoscopic_Company,_c1870

John Stuart Mill

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

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

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

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

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

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

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

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

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

4. An instance of the one-sidedness bias

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

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

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

(D2Ā ) then <n2 conflictual relationships will result>

(D3Ā ) then negative effects will result

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

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

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

(D2A) then <n2 collaborative relationships will result>

(D3A) then positive effects will result

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

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

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

while the conclusion of the parallel reasoning is as follows:

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

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

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

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

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

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

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

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

5. Dichotomic analysis and meta-philosophy

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

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

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

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

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

6. Application to the analysis of the philosophical paradoxes

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

6.1. Application to the analysis of Hempel’s paradox

Carl_Gustav_Hempel

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

(H) All ravens are black

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

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

Corvus_corax_(FWS)

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

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

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

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

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

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

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

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

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

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

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

6.2. Application to the analysis of the surprise examination paradox

chap4

A 7-day instance of the surprise examination paradox.

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

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

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

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

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

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

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

day

non-surprise

surprise

1

1

1

2

1

1

3

1

1

4

1

1

5

1

1

6

1

1

7

1

1

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

day

non-surprise

surprise

1

1

1

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

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

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

day

non-surprise

surprise

1

0

1

365

1

0

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

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

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

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

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

6.3. Application to the analysis of the Doomsday Argument

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Acknowledgements

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

References

Beck, AT. (1963) Thinking and depression: Idiosyncratic content and cognitive distortions, Archives of General Psychiatry, 9, 324-333.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Suber, E. (1998). The One-Sidedness Fallacy. Manuscript.

Walton, D. (1999) One-Sided Arguments: A Dialectical Analysis of Bias, Albany: State University of New York Press.

Williamson, T. (2000) Knowledge and its Limits, London & New York : Routledge.

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

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

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

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

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

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A Two-Sided Ontological Solution to the Sleeping Beauty Problem

eub-sbPreprint published on the PhilSci archive.

I describe in this paper an ontological solution to the Sleeping Beauty problem. I begin with describing the hyper-entanglement urn experiment. I restate first the Sleeping Beauty problem from a wider perspective than the usual opposition between halfers and thirders. I also argue that the Sleeping Beauty experiment is best modelled with the hyper-entanglement urn. I draw then the consequences of considering that some balls in the hyper-entanglement urn have ontologically different properties from normal ones. In this context, drawing a red ball (a Monday-waking) leads to two different situations that are assigned each a different probability, depending on whether one considers “balls-as-colour” or “balls-as-object”. This leads to a two-sided account of the Sleeping Beauty problem.

 

 

line_1

A Two-Sided Ontological Solution to the Sleeping Beauty Problem

1. The hyper-entanglement urn

Let us consider the following experiment. In front of you is an urn. The experimenter asks you to study very carefully the properties of the balls that are in the urn. You go up then to the urn and begin to examine its content carefully. You notice first that the urn contains only red or green balls. By curiosity, you decide to take a sample of a red ball in the urn. Surprisingly, you notice that while you pick up this red ball, another ball, but a green one, also moves simultaneously. You decide then to replace the red ball in the urn and you notice that immediately, the latter green ball also springs back in the urn. Intrigued, you decide then to catch this green ball. You notice then that the red ball also goes out of the urn at the same time. Furthermore, while you replace the green ball in the urn, the red ball also springs back at the same time at its initial position in the urn. You decide then to withdraw another red ball from the urn. But while it goes out of the urn, nothing else occurs. Taken aback, you decide then to undertake a systematic and rigorous study of all the balls in the urn.

At the end of several hours of a meticulous examination, you are now capable of describing precisely the properties of the balls present in the urn. The latter contains in total 1000 red balls and 500 green balls. Among the red balls, 500 are completely normal balls. But 500 other red balls have completely astonishing properties. Indeed, each of them is linked to a different green ball. When you remove one of these red balls, the green ball which is associated with it also goes out at the same time from the urn, as if it was linked to the red ball by a magnetic force. Indeed, if you remove the red ball from the urn, the linked green ball also disappears instantly. And conversely, if you withdraw from the urn one of the green balls, the red ball which is linked to it is immediately removed from the urn. You even try to destroy one of the balls of a linked pair of balls, and you notice that in such case, the ball of the other colour which is indissociably linked to it is also destroyed instantaneously. Indeed, it seems to you that relative to these pairs of balls, the red ball and the green ball which is linked to it behave as one single object.

The functioning of this urn leaves you somewhat perplexed. In particular, your are intrigued by the properties of the pairs of correlated balls. After reflection, you tell yourself that the properties of the pairs of correlated balls are finally in some respects identical to those of two entangled quantum objects. Entanglement (Aspect & al. 1982) is indeed the phenomenon which links up two quantum objects (for example, two photons), so that the quantum state of one of the entangled objects is correlated or anti-correlated with the quantum state of the other, whatever the distance where the latter is situated. As a consequence, each quantum object can not be fully described as an object per se, and a pair of entangled quantum objects is better conceived of as associated with a single, entangled state. It also occurs to you that perhaps a pair of correlated balls could be considered, alternatively, as a ubiquitous object, i.e. as an object characterised by its faculty of occupying two different locations at the same time, with the colours of its two occurrences being anti-correlated. Setting this issue aside for the moment, you prefer to retain the similarity with the more familiar quantum objects. You decide to call “hyper-entanglement urn” this urn with its astonishing properties. After reflection, what proves to be specific to this urn, is that it includes at the same time some normal and some hyper-entangled balls. The normal red balls are no different from our familiar balls. But hyper-entangled balls do behave in a completely different way. What is amazing, you think, is that nothing seemingly differentiates the normal red balls from the red hyper-entangled ones. You tell yourself finally that it could be confusing.

Your reflection on the pairs of hyper-entangled balls and their properties also leads you to question the way the balls which compose the pairs of hyper-entangled balls are to be counted. Are they to be counted as normal balls? Or do specific rules govern the way these pairs of hyper-entangled balls are to be counted? You add a normal red ball in a hyper-entanglement urn. It is then necessary to increment the number of red balls present in the urn. On the other hand, the total number of green balls is unaffected. But what when you add in the hyper-entanglement urn the red ball of a pair of hyper-entangled balls? In that case, the linked green ball of the same pair of hyper-entangled balls is also added instantly in the urn. Hence, when you add a red ball of a pair of hyper-entangled balls in the urn, it also occurs that you add at the same time its associated green ball. So, in that case, you must not only increment the total number of red balls, but also the total number of green balls present in the urn. In the same way, if you withdraw a normal red ball from the urn, you simply decrement the total number of red balls of the urn, and the number of green balls in the urn is unaffected. But if you remove the red ball (resp. green) of a pair of hyper-entangled balls, you must decrement the total number of red balls (resp. green) present in the urn as well as the total number of green balls (resp. red).

At this very moment, the experimenter happens again and withdraws all balls from the urn. He announces that you are going to participate in the following experiment:

The hyper-entanglement urn A fair coin will be randomly tossed. If the coin lands Heads, the experimenter will put in the urn a normal red ball. On the other hand, if the coin lands Tails, he will put in the urn a pair of hyper-entangled balls, composed of a red ball and a green ball, both indissociably linked. The experimenter also adds that the room will be put in absolute darkness, and that you will therefore be completely unable to detect the colour of the balls, no more that you will be able to know, when you will have withdrawn a ball from the urn, whether it is a normal ball, or a ball which is part of a pair of hyper-entangled balls. The experimenter tosses then the coin. While you catch a ball from the urn, the experimenter asks you to assess the likelihood that the coin felt Heads.

2. The Sleeping Beauty problem

Consider now the well-known Sleeping Beauty problem (Elga 2000, Lewis 2001). Sleeping Beauty learns that she will be put into sleep on Sunday by some researchers. A fair coin will be tossed and if the coin lands Heads, Beauty will be awakened once on Monday. On the other hand, if the coin lands Tails, Beauty will be awakened twice: on Monday and Tuesday. After each waking, she will be put into sleep again and will forget that waking. Furthermore, once awakened, Beauty will have no idea of whether it is Monday or Tuesday. On awakening on Monday, what should then be Beauty’s credence that the coin landed Heads?

At this step, one obvious first answer (I) goes as follows: since the coin is fair, the initial probability that the coin lands Head is 1/2. But during the course of the experiment, Sleeping Beauty does not get any novel information. Hence, the probability of Heads still remains 1/2.

By contrast, an alternative reasoning (II) runs as follows. Suppose the experiment is repeated many times, say, to fix ideas, 1000 times. Then there will be approximately 500 Heads-wakings on Monday, 500 Tails-wakings on Monday and 500 Tails-wakings on Tuesday. Hence, this reasoning goes, the probability of Heads equals 500/1500 = 1/3.

The argument for 1/2 and the argument for 1/3 yield conflicting conclusions. The Sleeping Beauty problem is usually presented accordingly as a problem arising from contradicting conclusions resulting from the two above-mentioned competing lines of reasoning aiming at assigning the probability of Heads once Beauty is awakened. I shall argue, however, that this statement of the Sleeping Beauty problem is somewhat restrictive and that we need to envisage the issue from a wider perspective. For present purposes, the Sleeping Beauty problem is the issue of calculating properly (i) the probability of Heads (resp. Tails) once Beauty is awakened; (ii) the probability of the day being Monday (resp. Tuesday) on awakening; and (iii) the probability of Heads (resp. Tails) on waking on Monday. From the halfer perspective, the probability of the day being Monday on awakening equals 3/4, and the probability of the day being Tuesday on awakening is 1/4. By contrast, from the thirder’s perspective, the probability of the day being Monday on awakening equals 2/3 and the probability of the day being Tuesday on awakening is 1/3.

But the argument for 1/2 and for 1/3 also have their own account of conditional probabilities. To begin with, the probability of Heads on waking on Tuesday is not a subject of disagreement, for it equals 0 in both accounts. The same goes for the probability of Tails on waking on Tuesday, since it equals 1 from the halfer’s or from the thirder’s viewpoint. But agreement stops when one considers the probability of Heads on waking on Monday. For it equals 2/3 from a halfer’s perspective. However, from a thirder’s perspective, it amounts to 1/2. On the other hand, the probability of Tails on waking on Monday is 1/3 from a halfer standpoint, and 1/2 for a thirder.

3. The urn analogy

In what follows, I shall present an ontological solution to the Sleeping Beauty problem, which rests basically on the hyper-entanglement urn experiment. A specific feature of this account is that it incorporates insights from the halfer and thirder standpoints, a line of resolution initiated by Nick Bostrom (2007) that has recently inspired some new contributions (Groisman 2008, Delabre 2008)1.

The argument for 1/3 and the argument for 1/2 rest basically on an urn analogy. This analogy is made explicit in the argument for 1/3 but is less transparent in the argument for 1/2. The argument for 1/3, to begin with, is based on an urn analogy which associates the situation related to the Sleeping Beauty experiment with an urn that contains, in the long run (assuming that the experiment is repeated, say, 1000 times), 500 red balls (Heads-wakings on Monday), 500 red balls (Tails-wakings on Monday) and 500 green balls (Tails-wakings on Tuesday), i.e. 1000 red balls and 500 green balls in total. In this context, the probability of Heads upon awakening is determined by the ratio of the number of Heads-wakings to the total number of wakings. Hence, P(Heads) = 500/1500 =1/3. The balls in the urn are normal ones and for present purposes, it is worth calling this sort of urn a standard urn.

On the other hand, the argument for 1/2 is also based on an urn analogy, albeit less transparently. The main halfer proponent grounds his reasoning on calculations (Lewis 2001), but for the sake of clarity, it is worth rendering the underlying associated analogy more apparent. For this purpose, let us recall how the calculation of the probability of drawing a red ball is handled by the argument for 1/2. If the coin lands Heads then the probability of drawing a red ball is 1, and if the coin lands Tails then this latter probability equals 1/2. We get then accordingly the probability of drawing a red ball (Monday-waking): P(R) = 1 x 1/2 + 1/2 x 1/2 = 3/4. By contrast, if the coin lands Tails, we calculate as follows the probability of drawing a green ball (Tuesday-waking): P(G) = 0 x 1/2 + 1/2 x 1/2 = 1/4. To sum up, according to the argument for 1/3: P(R) = 3/4 and P(G) = 1/4. For the sake of comparison, it is worth transposing this reasoning in terms of an urn analogy. Suppose then that the Sleeping Beauty experiment is iterated. It proves then that the argument for 1/2 is based on an analogy with a standard urn that contains 3/4 of red balls and 1/4 of green ones. These balls are also normal ones and the analogy underlying the argument for 1/2 is also with a standard urn. Now assuming as above that the experiment is repeated 1000 times, we get accordingly an urn that contains 500 red balls (Heads-wakings on Monday), 250 red balls (Tails-wakings on Monday) and 250 green balls (Tails-wakings on Tuesday), i.e. 750 red balls and 250 green balls in total. Such content of the urn results directly from Lewis’ calculation. However, as it stands, this analogy would arguably be a poor argument in favour of the halfer’s viewpoint. But at this step, we should pause and consider that Lewis’ argument for 1/2 did not rely on this urn analogy, though the latter is a consequence of Lewis’ calculation. We shall now turn to the issue of whether the standard urn is the correct analogy for the Sleeping Beauty experiment.

In effect, it turns out that the argument for 1/3 and the argument for 1/2 are based on an analogy with a standard urn. But at this stage, a question arises: is the analogy with the standard urn well-suited to the Sleeping Beauty experiment? In other terms, isn’t another urn model best suited? In the present context, this alternative can be formulated more accurately as follows: isn’t the situation inherent to the Sleeping Beauty experiment better put in analogy with the hyper-entanglement urn, rather than with the standard urn? I shall argue, however, that the analogy with the standard urn is mistaken, for it fails to incorporate an essential feature of the experiment, namely the fact that Monday-Tails wakings are indissociable from Tuesday-Tails wakings. For in the Tails case, Beauty cannot wake up on Monday without also waking up on Tuesday and reciprocally, she cannot wake up on Tuesday without also waking up on Monday.

When one reasons with the standard urn, one feels intuitively entitled to add red-Heads (Heads-wakings on Monday), red-Tails (Tails-wakings on Monday) and green-Tails (Tails-wakings on Tuesday) balls to compute frequencies. But red-Heads and red-Tails balls prove to be objects of an essentially different nature in the present context. In effect, red-Heads balls are in all respects similar to our familiar objects, and can be considered properly as single objects. By contrast, it turns out that red-Tails balls are quite indissociable from green-Tails balls. For we cannot draw a red-Tails ball without picking up the associated green-Tails ball. And conversely, we cannot draw a green-Tails ball without picking up the associated red-Tails ball. In this sense, red-Tails balls and the associated green-Tails balls do not behave as our familiar objects, but are much similar to entangled quantum objects. For Monday-Tails wakings are indissociable from Tuesday-Tails wakings. On Tails, Beauty cannot be awakened on Monday (resp. Tuesday) without being also awakened on Tuesday (resp. Monday). From this viewpoint, it is mistaken to consider red-Tails and green-Tails balls as separate objects. The correct intuition, I shall argue, is that the red-Tails and the associated green-Tails ball can be assimilated to a pair of hyper-entangled balls and constitute but one single object. In this context, red-Tails and green-Tails balls are best seen intuitively as constituents and mere parts of one single object. In other words, red-Heads balls and, on the other hand, red-Tails and green-Tails balls, cannot be considered as objects of the same type for probability purposes. And this situation justifies the fact that one is not entitled to add unrestrictedly red-Heads, red-Tails and green-Tails balls to compute probability frequencies. For in this case, one adds objects of intrinsically different types, i.e. one single object with the mere part of another single object.

Given what precedes, the correct analogy, I contend, is with a hyper-entanglement urn rather than with a normal urn. As will become clearer later, this new analogy incorporates the strengths of both above-mentioned analogies with the standard urn. And we shall now consider the Sleeping Beauty problem in light of this new perspective.

4. Consequences of the analogy with the hyper-entanglement urn

At this step, it is worth drawing the consequences of the analogy with the hyper-entanglement urn, that notably result from the ontological properties of the balls. Now the key point proves to be the following one. Recall that nothing seemingly distinguishes normal balls from hyper-entangled ones within the hyper-entanglement urn. And among the red balls, half are normal ones, but the other half is composed of red balls that are each hyper-entangled with a different green ball. If one considers the behaviour of the balls, it turns out that normal balls behave as usual. But hyper-entangled ones do behave differently, with regard to statistics. Suppose I add the red ball of a hyper-entangled pair into the hyper-entanglement urn. Then I also add instantly in the urn its associated green ball. Suppose, conversely, that I remove the red ball of a hyper-entangled pair from the urn. Then I also remove instantly its associated green ball.

At this step, we are led to the core issue of calculating properly the probability of drawing a red ball from the hyper-entanglement urn. Let us pause for a moment and forget temporarily the fact that, according to its classical formulation, the Sleeping Beauty problem arises from conflicting conclusions resulting from the argument for 1/3 and the argument for 1/2 on calculating the probability of Heads once Beauty is awakened. For as we did see it before, the problem also arises from the calculation of the probability of the day being Monday on awakening (drawing a red ball), since conflicting conclusions also result from the two competing lines of reasoning. In effect, Elga argues for 2/3 and Lewis for 3/4. Hence, the Sleeping Beauty problem could also have been formulated alternatively as follows: once awakened, what probability should Beauty assign to her waking on Monday? In the present context, this is tantamount to the probability of drawing a red ball from the hyper-entanglement urn.

What is then the response of the present account, based on the analogy with the hyper-entanglement urn, to the issue of calculating the probability of drawing a red ball? In the present context, “drawing a red ball” turns out to be somewhat ambiguous. For according to the ontological properties of the balls within the hyper-entanglement urn, one can consider red balls either from the viewpoint of colour-ness, or from the standpoint of object-ness2. Hence, in the present context, “drawing a red ball” can be interpreted in two different ways: either (i) “drawing a red ball-as-colour”; or (ii) “drawing a red ball-as-object”. Now disambiguating the notion of drawing a red ball, we should distinguish accordingly between two different questions. First, (i) what is the probability of drawing a red ball-as-colour (Monday-waking-as-time-segment)? Let us denote by P(R) the latter probability. Second, (ii) what is the probability of drawing a red ball-as-object (Monday-waking-as-object)? Let us denote it by P(R). This distinction makes sense in the present context, since it results from the properties of the hyper-entangled balls. In particular, this richer semantics results from the case where one draws a green ball of a hyper-entangled pair from the urn. For in the latter case, this green ball is not a red one, but it occurs that one also picks up a red ball, since the associated red ball is withdrawn simultaneously.

Suppose, on the one hand, that we focus on the colour of the balls, and that we consider the probability P(R) of drawing a red ball-as-colour. It occurs now that there are 2/3 of red balls-as-colour and 1/3 of green balls-as-colour in the urn. Accordingly, the probability P(R) of drawing a red ball-as-colour equals 2/3. On the other hand, the probability P(G) of drawing a green ball-as-colour equals 1/3.

Assume, on the other hand, that we focus on balls as objects, considering that one pair of hyper-entangled balls behaves as one single object. Now we are concerned with the probability P(R) of drawing a red ball-as-object. On Heads, the probability of drawing a red ball-as-object is 1. On Tails, we can either draw the red or the green ball of a hyper-entangled pair. But it should be pointed out that if we draw on Tails the green ball of a hyper-entangled pair, we also pick up instantly the associated red ball. Hence, the probability of drawing a red ball on Tails is also 1. Thus, P(R) = 1 x 1/2 + 1 x 1/2 = 1. Conversely, what is the probability P(G) of drawing a green ball-as-object (a waking on Tuesday)? The probability of drawing a green ball-as-object is 0 in the Heads case, and 1 in the Tails case. For in the latter case, we either draw the green or the red ball of a hyper-entangled pair. But even if we draw the red ball of the hyper-entangled pair, we draw then instantly its associated green ball. Hence, P(G) = 0 x 1/2 + 1 x 1/2 = 1/2. To sum up: P(R) = 1 and P(G) = 1/2. The probability of drawing a red ball-as-object (a waking on Monday) is then 1, and the probability of drawing a green ball-as-object (a waking on Tuesday) is 1/2. Now it turns out that P(R) + P(G) = 1 + 1/2 = 1.5. In the present account, this results from the fact that drawing a red ball-as-object and drawing a green ball-as-object from a hyper-entangled pair are not exclusive events for probability purposes. For we cannot draw the red-Tails (resp. green-Tails) ball without drawing the associated green-Tails (resp. red-Tails) ball.

To sum up now. It turns out that the probability P(R) of drawing a red ball-as-colour (Monday-waking-as-time-segment) equals 2/3. And the probability P(G) of drawing a green ball-as-colour (Tuesday-waking-as-time-segment) equals 1/3. On the other hand, the probability P(R) of drawing a red ball-as-object (Monday-waking-as-object) equals 1; and the probability P(G) of drawing a green ball-as-object (Tuesday-waking-as-object) equals 1/2.

At this step, we are led to the issue of calculating properly the number of balls present in the urn. Now we should distinguish, just as before, according to whether one considers balls-as-colour or balls-as-object. Suppose then that we focus on the colour of the balls. Then we have grounds to consider that there are in total 2/3 of red balls and 1/3 of green balls in the hyper-entanglement urn, i.e. 1000 red ones and 500 green ones. This conforms with the calculation that results from the thirder’s standpoint. Suppose, that we rather focus on balls as single objects. Things go then differently. For we can consider first that there are 1000 balls as objects in the urn, i.e. 500 (red) normal ones and 500 hyper-entangled ones. Now suppose that the 500 (red) normal balls are removed from the urn. Now there only remain hyper-entangled balls within the urn. Suppose then that we pick up one by one the remaining balls from the urn, by removing alternatively one red ball and one green ball from the urn. Now it turns out that we can draw 250 red ones and 250 green ones from the urn. For once we draw a red ball from the urn, its associated green ball is also withdrawn. And conversely, when we pick up a green ball from the urn, its associated red ball is also withdrawn. Hence, inasmuch as we consider balls as objects, there are in total 750 red ones and 250 green ones in the urn. At this step, it should be noticed that this corresponds accurately to the composition of the urn which is associated with Lewis’ halfer calculation. But this now makes sense, as far as the analogy with the hyper-entanglement urn is concerned. The above-mentioned analogy with the urn associated with Lewis’ halfer calculation was a poor argument inasmuch as the urn was a standard one, but things go differently when one considers now the analogy with the hyper-entanglement urn.

5. A two-sided account

From the above, it results that the line of reasoning which is associated with the balls-as-colour standpoint corresponds to the thirder’s reasoning. And conversely, the line of thought which is associated with the balls-as-object viewpoint echoes the halfer’s reasoning. Hence, the balls-as-colour/balls-as-object dichotomy parallels the thirder/halfer opposition. Grounded though they are on an unsuited analogy with the standard urn, the argument for 1/3 and the argument for 1/2 do have, however, their own strengths. In particular, the analogy with the urn in the argument for 1/3 does justice to the fact that the Sleeping Beauty experiment entails that 2/3 of Monday-wakings will occur in the long run. On the other hand, the analogy with the urn in the argument for 1/2 handles adequately the fact that one Heads-waking is put on a par with two Tails-wakings. In the present context however, these two analogies turn out to be one-sided and fail to handle adequately the probability notion of drawing a red ball (waking on Monday). But in the present context, the probability P(R) of drawing a red ball-as-colour corresponds to the thirder’s insight. And the probability P(R) of drawing a red ball-as-object corresponds to the halfer’s line of thought. At this step, it turns out that the present account is two-sided, since it incorporates insights from the argument for 1/3 and from the argument for 1/2.

Finally, it turns out that the standard urn which is classically used to model the Sleeping Beauty problem does not allow for two possible interpretations of the probability of drawing a red ball. Rather, in the standard urn model, the two interpretations are exclusive of one another and this yields the classical contradiction between the argument for 1/3 and the argument for 1/2. But as we did see it, with the hyper-entanglement urn model, this contradiction dissolves, since two different interpretations of the probability of drawing a red ball (waking on Monday) are now allowed, yielding then two different calculations. In the latter model, these probabilities are no more exclusive of one another and the contradiction dissolves into complementarity.

Now the same ambiguity plagues the statement of the Sleeping Beauty problem, and its inherent notion of “waking”. For shall we consider “wakings-as-time-segment” or “wakings-as-object”? The initial statement of the Sleeping Beauty problem is ambiguous about that, thus allowing the two competing viewpoints to develop, with their respective associated calculations. But once we diagnose accurately the source of the ambiguity, namely the ontological status of the wakings, we allow for the two competing lines of reasoning to develop in parallel, thus dissolving the initial contradiction3.

In addition, what precedes casts new light on the argument for 1/3 and the argument for 1/2. For given that the Sleeping Beauty experiment, is modelled with a standard urn, both accounts lack the ability to express the difference between the probability P(R) of drawing a red ball-as-colour (a Monday-waking-as-time-segment) and the probability P(R) of drawing a red ball-as-object (a Monday-waking-as-object), for it does not make sense with the standard urn. Consequently, there is a failure to express this difference with the standard urn analogy, when considering drawing a red ball. But such distinction makes sense with the analogy with the hyper-entanglement urn. For in the resulting richer ontology, the distinction between P(R) and P(R) yields two different results: P(R) = 2/3 and P(R) = 1.

At this step, it is worth considering in more depth the balls-as-colour/balls-as-object opposition, that parallels the thirder/halfer contradiction. It should be pointed out that “drawing a red ball-as-colour” is associated with an indexical (“this ball is red”), somewhat internal standpoint, that corresponds to the thirder’s insight. Typically, the thirder’s viewpoint considers things from the inside, grounding the calculation on the indexicality of Beauty’s present waking. On the other hand, “drawing a red ball-as-object” can be associated with a non-indexical (“the ball is red”), external viewpoint. This corresponds to the halfer’s standpoint, which can be viewed as more general and external.

As we did see it, the calculation of the probability of drawing a red ball (waking on Monday) is the core issue in the Sleeping Beauty problem. But what is now the response of the present account on conditional probabilities and on the probability of Heads upon awakening? Let us begin with the conditional probability of Heads on a Monday-waking. Recall first how the calculation goes on the two concurrent lines of reasoning. To begin with, the probability P(Heads|G) of Heads on drawing a green ball is not a subject of disagreement for halfers and thirders, since it equals 0 on both accounts. The same goes for the probability P(Tails|G) of Tails on drawing a green ball, since it equals 1 from the halfer’s or the thirder’s viewpoint. But agreement stops when one considers the probability P(Heads|R) of Heads on drawing a red ball. For P(Heads|R) = 1/2 from the thirder’s perspective and P(Heads|R) = 2/3 from the halfer’s viewpoint. On the other hand, the probability P(Tails|R) of Tails on drawing a red ball is 1/2 for a thirder and 1/3 for a halfer.

Now the response of the present account to the calculation of the conditional probability of Heads on drawing a red ball (waking on Monday) parallels the answer made to the issue of determining the probability of drawing a red ball. In the present account, P(Heads|G) = 0 and P(Tails|G) = 1, as usual. But we need to disambiguate how we interpret drawing a red ball by distinguishing between P(Heads|R) and P(Heads|R), to go any further. For P(Heads|R) is the probability of Heads on drawing a red ball-as-colour. And P(Heads|R) is the probability of Heads on drawing a red ball-as-object. P(Heads|R) is calculated in the same way as in the thirder’s account. Now we get accordingly: P(Heads|R) = 1/2. On the other hand, P(Heads|R) is computed in the same way as from the halfer’s perspective, and we get accordingly: P(Heads|R) = [P(Heads) x P(R|Heads)] / P(R) = [1/2 x 1] / 1 = 1/2.

Now the same goes for the probability of Heads upon awakening. For there are two different responses in the present account, depending on whether one considers P(R) or P(R). If one considers balls-as-colour, the probability of Heads upon awakening is calculated in the same way as in the argument for 1/3, and we get accordingly: P(Heads) = 1/3 and P(Tails) = 2/3. On the other hand, if one is concerned with balls-as-object, it ensues, in the same way as with the halfer’s account, that there is no shift in the prior probability of Heads. As Lewis puts it, Beauty’s awakening does not add any novel information. It follows accordingly that the probability P(Heads) of Heads (resp. Tails) on awakening still remains 1/2.

Finally, the above results are summarised in the following table:

halfer

thirder

present account

P(Heads)

1/3

1/3

P(Tails)

2/3

2/3

P(Heads)

1/2

1/2

P(Tails)

1/2

1/2

P(drawing a red ball-as-colour) P(R)

2/3

2/3

P(drawing a green ball-as-colour) P(G)

1/3

1/3

P(drawing a red ball-as-object) P(R)

3/4

1

P(drawing a green ball-as-object) P(G)

1/4

1/2

P(Heads| drawing a red ball-as-colour) P(Heads|R)

1/2

1/2

P(Tails| drawing a red ball-as-colour) P(Tails|R)

1/2

1/2

P(Heads| drawing a red ball-as-object) P(Heads|R)

2/3

1/2

P(Tails| drawing a red ball-as-object) P(Tails|R)

1/3

1/2

At this step, it is worth recalling the diagnosis of the Sleeping Beauty problem put forth by Berry Groisman (2008). Groisman attributes the two conflicting responses to the probability of Heads to an ambiguity in the protocol of the Sleeping Beauty experiment. He argues that the argument for 1/2 is an adequate response to the probability of Heads on awakening, under the setup of coin tossing. On the other hand, he considers that the argument for 1/3 is an accurate answer to the latter probability, under the setup of picking up a ball from the urn. Groisman also considers that putting a ball in the box and picking up a ball out from the box are two different events, that lead therefore to two different probabilities. Roughly speaking, Groisman’s coin tossing/picking up a ball distinction parallels the present balls-as-colour/balls-as-object dichotomy. However, in the present account, putting a ball in the urn is no different from picking up a ball from the urn. For if we put in the urn a red ball of a hyper-entangled pair, we also immediately put in the urn its associated green ball. Rather, from the present standpoint, drawing (resp. putting in the urn) a red ball-as-colour from the urn is probabilistically different from picking up a red ball-as-object. The present account and Groisman’s analysis share the same overall direction, although the details of our motivations are significantly different.

Finally, the lesson of the Sleeping Beauty Problem proves to be the following: our current and familiar objects or concepts such as balls, wakings, etc. should not be considered as the sole relevant classes of objects for probability purposes. We should bear in mind that according to an unformalised axiom of probability theory, a given situation is classically modelled with the help of urns, dices, balls, etc. But the rules that allow for these simplifications lack an explicit formulation. However in certain situations, in order to reason properly, it is also necessary to take into account somewhat unfamiliar objects whose constituents are pairs of indissociable balls or of mutually inseparable wakings, etc. This lesson was anticipated by Nelson Goodman, who pointed out in Ways of Worldmaking that some objects which are prima facie completely different from our familiar objects also deserve consideration: we do not welcome molecules or concreta as elements of our everyday world, or combine tomatoes and triangles and typewriters and tyrants and tornadoes into a single kind.4 As we did see it, in some cases, we cannot add unrestrictedly an object of the Heads-world with an object of the Tails-world. For despite the appearances, objects of the Heads-world may have ontologically different properties from objects of the Tails-world. And the status of our probabilistic paradigm object, namely a ball, proves to be world-relative, since it can be a whole in the Heads-world and a part in the Tails-world. Once this goodmanian step accomplished, we should be less vulnerable to certain subtle cognitive traps in probabilistic reasoning.

Acknowledgements

I thank Jean-Paul Delahaye and Claude Panaccio for useful discussion on earlier drafts. Special thanks are due to Laurent Delabre for stimulating correspondence and insightful comments.

References

Arntzenius, F. (2002). Reflections on Sleeping Beauty. Analysis, 62-1, 53-62

Aspect, A., Dalibard, J. & Roger, G. (1982). Physical Review Letters. 49, 1804-1807

Black, M. (1952). The Identity of Indiscernibles. Mind 61, 153-164

Bostrom, N. (2002). Anthropic Bias: Observation Selection Effects in Science and Philosophy. (New York: Routledge)

Bostrom, N. (2007). Sleeping Beauty and Self-Location: A Hybrid Model. Synthese, 157, 59-78

Bradley, D. (2003). Sleeping Beauty: a note on Dorr’s argument for 1/3. Analysis, 63, 266-268

Delabre, L. (2008). La Belle au bois dormant : débat autour d’un paradoxe. Manuscript, http://philsci-archive.pitt.edu/archive/00004342

Elga, A. (2000). Self-locating Belief and the Sleeping Beauty Problem. Analysis, 60, 143-147

Goodman, N. (1978). Ways of Worldmaking. (Indianapolis: Hackett Publishing Company)

Groisman, B. (2008). The End of Sleeping Beauty’s Nightmare. British Journal for the Philosophy of Science, 59, 409-416

Leslie, J. (2001). Infinite Minds (Oxford & New York: Oxford University Press)

Lewis, D. (2001). Sleeping Beauty: Reply to Elga. Analysis, 61, 171-176

Monton, B. (2002). Sleeping Beauty and the Forgetful Bayesian. Analysis, 62, 47-53

White, R. (2006). The generalized Sleeping Beauty problem: A challenge for thirders. Analysis, 66, 114-119

1 Bostrom opens the path to a third way out to the Sleeping Beauty problem: “At any rate, one might hope that having a third contender for how Beauty should reason will help stimulate new ideas in the study of self-location”. In his account, Bostrom sides with the halfer on P(Heads) and with the thirder on conditional probabilities, but his treatment has some counter-intuitive consequences on conditional probabilities.

2 This issue relates to the identity of indiscernibles and is notably hinted at by Max Black (1952, p. 156) who describes a universe composed of two identical spheres: Isn’t it logically possible that the universe should have contained nothing but two exactly similar spheres? We might suppose that each was made of chemically pure iron, had a diameter of one mile, that they had the same temperature, colour, and so on, and that nothing else existed. Then every quality and relational characteristic of the one would also be a property of the other. In the present context, it should be pointed out that the colours of the hyper-entangled balls are anti-correlated. John Leslie (2001, p. 153) also raises a similar issue with his paradox of the balls: Here is a yet greater paradox for Identity of Indiscernibles to swallow. Try to picture a cosmos consisting just of three qualitatively identical spheres in a straight line, the two outer ones precisely equidistant from the one at the centre. Aren’t there plain differences here? The central sphere must be nearer to the outer spheres than these are to each other. Identity of Indiscernibles shudders at the symmetry of the situation, however. It holds that the so-called two outer spheres must really be only a single sphere. And this single sphere, which now has all the same qualities as its sole surviving partner, must really be identical to it. There is actually just one sphere!.

3 It is worth noting that the present treatment of the Sleeping Beauty problem, is capable of handling several variations of the original problem that have recently flourished in the literature. For the above solution to the Sleeping Beauty problem applies straightforwardly, I shall argue, to these variations of the original experiment. Let us consider, to begin with, a variation were on Heads, Sleeping Beauty is not awakened on Monday but instead on Tuesday. This is modelled with a hyper-entanglement urn that receives one normal green ball (instead of a red one in the original experiment) in the Heads case.

Let us suppose, second, that Sleeping Beauty is awakened two times on Monday in the Tails case (instead of being awakened on both Monday and Tuesday). This is then modelled with a hyper-entanglement urn that receives one pair of hyper-entangled balls which are composed of two red balls in the Tails case (instead of a pair of hyper-entangled balls composed of a red and a green ball in the original experiment).

Let us imagine, third, that Beauty is awakened two times on Monday and Tuesday in the Heads case, and three times on Monday, Tuesday and Wednesday in the Tails case. This is then modelled with a hyper-entanglement urn that receives one pair of hyper-entangled balls composed of one red ball and one green ball in the Heads case; and in the Tails case, the hyper-entanglement urn is filled with one triplet of hyperentangled balls, composed of one red, one green and one blue ball.

4 Goodman (1978, p. 21).

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eub-sep

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.

 

 

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A Dichotomic Analysis of the Surprise Examination Paradox

 

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

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

1. The paradox

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

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

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

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

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

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

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

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

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

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

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

2. Monist or dichotomic analysis of the paradox

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

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

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

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

(D1)

S[k, 0]

S[k, 1]

S[7,s]

S[6,s]

S[5,s]

S[4,s]

S[3,s]

S[2,s]

S[1,s]

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

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

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

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

(D2)

S[k, 0]

S[k, 1]

S[7,s]

S[6,s]

S[5,s]

S[4,s]

S[3,s]

S[2,s]

S[1,s]

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

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

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

(C4) k (1 k n) such that S[k, 1] = 1 (validation step)

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

(D5)

S[k, 0]

S[k, 1]

S[7,s]

S[6,s]

S[5,s]

S[4,s]

S[3,s]

S[2,s]

S[1,s]

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

(D6)

S[k, 0]

S[k, 1]

S[7,s]

S[6,s]

S[5,s]

S[4,s]

S[3,s]

S[2,s]

S[1,s]

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

3. The surprise notion corresponding to the conjoint structure

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

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

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

(D7)

S[k, 0]

S[k, 1]

S[7,s]

S[6,s]

S[5,s]

S[4,s]

S[3,s]

S[2,s]

S[1,s]

and the associated professor’s announcement is the following:

(S7)

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

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

(D8)

S[1, 0]

S[1, 1]

S[1,s]

which corresponds to the following professor’s announcement:

(S8)

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

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

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

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

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

(D9)

S[1, 0]

S[1, 1]

S[1,s]

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

(D8)

S[1, 0]

S[1, 1]

S[1,s]

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

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

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

(D10)

S[k, 0]

S[k, 1]

S[7,s]

S[6,s]

S[5,s]

S[4,s]

S[3,s]

S[2,s]

S[1,s]

and to the following announcement of the professor

(S10)

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

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

(D11)

S[k, 0]

S[k, 1]

S[n,s]

S[n-1,s]

…………

…………

…………

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

(D12)

S[k, 0]

S[k, 1]

S[2,s]

S[1,s]

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

(S12)

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

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

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

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

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

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

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

4. The surprise notion corresponding to the disjoint structure

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

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

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

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

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

(D13)

S[1, 0]

S[1, 1]

S[2,s]

S[1,s]

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

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

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

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

(S14)

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

The corresponding definition presents then the following structure :

(D14)

S[1, 0]

S[1, 1]

S[365,s]

…………

…………

…………

S[1,s]

which is an instance of the following general form :

(D15)

S[1, 0]

S[1, 1]

S[n,s]

…………

…………

…………

S[1,s]

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

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

(S7)

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

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

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

(D16)

S[k, 0]

S[k, 1]

S[7,s]

S[6,s]

S[5,s]

S[4,s]

S[3,s]

S[2,s]

S[1,s]

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

(D17)

S[k, 0]

S[k, 1]

S[7,s]

S[6,s]

S[5,s]

S[4,s]

S[3,s]

S[2,s]

S[1,s]

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

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

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

(D18)

S[k, 0]

S[k, 1]

S[7,s]

1

0

S[6,s]

0,75

0,25

S[5,s]

0,5

0,5

S[4,s]

0,25

0,75

S[3,s]

0

1

S[2,s]

0

1

S[1,s]

0

1

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

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

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

(D19)

S[k, 0]

S[k, 1]

S[7,s]

S[6,s]

S[5,s]

S[4,s]

S[3,s]

S[2,s]

S[1,s]

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

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

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

5. The solution to the paradox

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

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

The professor: Yes, it is exact. Continue.

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

The professor: Thus let us see that.

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

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

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

The professor: I see. Conclude now.

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

6. Conclusion

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

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

REFERENCES

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

1 I simplify here considerably.

2 Without pretending to being exhaustive.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 

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intro-phi-a-bookIn this book, Paul Franceschi provides us with an introduction to analytic philosophy. In a concrete way, he chooses to describe forty paradoxes, arguments or philosophical issues that represent so many challenges for contemporary philosophy and human intelligence, for some paradoxes of millennial origin—such as the Liar or the sorites paradox—are still unresolved in the present day. Some other philosophical puzzles, however—such as the Doomsday argument—appeared only recently in the literature. The author strives to introduce us clearly to each of these problems as well as to major attempts that have been formulated to solve them.

“I’m really impressed by this very neat and stimulating book. I highly recommend it both to students for pedagogy and general culture (prisoner’s dilemma, twin-earth, etc.), and to professionals as well for the reference tool and even more generally to those who like to think.”

Julien Dutant, Philotropes, Philosophical blog

 


The Kindle version is also available.


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An Introduction to Analytic Philosophy

Paul Franceschi

Copyright (c) Paul Franceschi

All rights reserved

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From P. to T.

 Introduction to the English edition

 

introductionThis book aims to provide an introduction to analytic philosophy. It is primarily intended for readers who want to be initiated into this philosophical style. The approach that I have chosen to offer the reader for that introduction is the same as that by which I myself was introduced to analytic philosophy: the presentation of the most famous problems and paradoxes. An advantage of this approach is that there is no need for any prerequisites. This book aims thus at presenting a significant number of contemporary issues in analytic philosophy. It consists then of illustrating how the approach that is pursued involves the accurate description of problems, clearly identified, and whose presentation does not suffer from ambiguity. The approach adopted throughout this book will therefore consist of the description of a large number of contemporary philosophical problems, thus illustrating the methodology used in analytic philosophy, which consists in describing with precision—often step by step—a number of problems for which there exists, at present, no consensual solution. It may be useful for this purpose to classify contemporary philosophical problems into three distinct categories: paradoxes, arguments and problems per se. Each of these three types of problem is exposed, in what follows, and usually accompanied by one or more solutions that have been proposed in the contemporary literature.

I will endeavor first to describe a number of paradoxes. The most famous of them are rooted in antiquity and remain still unresolved: the Liar, the sorites paradox, etc. Paradoxes are arguments based on premises and reasoning that seem well-founded, but whose conclusion leads to a contradiction. An excellent definition is provided by Mark Sainsbury, in his book Paradoxes, published in 1995: “paradoxes are unacceptable conclusions drawn from seemingly true premises and correct reasoning”.

I shall also present a number of arguments that are widely discussed within contemporary philosophical literature. Such arguments are often made up of reasoning whose premises and the deductions that accompany them seem quite acceptable, but whose conclusion proves counter-intuitive. Problems of this type distinguish themselves from the paradoxes in that they do not truly lead to a contradiction. Unlike the paradoxes, we do not observe in this type of argument a contradiction per se, but only a conclusion that proves contrary to common sense. Arguments whose conclusion appears counter-intuitive are close to the paradoxes in the sense that it is very likely that the underlying reasoning is misleading. However, they differ from the paradoxes in the sense that one cannot rule out at the start the possibility that it is our intuition that is at fault. In this case, the solution to the problem posed by this type of argument has to explain why one’s conclusion appears at first sight counter-intuitive.

Lastly, I shall describe a number of problems per se that have led to recent discussions in analytic philosophy. Among these problems based on reasoning, some have a very ancient origin, while others have only recently been described.

Analytic philosophy is essentially characterized by a requirement for clarity in the exposition of ideas and a marked concern for rigor at the stage of the argument. Clarity of ideas is intended to avoid ambiguity and difficulties in the interpretation of texts. It also allows a better critical evaluation of the ideas. This necessity for rigor may sometimes require the use of a mathematical formalism, which should not, however, go so far as to require advanced knowledge in mathematics. As we can see, analytic philosophy is primarily a philosophical style.

It is customary to oppose analytic philosophy and continental philosophy. Continental philosophy refers to the philosophical writings of French and German authors of the nineteenth and twentieth centuries, among whom are, without being exhaustive: Friedrich Hegel, Sören Kierkegaard, Friedrich Nietzsche, Karl Marx, Herbert Marcuse, Martin Heidegger, Jean-Paul Sartre, Maurice Merleau-Ponty and Michel Foucault. The writings of these philosophers are characterized by a greater literary involvement and often a stronger political commitment.

Analytic philosophy is sometimes associated with Anglo-Saxon countries and continental philosophy with the European continent. Such a viewpoint is, however, somewhat simplistic. Indeed, it is true that analytic philosophy is currently the dominant style in the United Kingdom, the United States, Canada, Australia and New Zealand, for example. However, it is also represented in France, Italy, Germany, Spain, Portugal, Greece, Belgium, etc. Moreover, if one takes into account ancient and classical philosophers, it is clear that such a viewpoint proves to be incorrect, because one can find a particularly pure analytical style on the banks of the Mediterranean, in the writings of several philosophers of antiquity. The classical Greek philosophers, inventors of famous unresolved paradoxes such as the Liar, the sorites paradox, but also the paradoxes of Zeno of Elea, constitute outstanding examples. With Plato in particular, one can also find the clarity of the argument in the famous Allegory of the cave.

Moreover, one will find in Pascal, with the Wager argument, all the criteria of a detailed, accurate and clear argumentation. And most importantly, we can notice that Descartes was practicing an astonishingly pure and avant-gardist analytical style. Many of the arguments of Descartes could have been included without change in the contemporary analytic philosophical literature. In the present work, one will thus find the famous cogito argument, the evil demon argument, the ontological argument of Descartes, as well as an argument in favor of mind/body dualism.

It would be rather awkward and Manichean to oppose the two styles—analytical and continental—by considering that one is better than the other. So less overtly subjectively, we can estimate that these are two different styles of practicing philosophy, which each have their own advantages and disadvantages. It is certainly necessary to preserve both styles, given their respective merits and complementarity. Finally, it turns out that the coexistence of the two styles essentially constitutes an expression of cultural diversity that proves to be synonymous with wealth.

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1. The Liar Paradox

chap1The Liar paradox is one of the oldest and deepest of the known paradoxes. It is attributed to the Greek philosopher Eubulides of Miletus, who lived in the fourth century BC. The Liar paradox can be expressed very simply, since it arises directly from the consideration of the following statement: “This sentence is false.” The paradox stems from the fact that if this last sentence is true, then it follows that it is false, but if the sentence is false, then it is false that it is false and therefore it is true. Thus “This sentence is false” is false if it is true, and true if it is false. In conclusion, “This sentence is false” is true if and only if it is false, and the latter conclusion turns out to be paradoxical.

One often denotes “This sentence is false” by (λ). At this stage, it is worth describing in detail the various steps of reasoning that lead to the Liar paradox (the symbol λ denotes the conclusion):

(λ) (λ) is false
(1) (λ) is either true or false bivalence
(2) if (λ) is true hypothesis 1
(3) then it is true that (λ) is false from (λ),(2)
(4) then (λ) is false from (3)
(5) if (λ) is false hypothesis 2
(6) then it is false that (λ) is false from (λ),(5)
(7) then (λ) is true from (6)
(8) ∴ (λ) is neither true nor false from (4),(7)

The conclusion (8) is paradoxical here, since it follows that (λ) is neither true nor false, in contradiction of the principle (1) of bivalence. The problem with the Liar is thus the following: what is the truth value of the proposition (λ), given that we can not assign it, without contradiction, the truth value true or false?

A first attempt at a solution to the Liar is to consider that the truth value of (λ) is neither true nor false, but a third truth value: indeterminate. One can then consider a three-valued logic, which thus includes three truth values: true, false, indeterminate. The Liar is then reintroduced under the following form:

3) 3) is false or indeterminate

In this new context, a proposition may now be assigned three different truth values: true, false or indeterminate. The principle of trivalence then states that (λ3) is either true or false, or indeterminate. However, the fact of considering in turn that (λ3) is true, false, or indeterminate still does not lead to a satisfactory solution, since by the same reasoning as with the simple Liar, the conclusion follows that (λ3) is neither true nor false nor indeterminate. This results in the impossibility of properly assigning a truth value to the proposition (λ3).

Moreover, it turns out that the problem resurfaces in the same way if we consider not three, but four truth values: true, false, indeterminate1 and indeterminate2. We must then make use of a four-valued logic. However, the following variation of the Liar emerges:

4) 4) is false or indeterminate1 or indeterminate2

which leads, as previously, to the impossibility of assigning a truth value to (λ4).

Another attempt at a solution is to reject the principle of bivalence, of trivalence, and more generally of n-valence on which the reasoning that leads to the Liar is based. However, such a line of solution also fails, since it faces a more powerful variation of the Liar, namely the Strengthened Liar, which does not require us to appeal to any principle of bivalence, of 3-valence or of n-valence:

s) s) is non-true

This is because the Strengthened Liar leads to the following reasoning:

s) s) is non-true
(9) s) is either true or non-true dichotomy
(10) if (λs) is true hypothesis 1
(11) then it is true that (λs) is non-true from (λs),(10)
(12) then (λs) is non-true from (11)
(13) if (λs) est non- true hypothesis 2
(14) then it is non-true that (λs) is non-true from (λs),(13)
(15) then (λs) is true from (14)
(16) ∴ (λs) is neither true nor non-true from (12),(15)

Lastly, another attempt at a solution to the Liar paradox is to consider that the structure of the Liar is self-referential, since this very proposition makes direct reference to itself. According to this type of solution, it would suffice to prohibit the formation of self-referential propositions to prevent the emergence of the paradox. However, such a solution turns out to be too restrictive, since there exist many propositions whose structure is self-referential, but for which the attribution of a truth value does not pose any problem. It suffices to consider then the Contingent Liar:

c) either this proposition is false or 0 = 0

But it turns out that one can validly assign the truth value true to the Contingent Liar. Thus, although the Contingent Liar presents a self-referential structure, we can successfully and without contradiction—unlike the Liar—assign it a truth value. In this context, it turns out that simply prohibiting all self-referential propositions is too high a price to pay for solving the Liar paradox, and therefore does not constitute a satisfactory solution.

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2. The Sorites Paradox

chap2The sorites paradox is one of the oldest and most important of the known paradoxes. Its origin is usually attributed to Eubulides of Miletus, the Greek philosopher from antiquity to whom we also owe the Liar paradox. The paradox can be described informally as follows. Firstly, it is commonly accepted that a set comprising 100,000 grains of sand is a heap. In addition, it proves that if a set comprising a given number of grains of sand is a heap, then a set with a grain of sand less is also a heap. Given these premises, the conclusion follows that a set comprising a single grain of sand is also a heap. In effect, if a set comprising 100,000 grains of sand is a heap, it follows that a set with 99,999 grains of sand is a heap, and the same goes for a set comprising 99,998 grains of sand and 99,997, 99996, 99995, …, and so on, up to a single grain of sand. The paradox stems from the fact that the corresponding reasoning seems quite valid, whereas the resulting conclusion proves to be unacceptable.

The various steps leading to the sorites paradox can be detailed as follows:

(1) a set comprising 100000 grains of sand is a heap
(2) if a set comprising n grains of sand is a heap, then a set comprising n – 1 grains of sand is a heap
(3) if a set comprising 100000 grains of sand is a heap, then a set comprising 99999 grains of sand is a heap
(4) ∴ a set comprising 99999 grains of sand is a heap
(5) if a set comprising 99999 grains of sand is a heap, then a set comprising 99998 grains of sand is a heap
(6) ∴ a set comprising 99998 grains of sand is a heap
(7) if a set comprising 99998 grains of sand is a heap, then a set comprising 99997 grains of sand is a heap
(8) ∴ a set comprising 99997 grains of sand is a heap
(9)
(10) ∴ a set comprising 1 grain of sand is a heap

The conclusion of the paradox results from the repeated use of a logical and widely accepted principle, termed modus ponens, which has the form: p, if p then q, then q (where p and q denote two propositions).

Many variations of the sorites paradox can be found in the literature. Thus, another version of the paradox with the predicate tall is as follows:

(11) a man who measures 200 cm is tall
(12) if a man who measures n cm is tall, then a man who measures n1 cm is tall
(13)
(14) ∴ a man who measures 140 cm is tall

Likewise, we can also construct some variations of the paradox with other vague concepts such as: rich, old, red, etc. This highlights the structure of the paradox (where P denotes a vague predicate):

(15) P(100000) basis step
(16) if P(n) then P(n – 1) induction step
(17)
(18) ∴ P(1)

It should be pointed out here that the structure of the paradox is reversible. Indeed, the previous versions of the paradox proceed decrementally. But the paradox can also operate incrementally, in the following way:

(19) a man who has 1 hair is bald basis step
(20) if a man who has n hairs is bald, then a man who has n + 1 hairs is bald induction step
(21)
(22) ∴ a man who has 100000 hairs is bald

The structure of the paradox is then as follows (P denotes a vague predicate):

(23) P(1) basis step
(24) if P(n) then P(n + 1) induction step
(25)
(26) ∴ P(100000)

Numerous solutions have been proposed to solve the sorites paradox. However, none of them has proved satisfactory so far. Thus, the sorites paradox remains one of the most widely studied contemporary paradoxes.

A solution that calls into question the induction step has been proposed to solve the paradox. This type of solution is based on an approach by degrees and argues thus that the induction step is only true for some instances—the proper instances—of the notion of heap. Such analysis is based on the fact that the notion of heap is a vague concept. Such a notion is thus characterized by the existence of proper instances (e.g. a value of n equal to 1 million), of proper counter-instances (e.g. a value of n equal to 2), but also of borderline cases (e.g. a value of n equal to 100), which constitute a penumbra zone between the notions of heap and non-heap. According to the approach by degrees, the truth value of the induction step is 1 when one is in the presence of proper instances. But when it comes to borderline cases, its truth value is less than 1. It follows finally that the truth value of the induction step, when considering all possible values of n, is slightly less than 1. And this is sufficient to partially block the deductive process and finally prevent us from reaching the final conclusion.

The induction step is also referred to in another type of solution, which considers that the induction step is not necessarily true. It suffices, for example, to consider a stack consisting of cubes stacked on top of each other. Such a stack may comprise, for example, up to twenty stacked cubes. Now, the reasoning that leads to the sorites paradox can also be applied to this stack, because intuitively, if we remove the cubes one by one from the top, we are still in the presence of a stack. Yet in fact, we cannot remove some cubes of strategic importance without all the others falling simultaneously, thus destroying the entire stack. Conversely, some cubes—especially those from the top—are less critical, so they can be removed without jeopardizing the very existence of the stack. Such an analysis of the sorites paradox suggests that there are other factors that should be taken into account, such as the position of each cube, their alignment, etc.. However, this type of solution also fails because it faces a purely numerical variation of the same problem, which consists of Wang’s paradox:

(27) 100000000 is large basis step
(28) if n is large then n – 1 is large induction step
(29)
(30) ∴ 1 is large

Indeed, such a problem is an instance of the sorites paradox, to which the preceding type of solution fails to apply.

Lastly, according to another approach, of an epistemological nature, there exists a precise boundary in the number of grains that allows us to differentiate a heap from a non-heap, but we are unable to determine accurately where such boundary is located. The cause of the paradox lies in a deficiency in our knowledge, which is a kind of blind spot. Such a precise boundary also exists, according to this type of approach, at the level of notions such as young/non-young, small/non-small, bald/non-bald, etc., thus allowing us to distinguish them. As we can see, this type of solution tends to reject the induction step as false. However, such a solution does not prove satisfactory, since the existence, for each vague notion, of a precise numerical cut-off allowing us to distinguish the instances from the proper counter-instances seems rather counter-intuitive. And this type of solution does not do justice to the intuition that there exists, for each vague concept, a penumbra zone corresponding to borderline cases.

 

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3. Russell’s Paradox

chap3Russell’s paradox is one of the most famous paradoxes of mathematical set theory. The paradox, stated by Bertrand Russell, results informally from the fact of taking into consideration the set of all sets that do not contain themselves. The very existence of this set leads straightforwardly to a contradiction. Indeed, it follows, on the one hand, that if this set belongs to itself, then it does not belong to itself. And if it does not belong to itself, on the other hand, then it belongs to itself. Thus, such a set both does not belong to itself and does belong to itself.

A classical variation of Russell’s paradox is the barber problem. Such a barber shaves all men who do not shave themselves, and only those ones. The following question then ensues: does the barber shave himself? If the barber shaves himself, then by definition, he belongs to the class of the men who shave themselves, and therefore he does not shave himself. On the other hand, if the barber does not shave himself, then by definition, he belongs to the class of the men who do not shave themselves, and therefore he shaves himself. In conclusion, if the barber shaves himself, then he does not shave himself, and if he does not shave himself, then he shaves himself. Thus, whatever assumption we consider, a contradiction ensues.

Another version of Russell’s paradox arises under the following form: let us consider the catalog of all catalogs that do not mention themselves. The following question then ensues: does this catalog make mention of itself? If it mentions itself, then it is not part of this catalog and does not mention itself, and if it does not mention itself then it is part of the catalog and then it does mention itself. In both cases, one is faced with a contradiction.

Russell’s paradox can be stated more formally as follows. Let R be the set of all sets that do not contain themselves. We then have the following definition of R (where ∈ denotes set membership and ∉ denotes set non-membership):

(1) x ∈ R | x ∉ x

Now, given this general definition, let us consider the specific case of the set R. Two cases are now possible: either R belongs to itself, or R does not belong to itself. On the assumption that R does not belong to itself, the reasoning can be stated as follows:

(2) R ∈ R hypothesis 1
(3) R ∉ R from (2)

And likewise, under the assumption that R does not belong to itself, it follows by definition that:

(4) R ∉ R hypothesis 2
(5) R ∈ R from (4)

The resulting conclusion is that the set R belongs to itself if and only if it does not belong to itself. The different steps of the reasoning can be detailed as follows:

(6) x ∈ R | x ∉ x definition
(7) R ∈ R hypothesis 1
(8) R ∉ R from (6),(7)
(9) ∴ if (R ∈ R) then (R ∉ R) from (7),(8)
(10) R ∉ R hypothesis 2
(11) R ∈ R from (6),(10)
(12) ∴ if (R ∉ R) then (R ∈ R) from (10),(11)
(13) ∴ R ∉ R and R ∈ R from (9),(12)

Thus, the fact of taking into account the very existence of the set R of all sets that do not contain themselves leads straightforwardly to a contradiction.

The paradox stems from naive set theory, into which it is possible to define a set unrestrictedly. Naive set theory is proved thus to be too liberal, by allowing the construction of some sets the nature of which proves finally to be contradictory, such that the set R. In particular, it turns out that the axiom of comprehension of naive set theory was responsible for the emergence of Russell’s paradox. The axiom of comprehension, in fact, allowed the construction of any set that conformed to the following schema:

(14) x ∈ E | P(x)

where P(x) denotes any property of an object x, such that any x with property P belongs to the set E. Thus, the solution to Russell’s paradox consisted in restricting the expressive power of set theory. The axioms of set theory were thus modified to prohibit the construction of the set R of all sets that do not contain themselves. In 1908, Ernst Zermelo proposed a set theory with an axiom of comprehension thus amended, which did not allow for the construction of the set R. This resulted in Zermelo-Fraenkel’s set theory, which is still in use nowadays, and whose axioms render impossible the construction of the set R, thus avoiding the ensuing contradiction.

 

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4. The Surprise Examination Paradox

chap4The surprise examination paradox originated, it is said, in an announcement by the Swedish authorities during the Second World War. According to this announcement, a civil defense exercise was scheduled for the following week, but the specific day was not revealed, so that the exercise would truly take place by surprise. Professor Lennart Elkbom understood the subtle problem that resulted from the announcement and told his students. Subsequently, the problem spread in academic circles and then gave rise to many discussions.

The surprise examination paradox is classically described as follows. A professor tells his students that an examination will take place on the following week. However, the professor adds that it will not be possible to the students to know in advance the date of the examination, because it will occur surprisingly. A clever student then reasons as follows: the examination can not take place on the last day of the week—Friday—because otherwise he would know, with certainty, that the examination would take place on Friday. So Friday can be eliminated. Similarly the student reasons, the examination cannot take place on the penultimate day of the week—Thursday—because otherwise he would know that the examination would take place on Thursday. Thus Thursday is also eliminated. By the same reasoning, the student concludes that the examination cannot take place either on Wednesday or Tuesday or Monday. Finally, the student concludes that the examination cannot take place on any day of the week. However, this does not prevent the examination from occurring surprisingly, for example on Wednesday. The paradox arises here because the student’s reasoning seems valid, but turns out to be ultimately in contradiction with the facts, since the examination finally occurs totally unexpectedly.

The student’s reasoning that leads to the surprise examination paradox can be detailed as follows:

(1) if the examination takes place on Friday hypothesis 1
(2) then I will know that the examination will take place on Friday from (1)
(3) then the examination will not occur surprisingly from (2)
(4) ∴the examination cannot take place on Friday from (1),(3)
(5) if the examination takes place on Tuesday hypothesis 2
(6) then I will know that the examination will take place on Tuesday from (5)
(7) then the examination will not occur surprisingly from (6)
(8) ∴the examination cannot take place on Tuesday from (5),(7)
(9) if the examination takes place on Wednesday hypothesis 3
(10) then I will know that the examination will take place on Wednesday from (9)
(11) then the examination will not occur surprisingly from (10)
(12) ∴the examination cannot take place on Wednesday from (9),(11)
(13) if the examination takes place on Tuesday hypothesis 4
(14) then I will know that the examination will take place on Tuesday from (13)
(15) then the examination will not occur surprisingly from (14)
(16) ∴the examination cannot take place on Tuesday from (13),(15)
(17) if the examination takes place on Monday hypothesis 5
(18) then I will know that the examination will take place on Monday from (17)
(19) then the examination will not occur surprisingly from (18)
(20) ∴the examination cannot take place on Monday from (17),(19)
(21) ∴ the examination can take place on no day of the week from (4),(8),(12),(16),(20)

Several solutions have been proposed to solve the surprise examination paradox. None of them is currently, however, the subject of a consensus. A first attempt at a solution was put forward by O ‘Connor, in an article published in the Mind journal in 1948. According to O ‘Connor, the paradox is due to the contradictory nature that results from the professor’s announcement and its implementation. For O’Connor, the professor’s announcement, according to which the examination should occur unexpectedly, stands in contradiction with the known data from the implementation of the examination. Thus, the statement of the surprise examination paradox is, according to O’Connor, self-refuting. However, such an analysis has not proven satisfactory because it became apparent that the examination could finally occur by surprise, without contradiction, for example on Wednesday. And the fact that the examination could finally occur surprisingly clearly confirmed the professor’s announcement, without refuting it.

A second type of solution has also been proposed by Quine, who highlighted the fact that four possibilities exist (by denoting the last day of the week by n):

(i) the examination will take place on day n and the student will know that the examination will take place on day n
(ii) the examination will take place on day n and the student will know that the examination will not take place on day n
(iii) the examination will not take place on day n and the student will know that the examination will take place on day n
(iv) the examination will not take place on day n and the student will know that the examination will not take place on day n

According to Quine, the problem is that the student, when he develops his reasoning, only considers the cases (i) and (iv), disregarding the possibilities (ii) and (iii). In particular, he does not consider the case (ii), which is the actual situation in which he eventually finds himself, thus allowing the examination to take place finally as a surprise. But if the student had considered this possibility from the beginning, Quine concludes, he would not have reached an erroneous conclusion.

As part of the solutions, it was also proposed that the surprise examination paradox reduces to the sorites paradox. Such a view has been particularly endorsed, though with different nuances, by P. Dietl in 1973 and by J. W. Smith in 1984. Both authors argue that the two paradoxes exhibit a common structure, so that the surprise examination paradox ultimately proves equivalent to the sorites paradox. According to this analysis, the various stages of the two paradoxes are equivalent and the surprise examination paradox thus finds its origin in the fact that the notion of surprise is a vague concept. But such an analysis has been criticized by Roy Sorensen, in his book Blindspots, published in 1988, where he argued that the two problems are not really similar. Sorensen argues first, that the version of the sorites paradox that is equivalent to the surprise examination paradox would be too fast. And second, Sorensen argues, the basic premises of the two paradoxes cannot truly be considered equivalent.

 

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5. Goodman’s Paradox

chap5Goodman’s paradox was introduced by Nelson Goodman in an article published in 1946 in the Journal of Philosophy. Goodman exposes his paradox as follows (with some slight adaptations). Consider an urn containing 100 balls. Every day, a ball is drawn from the urn; this is repeated for 99 days, until today. At each draw, it turns out that the ball taken from the urn is red. At this stage, one expects intuitively that the 100th ball drawn will also be red. This prediction is based on the generalization according to which all balls present in the urn are red. The reasoning on which the latter conclusion is based consists in an enumerative induction.

We can translate the previous inductive reasoning more formally as follows. Let R be the predicate red. Also, let b1, b2, b3, …, b100 be the 100 balls in the urn (∧ denoting the logical connector and).

(1) Rb1 ∧ Rb2 ∧ Rb3 ∧ … ∧ Rb99 enumeration
(2) Rb1∧ Rb2 ∧ Rb3 ∧ … ∧ Rb99 ∧ Rb100 from (1), induction
(3) ∴ Rb100 from (2)

At this stage, if we consider the property S, “drawn before today and red, or drawn after today and non-red,” we can see that this property is also verified by the 99 instances already observed. But this time the resulting prediction, based on the generalization that all balls are S, is that the 100th ball will not be red. This contradicts the previous conclusion, which is itself, however, consistent with our intuition. The corresponding reasoning can be detailed thus:

(4) Sb1 ∧ Sb2 ∧ Sb3 ∧ … ∧ Sb99 enumeration
(5) Sb1∧ Sb2 ∧ Sb3 ∧ … ∧ Sb99 ∧ Sb100 from (4), induction
(6) ∴ Sb100 from (5)

But here, the conclusion that the 100th ball is S is equivalent to the fact that the latter will be non-red. Now this is at odds with the conclusion resulting from the previous inductive reasoning that the 100th ball will be red. The paradox arises here because of the fact that the two conclusions (3) and (6) are contradictory. Intuitively, the application of the inductive enumeration to (4) seems erroneous. But the difficulty lies here in the fact of diagnosing accurately the flaw in the reasoning at the origin of this false conclusion.

Goodman also gives, in his book Facts, fictions and predictions, published in its original version in 1954, a slightly different version of his paradox, this time applied to emeralds:

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

This version of Goodman’s paradox is well known and based on the predicate “grue”. The definition of “grue” is the following: green and observed before T or non-green and observed after T. This results in two types of competing lines of reasoning. A first reasoning implements a classical inductive enumeration: from the observation that all emeralds observed before T were green, we conclude that the next observed emerald will also be green (V denoting green, and e1, e2, e3, …, e100 denoting the emeralds):

(7) Ve1 ∧ Ve2 ∧ Ve3 ∧ … ∧ Ve99 enumeration
(8) Ve∧ Ve2 ∧ Ve3 ∧ … ∧ Ve99 ∧ Ve100 from (7), induction
(9) ∴ Ve100 from (8)

The alternative reasoning is based on the same type of inductive enumeration applied to the predicate “grue.” From the fact that all emeralds observed before T were “grue,” we conclude this time that the next observed emerald will be “grue” (“grue” being denoted by G):

(10) Ge1 ∧ Ge2 ∧ Ge3 ∧ … ∧ Ge99 enumeration
(11) Ge1∧ Ge2 ∧ Ge3 ∧ … ∧ Ge99 ∧ Ge100 from (10), induction
(12) ∴ Ge100 from (11)

A contradiction then follows, since according to (9), the 100th emerald will be green, whereas it follows from (11) that the 100th emerald will be non-green. The two problems described by Goodman are two variations of the same paradox, since the predicate S used by Goodman in his article of 1946 shares with “grue” a common structure. P and Q being two predicates, the latter structure corresponds to the definition: (P and Q) or (not-P and not-Q).

Goodman’s paradox has generated a huge literature and many different types of solution have been proposed to solve it. Goodman himself has proposed a solution based on the notion of entrenchment. In Fact, Fiction and Forecast, he considers thus that the problem reduces to that of drawing a distinction between predicates that are projectible and those that do not. Projectible predicates can validly be used to support an enumerative induction, while others, among which is “grue,” are not suitable for this. According to Goodman, projectible predicates are those that are integrated, embedded in our current inductive practice. They consist then of an inductive use, which is thus validated by practice. Projectible predicates are those that are somehow validated by current usage, present and past. Conversely, non-projectible predicates such as “grue” are not suitable for inductive use. However, Goodman’s solution based on entrenchment into the language and common usage did not prove satisfactory, as it turns out that new predicates appear every day. Many neologisms are indeed created, that quickly integrate into everyday language and practice. Even the predicate “grue,” which was originally much criticized, has become somewhat familiar.

Another notable solution that has been proposed to solve Goodman’s paradox is based on the fact that the predicate “grue” has a time reference, unlike the predicate “green.” According to this type of solution, predicates such as “grue”, which include such temporal clauses, should not be used for induction. Nevertheless, this type of solution has proved too restrictive, as there are predicates that have a time reference but whose inductive projection is not a problem. Consider then a tomato: it is green when immature and red after. This property applies to the 99 tomatoes that I have just found in my garden, but also to the 100th tomato that is located in my neighbor’s garden. Second, it is quite possible to build a version of Goodman’s paradox that is devoid of such a temporal clause. It suffices then to build a predicate G based, for example, on a color-space combination, replacing the color-time association, to create a variation of Goodman’s paradox that overcomes a time reference. Finally, the response of Nelson Goodman himself against this type of objection is that the predicate “green” can also be defined with a time reference when using “grue” as a primitive concept. It suffices thus to draw a parallel between, on the one hand, the predicates “green” and “blue” and on the other hand, “grue” (green before T and blue after T) and “bleen” (blue before T and green after T). In this case, it is quite possible to define “green” and “blue” with the primitive notions of “grue” and “bleen”. A “green” object is then defined as “grue” before T and “bleen” after T, and similarly, a “blue” object is defined as “bleen” before T and “grue” after T. Thus, the definitions of “green” and “blue,” and on the other hand, of “grue” and “bleen,” turn out to be perfectly symmetrical and present identically a time reference.

 

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6. Newcomb’s Problem

chap6Newcomb’s Problem was described in 1960 by physicist William Newcomb, and was then introduced into the philosophical literature in an essay published in 1969 by Robert Nozick. We can describe the problem as follows. Two boxes, A and B, are placed in front of you. One of them—box A—is transparent and contains 1000 dollars. You will be faced with a choice: either take only the contents of box B or take the contents of both boxes A and B. You also know that a diviner, whose predictions have been extremely reliable so far, will put one million dollars in box B if he predicts that you will only take the latter. However, if he predicts that you will take both boxes A and B, the diviner will leave box B empty. Now, do you choose to take only box B, or to take boxes A and B? By virtue of a first argument (I), it turns out that the predictions made in the past by the diviner have proved very reliable and there is no reason that the prediction that he will make for you will not hold true once again. Therefore, it seems prudent to take only box B in order to collect one million dollars, which is already a very nice amount of money. At this point, however, it turns out that an alternative argument (II) can also be held, for at the moment when you prepare to open box B or both boxes, the diviner has already made his choice. Therefore, if the diviner predicted that you will only open box B, he has then placed one million dollars in the box. Wouldn’t it then be absurd to leave the 1000 dollars that are in box A, for the latter box is transparent, and you can observe its content. You reason, and you find that it can no longer affect the choice of the diviner. Therefore, it is better to open both boxes, and thus collect 1001000 dollars. At this point, it turns out that each of the two arguments (I) and (II) seems to be valid. However, both lead to conflicting conclusions. And the riddle posed by Newcomb’s problem is precisely that of knowing which of the arguments (I) and (II) is valid.

It is worth somewhat formalizing some aspects of Newcomb’s problem, in order to highlight some elements of its internal structure. It thus appears that the structure of the statement is that of a double conditional:

(1) if < the diviner predicts that the subject will open box B> then < the diviner will put 1000000 dollars in box B>
(2) if < the diviner predicts that the subject will open both boxes A and B> then < the diviner will put 0 dollars in box B >

Similarly, the reasoning (I) can be described in detail in the following way:

(3) the predictions made in the past by the diviner proved very reliable premise
(4) the predictions made by the diviner are very reliable generalization
(5) this time also, the diviner should predict my choice from (4), induction
(6) if the diviner predicted that I would only open box B, then he placed 1000000 dollars in box B from (1)
(7) if the diviner predicted that I would open boxes A and B, then he has placed 0 dollar in the box B from (2)
(8) ∴ I have interest in opening up box B from (6),(7)

And we can also formalize thus the reasoning (II):

(9) by the time I make my choice, the amounts of money are already placed within the boxes, and they will not be affected by my choice premise
(10) if the diviner has put 1000000 dollars in box A, then by also taking box B, I shall win 1001000 dollars in place of 1000000 dollars from (9)
(11) if the diviner has put 0 dollars in box A, then by also taking box B, I shall win 1000 dollars from (9)
(12) in both cases, I get a higher gain by also taking box A from (10),(11)
(13) ∴ I have interest in opening up boxes A and B from (12)

Newcomb’s paradox has gained tremendous popularity and has engendered a huge literature. Among the solutions that have been proposed to solve the paradox, one of them focuses on the fact that the situation corresponding to the paradox is actually impossible and is such that we cannot meet it in practice. According to this analysis, the part of the statement according to which the diviner can accurately predict the choice of the agent is unlikely. Under this analysis, such a clause relies on extravagant properties that are not those of our physical world, such as retroactive causality (the fact that an effect can act on its own cause) or the lack of free will in individuals. Such a solution, however, has not proved satisfactory, for if it is allowed to question the existence of retroactive causation or the lack of free will, we can nevertheless highlight some other variations of the paradox that do not make use of such singular properties. It suffices for this to consider a probabilistic version of the paradox where the diviner’s prediction is most often accurate, for the diviner may well rely on merely psychological considerations. A study of Newcomb’s paradox showed in effect that 70% of people choose to take only box B. The diviner may thus have a computer program simulating human behavior in a very powerful way when faced with this type of situation and make its forecasts accordingly. In this context, the clause of the statement according to which the predictions of the diviner are very often accurate would be entirely respected.

 

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7. The Prisoner’s Dilemma

chap7The Prisoner’s Dilemma was described by Merrill Flood and Melvin Dresher in 1950. It can be formulated as follows. Two prisoners, John and Peter, are questioned by a judge who suspects them of having committed a crime. The judge offers to each of them the following options: “You have two options: either confess or not confess. But beware, the choice you make will have a very important consequence on the sentence you will be imposed. Thus, if one of you confesses but the other does not confess, the one who has confessed will be free, while the one who has refused to confess will be sentenced to ten years in prison. However, if you both confess, each of you will have only five years in prison. Finally, if neither of you confess, I would inflict on you both one year in prison. Now, think and then decide. I will then let you know my sentence.”

At this point, it is worth describing in more detail the structure of the prisoner’s dilemma. It turns out then that the four following cases are possible:

(i) John confesses and Peter confesses
(ii) John confesses and Peter doesn’t confess
(iii) John doesn’t confess and Peter confesses
(iv) John doesn’t confess and Peter doesn’t confess

In addition, the judge’s announcement can be described with the help of the following matrix, which defines the penalties assigned to each of the two prisoners depending on their attitude:

(i) John confesses and Peter confesses John (5 years) and Peter (5 years)
(ii) John confesses and Peter doesn’t confess John (0 year) and Peter (10 years)
(iii) John doesn’t confess and Peter confesses John (10 years) and Peter (0 year)
(iv) John doesn’t confess and Peter doesn’t confess John (1 year) and Peter (1 year)

The problem inherent to the prisoner’s dilemma stems from the fact that two different types of reasoning both seem valid. Indeed, by virtue of the first type (I) of reasoning, it turns out that the fact of not confessing is what gives to each the best chance of being free. Indeed, if one of the prisoners confesses, there ensues a sentence of 5 years (if the other confesses as well) or null (if the other does not confess); thus, the resulting sentence is an average of 2.5 years (5+0)/2. However, if the prisoner does not confess, there follows a sentence of 10 years (if the other confesses) or 1 year (if the other does not confess as well), so the result is a sentence which is on average 5.5 years (10+1)/2. It therefore seems much more rational to confess. However, another type of reasoning is also possible. In effect, according to another viewpoint (II), it turns out that the fact of not confessing proves to be very interesting for both prisoners, for it only results in a one-year sentence for each of them. Finally, one is faced with a dilemma, since each of the options resulting from the two competing arguments (I) and (II) proves to be, from a certain standpoint, optimal.

The prisoner’s dilemma corresponds to a concrete, practical situation, which has implications in the fields of game theory, economics, political science, biology, etc. In terms of game theory, one distinguishes thus classically between zero-sum games and non-zero sum games. For zero-sum games, there is a winner and a loser, but no intermediate situation (this is the case, for example, for tennis). In contrast, for non-zero sum games, there is a winner, a loser, and one or more intermediate situations (chess, where the possibility of a draw exists, is one example). In this context, the prisoner’s dilemma proves to be a non-zero-sum game, since there are two cases where both prisoners receive the same sentence: (1) if both confess, and (2) if both do not confess.

One can observe that the prisoner’s dilemma leads to a significant variation when the dilemma is repeated. This is termed the iterated prisoner’s dilemma. In this context, several strategies then turn out to be possible. This results in the following basic strategies: always confess or never confess. But other more complex strategies are possible, based notably on the option chosen by the other prisoner in the previous moves. In this case, the iterations then lead us to analyze the sequence of moves by the prisoner as a type of behavior. At this point, the possibilities become numerous. A strategy that has proven very successful has been called tit-for-tat. The strategy on which it is based is the following: confess at first move, then play at move n+1 what the other prisoner has played at move n. For the iterated prisoner’s dilemma, there is no strategy that we can say with certainty is better than any other.

 

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8. Cantor’s Paradox

chap8Cantor’s paradox was discovered by Georg Cantor in 1899, but was not published until 1932. The main idea of the paradox resides in the fact that the consideration of the set of all sets leads to a contradiction. In effect, if we call C the set of all sets, then it follows that there exists a set C*, which is itself defined as the set composed of the parts of the set C. By definition, the set C, which is the set of all sets, therefore also includes the set C*. This implies that the cardinal—that is to say the number of elements—of the set C is greater than or equal to the cardinal of the set C*. Now a theorem established by Cantor states that given a set E, the cardinal of E is less than the cardinal of the set E*, which consists of all parts of E. Thus, by virtue of Cantor’s theorem, it follows that the cardinal of the set C*, which includes all the parts of C, is necessarily greater than the cardinal of the set C. This results in a contradiction.

The reasoning corresponding to Cantor’s paradox can be detailed thus more formally (card denotes here the cardinal of a given set):

(1) C is the set of all sets definition
(2) C* is the set of all parts of the set C premise
(3) card (C) ≥ card (C*) from (1)
(4) for any set E, the set E* of all parts of E is such that card (E) < card (E*) Cantor’s theorem
(5) for the set C, the set C* of all parts of C is such that card (C) < card (C*) from (4)
(6) ∴ card (C) ≥ card (C*) and card (C) < card (C*) from (3),(5)

Cantor’s paradox, like Russell’s paradox, belongs to the category of the paradoxes of set theory. Just like Russell’s paradox, it emerges within the naive set theory, where the construction of the set C of all sets is allowed. In the present set theory, that of Zermelo-Fraenkel, the paradox is avoided because one cannot construct the set C. In effect, one of the axioms of Zermelo-Fraenkel theory, the axiom of comprehension, was designed more restrictively than in naive set theory in order to prohibit the construction of the set C of all sets. But such an approach may seem ad hoc, in the sense that it consists of a restriction of set theory which has the sole purpose of avoiding the paradoxes and the resulting contradiction. In this context, as with Russell’s paradox, one cannot truly consider that we now have a genuine solution to Cantor’s paradox.

 

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9. Grelling’s Paradox

This paradox was invented by Kurt Grelling. It is also termed the paradox of heterological words. Grelling’s paradox can be stated as follows: some adjectives describe properties that apply to themselves, such as “polysyllabic”, “French.” Such adjectives can be referred to as autological. Some other adjectives, in contrast, describe properties that do not apply to themselves. For example, “long”, “Italian”, “monosyllabic”. Such words can be termed heterological. This leads us to classify words into two categories: (i) autological and (ii) heterological. Such a distinction, however, leads to a paradox. Given the above definitions, the paradox emerges, in fact, when one questions the status of the predicate heterological itself. Is “heterological” autological or heterological? For if “heterological” is heterological, then by definition, “heterological” is autological. And conversely, if “heterological” is autological, it follows that it is heterological. The conclusion is paradoxical, since it follows that “heterological” is heterological if and only if it is autological.

The definitions and the reasoning that lead to Grelling’s paradox can be described in more detail in the following way (H and ~H denote respectively heterological and non-heterological—that is to say autological—and  denotes a given property):

(1) H(“Φ”) if and only if ~Φ(“Φ”) definition 1
(2) ~H(“Φ”) if and only if Φ(“Φ”) definition 2
(3) if H(“H”) hypothesis 1
(4) then ~H(“H”) from (1)
(5) if ~H(“H”) hypothesis 2
(6) then H(“H”) from (2)
(7) ∴ H(“H”) if and only if ~H(“H”) from (3),(4),(5),(6)

It turns out that one cannot validly assign the predicate “heterological” either to the property heterological or to the property autological.

At this stage, it is also interesting to examine the status of the word “autological” itself. Is “autological” heterological or autological? The rationale relating to “autological” goes as follows:

(1) H(“Φ”) if and only if ~Φ(“Φ”) definition 1
(2) ~H(“Φ”) if and only if Φ(“Φ”) definition 2
(8) if H(“~H”) hypothesis 1
(9) then ~~H(“~H”) from (1)
(10) then H(“~H”) from (9)
(11) if ~H(“~H”) hypothesis 2
(12) then ~H (“~H”) from (2)

Here, the specific stage (10) is justified by the elimination of the double negation. In this case, it turns out that if “autological” is heterological then it is heterological, and in the same way, if “autological” is autological then it is autological. Hence, it turns out that one cannot properly determine whether “autological” is heterological or not.

Among the solutions that have been proposed to solve the paradox of heterological words, one of them leads us to notice that the structure of the paradox is very similar to that of Russell’s paradox. Thus, the two paradoxes would present a common structure and would then lead to a solution of the same nature.

Another solution leads us, as with the Liar paradox, to reject the definitions of all predicates that exhibit self-referential structure. However, such a solution does not prove to be satisfactory. In effect, it is far too restrictive, since it turns out that one succeeds quite safely to determine the status of numerous self-referential predicates such as polysyllabic. To merely outlaw all predicates of which the structure is self-referential would be to pay much too high a price for merely eliminating the paradox.

 

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10. The Two-Envelope Paradox

chap10The two-envelope paradox can be enunciated as follows: in front of you are two envelopes that each contain a given amount of money and you know with certainty that one of them contains twice as much as the other. You take one of the two envelopes at random. Now you have the choice of keeping the envelope in your hand or switching with the other envelope. What do you decide to do? A first type of reasoning (I) comes immediately to your mind: the situation with regard to each of the two envelopes is quite similar. By choosing only one of the two envelopes, you do not get any new information. Therefore, the choice of either is equivalent, so you decide to keep the envelope that you initially took. However, it turns out that another type of reasoning (II) also proves to be possible: let x be the amount contained in the envelope that you have in your hands. The other envelope then contains an amount that is equal to either 2x or 1/2x. Both situations are equally probable and each can be assigned a probability of 1/2. Therefore, the general probability can be calculated as follows: 2x x 1/2 + 1/2x x 1/2 = 5/4x. It follows that in the general case, the other envelope contains a sum equal to 5/4x: that is to say, 1.25 x. Thus, it turns out that the other envelope contains an amount that is a quarter greater than the one that you have in your hands. Therefore, it is in your interest to switch with the other envelope. However, once the envelope has been exchanged, a similar reasoning leads you to switch the envelope again, and so on ad infinitum.

In the two-envelope paradox, it is clearly the reasoning (II) that is at issue, since it leads to the absurd conclusion that one should exchange the envelopes ad infinitum. Yet the task of determining accurately the fallacious step in the reasoning (II) proves to be very difficult. To this end, it is helpful to further formalize the various steps involved in reasoning (II):

(1) the other envelope contains either (i) the amount 2x or (ii) the amount 1/2x premise
(2) the probability of each situation (i) and (ii) is 1/2 premise
(3) the general probability is that the other envelope contains: 2x x 1/2 + 1/2x x 1/2 from (1),(2)
(4) the general probability is that the other envelope contains 1,25x from (3)
(5) ∴ I have interest to switch with the other envelope from (4)

Among the solutions that have been proposed to solve the paradox, one of them argued that the assertion (2), according to which the second envelope contains 2x or 1/2x with a probability equal to 1/2, is not true in all cases. Thus, Frank Jackson and his co-authors argued in an article published in 1994 that in reality, the values of x and the resulting pairs of values do not all have the same probability of being in the envelopes. In effect, there are certain limit values—either very small or very large—that one has very few chances to encounter for practical reasons. Thus, the two values that can be in one envelope are not equally probable and therefore the premise (2) is not exact. However, such a solution did not prove to be satisfactory. Indeed, as pointed out by McGrew and his co-authors in an article published in 1997, one succeeds in making the paradox reappear by considering a variant of the latter, wherein the envelopes do not contain money, but merely pieces of paper on which are inscribed numbers.

 

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11. Moore’s Paradox

chap11Moore’s paradox was described by G. E. Moore in a text published in 1942. Let us consider then the following proposition:

(1) It’s raining and I do not believe that it’s raining

It follows that such a proposition is in principle absurd. Intuitively, such a proposition presents a contradictory nature. However, it turns out that there are some situations where an assertion such as (1) may be validly expressed. Thus, that person may firmly believe that it is not raining today on the basis of the weather forecast that he/she heard the day before, while it is raining in reality. In this context, the assertion (1) will seem plausible again. Such a situation corresponds, for example, to a situation where a person has a justified belief that a given event will not occur, but this event eventually occurs, thus ultimately rendering the initial belief false. Hence, this person may strongly believe that it is not raining today, based on weather forecasts heard the day before, whereas it is raining in reality. In this context, the assertion (1) regains plausibility.

It is worth analyzing here in greater detail the structure of (1). If we consider then any proposition P, it follows that the assertion (1) has the following structure:

(2) P and I do not believe that P

The logical structure of (2) is the following (Q denoting “I believe” and ~ negation):

(3) P ∧ ~Q(P)

One classically distinguishes two variations of Moore’s paradox: Moore’s paradox of Hintikka and Moore’s paradox of Wittgenstein. Moore’s paradox of Hintikka presents a structure which is that of (2) and corresponds to the original version of Moore’s paradox. In contrast, Moore’s paradox deals with Wittgenstein’s proposition

(4) P and I do believe non-P

which presents the logical structure:

(5) P ∧ Q(~P)

According to some authors, the surprise examination paradox assimilates itself to Moore’s paradox. This was notably the view expressed by Robert Binkley, in an article published in 1968, where he argued that if the period in which the examination can take place is only one day, then the professor’s announcement presents the structure of Moore’s paradox, as the professor’s announcement made to the students is the following: “There will be an examination tomorrow, but you will not know that this examination will take place tomorrow.” Once the students conclude that the examination can take place, they are then on the very day of the examination, in a situation that permits the professor’s announcement to be validated. The result is then a real situation which corresponds, without contradiction, to proposition (1).

 

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12. Löb’s Paradox

chap12Löb’s paradox is mentioned in Jon Barwise and John Etchemendy’s book The Liar, published in 1987. The authors point out that the paradox was brought to their attention by Dag Westerstahl. Löb’s paradox, starting from one proposition that seems harmless, leads to the devastating conclusion that any proposition is true. The proposition that constitutes the starting point of the reasoning is as follows:

(1) if proposition (1) is true, then 0 = 1 premise

Such a proposition presents the structure of a conditional proposition (that is to say, it takes the form: if <antecedent> then <consequent>) whose antecedent is “proposition (1) is true” and the consequent is “0 = 1.” The paradox emerges when one considers the hypothesis that the antecedent of (1), that is to say “proposition (1) is true”, is true. If the antecedent of (1) is true, then it follows that 0 = 1. But the latter proposition is nothing but (1) itself. The result is, by application of modus ponens (a logical principle in virtue of which if P, P  Q, then Q), that proposition (1) itself is true. Consequently, the proposition (1) has just been proven. It consists here of a case of application of conditional proof. However, if (1) is true, another application of modus ponens leads finally to the fact that 0 = 1.

One can describe in more detail the various steps of reasoning leading to Löb’s paradox:

(1) if proposition (1) is true, then 0 = 1 premise
(2) if proposition (1) is true hypothesis
(3) then 0 = 1 from (1),(2)
(4) if proposition (1) is true, then 0 = 1 from (2),(3)
(5) ∴ (1) is true from (4)
(6) ∴ 0 = 1 from (1),(5)

Löb’s paradox thus goes on to prove, starting from one proposition that seems harmless, any proposition. As with other contemporary paradoxes, the task of diagnosing the exact cause of the paradox proves to be very difficult.

An attempted solution leads us to notice that the structure of (1) is self-referential. It consists here of a common feature with other paradoxes, especially the Liar paradox. But the solution that consists in prohibiting the propositions with a self-referential structure is not appropriate here either. Indeed, this is far too drastic and restrictive, thus leading to the elimination of the propositions whose structure is self-referential, but which do not, however, present a problem in being assigned a truth value. Here again, the problem arises of defining the criteria for distinguishing between: (i) self-referential propositions that admit a valid truth value, and (ii) self-referential propositions that cannot be validly assigned a truth value.

 

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13. The Race Course Paradox

chap13The race course paradox is one of the famous paradoxes attributed to Zeno of Elea. It is mentioned very clearly in Aristotle’s Physics:

[The race course paradox] asserts the non-existence of motion on the ground that that which is in locomotion must arrive at the half-way stage before it arrives at the goal.

Informally, the paradox—which is also termed the Dichotomy paradox—can be described as follows. A runner wants to travel the distance from point A to point B. To get to B, the runner must first traverse half the distance that separates point A from point B. But once he has traveled half the distance, the runner still has to go through half the distance that separates him from arrival at B. Once at this point, the runner will have traveled three quarters of the distance that separates him from B. But from there, he will still have to go through half the distance to the arrival, and so on ad infinitum. Thus, the runner must have to go through an infinite number of times the distances that are themselves finite. Now this should take an infinite time. Therefore the runner will never succeed in reaching B, and it follows that all motion is impossible.

The paradox can be described somewhat more formally. Let d be the distance from A to B. In this case, the runner must first have to go through half of d, then 1/4 d, then 1/8 d and 1/16 d, and so on ad infinitum. The reasoning that leads to the race course paradox can be described as follows:

(1) to travel from one point to another, a runner must first go through half the distance between two points premise
(2) the runner wants to travel the distance d which separates point A from point B premise
(3) to go from A to B, the runner must first travel 1/2 d from (1),(2)
(4) once reached 1/2 d, the runner must then travel 1/4 d from (1),(2),(3)
(5) once reached 3/4 d, the runner must then travel 1/8 d from (1),(2),…,(4)
(6) from (1),(2),…,(5)
(7) the runner shall have to travel an infinite number of times a fraction of d from (3),(4),…,(6)
(8) it is impossible to traverse an infinite number of distances in a finite time premise
(9) ∴ the runner will never reach point B from (7),(8)

A first type of response that can be made with respect to the paradox was formulated by Aristotle via Simplicius: everyone knows from personal experience that one can move from one point to another. Therefore, one can also move from point A to point B in the case corresponding to the statement of the paradox. So the runner will reach point B, in the same way that we are able to reach a point to which we want to move in real life. This objection, however, did not prove to be convincing. In effect, the empirical finding that it highlights turns out indeed to be true. However, it consists precisely of one component of the paradox, for what represents here the heart of the paradox is that the reasoning inherent to the race course paradox leads to a conclusion that contradicts the current data that result from the experiment. Thus, this objection only mentions one inner element of the paradox. What is required, however, is to determine accurately the fallacious step in the reasoning described by Zeno.

Another answer, regarded by many as a compelling resolution of the race course paradox, results directly from the work of Cauchy and his theory of infinite series. In effect, Cauchy has shown that the sum of an infinite series was sometimes finite. In this case, it turns out that the sum of the infinite series 1/2 + 1/4 + 1/8 + 1/16 +… + 1/2n equals 1. Under these circumstances, each intermediate distance is thus covered in a finite time. The distance d is therefore traveled in a finite time, which is equal to the sum of the intermediate laps.

 

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14. The Stone Paradox

chap14The stone paradox was described by W. Savage in 1967 in an article published in the Philosophical Review journal. It can be stated as follows: let be first the definition that God is an omnipotent being. Then consider a stone that has the following characteristic: it is so heavy that God cannot lift it. At this stage, there are two possibilities: either God can create it, or God cannot create it. Let us consider first the first hypothesis. If God can create such a stone, it follows that God cannot lift it. Therefore, if God can create such a stone, then there exists a task that God cannot accomplish. Now consider the second hypothesis, by which God cannot create such a stone. In this case, it also follows that there is a task that God cannot accomplish. Thus, the fact of taking into account each of the two assumptions leads to the conclusion that in each case there exists a task that God cannot accomplish. And this proves to be inconsistent with the fact that God is omnipotent. It follows then that God does not exist.

The steps of the argument can be decomposed as follows:

(1) God is an omnipotent being definition
(2) either God can create a stone that he cannot lift, or God cannot create it dichotomy
(3) God can create a stone that he cannot lift hypothesis 1
(4) God cannot lift a stone from (3)
(5) there exists a task that God cannot do from (4)
(6) God cannot create a stone that he cannot lift hypothesis 2
(7) there exists a task that God cannot do from (6)
(8) there exists a task that God cannot do from (5),(7)
(9) ∴ God is not an omnipotent being from (8)

A solution that has been formulated to solve the stone paradox rests on the fact that the concept of a stone that God can not lift has an inherently contradictory nature. The status of such a stone, if it existed, would be contradictory in essence. It is therefore not surprising that the use of a contradictory notion in an argument yields illogical consequences. The concept of a stone that God can not lift can then be compared with a “square circle” or a “single married,” for one can in effect have the exact same type of argument with a “square circle,” thus leading in the same way to an inconsistent consequence.

According to another point of view, resulting from the writings of St. Thomas Aquinas, the concept of omnipotence cannot be used without restriction, for the notion of divine omnipotence should only be considered in relation to things that are actually possible. In no case does the notion of omnipotence entail the ability to do impossible things. Such a perspective can be applied directly to the stone paradox. It follows then that the fact of lifting a stone that no one can lift is precisely an impossible task.

 

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15. The Doomsday Argument

chap15The Doomsday Argument is an argument that was enunciated by astrophysicist Brandon Carter in the early 1990s. This type of reasoning was also discovered independently by Richard Gott and H. Nielsen. The Doomsday Argument was then developed in detail and defended by the Canadian philosopher John Leslie in a series of publications. The main feature of the argument is that the premises of the corresponding reasoning seem quite acceptable, whereas the corresponding conclusion proves to be unacceptable to most people.

The reasoning on which the argument is based is as follows. One considers first an urn that contains either 10 or 1000 balls. The balls are numbered 1, 2, 3, 4, 5,…. The competing hypotheses are thus as follows:

(H1) the urn contains 10 numbered balls
(H2) the urn contains 1000 numbered balls

Let us consider that the initial probability that the urn contains 10 balls or 1000 balls is 1/2. Now, you pull a ball at random from the urn and you find out that it has the number 5. Does this draw make more likely the hypothesis that the urn contains 10 balls, or the one according to which it contains 1000 balls? Given the novel information that the ball retrieved from the urn is numbered #5, it turns out that an upward revision of the initial probability of the hypothesis that the urn contains only ten balls must be carried out. In effect, the random draw of the ball # 5 renders much more likely this last hypothesis. For if the urn contains only 10 balls, there is one chance out of 10 that you will pick ball # 5. In contrast, if the urn contains 1000 balls, there is one chance in 1000 of picking ball # 5. An accurate calculation by using Bayes’ theorem leads to a revision to 0.99 of the initial probability that the urn contains 10 balls. Such reasoning, based on the contents of an urn, proves to be consensual.

At this stage, we can now draw a parallel with the human situation. One considers two assumptions regarding the evolution of humankind. It can be envisaged that the total population of humans that have ever existed will reach 100 billion or 10 trillion. We thus formulate the following two assumptions regarding the future of humanity:

(H3) humankind will have a total of 100 billion humans
(H4) humankind will have a total of 10000 billion humans

The first hypothesis corresponds to a fast and imminent extinction of humanity, whereas the second corresponds to a very long lifetime of humanity, which could then colonize other planets and spread across the galaxy. We assign, for simplicity, a probability of 1/2 to each of these two hypotheses. At this stage I am led to consider my position since the birth of humanity. Considering then that I am the 70 billionth human, I am led to reason in the same way as I did before with the urn. This leads to an upward revision of the initial probability that the total population of humans who ever lived will only reach 100 billion. Finally, this argues for the probability—much greater than we could have imagined at first glance—of a relatively imminent extinction of humanity. But unlike the previous case concerning the urn, the latter conclusion turns out this time to be quite unacceptable and contrary to intuition. In the reasoning that led to the conclusion that humankind should face imminent extinction, a step proves to be defective. But the task of assessing accurately the weakness in the Doomsday argument proves to be a very difficult one, about which opinions differ widely.

A first approach to try to solve the problem raised by the argument is simply to accept its conclusion. According to some authors, especially John Leslie, the argument is correct and the conclusion that results should be accepted (with an important caveat, however, regarding the case where our universe is not completely deterministic). Leslie bases this position on the fact that in two articles published in 1992 in the journal Mind and in his book The End of the World published in 1996, he has refuted, often convincingly, an impressive number of objections to the Doomsday argument. However, accepting the conclusion of the argument remains quite counterintuitive. On the other hand, the acceptance that the mere knowledge of our birth rank leads to an upward shift in the probability of an imminent extinction of humanity leads to a similar conclusion in many common analog situations. It follows, for example, an upward revision in the probability of an imminent disappearance of the association that I have just joined, and so on.

Another type of solution that I developed in an article published in 1999 by the Canadian Journal of Philosophy is to consider that the reference class which bears on the Doomsday argument – that is to say, the human species – is not precisely defined, for must we assimilate it to the subspecies homo sapiens sapiens, to the species homo sapiens, to the homo gender, etc.? We can choose the reference class differently by operating by restriction or extension. In the statement of the Doomsday argument, no objective criteria allowing us to choose the reference class are present. It therefore follows an arbitrary choice of the latter. Suppose then that I assimilate, in an arbitrary way, the reference class to the subspecies homo sapiens sapiens. There then follows, by applying the Doomsday argument, a bayesian shift in favor of the hypothesis that the subspecies homo sapiens sapiens is promised future extinction. However, the extinction of the subspecies homo sapiens sapiens could also be accompanied by the appearance of one or more new sub-species, such as homo sapiens supersapiens. In this case, the disappearance of the reference class that identifies, by restriction, to the subspecies homo sapiens sapiens is accompanied by the survival of a broader reference class, which assimilates itself with the species homo sapiens. Such reasoning has the effect of rendering the Doomsday argument harmless and of neutralizing its initially devastating conclusion. It may be objected, however, that such a solution always admits the validity of the argument with regard to a limited reference class such as homo sapiens sapiens, even though such a conclusion—although inoffensive—seems counterintuitive.

Another solution that has been proposed recently by George Sowers, in an article published in 2002 in the Mind journal, is as follows. According to the author, the analogy with the urn behind the Doomsday argument is not valid, because our individual birth rank is not obtained randomly as are the numbers of the balls extracted from the urn. Indeed, our birth rank is indexed on the temporal position which corresponds to our birth. Therefore, Sowers concludes, the reasoning underlying the Doomsday argument is misleading, since it is based on a false analogy. However, Sowers’ analysis is not fully convincing. Indeed, one can easily imagine an analogy with a slightly different urn, where the drawing of the ball occurs randomly, but where the number of the ball is indexed on the corresponding time position. It suffices for this to consider a device with an urn from which ball #n for example is extracted at random. Then the mechanism expels the ball #1 at time T1, the ball #2 at time T2, the ball #3 at time T3, the ball #4 time T4, … and finally the ball #n at time Tn. The device then stops. In this case, it seems clear that the draw of the ball was made randomly, even though the number of the ball is indexed on the corresponding time position.

 

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16. The Ship of Theseus Problem

chap16In the literature, one finds the first trace of the problem of the ship of Theseus in the works of Plutarch. The problem can be described as follows. Theseus has a ship with which he goes overseas on a given day, accompanied by several of his companions. Let A be this ship, which is thus the “ship of Theseus.” During the trip, multiple damages necessitate many repairs and thus quite often, parts of the ship must be replaced with new ones. Many years go by, and as the return time approaches, it transpires that all parts of the ship have finally been replaced. Thus, upon the return of Theseus to Greece, the ship does not include any of its original parts. Let us call B the ship of Theseus upon his return to Greece. Now the question arises: Is ship A identical to ship B? In other words, is ship B still the ship of Theseus?

It is interesting to model this problem more accurately. Let us consider thus that ship A possesses n parts (boards, metal, rope, etc.), which are so many parts that can be denoted by a1, a2, a3,…, an-1, an. Similarly, the parts of ship B are b1, b2, b3,…, bn-1, bn. We can then denote ship A by a1a2a3… an-1an and ship B by b1b2b3… bn-1bn. Over the years, i.e. from time T0 to time Tn, the replacement process of the n parts comprises the following successive steps:

(1) a1a2a3… an-1an in T0
(2) b1a2a3… an-1an in T1
(3) b1b2a3… an-1an in T2
(4) b1b2b3… an-1an in T3
(…)
(5) b1b2b3… bn-1an in Tn-1
(6) b1b2b3… bn-1bn in Tn

It turns out at this stage that two hypotheses can be made:

(7) ship B is identical to ship A
(8) ship B is not identical to ship A

Intuitively, what justifies the fact that ships A and B are identical is that in everyday life, the mere fact of changing a piece of a device does not entail that this device is different. In the same way, intuitively, the ship’s identity remains identical each time a board or a metal part is replaced. On this basis, it can be concluded that ship B is identical to ship A.

However, another strong argument favors, conversely, the hypothesis that ships A and B are not identical. In effect, all parts of the ship have been changed over the years. Thus, ship B has no original parts of ship A. How, under these conditions, can we still consider that ships A and B are identical? Under the principle that two objects that have no part in common are different, the conclusion follows that the two vessels are different.

The statement of the problem of the ship of Theseus is often associated with a second part, which is as follows. As the vessel moves away from Greece at the time of departure, it is accompanied by a second ship, in charge of assistance. Each time a repair is carried out on the ship of Theseus, the assistance ship gets the old piece that has been changed. And the master of the assistance ship decides, with the help of his crew, to rebuild identically the original ship of Theseus. This way, when it reaches Greece upon his return, the second ship has all the planks of the original ship. Let C be the assistance ship. The question is then: Is ship C identical to ship A? Now it turns out even more clearly than before that ship C is identical to ship A, since both are made up of exactly the same boards. In the latter variation of the problem of the ship of Theseus, we now face four hypotheses:

(i) ship B is identical to ship A and ship C is identical to ship A
(ii) ship B is identical to ship A and ship C is not identical to ship A
(iii) ship B is not identical to ship A and ship C is identical to ship A
(iv) ship B is not identical to ship A and ship C is not identical to ship A

A first type of solution that was proposed to solve the problem of the ship of Theseus is the idea that it is but a variation of the sorites paradox. Yet a closer examination reveals that the problem of the ship of Theseus is based on defining the criteria of identity between two objects. The crucial question that arises here is the following: under which conditions is an object A identical to an object B, and in particular under which conditions does the identity of an object persist over time? Given the lack of a consensual response to the latter question, we can consider that we do not have a satisfactory solution to the problem of the ship of Theseus.

Another type of solution has been put forth by Derek Parfit, in his book Reasons and Persons, published in 1984. According to Parfit, it is the fact of formulating the two hypotheses in terms of an identity relationship that is at the origin of the problem, because it would require us to reformulate the problem with regard to a different type of relationship, which can be denoted by R. The conclusion thus results that the original ship of Theseus is in relation R with the two ships, A and B. However, such an analysis does not prove entirely convincing, because while the fact of replacing the identity relation with another relation eliminates the problem, such a solution does not veritably handle the pressing issue raised by the problem of the ship of Theseus, which deals specifically with our intuitive notion of identity and the conditions of its persistence through time.

 

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17. Hempel’s Problem

chap17Hempel’s problem was described by Carl Hempel in an article published in 1945 in the journal Mind, in the context of the study of confirmation theory. The starting point is the following assertion: “All ravens are black.” Clearly, the discovery of a black raven confirms this hypothesis. Similarly, this hypothesis would also be invalidated by the discovery of a blue raven. However, it turns out that the assertion that “All ravens are black” is equivalent to the assertion that “All non-black objects are non-ravens.” Similarly, we can consider validly that everything which confirms a given proposition P also confirms a proposition P* which is equivalent to it. But this then entails that the discovery of a pink flamingo or a blue umbrella, which confirms the proposition that “All non-black objects are non-ravens,” also confirms the assertion that “All ravens are black.” And the latter conclusion proves to be paradoxical.

The reasoning on which Hempel’s problem is based can thus be described in detail:

(1) All ravens are black hypothesis 1
(2) All non-black objects are non-ravens hypothesis 2
(3) (2) is equivalent to (1) contraposition
(4) the instances that confirm a proposition P also confirm a proposition P* which is equivalent premise
(5) the discovery of a pink flamingo confirms (2) from (3),(4)
(6) ∴ the discovery of a pink flamingo confirms (1) from (4),(5)

One can observe here that the logical structure of the proposition (1) according to which “All ravens are black” has the form:

(7) All X are Y

whereas that of (2) according to which “All non-black objects are non-ravens” is the following:

(8) All non-Y are non-X

In fact, the structure of the contrapositive form (8) is clearly equivalent to (7). As we see it, the propositions (1) and (2) are based on four properties, which correspond respectively to: raven, non-raven, black, and non-black. These four properties determine four categories of objects: black ravens, non-black ravens, black non-ravens and non-black non-ravens.

It should be noticed here that Hempel’s problem is not, strictly speaking, a paradox, because it does not entail a genuine contradiction. However, the conclusion resulting from the reasoning inherent to Hempel’s problem proves highly counter-intuitive. However, one solution that has been proposed to solve Hempel’s problem is based on the acceptance of its conclusion (6). According to this type of solution, the discovery of a pink flamingo actually confirms that all ravens are black but only to an infinitesimal degree, for the class of non-ravens contains an extremely high number of objects. Thus, under this type of solution, the discovery of a non-raven confirms the proposition (1) according to which “All ravens are black,” but only at an infinitesimal degree.

Paul Feyerabend, in an article published in 1968 in the British Journal for the Philosophy of Science, believes that Hempel’s paradox and Goodman’s paradox admit the same type of solution. According to Feyerabend, one should only consider valid, from a scientific perspective, the negative instances (those that refute a given hypothesis), thus leading one to purely and simply ignore the positive instances (those that confirm a hypothesis). Once one ignores the latter, the step that leads one to put on the same level the instances confirming (2) and those confirming (1) is then blocked, and therefore, the paradox disappears. Nevertheless, Feyerabend’s approach proved too radical, because it turns out that to confirm a hypothesis H1 is also to refute the opposite hypothesis H2, and conversely, to refute the hypothesis H1 is also to confirm the opposite hypothesis H2. Thus, a given instance constitutes a positive instance for a given hypothesis as well as a negative instance for the opposite hypothesis. For this reason, Feyerabend’s approach did not prove to be truly convincing.

Another type of solution that has been proposed to solve Hempel’s problem is that a predicate such as “black” should not be used unrestrictedly in inductive practice. According to this type of solution, one should restrict oneself to those predicates that are projectible, since every predicate is likely to give rise to many variations built on the model of “grue.” According to this type of analysis, Hempel’s problem and Goodman’s paradox are the result of the unrestricted application of all predicates for inductive processes. However, such an analysis does not prove convincing. Indeed, “black”, unlike “grue,” does not include any temporal clause. It is “black” that is projected here, not “black before T,” for the projection of a predicate such as “black” is carried out regularly in common practice in an entirely satisfactory way, so that it cannot reasonably be considered to renounce any logical inference relating to the predicate “black”.

 

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18. McTaggart’s Argument

chap18In an article that remains famous, published in 1908 in the Mind journal, John Ellis McTaggart described an argument intended to prove that time is not real. McTaggart begins by distinguishing two types of properties of time positions:

Positions in time, as time appears to us prima facie, are distinguished in two ways. Each position is earlier than some, and later than some, of the other positions. And each position is either past, present, or future. The distinctions of the former class are permanent, while those of the latter are not. If M is ever earlier than N, it is always earlier. But an event, which is now present, was future and will be past.

McTaggart terms B series the first distinction, by virtue of which any temporal position M is placed before but also after some other time positions. He also points out a constant property of the B Series: when an event M is earlier than an event N at a given time, it turns out to be earlier than N permanently. McTaggart also terms A Series the second distinction, by virtue of which any temporal position M belongs either to the past or the present or the future. McTaggart notices that the A series are such that each event M is in turn past, present and future. Thus, an event that is present was future and will be past. Similarly, an event that is past was present and future. Finally, an event that is future will be present and past. Hence, the second distinction highlights a non-permanent feature of time.

McTaggart then continues his argument by showing how time must necessarily have all properties of the A series. Suppose, McTaggart says, that time is defined only by means of the B series. In this case, we are not in a position to account for an essential element of time, namely change. Thus, McTaggart continues, it proves to be necessary to resort to the A series to account for the essential properties of time.

Finally, McTaggart sets out to demonstrate how the properties of the A series lead to a contradiction, for the A series are mutually exclusive: an event cannot be past, present and future. The intuition which governs our notion of time is that a given event cannot be past, present and future simultaneously. However, McTaggart considers a given time position M: the latter is present, will be past and was future. But “will be past” is tantamount to “is past at a future time position” and similarly, “was future” is tantamount to “is future at a past time position.” Thus, we define past with regard to future, and future with regard to past. This results in a circular definition. This shows the inconsistency of the A series. Therefore, no event can have all properties of the A series. It follows that time cannot present all the properties of the A series. Thus, concludes McTaggart, time has no reality.

The structure of McTaggart’s argument can thus be described in detail as follows:

(1) any time position has two distinct properties: the A series and the B Series premise
(2) the B Series cannot account for change premise
(3) change is an essential element of time premise
(4) the B series cannot account for an essential element of time from (2),(3)
(5) time must have the properties of the A Series in order to account for an essential element: change from (1),(4)
(6) time possesses the properties of the A Series hypothesis
(7) in the A Series, a future event is defined with respect to the past
(8) in the A Series, a present event is defined with respect to the present
(9) in the A Series, a past event is defined with respect to the future
(10) ∴ in the A Series, the definitions are circular from (7),(8),(9)
(11) time cannot have the properties of the A series from (10)
(12) ∴ time is unreal from (5),(11)

An objection that can be raised against McTaggart’s argument is that the fact that the B series are not sufficient to account for the essential properties of time does not prove that it is indispensable to have recourse to the A series. For perhaps could one find another series—let us call it the D series—which would account for the properties of time, in combination with the B series, but without exhibiting the disadvantages of the A series. In other words, there may be other alternatives to the A series, which would adequately account for the intrinsic properties of time.

Another objection that was made against McTaggart’s argument, especially by Bertrand Russell, is that the A series may be obtained logically starting from the B series. Thus, according to Russell, the concepts of past, present and future can be defined starting from the relationships before, during and after, which are then the primitive terms. Thus, past, present and future are respectively defined as: before T, during T, after T. Russell’s objection is intended to show how the A series are ultimately not necessary to describe the properties of time. However, Russel’s definition has the disadvantage of including a reference to time T. And it is allowed to think that this implicit reference to T assimilates itself to the “present time”. This leads us finally to define the present as “during present time” in a manner which proves, however, also circular.

 

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19. The Ontological Argument

chap19An ontological argument is an argument that concludes as to the existence of God from a priori considerations only: that is to say, from premises that are not based on empirical data or physical evidence. An ontological argument is intended to constitute a proof of the existence of God. However, unlike classical proofs resulting from mere observation of the reality, such evidence is based only on reasoning. There exist, then, several types of ontological argument. The earliest is due to St. Anselm of Canterbury (1077). The starting point is the consideration of a being of which we cannot conceive of a greater being. If it does not exist, we can therefore conceive of a being of which one cannot conceive of a greater being and which moreover exists. But this implies that we can conceive of a greater being that the being of which we cannot conceive of a greater being, and this latter conclusion turns out to be contradictory. Thus, taking into account the assumption according to which the being of which we cannot conceive of a greater being does not exist, leads to a contradiction. Therefore the being of which we cannot conceive of a greater being exists. The ontological argument of St. Anselm can thus be described in detail as follows:

(1) I can conceive of a being of which one cannot conceive a greater being premise
(2) either a being of which one cannot conceive a greater being exists or he does not exist dichotomy
(3) if a being of which one cannot conceive a greater being does not exist hypothesis 1
(4) then I can conceive of a being of which one cannot conceive a greater being but who exists from (3)
(5) I can conceive a being greater than the being of which one cannot conceive a greater being from (3),(4)
(6) ∴ a being of which one cannot conceive a greater being exists from (2),(4)

A slightly different ontological argument is due to Descartes, who describes it in his Meditations. According to Descartes, God, by definition, is a perfect being. It therefore has all qualities. Hence, it also has that of existing. Therefore God exists. Descartes’ ontological argument emphasizes the definition of God as a perfect being. The original excerpt of the Meditations that contains Descartes’ ontological argument is the following:

But now, if just because I can draw the idea of something from my thought, it follows that all which I know clearly and distinctly as pertaining to this object does really belong to it, may I not derive from this an argument demonstrating the existence of God? It is certain that I no less find the idea of God, that is to say, the idea of a supremely perfect Being, in me, than that of any figure or number whatever it is; and I do not know any less clearly and distinctly that an [actual and] eternal existence pertains to this nature than I know that all that which I am able to demonstrate of some figure or number truly pertains to the nature of this figure or number, and therefore, although all that I concluded in the preceding Meditations were found to be false, the existence of God would pass with me as at least as certain as I have ever held the truths of mathematics (which concern only numbers and figures) to be.

This indeed is not at first manifest, since it would seem to present some appearance of being a sophism. For being accustomed in all other things to make a distinction between existence and essence, I easily persuade myself that the existence can be separated from the essence of God, and that we can thus conceive God as not actually existing. But, nevertheless, when I think of it with more attention, I clearly see that existence can no more be separated from the essence of God than can its having its three angles equal to two right angles be separated from the essence of a [rectilinear] triangle, or the idea of a mountain from the idea of a valley; and so there is not any less repugnance to our conceiving a God (that is, a Being supremely perfect) to whom existence is lacking (that is to say, to whom a certain perfection is lacking), than to conceive of a mountain which has no valley.

More precisely, the structure of Descartes’ ontological argument can be thus defined:

(1) God is a perfect being definition
(2) God is a being who possesses all qualities from (1)
(3) existence constitutes a quality premise
(4) ∴ God exists from (2),(3)

Ontological arguments have been the subject of multiple objections in the literature. Famous criticism notably emanates in particular from Kant, in his Critique of Pure Reason, who considers that existence does not constitute a genuine property. This has the effect of blocking premise (3) of Descartes’ ontological argument, thereby neutralizing the reasoning that leads to the conclusion that God exists. According to Kant, we cannot consider that the mere fact of existing is a property, in the same way as red is the property of a tomato or hard is the property of a stone. For Kant, it is the very existence of a thing x which is a necessary condition for the attribution of its properties (color, size, density, roughness, hardness, etc.).

In a general way, ontological arguments are usually not regarded as truly convincing evidence of God’s existence, and they usually prove insufficient to convince non-theists of God’s existence.

 

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20. The Fine-Tuning Argument

chap20The fine-tuning argument belongs to the category of arguments which aim to prove the existence of God. The argument rests on the fact that a large number of cosmological constants governing our universe are such that if they had been slightly different, the emergence of intelligent life based on carbon chemistry as we observe it on Earth would not have been possible. Among these constants are: the ratio of the respective masses of the electron and the proton, the age of the universe, the neutrino mass, the average distance between the stars, the speed of light, the universal cosmological constant, Planck’s constant, etc. The argument is underpinned by the fact that each of these parameters could have had a slightly different value, which then would not have allowed the emergence of life. Consider, for example, the speed of light in a vacuum (v = 299,792.458 km/s): if it had been even slightly higher, then the stars would have emitted too much light to allow for the emergence of life. And even if the speed of light had been fractionally lower, light emission by the stars would have been insufficient to permit the emergence of life. The same goes for the gravitational constant (G = 6,672.10-11 Nm2kg-2): if the latter had had a slightly higher value, the stars would have had too high a temperature and would have been consumed far too quickly to allow the emergence of life based on carbon chemistry. Similarly, if the gravitational constant had been slightly lower, the stars’ temperature would have been too low to allow for the formation of many of the chemical elements required for the appearance of life. One can also consider the ratio of the mass of the electron relative to the proton (me/mp = 5.446170232x10-4): if it had been slightly different, the chemical bonds that would have resulted would have been insufficient to permit the emergence of life. Finally, if the rate of expansion of the universe had been slightly higher, no galaxy would have formed, and likewise, if it had been slightly lower, the universe would have collapsed even before the formation of stars, etc.

Thus, these different parameters, in virtue of the fine-tuning argument, were not determined randomly, but according to a specific purpose: the emergence of intelligent life in the universe. This particular purpose reflects the presence of a divine plan and thus finally the existence of God.

We can detail as follows the different steps of the fine-tuning argument:

(1) several cosmological constants governing our universe have values such that they permit the emergence of intelligent carbon-based life premise
(2) the cosmological constants that govern our universe could have had many different values hypothesis
(3) if the values of these cosmological constants were slightly different, then the emergence of intelligent carbon-based life would not have been possible from (2)
(4) if the cosmological constants had been obtained at random, then the probability that their setting was optimal would have been extremely low from (2)
(5) the optimal setting of the cosmological constants is not due to chance from (4)
(6) the optimal setting of the cosmological constants was done purposely in order to allow the emergence of intelligent life from (3),(5)
(7) the optimal setting of the cosmological constants was made by God from (6)
(8) ∴ God exists from (7)

Several objections have been raised against the fine-tuning argument. One of them in particular relies on the speculative idea, defended by a number of cosmologists, that the universe we observe is not the only one, but is only one universe among many others, in a system composed of multiple causally independent universes. In this context, there exist many other universes, entirely different from ours, which have quite distinct cosmological parameters. As we can see, this objection targets directly step (5) of the reasoning underlying the fine-tuning argument, according to which the optimal setting of cosmological constants is not due to chance, for the hypothesis of multiple universes is quite consistent with the fact that the parameters of our universe may have been obtained randomly.

 

21. The Dreaming Argument

chap21The Dreaming argument is due to Descartes. It can be formulated very simply. It consists of an argument which leads to the conclusion that our current perceptions may well be illusory and misleading because they are in all respects similar to those that we have when we dream. When we are indeed in a dream state, our perceptions are in effect realistic enough that they are able to create the illusion of reality. The Dreaming argument is described in the following passage (First Meditation) of the Metaphysical Meditations:

But it may be that although the senses sometimes deceive us concerning things which are hardly perceptible, or very far away, there are yet many others to be met with as to which we cannot reasonably have any doubt, although we recognize them by their means. For example, there is the fact that I am here, seated by the fire, attired in a dressing gown, having this paper in my hands and other similar matters. And how could I deny that these hands and this body are mine, were it not perhaps that I compare myself to certain persons, devoid of sense, whose cerebella are so troubled and clouded by the violent vapors of black bile, that they constantly assure us that they think they are kings when they are really quite poor, or that they are clothed in purple when they are really without covering, or who imagine that they have an earthenware head or are nothing but pumpkins or are made of glass. But they are mad, and I should not be any the less insane were I to follow examples so extravagant.

At the same time I must remember that I am a man, and that consequently I am in the habit of sleeping, and in my dreams representing to myself the same things or sometimes even less probable things, than do those who are insane in their waking moments. How often has it happened to me that in the night I dreamt that I found myself in this particular place, that I was dressed and seated near the fire, whilst in reality I was lying undressed in bed! At this moment it does indeed seem to me that it is with eyes awake that I am looking at this paper; that this head which I move is not asleep, that it is deliberately and of set purpose that I extend my hand and perceive it; what happens in sleep does not appear so clear nor so distinct as does all this. But in thinking over this I remind myself that on many occasions I have in sleep been deceived by similar illusions, and in dwelling carefully on this reflection I see so manifestly that there are no certain indications by which we may clearly distinguish wakefulness from sleep that I am lost in astonishment. And my astonishment is such that it is almost capable of persuading me that I now dream.

The Dreaming argument can be detailed as follows:

(1) when I am awake, I have some perceptions premise
(2) when I dream, I also have some perceptions premise
(3) the perceptions that I have when I am awake are identical in all respects to those that I have when I dream premise
(4) I do not have a criterion that allows me to distinguish my perceptions when I am awake or when I dream from (3)
(5) I have no proof that I am not currently in a state of dream from (4)
(6) ∴ it is possible that I am currently in a state of dream from (5)
(7) when I dream, my perceptions are false premise
(8) ∴ it is possible that all my current perceptions are false from (6),(7)

Descartes’ Dreaming argument has given rise to several contemporary variations. One of these modern variations rests on the idea that we are “brains in a vat.” The film Matrix, by Larry and Andy Wachowski, also develops a variation of this idea.

An objection to the Dreaming argument was raised by Barry Stroud, in a book published in 1989. According to this objection, premise (4) is false, because it is quite possible to carry out a test to determine whether each of us is or is not in a dream state. By using sensors that determine whether brain waves characteristic of the dream state are produced by the brain, we can determine whether or not a person dreams, and thus provide a definitive and reliable answer to this question. However, this objection has failed to convince many authors, who have argued that such a response presupposes that you are not dreaming when you perform the test. Under this assumption, the fact of carrying out a test proves indeed successful. But suppose on the contrary that we are in a dream state when we perform the test. In this case, the test is part of our dream and we cannot validly rely on it trustfully. Thus, the idea underlying this objection presupposes ultimately that we are not dreaming, whereas it is this very question which is at hand here.

Another type of objection may also be raised against the Dreaming argument. Suppose that this last argument is perfectly valid and that its conclusion is irrefutable. In this case, we then have compelling evidence that we are in a dream state. But if this were the case, would it not follow that the Dreaming argument itself is a pure product of our dreams, and therefore something illusory? Thus, in no case could it be an argument on which we could base our knowledge. As we see it, such a property has the effect of rendering the Dreaming argument self-refuting.

 

22. The “Brains in a Vat” Experiment

chap22The “brains in a vat” experiment was enunciated by Hilary Putnam in his book Reason, Truth and History, published in 1982. The argument begins with the following question: am I not a brain in a vat? In other words, am I certain that some mad scientist did not abduct me, and did not then remove my brain and place it in a nutrient liquid, and did not finally simulate all information usually reaching my brain with an especially sophisticated device. In this way, my feelings, my perceptions, my thoughts, etc. would be the effect of stimulations that the mad scientist sends to all of my neurons using his device. Am I quite sure that I do not find myself in a situation like this? If this were the case, stimulations sent to my brain would produce exactly the impressions that are mine as they occur when I have sensations, perceptions, emotions or thoughts, in normal conditions. How then can I be absolutely sure that I am I not a brain in a vat?

However, Putnam’s argument does not aim to suggest that we are actually brains in vats. For Putnam, it is indeed clear, however, that we are not “brains in vats.” For him, this results from the simple consideration of the assertion that “we are brains in vats.” Putnam proposes to prove that the latter assertion is always false. He distinguishes two hypotheses: if (i) we are not brains in vats, then it is false that we are brains in vats, and if (ii) we are brains in vats, then the concepts and words that we use on a daily basis do not refer to real objects, but to virtual objects, which are the result of a simulation. Such is the case when we use concepts such as “table”, “chair”, “umbrella”, etc. In this case, our concepts of “table” or “umbrella” refer not to a table or an umbrella, but to a simulation of table or of umbrella that comes from electrical impulses sent to our brains by a sophisticated electronic device. And this is also the case when we make use of words like “brain” or “vat”. In this case, we are then referring ourselves to a simulated brain or vat. Thus, when we affirm that “we are brains in vats”, we state that “we are simulated brains in simulated vats.” But this does not correspond to reality. Thus, if we consider the hypothesis that we are brains in vats, it is also true that it is false that we are brains in vats. In conclusion, whatever the hypothesis, it is false that we are brains in vats.

Putnam’s argument can be described more precisely as follows:

(1) there does not exist any internal criterion to determine whether or not our sensations, our perceptions, our emotions and our thoughts are stimulated by a device premise
(2) if we are brains in vats hypothesis 1
(3) then our sensations, our perceptions, our emotions and our thoughts are stimulated by a device from (2)
(4) then “we are brains in vats” means “we are simulated brains in simulated vats” from (3)
(5) if we are not brains in vats hypothesis 2
(6) then our sensations, our perceptions, our emotions and thoughts are not stimulated by a device from (5)
(7) then it is false that “we are brains in vats” from (5)
(8) ∴ it is necessary to resort to an external criterion to determine whether or not we are brains in vats from (1),(4),(7)

Putnam’s thought experiment aims to emphasize that the internal states resulting from stimulation by, on the one hand, an external device of a brain in a vat, and on the other hand, the thoughts and perceptions of a normal person, cannot be distinguished, for the resulting internal mental states are identical in both cases. Therefore, it is necessary to resort to external criteria to differentiate them. Thus, the point of view expressed by Putnam turns out to be fundamentally externalist. Putnam’s argument stresses that the meaning of words or phrases depends not only on internal content: that is to say, our thoughts, our emotions, etc. According to Putnam’s famous formula, “Meanings just are not in the head.”

 

23. The Teleological Argument

chap23The teleological argument or argument from design belongs, like the ontological argument, to a family of arguments that aim to prove the existence of God. The teleological argument is based on the simple idea that our universe is so complex and so well-arranged that this can only be the manifestation of the design of an intelligent entity. The complex scheduling of our universe thus demonstrates that it has a Creator.

The argument from design can be described in more detail as follows:

(1) our universe is very complex and well organized premise
(2) the complexity and settlement of our universe can only be the manifestation of the design of an intelligent being from (1)
(3) an intelligent being is the Creator of our universe from (2)
(4) ∴ God is the Creator of our universe from (3)

A famous formulation of the argument from design is notably due to William Paley (1743-1805), in his book Natural Theology, published in 1802. Paley describes the argument in the following terms:

In crossing a heath, suppose I pitched my foot against a stone, and were asked how the stone came to be there; I might possibly answer, that, for anything I knew to the contrary, it had lain there forever: nor would it perhaps be very easy to show the absurdity of this answer. But suppose I had found a watch upon the ground, and it should be inquired how the watch happened to be in that place; I should hardly think of the answer which I had before given, that, for anything I knew, the watch might have always been there.

According to Paley, the reason why we cannot conceive that the watch has been there forever is that its different parts have been assembled purposely, and that this purpose can only be the work of an intelligent being. Paley’s argument is based on an analogy between the watch and the universe, and leads to the conclusion that the universe was created purposely, and that design is that of God.

An objection that has been raised against the design argument is targeted against premise (1), that our universe is very well arranged. But to this, it can be retorted that this is only the expression of a specific viewpoint regarding our universe, because from another standpoint, our universe could seem poorly arranged. It would suffice for that to consider that disorder is present everywhere in the world, as it could be observed that our world is troubled by frequent earthquakes, devastating tidal waves, destructive cyclones, etc. and undergoes, in general, many natural disasters. From this point of view, we cannot really consider the universe as well-arranged.

Another objection targets directly step (2), according to which the scheduling of our beautiful universe can only be the work of an intelligent being. Under this objection, the complexity of our universe and its sophisticated layout are well proven, but that does not mean that it results from the work of a creator, for one could also imagine that many universes coexist, some being very simple and basic, while others are complex and sophisticated. As observers, we obviously can only find ourselves in a complex and well-ordered universe, allowing in particular the emergence of life based on carbon chemistry. However, there could well exist many universes very different from ours, some of which are very crude and rudimentary, and devoid of any observers.

 

24. Pascal’s Wager

chap24Pascal’s wager is an argument contained in paragraph 233 of the Pensées. It is one of the most famous arguments in the philosophy of religion, and aims to provide the reader with solid reasons to believe in the existence of God. Pascal exposes therein the alternative in front of which we find ourselves placed: either God exists or he does not exist. Faced with such a situation, we can bet in favor of the existence of God, or in favor of his non-existence. Pascal then analyzes the consequences that follow from a bet in favor of either option. He then considers the four cases that are thus determined. If I bet in favor of the existence of God and God exists (i), then I receive an infinite gain. If I bet in favor of the existence of God and God does not exist (ii), then this results in a zero loss. If I bet for the non-existence of God and God exists (iii), there follows an infinite loss. Lastly, if I bet for the non-existence of God and God does not exist (iv) then I receive no gain or loss. Thus, it turns out that if I bet for the non-existence of God, I am exposed to an infinite loss. Therefore, concludes Pascal, it is wiser to bet in favor of the existence of God, because there follows either an infinite gain or a zero loss.

The excerpt from Pascal’s Pensées that contains the Wager’s argument is the following:

If there is a God, He is infinitely incomprehensible, since, having neither parts nor limits, He has no affinity to us. We are then incapable of knowing either what He is or if He is. This being so, who will dare to undertake the decision of the question? Not we, who have no affinity to Him.

Who then will blame Christians for not being able to give a reason for their belief, since they profess a religion for which they cannot give a reason? They declare, in expounding it to the world, that it is a foolishness, stultitiam; and then you complain that they do not prove it! If they proved it, they would not keep their word; it is in lacking proofs, that they are not lacking in sense. “Yes, but although this excuses those who offer it as such, and takes away from them the blame of putting it forward without reason, it does not excuse those who receive it.” Let us then examine this point, and say, “God is, or He is not.” But to which side shall we incline? Reason can decide nothing here. There is an infinite chaos which separated us. A game is being played at the extremity of this infinite distance where heads or tails will turn up. What will you wager? According to reason, you can do neither the one thing nor the other; according to reason, you can defend neither of the propositions.

Do not then reprove for error those who have made a choice; for you know nothing about it. “No, but I blame them for having made, not this choice, but a choice; for again both he who chooses heads and he who chooses tails are equally at fault, they are both in the wrong. The true course is not to wager at all.”

Yes; but you must wager. It is not optional. You are embarked. Which will you choose then? Let us see. Since you must choose, let us see which interests you least. You have two things to lose, the true and the good; and two things to stake, your reason and your will, your knowledge and your happiness; and your nature has two things to shun, error and misery. Your reason is no more shocked in choosing one rather than the other, since you must of necessity choose. This is one point settled. But your happiness? Let us weigh the gain and the loss in wagering that God is. Let us estimate these two chances. If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation that He is.—“That is very fine. Yes, I must wager; but I may perhaps wager too much.”—Let us see. Since there is an equal risk of gain and of loss, if you had only to gain two lives, instead of one, you might still wager. But if there were three lives to gain, you would have to play (since you are under the necessity of playing), and you would be imprudent, when you are forced to play, not to chance your life to gain three at a game where there is an equal risk of loss and gain. But there is an eternity of life and happiness. And this being so, if there were an infinity of chances, of which one only would be for you, you would still be right in wagering one to win two, and you would act stupidly, being obliged to play, by refusing to stake one life against three at a game in which out of an infinity of chances there is one for you, if there were an infinity of an infinitely happy life to gain. But there is here an infinity of an infinitely happy life to gain, a chance of gain against a finite number of chances of loss, and what you stake is finite. It is all divided; wherever the infinite is and there is not an infinity of chances of loss against that of gain, there is no time to hesitate, you must give all. And thus, when one is forced to play, he must renounce reason to preserve his life, rather than risk it for infinite gain, as likely to happen as the loss of nothingness.

For it is no use to say it is uncertain if we will gain, and it is certain that we risk, and that the infinite distance between the certainty of what is staked and the uncertainty of what will be gained, equals the finite good which is certainly staked against the uncertain infinite. It is not so, as every player stakes a certainty to gain an uncertainty, and yet he stakes a finite certainty to gain a finite uncertainty, without transgressing against reason. There is not an infinite distance between the certainty staked and the uncertainty of the gain; that is untrue. In truth, there is an infinity between the certainty of gain and the certainty of loss. But the uncertainty of the gain is proportioned to the certainty of the stake according to the proportion of the chances of gain and loss. Hence it comes that, if there are as many risks on one side as on the other, the course is to play even; and then the certainty of the stake is equal to the uncertainty of the gain, so far is it from fact that there is an infinite distance between them. And so our proposition is of infinite force, when there is the finite to stake in a game where there are equal risks of gain and of loss, and the infinite to gain. This is demonstrable; and if men are capable of any truths, this is one.

“I confess it, I admit it. But, still, is there no means of seeing the faces of the cards?”—Yes, Scripture and the rest, etc. “Yes, but I have my hands tied and my mouth closed; I am forced to wager, and am not free. I am not released, and am so made that I cannot believe. What, then, would you have me do?”

True. But at least learn your inability to believe, since reason brings you to this, and yet you cannot believe. Endeavor then to convince yourself, not by increase of proofs of God, but by the abatement of your passions. You would like to attain faith, and do not know the way; you would like to cure yourself of unbelief, and ask the remedy for it. Learn of those who have been bound like you, and who now stake all their possessions. These are people who know the way which you would follow, and who are cured of an ill of which you would be cured. Follow the way by which they began; by acting as if they believed, taking the holy water, having masses said, etc. Even this will naturally make you believe, and deaden your acuteness.—“But this is what I am afraid of.”—And why? What have you to lose?

But to show you that this leads you there, it is this which will lessen the passions, which are your stumbling-blocks.

The end of this discourse.—Now, what harm will befall you in taking this side? You will be faithful, honest, humble, grateful, generous, a sincere friend, truthful. Certainly you will not have those poisonous pleasures, glory and luxury; but will you not have others? I will tell you that you will thereby gain in this life, and that, at each step you take on this road, you will see so great certainty of gain, so much nothingness in what you risk, that you will at last recognize that you have wagered for something certain and infinite, for which you have given nothing.

“Ah! This discourse transports me, charms me,” etc.

If this discourse pleases you and seems impressive, know that it is made by a man who has knelt, both before and after it, in prayer to that Being, infinite and without parts, before whom he lays all he has, for you also to lay before Him all you have for your own good and for His glory, that so strength may be given to lowliness.

The Wager’s argument can be described more precisely as follows:

(1) either God exists or God does not exist dichotomy 1
(2) I can bet either for the existence of God or for its non-existence dichotomy 2
(3) if I bet if in favor of the existence of God and God exists case 1
(4) then I get an infinite gain from (3)
(5) if I bet if in favor of the existence of God and God does not exist case 2
(6) then it follows a zero loss from (5)
(7) if I bet if in favor of the non-existence of God and God exists case 3
(8) then it results in an infinite loss from (7)
(9) if I bet if in favor of the non-existence of God and God does not exist case 4
(10) then it does not follow any gain or loss from (9)
(11) it is rational to make a choice in order to maximize the expected gain and loss premise
(12) if I bet if in favor of the existence of God from (3),(5)
(13) then the maximum gain is infinite and the maximum loss is null from (4),(6)
(14) if I bet if in favor of the non-existence of God from (7),(9)
(15) then the maximum gain is null and the maximum loss is infinite from (8),(10)
(16) ∴ it is rational to bet in favor of the existence of God from (11),(13),(15)

Pascal’s wager argument has led to a number of objections. Some critics, such as Jeffrey in his book The Logic of Decision, published in 1983, or McClennen in an essay published in 1994, blamed steps (3)-(4) and argued that the infinite utility that results from the expected gain when betting in favor of the existence of God is not a realistic gain and therefore has no real practical interest.

In addition, some other authors have pointed out that the intrinsic attitude which underlies the wager is itself questionable. Voltaire particularly found this attitude unbecoming, as it consists of deciding on a matter as serious as the existence of God according exclusively to considerations of public interest. In the situation of the wager, Voltaire considers that one may well have some rational elements for deciding on the existence of God, but that one lacks, however, the moral elements.

 

25. The Argument from Evil

chap25The argument from Evil is an argument that tends to prove the non-existence of God. Its formulation is very simple. The argument from Evil is based on the fact that evil is present in the world. The presence of suffering and pain is an inherent feature of our world today. Even worse, atrocities and horrendous crimes unfortunately occur every day in the world. The argument from Evil takes into account these undeniable facts and concludes that this demonstrates that God does not exist. There are different formulations of the argument from Evil. According to one of them, God is by definition a perfect being. God, moreover, is the creator of all things. Yet the obvious evil that exists in the world is one of those things. Therefore, according to this variation of the argument, God is the creator of evil. If this is the case, the claim that God is perfect is thus contradicted. The latter contradiction leads to the conclusion that God does not exist. The different steps of the argument from Evil can thus be described as follows:

(1) God is perfect definition
(2) God is the creator of all that exists definition
(3) evil exists in the world premise
(4) God is the creator of evil that exists in the world from (2),(3)
(5) God is not perfect from (4)
(6) ∴ God does not exist from (1),(5)

Another formulation of the argument from Evil emphasizes the all-powerfulness of God, and especially the notion of omnipotence. The argument considers that if God exists, then God is all-powerful and as such has the power to eliminate evil. But it turns out that there is evil in the world, in contradiction with the assumption that God exists. This results in the conclusion that God does not exist. This variant of the argument from evil can be described as follows:

(7) if God exists hypothesis
(8) then God is all-powerful definition
(9) then God has the power to remove evil from (8)
(10) if God exists then God has the power to remove evil from (7),(9)
(11) evil exists in the world premise
(12) God has not the power to remove evil from (10),(11)
(13) ∴ God does not exist from (10),(12)

The argument from Evil has been the subject of both old and recent objections. According to a recent objection raised by Alvin Plantinga in his book God and Other Minds, published in 1967, the argument is not valid because it is based on the false premise that God creates evil, or has the power to remove evil. Plantinga considers in contrast that free will is a necessary virtue, and therefore God aims to enable the development of free will in humans. According to Plantinga, God is not responsible for evil (by creating it or making it possible) because evil results directly from the exercise of human choice. And those choices—whether good or bad—made by men themselves are essential to the development of free will.

 

26. Descartes’ Cogito

The Cogito argument is due to Descartes. It can be formulated very briefly and simply: “I think, therefore I am.” However, in order to understand the exact scope of the Cartesian cogito, it is necessary to delve more deeply into its structure and context.

The original formulation of the Cogito can be found in the Discourse on the Method (Part IV):

I am in doubt as to the propriety of making my first meditations in the place above mentioned matter of discourse; for these are so metaphysical, and so uncommon, as not, perhaps, to be acceptable to every one. And yet, that it may be determined whether the foundations that I have laid are sufficiently secure, I find myself in a measure constrained to advert to them. I had long before remarked that, in relation to practice, it is sometimes necessary to adopt, as if above doubt, opinions which we discern to be highly uncertain, as has been already said; but as I then desired to give my attention solely to the search after truth, I thought that a procedure exactly the opposite was called for, and that I ought to reject as absolutely false all opinions in regard to which I could suppose the least ground for doubt, in order to ascertain whether after that there remained aught in my belief that was wholly indubitable. Accordingly, seeing that our senses sometimes deceive us, I was willing to suppose that there existed nothing really such as they presented to us; and because some men err in reasoning, and fall into paralogisms, even on the simplest matters of geometry, I, convinced that I was as open to error as any other, rejected as false all the reasonings I had hitherto taken for demonstrations; and finally, when I considered that the very same thoughts (presentations) which we experience when awake may also be experienced when we are asleep, while there is at that time not one of them true, I supposed that all the objects (presentations) that had ever entered into my mind when awake, had in them no more truth than the illusions of my dreams. But immediately upon this I observed that, whilst I thus wished to think that all was false, it was absolutely necessary that I, who thus thought, should be somewhat; and as I observed that this truth, I think, therefore I am (COGITO ERGO SUM), was so certain and of such evidence that no ground of doubt, however extravagant, could be alleged by the skeptics capable of shaking it, I concluded that I might, without scruple, accept it as the first principle of the philosophy of which I was in search.

It is tempting at this stage to consider that the Cogito argument can be formulated very briefly: “I think, therefore I am” and that its structure can be described as follows:

(1) I think premise
(2) if I think therefore I am from (1)
(3) ∴ I am from (1),(2)

However, this is an interpretation of Descartes’ argument that proves to be restrictive. It appears preferable to describe the Cartesian cogito in a way that better captures its essence, by better taking into account the context of doubt in which occurs the Cogito argument itself, for the Cogito is an argument which aims at demonstrating the existence of self, by taking into account the possibility of being self-deceived about one’s thoughts or perceptions. Descartes goes so far as to consider the case where the object of his own thoughts is wrong, that is to say he is deceived about the existence of sensible things that surround him, for example because he is dreaming. But even in this case, the conclusion that he exists also imposes itself to Descartes. The strength of the argument lies in the fact that even if I admit that I am currently deceived by my own thoughts because their object is false, it follows that I exist by the mere fact that my thoughts are erroneous. Therefore, what the Cogito argument ultimately shows is that I cannot be deceived about the mere fact that I exist, whether my thoughts are misleading or not. Thus, the Cogito argument can be rendered more accurately as follows:

(4) the object of my thoughts is either true or false dichotomy
(5) if the object of my thoughts is true hypothesis 1
(6) then I think consequence 1
(7) if the object of my thoughts is false hypothesis 2
(8) then I think consequence 2
(9) I think from (4),(6),(8)
(10) if I think then I do exist from (9)
(11) ∴ I do exist from (9),(10)

The Cogito argument is an application of the methodological doubt implemented by Descartes. He undertakes then to doubt the reality of all the knowledge he had previously acquired and that he had always considered as certain, not because he casts serious doubt on its existence, but because such a method allows him to achieve, in an optimum manner, some very certain and better assured knowledge. The Cogito argument is thus an illustration of this methodological doubt, which allows Descartes, in this context, to obtain a firm and steady knowledge, which corresponds to the certainty of his own existence.

 

27. Lewis Caroll’s Argument

chap27Lewis Carroll’s argument was published in 1895 in the Mind journal. The argument is presented there under the form of a dialogue between Achilles and the tortoise. The problem resulting from this argument can be stated as follows. Let us consider the following steps of reasoning:

(1) two things which are equal to a third one are themselves equal premise
(2) the sides AB and AC of a triangle ABC are both equal to the length DE premise
(Z) ∴ the sides AB and AC of the triangle ABC are equal from (1),(2)

At this stage, such reasoning seems to be quite valid. But now consider the following argument, which comprises an additional step (3):

(1) two things which are equal to a third one are themselves equal premise
(2) the sides AB and AC of a triangle ABC are both equal to the length DE premise
(3) if (1) and (2) are true then (Z) is true from (1),(2)
(Z) ∴ the sides AB and AC of the triangle ABC are equal from (1),(2),(3)

Before asserting the conclusion (Z), would it not be better to first acknowledge step (3) as true? Step (3) considers that the reasoning leading to (Z) is valid. It consists here of a necessary step to establish that (Z) is true, for if step (3) turned out to be false, we could not legitimately conclude that (Z) is true. Therefore, it is legitimate to include this step in the reasoning that leads to (Z). At this stage, however, it turns out that if we restore step (3), we must also take into account a new additional step (4), which leads us to consider the whole reasoning that follows:

(1) two things which are equal to a third one are themselves equal premise
(2) the sides AB and AC of a triangle ABC are both equal to the length DE premise
(3) if (1) and (2) are true then (Z) is true from (1),(2)
(4) if (1), (2) and (3) are true then (Z) is true from (1),(2),(3)
(Z) ∴ the sides AB and AC of the triangle ABC are equal from (1),(2),(3),(4)

But again, it turns out that the above reasoning can be extended by incorporating a new further step:

(1) two things which are equal to a third one are themselves equal premise
(2) the sides AB and AC of a triangle ABC are both equal to the length DE premise
(3) if (1) and (2) are true then (Z) is true from (1),(2)
(4) if (1), (2) and (3) are true then (Z) is true from (1),(2),(3)
(5) if (1), (2), (3) and (4) are true then (Z) is true from (1),(2),(3),(4)
(Z) ∴ the sides AB and AC of the triangle ABC are equal from (1),(2),(3),(4),(5)

Such reasoning can be extended ad infinitum and thus results in an infinite regress. Therefore, it follows that we never reach the conclusion (Z).

Lewis Carroll’s argument rests on the fact that, before reaching the conclusion (Z), it should be recognized that the reasoning leading to this conclusion is valid. In a general way, the argument—just like Zeno of Elea’s paradox of the race, but also the paradox of Achilles and the tortoise, which is another paradox from Zeno—emphasizes that before reaching the conclusion (Z), we must go through an infinite series of steps and in these conditions, we never succeed in formulating the conclusion (Z).

Lewis Carroll’s argument notably emphasizes the importance of modus ponens. This inference rule allows the reasoning which has the following structure (P and Q being two propositions):

(6) P is true premise
(7) if P is true then Q is true premise
(8) ∴ Q is true from (6),(7)

The argument points to the fact that before applying a rule of inference such as modus ponens, it is necessary to have available a second rule describing how to apply modus ponens, then a third rule describing how we should apply the rule that describes how to apply modus ponens, and so on. An infinite regress ensues.

An objection that has been traditionally raised against Caroll’s argument is that such problem does not occur within formal logic, where each rule is formalized. In this case, the deductive mechanism reduces to a manipulation of symbols. However, such a formal system has the disadvantage of not taking into account the semantic aspect of things, yet this is essential, for the latter aspect proves totally absent from what only reduces then to a manipulation of meaningless symbolic characters.

 

28. The Twin Earth Thought Experiment

chap28The Twin Earth thought experiment was introduced by Hilary Putnam in an essay published in 1975. Putnam presents three thought experiments, and one of them—the H2O-XYZ thought experiment—introduces the problem of the Twin Earth. Putnam features a planet, the Twin Earth, which proves to be identical in all respects to the Earth, with one difference. This difference relates to the compound body that is referred to as “water” on Earth and whose atomic structure is H2O. On the Twin Earth, there exists indeed a compound body that possesses all the properties of our water, such as being liquid, transparent, odorless, etc. but whose chemical composition is XYZ. Let us term such a compound body water*. On the Twin Earth, people also call the latter compound body “water”. At this stage, according to Putnam, it turns out that water refers to the compound body H2O and water* refers to the compound body XYZ. Thus, water and water* are respectively used in a quite identical manner by Earth people and by the inhabitants of Twin Earth. Moreover, the content of thoughts of an inhabitant of the Earth or of the Twin Earth is quite identical when they each think of water or of water*. Therefore, it follows that the semantic content of their respective thoughts cannot be determined in a purely internal way, and therefore can only be elucidated by resorting to external data. It is here, according to Putnam, that the lesson of the problem resulting from the Twin Earth experiment situates itself. One can indeed wonder whether or not the meaning, the semantic content of a word or concept, finds itself exclusively in our brain. According to Putnam, what the Twin Earth experiment demonstrates is that a negative answer should be given to this question, for only the use of external data in the Twin Earth thought experiment allows us to determine the semantic content of the thoughts of a human being and of an inhabitant of the Twin Earth when they think or speak respectively of water or of water*. Thus, Putnam concludes, it is necessary to adopt an externalist conception for the determination of mental content.

The reasoning to which the Twin Earth experiment leads can be thus detailed:

(1) on Earth, there exists a liquid, transparent, odorless compound body, etc. whose composition is H2O premise
(2) the compound body whose composition is H2O is water definition
(3) on Twin Earth, there exists a liquid, transparent, odorless compound body, etc. whose composition is XYZ premise
(4) the compound body whose composition is XYZ is water* definition
(5) the inhabitants of Earth call “water” the compound body whose composition is H2O premise
(6) the inhabitants of Twin Earth call “water” the compound body whose composition is XYZ premise
(7) the content of the thoughts of an inhabitant of Earth when he/she thinks to water is x from (1),(5)
(8) the content of the thoughts of an inhabitant of Twin Earth when he/she thinks to water* is x from (2),(6)
(9) the content of the thoughts of an inhabitant of Earth or of Twin Earth when they think respectively of water or to water* is identical from (7),(8)
(10) ∴ we must resort to external data in order to distinguish the semantic content of the thoughts of an inhabitant of Earth who thinks of water from the content of the thoughts of an inhabitant of Twin Earth who thinks of water* from (9)

At this stage, it turns out that the scope of the problem raised by Putnam extends beyond the sole Twin Earth thought experiment and our concept of water, for a similar reasoning can be applied to all categories of objects referred to by our ordinary language, such as a cloud, a mountain, a chair, etc. For each of our everyday and familiar objects, it proves necessary, by virtue of the Twin Earth experiment, to resort to an external criterion to understand the corresponding semantic content.

It has been objected that the corresponding situation is unrealistic. Indeed, if a compound body should possess properties very similar to those of our water, shouldn’t then its composition be the same as that of water, that is to say H2O? According to this objection, proposition (3), that there exists another planet with a compound body with identical properties to those of water and whose chemical composition is different, proves unrealistic and even contradictory.

 

29. The Argument against the Principle of Verifiability

chap29The argument against the principle of verifiability results from the work of a group of philosophers belonging to the school of thought of logical positivism. This line of thinking is part of the ideas put forward in 1920-1930 by the Vienna Circle, which notably included Rudolf Carnap and Kurt Gödel. Logical positivism distinguishes two types of meaningful propositions: some propositions (i) are analytical, while some others (ii) can be verified experimentally. Analytical propositions include, for example, mathematical propositions, such as “a triangle has two right angles” (which is analytically false) or “a dog is a mammal” (which is analytically true), of which one can determine the truth or falsity by deduction only. In contrast, experimentally verifiable propositions can be confirmed or disconfirmed empirically. Thus, “I measure 1.73 meters” or “Proxima Centauri is 4.23 light-years from Earth” are propositions that can be tested experimentally. Any other type of proposition, i.e. one that is neither analytically nor experimentally verifiable, is meaningless. Logical positivism, influenced by ideas put forth by Ludwig Wittgenstein, thus leads to the rejection of metaphysical propositions, considered insignificant because they do not meet one of the two above-mentioned criteria. According to this viewpoint, the metaphysical assertions do not have a logical basis, because they do not meet the criterion of verifiability, under which any claim must be experimentally verified. In this respect, a metaphysical statement should be subject to the possibility of confirmation or disconfirmation. Such is not the case, and therefore, metaphysical assertions must be discarded.

However, such an argument based on the principle of verifiability has been the subject of the following objection, notably due to Ewing, in his book The Fundamental Questions of Philosophy published in 1962: the principle of verifiability itself is not experimentally verifiable. Thus, the principle of verifiability itself does not satisfy the criterion of verifiability, for one does not have at one’s disposal a process allowing one to verify it experimentally. Thus, the principle of verifiability itself falls victim to the very principle that it claims to promote. This shows how this principle actually proves too restrictive. The argument against the principle of verifiability can be described step by step as follows:

(1) either the principle of verifiability prevails or it does not prevail dichotomy
(2) in virtue of the principle of verifiability, any claim must be verifiable premise
(3) to be verifiable, for a given proposition, consists in the fact that it is possible to confirm or to disconfirm it definition
(4) the principle of verifiability cannot be confirmed experimentally premise
(5) the principle of verifiability cannot be disconfirmed experimentally premise
(6) the principle of verifiability cannot be confirmed nor disconfirmed experimentally from (4),(5)
(7) the principle of verifiability is not verifiable from (3),(6)
(8) ∴ the principle of verifiability does not prevail from (7)
  1. 30. The Allegory of the Cave

chap30The celebrated Allegory of the Cave was described by Plato in the Republic (Book VII). Plato depicts humans who have been chained, since their childhood, to the walls of a cave. These prisoners are chained in such a way that they cannot move their heads and cannot see each other. However, the cave communicates with the outside via an opening. All that these prisoners can observe are merely reflections of people and animals that pass outside the cave and shadows of flowers, rocks, etc. as they appear on the walls of the cave. For the prisoners, reality itself is restricted to shadows and reflections that they observe on these walls. But one day, one of the prisoners manages to break his chains and escape the cave. He goes out of the cave for the first time, and in the light of day he discovers real people, authentic animals, genuine flowers, etc. in their original shapes and colors. His one idea is then to return to the cave and inform his former companions that what they see on the walls of the cave are mere reflections, shadows and images of another level of reality, which would appear if they also broke their ties and went into the light of day. Returning to the cave, he begins to explain to his chained companions that what they see is only a reflection of the true reality. But his former companions do not believe him, and eventually kill him. The allegory clearly has the structure of an analogy, since for Plato, the shadows that appear on the walls of the cave represent the world of appearances. In contrast, real objects such that one can observe in the daylight belong to the world of Ideas.

The extract of the Republic that includes the Allegory of the cave depicts the following dialogue between Socrates and Glaucon:

[Socrates:] And now, I said, let me show in a figure how far our nature is enlightened or unenlightened: — Behold! human beings living in a underground cave, which has a mouth open towards the light and reaching all along the cave; here they have been from their childhood, and have their legs and necks chained so that they cannot move, and can only see before them, being prevented by the chains from turning round their heads. Above and behind them a fire is blazing at a distance, and between the fire and the prisoners there is a raised way; and you will see, if you look, a low wall built along the way, like the screen which marionette players have in front of them, over which they show the puppets.

[Glaucon:] I see.

— And do you see, I said, men passing along the wall carrying all sorts of vessels, and statues and figures of animals made of wood and stone and various materials, which appear over the wall? Some of them are talking, others silent.

— You have shown me a strange image, and they are strange prisoners.

— Like ourselves, I replied; and they see only their own shadows, or the shadows of one another, which the fire throws on the opposite wall of the cave?

— True, he said; how could they see anything but the shadows if they were never allowed to move their heads?

— And of the objects which are being carried in like manner they would only see the shadows?

— Yes, he said.

— And if they were able to converse with one another, would they not suppose that they were naming what was actually before them?

— Very true.

— And suppose further that the prison had an echo which came from the other side, would they not be sure to fancy when one of the passers-by spoke that the voice which they heard came from the passing shadow?

— No question, he replied.

— To them, I said, the truth would be literally nothing but the shadows of the images.

— That is certain.

— And now look again, and see what will naturally follow if the prisoners are released and disabused of their error. At first, when any of them is liberated and compelled suddenly to stand up and turn his neck round and walk and look towards the light, he will suffer sharp pains; the glare will distress him, and he will be unable to see the realities of which in his former state he had seen the shadows; and then conceive someone saying to him, that what he saw before was an illusion, but that now, when he is approaching nearer to being and his eye is turned towards more real existence, he has a clearer vision, —what will be his reply? And you may further imagine that his instructor is pointing to the objects as they pass and requiring him to name them, —will he not be perplexed? Will he not fancy that the shadows which he formerly saw are truer than the objects which are now shown to him?

— Far truer.

— And if he is compelled to look straight at the light, will he not have a pain in his eyes which will make him turn away to take and take in the objects of vision which he can see, and which he will conceive to be in reality clearer than the things which are now being shown to him?

— True, he now.

— And suppose once more, that he is reluctantly dragged up a steep and rugged ascent, and held fast until he is forced into the presence of the sun himself, is he not likely to be pained and irritated? When he approaches the light his eyes will be dazzled, and he will not be able to see anything at all of what are now called realities.

— Not all in a moment, he said.

— He will require to grow accustomed to the sight of the upper world. And first he will see the shadows best, next the reflections of men and other objects in the water, and then the objects themselves; then he will gaze upon the light of the moon and the stars and the spangled heaven; and he will see the sky and the stars by night better than the sun or the light of the sun by day?

— Certainly.

— Last of he will be able to see the sun, and not mere reflections of him in the water, but he will see him in his own proper place, and not in another; and he will contemplate him as he is.

— Certainly.

— He will then proceed to argue that this is he who gives the season and the years, and is the guardian of all that is in the visible world, and in a certain way the cause of all things which he and his fellows have been accustomed to behold?

— Clearly, he said, he would first see the sun and then reason about him.

— And when he remembered his old habitation, and the wisdom of the cave and his fellow-prisoners, do you not suppose that he would felicitate himself on the change, and pity them?

— Certainly, he would.

— And if they were in the habit of conferring honors among themselves on those who were quickest to observe the passing shadows and to remark which of them went before, and which followed after, and which were together; and who were therefore best able to draw conclusions as to the future, do you think that he would care for such honors and glories, or envy the possessors of them? Would he not say with Homer, “Better to be the poor servant of a poor master,” and to endure anything, rather than think as they do and live after their manner?

— Yes, he said, I think that he would rather suffer anything than entertain these false notions and live in this miserable manner.

— Imagine once more, I said, such a one coming suddenly out of the sun to be replaced in his old situation; would he not be certain to have his eyes full of darkness?

— To be sure, he said.

— And if there were a contest, and he had to compete in measuring the shadows with the prisoners who had never moved out of the cave, while his sight was still weak, and before his eyes had become steady (and the time which would be needed to acquire this new habit of sight might be very considerable) would he not be ridiculous? Men would say of him that up he went and down he came without his eyes; and that it was better not even to think of ascending; and if any one tried to loose another and lead him up to the light, let them only catch the offender, and they would put him to death.

— No question, he said.

We can detail, at this stage, the different steps underlying the Allegory of the cave:

(1) the prisoners of the cave are convinced that the objects they observe daily are the actual objects premise
(2) the prisoners of the cave observe in reality on the walls shadows and reflections of the actual objects from (1)
(3) the situation of the prisoners of the cave is similar to our present situation analogy
(4) we are convinced that the objects we see every day are the actual objects premise
(5) ∴ the objects that we see are in reality the shadows and reflections of the actual objects from (2),(3),(4)

Plato’s conclusion is that the human situation is analogous to that of the prisoners of the cave. In this sense, the Allegory of the cave is clearly an argument by analogy. However, at this stage, the resulting conclusion can be diversely interpreted. We can thus distinguish two main interpretations. According to the first interpretation, the prisoners of the cave are men, and the objects they see are merely a pale reflection of real objects, which are the ideas or archetypes. There are therefore archetypes of the number “7”, of courage, of tolerance, of a lion and of the sun, etc. in the world of ideas. In this sense, humans believe that ultimate reality is the one which corresponds to their perceptions, while this is illusory, and that the true reality is located at the level of the Archetypes. Thus, we live every day in what constitutes only the second plane corresponding to the projection of the real objects which situate themselves in the foreground: that is to say, at the level of the Archetypes. In this sense, the Allegory of the cave proves to be close to the “brains in a vat” experiment and its modern illustration through the Matrix movie.

A second type of interpretation, however, can be applied to the Allegory of the cave. Such an interpretation is directly related to Plato’s theory of knowledge, for Plato distinguishes between the knowledge gained from opinion and genuine knowledge. Thus, the knowledge of the beings and objects that the prisoners of the cave possess is merely knowledge gained from opinion. It does not consist of true knowledge, because it is shaped, transformed and distorted by the education that was received. Basic knowledge that we possess is, according to Plato, perverted by the turmoil of human passions, ambition, competition, received ideas, etc. By contrast, authentic and true knowledge situates itself beyond passions, hatred, honors and established ideas. According to Plato, each human must rise well above the passions that bind him in chains in order to reach true knowledge.

 

31. The Simulation Argument

chap31The Simulation Argument has been described recently by Nick Bostrom, in an article published in 2003 in the Philosophical Quarterly journal. The argument is basically grounded on the fact that it appears quite likely that post-human civilization will carry out simulations of humans. Indeed, it seems likely that far advanced post-human civilizations will have both the ability and the willingness to carry out extremely realistic simulations of humans. If this were the case, the number of simulated humans would then greatly exceed the number of genuine humans. In such a case, it follows that taking into account the fact that each of us exists leads us to consider it more probable that we belong to the simulated humans rather than to the authentic humans. According to Bostrom, the conclusion that follows from the Simulation argument is that the probability of each of the three following statements is approximately 1/3:

(1) humanity will face a relatively imminent extinction
(2) a post-human civilization will not carry out human simulations
(3) we currently live within a simulation

These probabilities are not surprising regarding the assertions (1) and (2), but the inherent likelihood of the assertion (3) under which we now live in a simulation, turns out to be quite counter-intuitive.

The Simulation argument is also described more succinctly by Brian Weatherson, in a response to Bostrom’s original article, published in 2004. According to him, the real core of the Simulation argument can be described as follows. Firstly, it is very likely that post-human civilizations will be able to produce realistic simulations of human beings. Similarly, it is very likely that the number of simulated humans will greatly exceed the number of real humans. Thus, at a post-human age, the ratio between simulated humans and genuine human beings should be largely in favor of simulated humans. At this step, it turns out that the simple fact of taking into account our present existence leads to the conclusion that it is likely that we are simulated humans. This invites us to think that the probability that our thoughts, feelings, sensations, etc. are the result of a simulation is high.

The conclusion of the Simulation argument—in a somewhat similar way to the Doomsday argument—proves to be counter-intuitive and contrary to common sense. However, in the same way as for the Doomsday argument, the task that consists in determining accurately the fallacious step at the level of the Simulation argument proves to be very difficult.

A first objection that could be raised against the Simulation argument focuses on the need to appeal to a principle of indifference (in virtue of which there is no reason to favor a priori any of the assumptions here). For are we really human beings randomly chosen within the reference class that includes both humans and simulated human beings? It seems in fact that the Simulation argument is worthwhile only if we are chosen randomly within the reference class. Is there not here the same problem that emerges when faced with the Doomsday argument? Bostrom, however, responds to this objection by arguing that the principle of indifference used in the Simulation argument is not of the same nature as that referred to by the Doomsday argument. Indeed, in the Doomsday argument, an important premise is that every human being, given its birth rank, should be considered as randomly chosen within the reference class. The principle of indifference used in the Simulation argument proves to be weaker, since it is applied regardless of the birth rank (or any other criterion of a similar nature), but proceeds from the mere observation of our existence as members of the reference class.

Another objection that might be raised is that the simulation argument itself is self-refuting. Indeed, if its conclusion is true, it follows that the argument itself is the product of a simulation and that all our logic is in turn simulated. In this case, we cannot therefore accept as valid the conclusions resulting from the argument. However, we can see that such an objection also applies to the Dreaming argument, to the “brains in a vat” experiment, etc. Thus, such an objection turns out to be too general, and it seems that it does not respond accurately to the specific problem posed by the Simulation argument.

 

32. The Dualist Argument from Divisibility

chap32In the course of the Metaphysical Meditations (Sixth Meditation), Descartes develops an argument that aims to prove the existence of the mind/body duality. He proposes to show how body and mind are two vital components of human nature, which prove, however, fundamentally different in nature. This argument takes place within the debate opposing materialism to idealism. Materialism is the doctrine according to which only material and physical things exist. In this context, the phenomena of mental nature can be reduced to mere phenomena of material origin. Thus, according to materialism, everything that exists is material and can be characterized in purely physical terms. In contrast, idealism is the view that only things of a mental nature exist. In this context, material things only gain their existence through our own perceptions. According to the idealist standpoint, all that exists reduces thus to a purely mental existence. In essence, materialism and idealism are monistic views. In contrast, dualism is a pluralist perspective that considers that things of physical and mental nature both exist. According to this view, the mental and the physical, whose very nature is fundamentally different, coexist. The dualistic point of view was famously defended by Descartes. There exists, according to Descartes, a mind/body duality, which constitutes the counterpart relevant to human beings of the mental/physical dualism. Descartes based his reasoning on the respective properties of the body and the mind, which are fundamentally different. He thus considers that physical matter that constitutes our body has an extension in space and proves therefore divisible. Conversely, the mind, according to Descartes, has no spatial extension and therefore does not have the same property of divisibility. Thus, the body and the mind have at least one different property and are therefore, in virtue of Leibniz’s law—according to which two objects are identical if and only if all their properties are identical—fundamentally different.

The Dualist argument from Divisibility comes from the following passage of the Metaphysical Meditations:

In order to begin this examination, then, I here say, in the first place, that there is a great difference between mind and body, inasmuch as body is by nature always divisible, and the mind is entirely indivisible. For, as a matter of fact, when I consider the mind, that is to say, myself inasmuch as I am only a thinking thing, I cannot distinguish in myself any parts, but apprehend myself to be clearly one and entire; and although the whole mind seems to be united to the whole body, yet if a foot, or an arm, or some other part, is separated from my body, I am aware that nothing has been taken away from my mind. And the faculties of willing, feeling, conceiving, etc. cannot be properly speaking said to be its parts, for it is one and the same mind which employs itself in willing and in feeling and understanding. But it is quite otherwise with corporeal or extended objects, for there is not one of these imaginable by me which my mind cannot easily divide into parts, and which consequently I do not recognize as being divisible; this would be sufficient to teach me that the mind or soul of man is entirely different from the body, if I had not already learned it from other sources.

The different steps of Descartes’ Dualist Argument from Divisibility can be detailed as follows:

(1) my body has a space extension premise
(2) everything that has a space extension is divisible premise
(3) my body is divisible from (1),(2)
(4) my mind has no space extension premise
(5) my mind is not divisible premise
(6) my body and my mind have at least one different property from (3),(5)
(7) two things are identical if and only if they have identical properties Leibniz law
(8) if two things have different properties then these two things are distinct from (7)
(9) ∴ my body and my mind are two distinct things from (6),(8)

Descartes’ dualistic point of view gave rise to an important objection, which is the following: if there exists a mind/body duality, how do these two fundamentally different components of the same human being interact? The nature of the interaction resulting from the doctrine of mind/body duality has so far not been elucidated. This is an important gap in the dualist approach, since a comprehensive dualistic theory needs to explicitly describe the modalities of the interaction between the body and the mind.

 

33. The Sleeping Beauty Problem

chap33The Sleeping Beauty Problem has been the subject of a number of recent discussions, notably by Adam Elga and David Lewis in articles published respectively in 2000 and 2001 in the Analysis journal. The Sleeping Beauty problem was described as follows by Elga. Some researchers have planned an experiment in which they propose to put to sleep Sleeping Beauty. She will be asleep for two days, namely Monday and Tuesday. However, during her sleep, she will be awakened once or twice. The number of times she will be awakened will depend on the outcome of the throw of a perfectly balanced coin. If the coin lands on heads, Beauty will be awakened once on Monday. However, if the coin lands on tails, she will be awakened twice, on Monday and on Tuesday. In both cases, after being awakened on Monday, Beauty will be put to sleep again and will forget that she has been awakened. Given these elements, when Beauty awakens, to what extent should she believe that the coin fell heads?

At this stage, a first type (I) of reasoning suggests that the probability that the coin fell equals 1/2. Indeed, the coin is balanced, and therefore, if the experiment is repeated, it will result in a roughly equal number of tail draws. The initial likelihood of heads or tails is thus 1/2. But when Beauty awakens, she receives no new information. Therefore, she has no reason to change her initial belief. It would have been rational to modify the initial probabilities if new information had been provided, but such is not the case, and therefore, Beauty has no justification to change her initial probabilities. Such reasoning corresponds, in a simplified way, to that carried out by David Lewis.

It turns out, however, that a second type (II) of response is possible. The corresponding reasoning leads to the conclusion that the probability that the coin fell heads is 1/3. Let us suppose that the experiment is repeated many times. In this case, it will turn out that about 1/3 of the awakenings will occur when the coin fell heads. Similarly, about 2/3 of the awakenings will occur when the coin fell tails. So when Beauty awakes, she can legitimately consider that it is a heads-awakening with a probability of 1/3. Therefore, Beauty must conclude that the probability that the coin fell heads is 1/3.

It is worth formalizing the elements of the Sleeping Beauty problem, in order to cast light on its internal structure. The problem is based on the following two competing hypotheses:

(H1) Sleeping Beauty will be awakened once (HEADS)
(H2) Sleeping Beauty will be awakened twice (TAILS)

Similarly, it turns out that three cases are possible:

(i) the coin fell HEADS and Sleeping Beauty will be awakened on Monday
(ii) the coin fell TAILS and Sleeping Beauty will be awakened on Monday
(iii) the coin fell TAILS and Sleeping Beauty will be awakened on Tuesday

The problem resulting from the situation corresponding to the Sleeping Beauty problem is that both rationales (I) and (II) seem a priori correct, whereas they lead to contradictory conclusions. Thus, one of the two arguments must be false. But which one? And why? In the contemporary literature on the Sleeping Beauty problem, the two competing arguments have their advocates and detractors, and there is currently no consensual solution.

 

34. The Evil Demon Argument

chap34The Evil Demon argument is a famous argument described by Descartes in the Metaphysical Meditations. The evil demon argument is an argument in favor of skepticism. The argument itself is based on a thought experiment. Descartes considers thus the hypothesis that there exists an evil genius who is capable of deceiving him not only at the level of all his sensory perceptions, but also at the level of the whole of his knowledge, including that relating to mathematics. Considering that he does not have absolute certainty that this evil demon does not exist, Descartes concludes that it is possible that all his knowledge is false and that he is therefore entitled to question his overall knowledge.

The evil demon argument is mentioned in the following passage from the Meditations on First Philosophy:

I shall then suppose, not that God who is supremely good and the fountain of truth, but some evil genius not less powerful than deceitful, has employed his whole energies in deceiving me; I shall consider that the heavens, the earth, colors, figures, sound, and all other external things are nought but the illusions and dreams of which this genius has availed himself in order to lay traps for my credulity; I shall consider myself as having no hands, no eyes, no flesh, no blood, nor any senses, yet falsely believing myself to possess all these things; I shall remain obstinately attached to this idea, and if by this means it is not in my power to arrive at the knowledge of any truth, I may at least do what is in my power [i.e. suspend my judgment], and with firm purpose avoid giving credence to any false thing, or being imposed upon by this arch deceiver, however powerful and deceptive he may be.

The evil demon argument can be detailed as follows:

(1) it is possible that there exists an evil genius, capable of deceiving me on the whole of my sensory perceptions and my mathematical knowledge hypothesis
(2) if I am deceived at the level of the whole of my sensory perceptions and my mathematical knowledge (i.e. the fact that I am now in front of the fire in the fireplace or that the sum of the angles of a triangle is equal to a flat angle) then the whole of my beliefs are false premise
(3) it is possible that the whole of my beliefs are false from (1),(2)
(4) if I do not have the certainty that such evil genius does not exist, then I can not consider that the whole of my beliefs are true from (1),(3)
(5) I do not have the certainty that such an evil genius does not exist premise
(6) I cannot consider that the whole of my beliefs are true from (4),(5)
(7) ∴ I have good reason to doubt the whole of my beliefs from (6)

The argument clearly targets a posteriori knowledge related to physical objects (i.e. a table, a horse or the planet Saturn), but also a priori knowledge such as that resulting from mathematics (i.e. the fact that the sum of the angles of a triangle is equal to a flat angle, or 1 + 3 = 4).

It is doubtful, however, whether the evil demon argument allows generalized doubt: that is to say, if it applies to our overall knowledge. Indeed, as demonstrated by Descartes himself, it seems that a proposition such as “I think, therefore I am” escapes such a doubt of universal scope. In this sense, the conclusion of the evil demon argument proves too strong. However, it turns out that even if we restrict the scope of the argument’s conclusion, much of it remains and still permits us to conclude in favor of skepticism.

Another objection that can be raised against the argument is that the argument is self-refuting, for the latter applies to both a posteriori and a priori knowledge. But the conclusion that follows from the evil demon argument itself constitutes a priori knowledge. I am also entitled to doubt the latter conclusion. Thus, the argument itself is undermined by its own conclusion. Given that I am justified in doubting all my a priori knowledge, I am well founded to doubt that I can doubt the whole of my beliefs.

 

35. Searle’s Chinese Room Argument

chap35The Chinese Room argument was described by John Searle in an article published in 1980 in the Behavioral and Brain Sciences journal. This argument is based on a thought experiment, which can be described as follows. Suppose you have no knowledge of the Chinese language and you are locked alone in a room that contains the following items: (i) a set of typed text in Chinese, entitled “script”; (ii) a second set of documents in Chinese, entitled “the story”, accompanied by a set of rules in English to relate the initial documents with the latter; (iii) a third set of documents, entitled “questions”, with symbols in Chinese and instructions in English to relate the Chinese symbols with the first two sets of documents. At this point, a Chinese text is sent to you under the door. Consulting your four sets of documents, you then write another text in Chinese, entitled “answers” that you pass in your turn under the door of the room.

Searle’s thought experiment is based on an analogy. It draws a parallel between the situation of the person who is in the room and the situation corresponding to a computer program performing a translation. The person who is in the room receives a text written in Chinese, and then, after consulting a series of documents, writes in turn a new document in Chinese, which is a response to the first document received. Such a response is no different from the one that a person with a thorough understanding of the Chinese language would have made. And this stresses how the true understanding of the Chinese text that was submitted to him/her in fact completely escapes him/her, for the person who is in the room is able to respond competently to the question that is asked, but completely ignores the content of this answer. The experiment thus aims to highlight how the semantic content of the text escapes the machine, even though it possesses control of its syntactic content.

Searle’s argument is intended to constitute an objection to the view that a computer program is capable of thinking. This viewpoint represents the thesis called “strong AI” (strong artificial intelligence). According to this latter view, the computers have the ability to actually think in the same way that humans do. In this sense, a computer program can have a true understanding of a given situation. Strong AI is thus opposed to the thesis of weak AI, according to which computer programs are only simulations of the human mind. In this sense, the result of a computer program is not a genuine process of thought, but a mere simulation, however successful it may be, of the latter.

Searle’s argument itself, illustrated by the Chinese room experiment, can be detailed as follows:

(1) either strong AI prevails or weak AI prevails dichotomy
(2) computer programs make use of symbols premise
(3) the symbols correspond to the syntactic content of a text premise
(4) the human mind makes use of the semantic content of a text premise
(5) the Chinese room experiment shows that the syntactic content of a text does not suffice to determine its semantic content from (3),(4)
(6) the situation of the person within the Chinese room is analogous to that of a computer program performing a translation analogy
(7) ∴ computer programs do not reach to determine the semantic content of a text from (5),(6)
(8) strong AI does not prevail from (7)
(9) ∴ it is weak AI that prevails from (1),(8)

The Chinese room argument has generated a huge controversy. Although Searle responds in advance in his original article to a number of objections, the argument has left many authors unconvinced. However, to date, none of them has been able to indicate in a consensual way the accurate step in Searle’s reasoning that proves defective.

 

36. The Turing Test

chap36Alan Turing, in a famous paper published in 1950 in the journal Mind, proposes to clarify the question, “Can machines think?” Instead of trying to answer that in a conventional manner by defining the concepts of “machine” and “think”, Turing chooses a different path. He seeks then to describe the following game, which he calls the Imitation game:

The imitation game This game is played with three people, a man (A), a woman (B) and an interrogator (C) of either sex. The interrogator is located in a room separate from the other two. The goal of the game for the interrogator is to be able to determine which one among the two others is the man or the woman. The interrogator knows each of them by the names X and Y, and at the end of the game, he must say either “X is A and Y is B” or “X is B and Y is A”. For this purpose, the interrogator is allowed to ask questions to A and B.

Nowadays, the original version of the Imitation game described by Turing is usually replaced by a simplified experiment which is as follows:

The Imitation Game (modern version) This game is played with two people and one machine: a man (A), a machine (M) and an interrogator (C). A and C are of one or the other sex. The interrogator is connected to A and M with a terminal, through which they can communicate. However, the interrogator can see neither man nor machine and does not know who is human and which is the machine. Its mission is to endeavor to determine who is human and which is the machine by asking questions. The interrogator is located in a room separate from that where the other two are. The machine and the human are trying to convince the interrogator that each of them is human. The goal of the game for the interrogator is to be able to determine who is truly human. If the interrogator cannot distinguish the human from the machine, it is then considered that the machine is intelligent.

It should be noted that an earlier version of the Turing test can be traced back to Descartes, who, in his Discourse on the Method, imagines a situation of a similar nature in the following passage:

And here I specially stayed to show that, were there such machines exactly resembling organs and outward form an ape or any other irrational animal, we could have no means of knowing that they were in any respect of a different nature from these animals; but if there were machines bearing the image of our bodies, and capable of imitating our actions as far as it is morally possible, there would still remain two most certain tests whereby to know that they were not therefore really men. Of these the first is that they could never use words or other signs arranged in such a manner as is competent to us in order to declare our thoughts to others: for we may easily conceive a machine to be so constructed that it emits vocables, and even that it emits some correspondent to the action upon it of external objects which cause a change in its organs; for example, if touched in a particular place it may demand what we wish to say to it; if in another it may cry out that it is hurt, and such like; but not that it should arrange them variously so as appositely to reply to what is said in its presence, as men of the lowest grade of intellect can do. The second test is, that although such machines might execute many things with equal or perhaps greater perfection than any of us, they would, without doubt, fail in certain others from which it could be discovered that they did not act from knowledge, but solely from the disposition of their organs: for while reason is a universal instrument that is alike available on every occasion, these organs, on the contrary, need a particular arrangement for each particular action; whence it must be morally impossible that there should exist in any machine a diversity of organs sufficient to enable it to act in all the occurrences of life, in the way in which our reason enables us to act.

Secondly, on the basis of the Imitation game, Turing made the following prediction. He considered that by the year 2000, it would be quite possible to program a computer so that the average human interrogator would have not more than a 70/100 chance at the imitation game to correctly identify the human and the machine, after asking a series of questions for five minutes. In a general way, the Turing test aims to demonstrate that the time is not far off when it will be impossible to distinguish man from machine. According to Turing, this is a demonstration that human intelligence can be fully simulated by computer.

The argument underlying the Turing test can thus be presented in detail as follows:

(1) if a test is performed to distinguish human intelligence from the simulated intelligence of the machine hypothesis
(2) then one cannot define a criterion to make such a distinction from (1)
(3) it is almost impossible to discern human intelligence from the simulated intelligence of the machine from (2)
(4) ∴ human intelligence can be fully simulated from (3)

In this context, the argument based on the Turing test appears closely related to the Simulation argument recently described by Nick Bostrom.

It may be objected to the Turing experiment that the potentialities of the human brain and intelligence are barely known. Thus, new capabilities of human intelligence may well be discovered, which would then entirely escape the Turing test. In the same vein, we can also consider that what allows us to conclude the Turing test is that at present and in the near future, it will be difficult to distinguish a machine from a human being. However, this does not allow us to conclude that such differentiation will never be possible. Isn’t it too strong a conclusion? To conclude validly that human intelligence can be fully computer simulated, it would be necessary to have an absolute certainty that the differentiation between human and machine, under the conditions of the test, cannot be performed.

 

37. Gettier’s Problem

chap37The Gettier problem was exposed by Edmund Gettier, in an article published in 1963 in the Analysis journal. Traditionally, one considers that a given person S knows a given proposition P provided that three conditions are simultaneously met: (i) the proposition P is true, (ii) S believes that P is true, (iii) S is justified in his/her belief that P is true. Hence, S knows that P if S has a justified true belief that P. This threefold condition of knowledge is widely accepted. However, Gettier undertakes to demonstrate that this threefold condition of knowledge is ungrounded and that these three criteria do not constitute a sufficient condition.

The first concrete case described by Gettier is as follows. Two protagonists, Smith and Jones, both apply for a job. Smith has decisive elements that allow him to think that the following proposition, whose structure is that of a conjunction, is true:

(1) Jones is the one who will get the job and Jones has ten coins in his pocket

The key elements that are at the disposal of Smith are, on the one hand, the fact that the president of the company assured him that it would be Jones who would get the job, and on the other hand, the fact that Smith had previously counted the number of coins—that amount to ten—that were in Jones’ pocket. Hence, (1) has the following consequence:

(2) the one who will get the job has ten coins in his pocket

In this case, we can consider that Smith knows that (2), since the above-mentioned three conditions are satisfied: proposition (2) is true, Smith believes that (2) is true and Smith is justified by (1) in his belief that (2) is true. Now let us imagine that, unknown to Smith, it is Smith himself who has actually got the job and he also has ten coins in his pocket. In this case, (1) turns out to be false. In addition, it turns out that Smith did not really know that (2), even though the threefold condition of knowledge was, however, satisfied. Thus, it turns out in this case that Smith does not know P, although the three aforementioned conditions are met.

The Gettier’s second practical case is the following. Consider the following proposition:

(3) Jones owns a Ford

Moreover, Smith knows that Jones has always had a Ford and that the latter has recently made a trip with him. Smith thus possesses crucial elements in favor of (3). In addition, it turns out that Smith has another friend, Brown, about whom he ignores, however, a number of things. Let us consider now the three propositions:

(4) either Jones owns a Ford, or Brown is in Boston
(5) either Jones owns a Ford, or Brown is in Barcelona
(6) either Jones owns a Ford, or Brown is in Brest-Litovsk

At this stage, it proves that each of these three propositions represents a logical consequence of (3). However, we can consider that Smith knows that (4), (5) and (6), because each of these propositions is true, and secondly Smith has for each of them a justified belief. But let us suppose now that Brown does not have a Ford, but uses a rental Chrysler, and that Brown, unknown to Smith, is located, in Barcelona. In this case, it proves that Smith does not really know that (5) is true, even though the three conditions of knowledge relating to (5) are once again satisfied.

The two preceding examples, Gettier concludes, demonstrate that the aforementioned threefold condition does not constitute a sufficient condition to ensure that S knows that P. However, a number of answers have been given to Gettier’s problem. One of these responses stresses that the justification that is present in the two cases mentioned by Gettier proves insufficient. For shouldn’t knowledge be motivated by actual evidence, not by what merely constitutes a weak justification? Smith, in effect, grounds his belief on the mere fact that the president of the company assured him that Jones would get the job. However, at this step, Smith has the certainty of the President’s declarations but lacks evidence of the corresponding facts, for couldn’t the President change his mind later? Therefore, we can assume that the justification step proves insufficient. In this sense, the two examples described by Gettier are characterized by a weak justification, whereas a strong justification is clearly needed. According to this type of objection, as we can see, the threefold condition of knowledge remains acceptable, but the condition of justification must be replaced by a stronger condition, which corresponds to evidence. In this context, genuine knowledge corresponds to a true and proved belief. However, such type of response to the Gettier problem proves insufficient to dissipate its consequences, because this type of response has the disadvantage of being too radical. Its application leads us, then, to consider that many situations of everyday life where we lack such definitive and absolute proof do not lead to genuine knowledge.

Several proposed solutions to the Gettier problem are intended to prevent the emergence of the cases described by Gettier by adding a supplementary condition. Solutions of this type are based on the fact that knowledge is the result of a true and justified belief, but also that this threefold condition cannot be obtained accidentally. The latter condition is intended to prevent the cases described by Gettier from occurring. But such an approach has not proved entirely satisfactory, because the very definition of accidental conditions has proved problematic. In effect, in some cases, the accidental occurrence of the threefold aforementioned condition does not lead to true knowledge, whereas in other circumstances, the accidental occurrence of this threefold condition generates genuine knowledge.

 

38. Frege’s Puzzle about Identity Statements

chap38The puzzle about identity statements was described by Gottlob Frege in his essay On Sense and Reference, published in 1892. This puzzle is as follows. Let us consider first a statement such as “the morning star is the evening star.” In this case, it turns out that the expressions “the morning star” and “the evening star” refer to one and the same object: the planet Venus. We can see that the structure of the proposition “the morning star is the evening star” has the form “A” = “B”. In a general way, propositions that have such a structure are true if and only if “A” and “B” refer to the same object. This can also be formulated in terms of numbers. If we consider the terms “160 + 10” and “153 + 17”, it proves that these two expressions refer to the same natural number, which is 170. Frege endeavored thus to describe a theory of truth for propositions with the structure “A” = “B” by defining the conditions under which such propositions turn out to be true. However, Frege noticed that a problem emerged with this type of analysis. It occurred indeed that the conditions under which a proposition of the form “A” = “B” proved true (the truth conditions) were identical to those under which a proposition of the form “A” = “A” was also true. But a proposition of the form “A” = “A” as “the morning star is the morning star” proves to be, from a semantic point of view, very different from a proposition such as “the morning star is the evening star.” The conclusion follows that the truth conditions are identical for semantically very different propositions of the form “A” = “B” or “A” = “A”.

The reasoning that leads to Frege’s puzzle about identity statements can be formalized as follows:

(1) the Morning Star is the Evening Star premise
(2) “the Morning Star” and “the Evening Star” refer to the planet Venus definition
(3) (1) is true from (1),(2)
(4) the Morning Star is the Morning Star identity
(5) “the Morning Star” refer to the planet Venus definition
(6) (4) is true from (4),(5)
(7) (1) has the structure “A” = “B” from (1)
(8) (4) has the structure “A” = “A” from (4)
(9) a proposition which has the structure “A” = “B” is true if and only if “A” and “B” refer to the same object generalization
(10) a proposition which has the structure “A” = “A” is true if and only if “A” and “A” refer to the same object from (9)
(11) the truth conditions of a proposition which has the structure “A” = “B” and of a proposition which has the structure “A” = “A” are identical from (9),(10)
(12) from a semantic viewpoint, a proposition which has the structure “A” = “B” is very different from a proposition which has the structure “A” = “A” from (1),(4)
(13) ∴ the truth conditions of two propositions which are semantically very different are identical from (11),(12)

 

 

39. The Paradox of Analysis

chap39The paradox of analysis results from the work of George Edward Moore. The paradox is based on a methodological approach which consists in analyzing a given concept. Let us call α  such a concept. The analysis of this concept, then, presents the form: α = E. Here, α is the concept which is analyzed (the analysandum) whereas E is an expression (the analysans)—more or less complex—that defines and describes the semantic content of α. The paradox emerges when we consider the two possibilities that arise: (i) either the analysans accurately describes the content of the concept α or (ii) the analysans does not accurately describe the content of the concept α. In the first case, it follows that the analysis is trivial, and therefore has no interest. In the second case, it turns out that the analysans does not accurately describe the content of the concept α and therefore, the analysis is false. Thus, the analysans is either trivial or false. In both cases, the analysis is useless. However, this is in contradiction with the data resulting from our pretheoretical intuition, according to which the analysis of a given concept proves most often useful.

The reasoning corresponding to the paradox of analysis can be detailed as follows:

(1) either the analysans describes accurately the content of concept α, or the analysans does not describe accurately its content dichotomy
(2) if the analysans describes accurately the content of concept α hypothesis 1
(3) then the analysis is trivial from (2)
(4) if the analysans does not describe accurately the content of concept α hypothesis 2
(5) then the analysis is inexact from (4)
(6) the analysis of concept α is either trivial or inexact from (3),(5)
(7) ∴ the analysis of concept α is useless from (6)

A notable solution to the paradox of analysis that results from the ideas put forward by Gottlob Frege in his essay On Sense and Reference is as follows. This solution undermines the transition from step (2) to step (3), which leads to the conclusion that the analysis is trivial if the analysans accurately describes the content of the concept α. Frege distinguished two types of semantic content: first, the meaning, and secondly, the reference. In this context, it turns out that if the concept α and the analysans have the same reference, then the resulting analysis is exact. However, this does not preclude the analysans from having a different meaning of the concept α. And under such conditions, the analysis proves not trivial but useful, since it provides new information.

 

40. Heraclitus’ river puzzle

chap40The Problem of Heraclitus’ River originates from the Fragments of the work of Heraclitus that have reached us. Heraclitus affirms that it is not possible to cross the same river twice, because the waters that constitute the latter are constantly renewed. The underlying idea in this last problem is that between two crossings, the river has undergone changes such that it is no longer exactly the same river.

We can formulate more accurately the problem of Heraclitus’ river as follows:

(1) I cross the river r at time T1 premise
(2) I cross the river r at time T2 (with T1 < T2) premise
(3) the river r has undergone changes between T1 and T2 premise
(4) ∴ the river r at time T1 is different from the river at time T2 from (3)
(5) ∴ at time T2 I cross a river which is different from the river r that I crossed at time T1 from (1),(2),(4)

An objection that has been raised with regard to the problem of Heraclitus’ river is that the changes undergone by the river between T1 and T2 are not significant enough to transform the river in T1 into a different river in T2. According to this standpoint, the changes undergone by the river are minor and do not affect its identity as a river. This type of objection, as we can see, has the effect of blocking the passage from step (3) to step (4). It thus emphasizes the persistence of the identity of an object o through time, despite the changes of a secondary nature that are undergone by this object, for according to this standpoint, steps (3) and (4) should be replaced by:

(3*) the river r has undergone minor changes between T1 and in T2 premise
(4*) ∴ the river r at time T1 is no different from the river r at time T2 from (3*)

However, such an objection is insufficient to permanently resolve the problem of Heraclitus’ river. Indeed, the underlying distinction between substantive and non-substantive changes that can affect a given object proves difficult to apply. Thus, between two given time positions, the river water is completely new, so that the elements that compose the latter have been entirely changed. It is therefore difficult to consider that the totality of elements that compose an object at a given time are not essential to it.

 

Conclusion

conclusionThe aforementioned paradoxes, philosophical problems and arguments represent only a selection of the many issues addressed in the rich literature constituted by contemporary analytical philosophy, for it consists of a lively and evolving field, where every year, new arguments are created, and then exposed and discussed. As we have seen, millennial unresolved paradoxes coexist with philosophical arguments that have just been described.

On the other hand, the presentation of these contemporary problems of analytic philosophy is primarily designed to allow a better understanding of the analytic style with regard to the so-called continental philosophy, as both styles of philosophizing, as we have seen, are deserving of respect. The goal here was simply to introduce an often overlooked facet of contemporary philosophy. Some will immediately feel a natural affinity with the analytic style. Others will prefer the “continental” style. All, however, I hope, will benefit from a better understanding of the diversity of philosophical styles.

Of the statement of the foregoing paradoxes and arguments, it is also apparent, I believe, that human reasoning is perfectible and surprisingly vulnerable to error. For the pitfalls of reasoning that have been described, the contradictions to which paradoxes easily lead us indicate that the thinking of all of us proves to be vulnerable. It is quite fascinating to see how we are all prone to think in a way that leads to paradoxical conclusions, leaving us with the contradictions that result from reasoning that seemed quite valid. The reasoning that leads to error is common to us, and again, if a solution had to be provided to a particular problem or paradox, it should prove to be consensual to be validated. As we can see, such a field has a significant practical importance. It consists here of improving and enhancing the way of thinking that we share in common. In this context, the discovery of a consensual solution to a specific argument or an unresolved paradox should, then, be of benefit to everyone.

 

Acknowledgements

I thank Francis Antona, Christian Carayon and Eric Odin for very useful comments on earlier drafts.

 

Credits

The 3D illustrations were made with Blender software (http://www.blender.org/). Other illustrations are from Wikimedia commons.

 

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Oppy, G. (1990) On Rescher on Pascal’s Wager, International Journal for Philosophy of Religion, 30, 159-68

Oppy, G. (1995) Ontological Arguments and Belief in God, New York: Cambridge University Press

Oppy, G. (2002) Ontological Arguments, The Stanford Encyclopedia of Philosophy (Summer 2002 Edition), E. N. Zalta (ed.), http://plato.stanford.edu/archives/sum2002/entries/ontological-arguments/

Parfit, D. (1984) Reasons and Persons, Oxford: Oxford University Press

Pascal, B. (1958) Pascal’s Pensées, translation by W. F. Trotter, E. P. Dutton & Co, New York

Pettit, P. & Sugden, R. (1989) The Backward Induction Paradox, Journal of Philosophy, 86, 169-182

Plantinga, A. (1967) God and Other Minds—A Study of Rational Justification of Belief in God, Ithica, New York: Cornell University Press

Plantinga, A. (1974) The Nature of Necessity, Oxford: Oxford University Press

Plato, The Republic, translated by Benjamin Jowett, Vintage, 1991

Poundstone, W. (1988) Labyrinths of Reason, New York: Anchor Press/Doubleday

Putnam, H. (1975) The Meaning of ‘Meaning’, in Mind, Language, and Reality, Cambridge: Cambridge University Press, 215-271

Putnam, H. (1982) Reason, Truth and History, Cambridge: Cambridge University Press

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

Rosenthal, D.M. (2002) Moore’s paradox and Crimmins’s case, Analysis, 62, 167­171

Ross, H. (1998) Big Bang Refined by Fire, Pasadena, CA: Reasons to Believe

Russell, B. (1923) Vagueness, The Australian Journal of Philosophy and Psychology, 1, 84-92

Russell, B. (1980) Correspondence with Frege, dans Philosophical and Mathematical Correspondence, by Gottlob Frege, traduction par Hans Kaal, Chicago: University of Chicago Press

Sainsbury, M. (1995) (2nd ed.) Paradoxes, Cambridge: Cambridge University Press

Saint Anselme (1077), Proslogion, trad. B. Pautrat, éd. Garnier Flammarion, 1993

Salmon, N. (1986) Frege’s Puzzle, Cambridge MA: MIT Press

Salmon, W. C. (ed.) (1970) Zeno’s Paradoxes, Indianapolis & New York: Bobbs-Merrill

Savage, W. (1967) The paradox of the stone, Philosophical Review, 76, 74-79

Saygin, A.P., Cicekli, I. & Akman, V. (2000), Turing Test: 50 Years Later, Minds and Machines, 10(4), 463-518

Schrader, D. (1979) A solution to the stone paradox, Synthese, 42, 255-264

Scriven, M. (1951) Paradoxical announcements, Mind, 60, 403-407

Searle, J. (1980) Minds, brains, and programs, Behavioral and Brain Sciences, 3, 417-424

Searle, J. (1984) Minds, Brains, and Science, Cambridge: Harvard University Press

Searle, J. (1990) Is the Brain’s Mind a Computer Program?, Scientific American, 262, 26-31

Smith, J. W. (1984) The surprise examination on the paradox of the heap, Philosophical Papers, 13, 43-56

Smith, Q. & Oaklander, L. N. (1994) The New Theory of Time, Yale University Press, New Haven

Sorensen, R. A. (1982) Recalcitrant versions of the prediction paradox, Australasian Journal of Philosophy, 69, 355-362

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

Sorensen, R.A. (2000) Moore’s problem with iterated belief, Philosophical Quarterly, 50, 2843

Sorensen, R.A. (2003) A Brief History of the Paradox, Oxford: Oxford University Press

Sowers, G. F. (2002) The Demise of the Doomsday Argument, Mind, 111, 37-45

Stroud, B. (1989) The Significance of Philosophical Scepticism, Oxford: Oxford University Press

Sullivan, P. (2003) A note on incompleteness and heterologicality, Analysis, 63, 32-38

Suppes, P. (1972) Axiomatic Set Theory, New York: Dover

Swinburne, R. (1968) The Argument from Design, Philosophy, 43, 199-211

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Weatherson, B. (2004) Are You a Sim?, Philosophical Quarterly, 53, 425-431

Wiggins, D. (1982) Heraclitus’ Conceptions of Flux, Fire, and Material Persistence, in Schofield, M. & Nussbaum, M., eds., Language and Logos, Cambridge University Press

Williams, J.N. (1979) Moore’s paradox: One or Two, Analysis, 39, 141-142

Williams, J.N. (1999) Wittgensteinian accounts of Moorean absurdity, Philosophical Studies, 92, 283-306

Williamson, T. (1994) Vagueness. London: Routledge

Zadeh, L. (1975) Fuzzy logic and approximate reasoning, Synthese, 30, 407-428

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Internet sites

Adam Elga, http://www.princeton.edu/~adame/, author site

Brian Weatherson, http://brian.weatherson.net/papers.html, author site

David Chalmers, http://consc.net/chalmers/, author site and philosophical resources

Eliott Sober, http://philosophy.wisc.edu/sober/papers.htm, author site

Graham Oppy, www.arts.monash.edu.au/phil/department/Oppy/grahampapers.html, author site

Institut Jean Nicod, http://jeannicod.ccsd.cnrs.fr/, web site with the articles of the Jean Nicod Institute’s researchers

Nicholas J. J. Smith, www.personal.usyd.edu.au/~njjsmith/papers/, author site

Pascal Engel, http://www.unige.ch/lettres/philo/enseignants/pe/, author site

Roy Sorensen, http://www.dartmouth.edu/~rasoren/papers/papers.html, author site

Stanford Encyclopedia of Philosophy, edited by E. N. Zalta, http://plato.stanford.edu/, philosophical encyclopedia

The Anthropic Principle, by Nick Bostrom, www.anthropic-principle.com, web site related to the Doomsday argument and related arguments

The Internet Encyclopedia of Philosophy, http://www.utm.edu/research/iep/, philosophical encyclopedia

The Simulation Argument, by Nick Bostrom, www.simulation-argument.com, web site related to the Simulation argument

Paul Franceschi, http://www.paulfranceschi.com, the author’s site

 

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A Polar Concept Argument for the Existence of Abstracta

Preprint. In this paper, I present a polar concept argument for the existence of abstract objects. After recalling the fundamentals of the debate about the existence of abstracta, I present in a detailed way the argument for the existence of abstracta. I offer two different variations of the argument: one, deductive and the other, inductive. The argument rests primarily on the fact that our universe is well-balanced. This well-balanced property results from the fact that all instantiable polar dualities are instantiated. Hence, the abstract pole of the abstract/concrete duality must also be exemplified. Lastly, I review several objections that can be raised against the present argument.

 

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A Polar Concept Argument for the Existence of Abstracta

 

There are several famous problems about abstract entities. One of them consists of whether there exist any abstract objects. A second issue is concerned with the definition of which sorts of entities are genuinely abstract. A third issue relates to whether the abstract/concrete duality is exhaustive or not. The purpose of this paper is to address the first of these issues and to describe a polar concept argument that entails the existence of abstracta. Before stating the argument in detail and reviewing several objections that can be raised against it, it is worth recalling preliminarily the fundamentals of the debate about the existence of abstracta.

1. The debate about the existence of abstracta

Let us recall preliminarily the main lines of the issue of whether there exist abstract objects1. This latter problem rests primarily on the abstract/concrete distinction. Uncontroversially, the following objects are considered as abstracta: the natural numbers, the cosine function, the Pythagorean theorem. For this reason, I shall only be concerned in what follows with paradigm abstracta, i.e. natural numbers, setting aside other sorts of entities whose status remains controversial. On the other hand, an instance of a jay or of an oak-tree, the mountain in front of me, the sun, our galaxy, are paradigm concrete objects. Uncontroversially, concreta are considered as existent objects. But at this stage, agreement stops. In effect, by contrast, the mere existence of abstracta is at issue. Do abstract objects truly exist? There are two main philosophical answers to this question. On the one hand, some philosophers simply deny the existence of abstracta. According to the corresponding line of thought, only concrete objects exist in our universe, and abstract concepts are a mere product of our brain circuitry. Thus, natural numbers, the sine function, etc., which are considered as paradigm cases of abstracta, exist only in our mind. The view that denies the existence of abstract objets is nominalism.

On the other hand, according to a line of thought originating from Plato, abstract objects do truly exist. According to platonism, abstracta have an existence of their own, which parallels the existence of concreta. In addition, abstracta are classically considered as having no spatiotemporal position, in contrast to concreta which possess a given position in space and time.

The main argument for abstracta is the Quine-Putnam indispensability argument.2 According to its first premise, we should be committed to the existence of all entities that are indispensable to our best scientific theories. Given that natural numbers are indispensable to these scientific theories, it follows that we should be committed to the existence of natural numbers. The indispensability argument is controversial and has notably led to important objections raised by Hartry Field (1980), Penelope Maddy (1992) and Elliott Sober (1993). Without entering into the details of these criticisms, I will offer here a different line of argument for the existence of abstract objects.

2. The polar concept argument for the existence of abstracta

In what follows, I borrow the expression ‘polar concept argument’ from the characterisation of Ryle’s argument against scepticism3 (1960), put forth by Anthony Grayling (1995, pp. 49-50). Grayling describes Ryle’s argument in the following terms:

The point can be simply illustrated by a consideration of Gilbert Ryle’s attempt to refute the argument from error by a ‘polar concept’ argument. There cannot be counterfeit coins, Ryle observes, unless there are genuine ones, nor crooked paths unless there are straight paths, nor tall men unless there are short men. Many concepts come in pairs which are polar opposites of one another, and these conceptual polarities are such that one cannot grasp either pole unless one grasps its opposite at the same time. Now error and ‘getting it right’ are conceptual polarities. If one understands the concept of error, one understands the concept of getting it right. But to understand this latter concept is to be able to apply it. So our very grasp of the concept of error implies that we sometimes get things right.

Grayling characterises thus as a polar concept argument the argument used by Gilbert Ryle to refute the argument from error, which arises in the context of the debate against scepticism. The argument from error puts in parallel two types of situations. On the one hand, it appears that we often mistakenly have some knowledge that comes from perceptual experience. But these latter situations, the sceptic argues, are indistinguishable from present situations in which we have some knowledge that stem from our current perceptual experiences. Therefore, concludes the sceptic, our present knowledge could also be mistaken. According to Ryle, the argument from error is inconclusive, since ‘getting it right’ and ‘error’ are opposites and originate from the same duality. Hence, Ryle pursues, whenever one grasps the concept of ‘error’, one also grasps the opposite concept of ‘getting it right’. A further step states that to understand the corresponding concept is tantamount to being able to apply it in practice. And this finally undermines the conclusion of the argument from error. I shall not discuss here whether Ryle’s argument is valid or not. Rather, my concern will be with showing that a polar concept argument along the same lines can be used in support of the existence of abstracta.

The polar concept argument for the existence of abstracta can be sketched informally as follows. Begin with the fact that our universe is well-balanced. A rough analysis reveals that this well-balanced property is exemplified many times. Consider for example the existence, at a macroscopic level, of very large objects such as stars, supernovae or galaxies. Contrast now with the existence, at a microscopic level, of very small objects such as atoms or molecules. This illustrates how our universe is well-balanced with regard to the large/small duality. Now it appears that numerous other polar opposites are also instantiated in our universe. Consider then how both poles of the hot/cold duality are exemplified. It suffices to consider the existence, on the one hand, of hot objects such as stars, and on the other hand, of cold objects such as brown dwarfs, dead stars or asteroids. This shows that our universe is also well-balanced with regard to the hot/cold duality. Now consider how many dualities such as attractive/repulsive, static/dynamic, bright/dark, positive/negative, neutral/charged, visible/invisible, etc., are also instantiated. At this step, it is worth considering the case of the abstract/concrete duality. There is uncontroversial evidence that concrete objects do exist in our universe. Now it follows that the necessary well-balance of our universe with regard to the abstract/concrete duality also entails the necessary existence of abstract objects.

At this step, for the sake of accuracy, it is worth stating a few definitions. Let us begin with polar opposites. Intuitive though it is, the notion of polar opposites needs in effect clarifying. Paradigm examples of polar opposites are positive/negative, small/large, static/dynamic, internal/external, etc. But let us provide an explicit definition. To begin with, polar opposites are polar concepts, i.e. concepts which intuitively come in pairs (let us term them A and Ā), and are such that each one is defined as the opposite of the other. For example, internal can be defined as the opposite of external, while symmetrically, external can be defined as the opposite of internal. Both poles are the contrary of one another. In a sense, there is no primitive notion and neither poles of the A/Ā duality can be regarded as the primitive notion.

Let us stress, second, that both poles of a given duality are simple qualities, in opposition to composite qualities. The distinction between simple and composite qualities can be drawn as follows. Let A1, A2 be simple qualities. Now A1 A2, A1 A2 are composite qualities. To give an example, small, static, positive are simple qualities, while small and static, small and positive, static and positive, small and static and positive are composite qualities. A more general definition is then as follows: B1, B2 being simple or composite qualities, B1 B2, B1 B2 are composite qualities. Incidentally, this also casts light on the reason why red/non-red, blue/non-blue cannot be considered as polar opposites. For example, non-red can be defined as violet indigo blue green yellow orange white black.

It is worth mentioning, third, that polar opposites are neutral concepts, i.e. neither meliorative nor pejorative. Accordingly, large, small, external, internal, concrete, abstract, etc., are neutral polar concepts, to the difference of such concepts as beautiful, ugly, courageous, which are non-neutral.

Given this definition, we are notably in a position to distinguish polar opposites from vague objects. It should be noticed first that polar opposites and vague objects share some common features. In effect, vague objects come in pairs, in the same way as polar opposites. In addition, vague concepts are classically viewed as possessing an extension and an anti-extension, which are mutually exclusive. Such a feature is also shared by polar opposites. But let us stress now the differences between the two categories. A first difference (i) consists in the fact that the extension and the anti-extension of vague concepts are not jointly exhaustive, for they admit of borderline cases (and also borderline cases of borderlines cases, etc., thus giving rise to the hierarchy of higher-order vagueness), which constitute a penumbral zone. In contrast however, polar contraries do not necessarily possess this latter feature. In effect, polar opposites can be either exhaustive or non-exhaustive. For example, the abstract/concrete duality is intuitively exhaustive, for there does not seem to exist objects that are neither abstract nor concrete. Now the same goes for the vague/precise duality: intuitively, there does not exist objects which are neither precise nor vague and pertain to an intermediate category. Hence, there exists polar opposites which are not vague, e.g. both poles of the abstract/concrete duality. Now a second difference (ii) between polar opposites and vague objects is that the former are simple qualities, while the latter consist of either simple or composite qualities. For there exists so-called multi-dimensional vague concepts, such as vehicle, machine.4 Lastly, a third difference (iii) resides in the fact that some polar opposites are inherently precise. To take an obvious example, the positive/negative duality is entirely made up of precise constituents.

Let us also distinguish, second, polar contraries from the pair consisting of a given concept and its complement. To take an example, positive/negative are polar opposites, to the difference of positive/non-positive. For non-positive includes both neutral and negative. In the same way, negative/non-negative are not genuine polar opposites, since non-negative includes both neutral and positive. However, if a given duality A/Ā is exhaustive, it follows that the polar opposite of A (respectively Ā) identifies itself with non-A (respectively non-Ā). But as we have seen, not all polar dualities are exhaustive and this entails that the polar opposite of a given concept must be distinguished, from a general viewpoint, from its complement.

On the other hand, it is also worth defining the well-balanced property. For a given object o of which one part has a property A, the well-balanced property relative to the A/Ā duality results from the fact that there also exists another part of o which has the opposite property Ā. To give an example, protons have a positive charge, while on the other hand, electrons have a negative charge. Thus, an atom of hydrogen, which includes one electron and one proton, is well-balanced with regard to the positive/negative duality, since it has both a positive and a negative charge. More generally, being well-balanced relative to our universe results from a generalisation of this latter property to all polar dualities.

At this step, we are in a position to state the present argument more explicitly:

(1)

our universe is well-balanced

premise

(2)

in our universe the following polar dualities are instantiated: large/small, positive/negative, external/internal, absolute/relative, bright/dark, static/dynamic, attractive/repulsive, visible/invisible, cold/hot, etc.

evidence

(3)

the well-balanced property relative to the A/Ā duality results from the fact, for a given object having a polar property A, of also having the opposite property Ā

definition

(4)

∴ the well-balance of our universe results from the fact that all instantiable polar opposites are instantiated

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

(5)

concrete pertains to the abstract/concrete duality

definition

(6)

concrete objects exists in our universe

evidence

(7)

∴ the concrete pole of the abstract/concrete duality is instantiated

from (6)

(8)

∴ the abstract pole of the abstract/concrete duality is necessarily instantiated

from (3),(4),(7)

(9)

∴ abstract objects exist in our universe

from (8)

The argument being stated, it is worth highlighting now some of its distinctive features. It should be pointed out first that the argument is deductive. In effect, it starts from the consideration that our universe is well-balanced and derives the conclusion that abstract objects do exist. The well-balanced property is crucial to the argument and two different types of well-balanced properties can be distinguished: (i) well-balanced relative to a given polar duality A/Ā; (ii) well-balanced relative to our universe.

It is also useful to define accurately the scope of the argument. More generally, the argument postulates that for each pole observed in our universe, there exists an opposite pole. The argument is thus based on the fact that for every object that exists and exemplifies a pole, there also exist in our universe other objects that instantiate the opposite pole. The argument postulates that there do not exist things in our universe that instantiate a pole of a given duality without also instantiating the opposite pole.

Lastly, it should be pointed out that the above argument could be stated alternatively under the form of an inductive argument. It would then be recast as follows:

(1i)

in our universe the following polar dualities are instantiated: large/small, positive/negative, external/internal, absolute/relative, bright/dark, static/dynamic, attractive/repulsive, visible/invisible, cold/hot, etc.

evidence

(2i)

the well-balanced property relative to the A/Ā duality results from the fact, for a given object having a polar property A, of also having the opposite property Ā

definition

(3i)

∴ in our universe all instantiable polar dualities are instantiated

from (1i), induction

(4i)

the well-balance of our universe results from the fact that that all instantiable polar opposites are instantiated

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

(5i)

concrete pertains to the abstract/concrete duality

definition

(6i)

concrete objects exists in our universe

evidence

(7i)

the concrete pole of the abstract/concrete duality is instantiated

from (6i)

(8i)

the abstract pole of the abstract/concrete duality is necessarily instantiated

from (2i),(4i),(7i)

(9i)

∴ abstract objects exists in our universe

from (3i),(4i),(5i)

It should be noticed that the inductive form of the argument proceeds by enumerating all instantiable polar dualities and then generalising to all polar dualities. It follows then straightforwardly by induction that the abstract/concrete duality is also instantiated.

3. Response to objections

At this stage, it is worth considering several objections that can be pressed against the present argument. Let us review, first, a line of objection that stems from the issue of whether Ryle’s argument is a sound one. Grayling (1995, p. 50) mentions in effect that a sceptic critic could object to Ryle’s polar concept argument that a same line of reasoning applied to other dualities such as ‘perfect/imperfect’, ‘mortal/immortal’, ‘finite/infinite’ would entail the existence of perfect, immortal or infinite entities. An objection along the same lines could then be raised against the present argument for abstracta. However, in the present context, the above three dualities do not deserve the same type of response. For that reason, I shall consider them in turn. Begin then with the perfect/imperfect duality. From the above, it should be apparent that the perfect/imperfect duality does not fall under the scope of the present argument. For perfect is not a simple quality. In effect, perfect can be defined as the sum of all simple positive qualities. Thus, perfect can be characterised as a complex or composite quality. But as we have seen, the scope of the present argument is restricted to simple qualities. For that reason, the existence of perfect entities is not entailed by the present argument.

Let us turn now to the mortal/immortal duality. At this step, it should be pointed out that it is not clear whether mortal can be considered as a simple quality, in the sense defined above. For that reason, I shall replace it by the temporal/atemporal duality. This latter pair is made up of two conceptual polarities that can be regarded unambiguously as simple qualities. Now it should be acknowledged that the temporal/atemporal duality also falls under the scope of the above argument. And a same line of reasoning yields the existence of atemporal entities. I shall endorse such a consequence here. In effect, the present argument is also for the existence of atemporal entities. But is there something counter-intuitive here? It seems that some objects such as numbers are obvious candidates for this definition. In this context, natural numbers can be considered consistently as both abstract and atemporal entities.

Now the same goes for the application of the present argument to the infinite/finite duality. For it should be acknowledged that infinite and finite are simple qualities in the sense defined above. Thus the argument also applies to these latter concepts and entails the existence of infinite objects. But such inference should not be very disturbing, I think, for it is much in line with our current intuitions. Just as in the previous case regarding the temporal/atemporal duality, there exist immediate candidates for the definition of infinite entities. Natural numbers, for example, straightforwardly instantiate the property of being both abstract and infinite.

Let us consider, second, another line of objection. It could be opposed to the present argument that a similar line of reasoning leads to the existence of impossible objects. Few would doubt, in effect, that we currently have a large body of evidence in favour of the existence of possible objects. Hence, according to the above argument, from the possible/impossible duality, we can infer the existence of impossible objects. But as this latter notion is self-contradictory, the objection goes, the whole enterprise is vowed to inconsistency. However, this line of objection does not undermine the force of the argument, I think. For the present argument is only concerned with instantiable objects. It begins with the observation that many objects exemplifying both poles of a given duality do exist. It pursues by inferring the existence of objects that instantiate a pole of the abstract/concrete duality. But in all cases, the present argument is only concerned with pairs of polar contraries that are compatible with existence. Perhaps, some would agree that certain objects possess the property of being impossible, contradictory or imaginary. But such inferences don’t fall under the scope of the present argument. For the dualities which are concerned with the present argument need at least to be instantiable. As a consequence, predicates such as impossible, inexistent, imaginary, contradictory, inconceivable, etc., should be discarded from the beginning. And all non-instantiable dualities (i.e. dualities which contain at least one non-instantiable pole) such as possible/impossible, existent/inexistent, coherent/incoherent, etc., are not targeted by the current argument. In addition, the same response prevails for a similar line of objection that would respectively infer the existence of inexistent, inconceivable, imaginary objects, from the existent/inexistent, conceivable/inconceivable, real/imaginary dualities.

It is worth examining, third, a different line of objection. It runs as follows. The present argument rests on the necessity of instantiating both poles of all dualities. But it could be retorted that certain poles of some dualities need not being instantiated. And such is the case, the objection runs, for the abstract pole of the abstract/concrete duality. It should be apparent that this latter objection challenges premise (4), according to which, due to the well-balance requirement of our universe, all instantiable polar dualities are exemplified. But this latter objection is not very promising, I think. For the present polar concept argument is concerned with our universe’s well-balanced requirement. And this well-balance results precisely from the instantiation of both poles of each duality. Consider the case of the bright/dark duality. Imagine our universe containing only dark objects, with all bright objects missing. Would we expect to find ourselves in such a universe? No. For the emergence of carbon-based life would be impossible in such one-sided (from the viewpoint of the dark/bright duality) universe. Or consider alternatively the situation if all objects in our universe were static and no objects were dynamic. Or else imagine if our present universe only contained cold objects, and were entirely devoid of hot ones. All such universes would be very unfriendly, to say the least. Now the same applies to the abstract pole of the abstract/concrete duality. Perhaps it could be helpful to recall, at this step, one major premise of the fine-tuning argument. From the fact that many cosmological constants are fine-tuned for the emergence of carbon-based life, the fine-tuning argument derives the conclusion that this latter feature of our universe is non-random and due to the intention of its Creator. Now setting aside the controversial conclusion of the fine-tuning argument, it appears that the premise according to which the cosmological constants are fine-tuned for further emergence of carbon-based life can be replaced by the two following steps:

(1f)

the cosmological constants of our universe are fine-tuned for the emergence of carbon-based life

(2f)

∴ the cosmological constants of our universe are fine-tuned for the instantiation of the following dualities: large/small, positive/negative, external/internal, bright/dark, static/dynamic, visible/invisible, cold/hot, etc.

from (1f)

At this point, it is worth noting that the assertion according to which our universe is well-balanced is even weaker than the uncontroversial premise (1f) of the fine-tuning argument. For consider the following instance of anthropic coincidence related to the gravitational force constant which is part of (1f):5

(3f)

if the gravitational force constant had been larger then stars would be have been too hot to allow for carbon-based life chemistry6

(4f)

if the gravitational force constant had been smaller then stars would have been be too cool to permit carbon-based life chemistry

Now it appears that these two propositions can be recast as follows:

(5f)

if the gravitational force constant had been larger then the cold pole of the cold/hot duality would have not been instantiated

(6f)

if the gravitational force constant had been smaller then the hot pole of the cold/hot duality would have not been instantiated

To take another example:

(7f)

if the velocity of light had been faster then stars would have been be too luminous for life support

(8f)

if the velocity of light had been slower then stars would have been insufficiently luminous for life support

can be restated into the weaker:

(9f)

if the velocity of light had been faster then the dark pole of the bright/dark duality would have not been exemplified

(10f)

if the velocity of light had been slower then the bright pole of the bright/dark duality would have not been exemplified

More generally, every anthropic coincidence can be restated into the weaker couple of propositions:

(11f)

if the <cosmological constant> had been larger then the A pole of the A/Ā duality would have not been instantiated

(12f)

if the <cosmological constant> had been smaller then the Ā pole of the A/Ā duality would have not been instantiated

At this step, it should be apparent that challenging (2f) also implies being committed to doubt (1f), while on the other hand it is a widely accepted premise of the fine-tuning argument.

The above argument could be attacked, fourth, on the grounds that is not deductive, but rather inductive. According to this line of objection, the present argument is a disguised inductive argument. If the argument were inductive instead of deductive, it would be probabilistic and as such, its impact would be weaker than in its original deductive presentation. As mentioned above, it should be acknowledged that the argument can be presented alternatively as an inductive argument. The inductive form of the argument begins with an enumeration of all exemplified polar dualities. From this, it derives a generalisation to all polar dualities. The conclusion that the abstract/concrete duality and in particular its abstract pole is also instantiated ensues. If the above argument is to be considered as essentially inductive, this has the effect of weakening the argument by making it inductive rather than deductive. However, the present argument is not intended to count as a proof yielding absolute certainty. Then choose whatever variation – deductive or inductive – of the argument you prefer. In either case, the essence of the argument remains in force.

It would also be tempting to challenge, fifth, premise (1), namely the fact that our universe is well-balanced. But such an objection is not very promising, I think. In effect, our universe is about 14 billion years old. How could our universe have lasted so long if it hadn’t been well-balanced? Considering now its spatial extension, a question of the same nature arises. For our universe extends billions of light years in any directions. How could our universe have occupied such huge spatial region without being well-balanced? And again: How could our universe contain so much objects such as neutrons, monkeys, stars, galaxies, etc., and a total number of atoms amounting to 1080, without being well-balanced?

Another line of objection that could be pressed, sixth, against the present argument is that it is simply a generalisation of Ryle’s argument. Although both arguments share a common ground, I shall stress that Ryle’s argument is slightly differently motivated. In effect, the key concept in the current polar concept argument is the well-balanced property. A key premise is in effect that being well-balanced is a prominent feature of our universe. And this last premise is reinforced by the additional premise based on the empirical fact that some properties of our universe currently instantiate this well-balanced property. Thus, the present argument is not entirely a priori as could be characterised Ryle’s argument. The present argument incorporates in effect some empirical features of our universe. In addition, I should be stressed that Ryle’s argument contrasts ‘error’ and ‘getting it right’. But it worth emphasizing that ‘error’ has a pejorative connotation and ‘getting it right’ reveals a meliorative nuance. Hence, ‘error’ is a negative concept and ‘getting it right’ is a positive one. By contrast, the current argument is only concerned with neutral concepts and pairs of neutral opposites. Consequently, it is worth stressing that the present argument would be inapplicable to the ‘error/getting it right’ pair of concepts.

In conclusion, it is worth stating accurately the scope of the above argument. It is not designed in effect to serve as a proof of the existence of abstracta.7 For given our current high standards, it should be acknowledged that it does not meet our present criteria of proof. Rather, the present argument aims at reinforcing an initial credence that abstract objects could exist. The above argument is simply designed to increase our a priori belief about the existence of abstracta. As such, it is consistent with the Quine-Putnam indispensability argument. It is also consistent with recent trends in cosmology and in particular with the level IV of the kind of multiverse described in Tegmark (2003). In this context, the purpose of the present argument for abstracta is to provide some additional grounds in support of the hypothesis that abstract objects do exist.

References

Colyvan, M. (2001) “Indispensability Arguments in the Philosophy of Mathematics”, The Stanford Encyclopedia of Philosophy (Fall 2001 Edition), E. Zalta (ed.), http://plato.stanford.edu/archives/fall2001/entries/mathphil-indis.

Field, H. (1980) Science Without Numbers: A Defence of Nominalism, Oxford: Blackwell.

Grayling, A. C. (1995) “Scepticism”, Philosophy A guide through the subject, Grayling, G. (ed.), Oxford: Oxford University Press.

Lowe, E. J. (1995) “The Metaphysics of Abstract Objects”, Journal of Philosophy 92, 509-24.

Maddy, P. (1992) “Indispensability and Practice”, Journal of Philosophy, 89-6, 275-289.

Rosen, G. (2001) “Abstract Objects”, The Stanford Encyclopedia of Philosophy (Fall 2001 Edition), E. Zalta (ed.), http://plato.stanford.edu/archives/fall2001/entries/abstract-objects.

Ross, H. (1995) The Creator and the Cosmos, Colorado: Navpress Colorado Springs.

Ryle, G. (1960) Dilemmas, Cambridge: Cambridge University Press.

Soames, S. (1999) Understanding Truth, New York, Oxford: Oxford University Press.

Sober, E. (1993) “Mathematics and Indispensability”, Philosophical Review, 102-1, 35-57.

Tegmark, M. (2003) “Parallel Universes”, to appear in Science and Ultimate Reality: From Quantum to Cosmos, J.D. Barrow, P.C.W. Davies, & C.L. Harper eds., Cambridge: Cambridge University Press (2003), at arXiv:astro-ph/0302131 v1 7 Feb 2003.

1 E. J. Lowe (1995) distinguishes three different conceptions of abstract objects. In what follows, my concern will be with what he terms abstract1 objects, i.e. in opposition to concrete objects.

2 For a survey of the indispensability argument, see Colyvan (2001).

3 Cf. Ryle (1960, pp. 94-95): “A country which had no coinage would offer no scope to counterfeiters. There would be nothing for them to manufacture or pass counterfeits of. They could, if they wished, manufacture and give away decorated disks of brass or lead, which the public might be pleased to get. But these would not be false coins. There can be false coins only where there are coins made of the proper materials by the proper authorities. In a country where there is a coinage, false coins can be manufactured and passed; and the counterfeiting might be so efficient that an ordinary citizen, unable to tell which were false and which were genuine coins, might become suspicious of the genuineness of any particular coin that he received. But however general his suspicions might be, there remains one proposition which he cannot entertain, the proposition, namely, that it is possible that all coins are counterfeits. For there must be an answer to the question `Counterfeits of what?'”.

4 From Soames (1999, p. 217).

5 From Ross (1999).

6 Paraphrasing Ross.

7 This raises the interesting question of what could count (if any) as a proof of the existence of abstracta. Or alternatively put: Is the existence of abstracta a testable hypothesis?

 

 

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

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

 

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A Third Route to the Doomsday Argument

 

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

I. The Doomsday Argument and the Carter-Leslie model

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

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

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

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

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

(H1few)

the urn contains 10 balls

(H2many)

the urn contains 1000 balls

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

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

(H3few)

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

(H4many)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

IV. Construction of the preliminary space of solutions

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

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

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

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

Case

Model

T0

T2

P

Nature of the scenario

1

synchronic two-urn case

1/2

terrifying

2

diachronic two-urn case

1/2

reassuring

Figure 1.

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

V. The reference class problem

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

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

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

A. The two-color diachronic two-urn case

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

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

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

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

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

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

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

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

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

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

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

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

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

 

Case

T0

T2

P

1

1a

1/4

1b

1/4

2

2a

1/4

2b

1/4

 

Figure 2.

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

 

Case

T0

T2

P

Nature of the scenario

P

1

1a

1/4

terrifying

1/4

1b

1/4

reassuring

2

2a

1/4

reassuring

3/4

2b

1/4

reassuring

 

Figure 3.

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

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

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

REFERENCES

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

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

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

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

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

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

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

———. 1999. Comment l’urne de Carter et Leslie se déverse dans celle de Hempel. Canadian Journal of Philosophy 29: 139-156, English translation under the title The Doomsday Argument and Hempel’s Problem. http://cogprints.org/2172/.

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

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

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

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

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

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

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

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

ii Cf. Korb & Oliver (1998).

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

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

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

vi Cf. (1997: 251).

vii Cf. (2002: 39).

viii I borrow this terminology from Chambers (2001).

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

x Cf. Sowers (2002: 40).

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

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

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

xiv Cf. 1996: 260-261.

xv Cf. Leslie (1996: 259).

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

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

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

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This site presents my work in philosophy. It contains published articles, preprints , as well as books. The texts relate to analytic philosophy, semiotics, the study of concepts, cognition and psycho-pathological philosophy.

My works are mainly in analytic philosophy and consist of proposed solutions to some philosophical paradoxes : the Doomsday argument, Hempel’s paradox, Goodman ‘s paradox, the surprise examination paradox, the Sleeping Beauty problem, but also the Black-Leslie paradox of the spheres, etc.. A conceptual tool, the n-universes, which are useful for the study of philosophical problems is also presented.

There are also texts on semiotics and the study of concepts. These texts are based on a specific conceptual tool : the matrices of concepts. Recent applications to the dialectical plan, to  paradigm analysis of a corpus of proverbs , to the analysis of the love-hate indifference triplet of concepts are also presented.

Finally, several texts relate to cognition and cognitive distortions. Additions to the theory of cognitive distortions are exposed, and their applications in the field of psycho-pathological philosophy.

The work involves some fields that are apparently very different. However, all texts are underpinned by a particular philosophical doctrine , which can be defined as “dialectical contextualism”. And this doctrine has applications in analytical philosophy, semiotics and psycho-pathological philosophy. Its role as a methodological tool aimed at solving philosophical paradoxes is notably explained in more detail in my “Elements of dialectical contextualism“.

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November 2014: The English translation of a paper published in french (under the title “Traitement cognitif différentiel des délires polythématiques et du trouble anxieux généralisé”) in the Journal de Thérapie Comportementale et Cognitive, 2011, vol. 21-4, pp. 121-125, is available online under the title “Differential Cognitive Treatment of Polythematic Delusions and Generalised Anxiety Disorder“.

 

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intro-phi-a-bookOctober 2014: my book An Introduction to Analytic Philosophy is freely online.

 

 

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June 2014: New addition : The Simulation Argument and the Reference Class Problem: the dialectical contextualist‘s standpoint on the Simulation Argument. This preprint supersedes my previous work on this topic.

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