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.
The classical version of the surprise examination paradox goes as follows: a teacher tells his students that an examination will take place on the next week, but they will not know in advance the precise date on which the examination will occur. The examination will thus occur surprisingly. The students reason then as follows. The examination cannot take place on Saturday, they think, otherwise they would know in advance that the examination would take place on Saturday and therefore it could not occur surprisingly. Thus, Saturday is eliminated. In addition, the examination can not take place on Friday, otherwise the students would know in advance that the examination would take place on Friday and so it could not occur surprisingly. Thus, Friday is also ruled out. By a similar reasoning, the students eliminate successively Thursday, Wednesday, Tuesday and Monday. Finally, every day of the week is eliminated. However, this does not preclude the examination of finally occurring by surprise, say on Wednesday. Thus, the reasoning of the students proved to be fallacious. However, such reasoning seems intuitively valid.
The paradox lies here in the fact the students’ reasoning is apparently valid, whereas it finally proves inconsistent with the facts, i.e. that the examination can truly occur by surprise, as initially announced by the professor.
Franceschi, P., “Éléments d’un contextualisme dialectique” (in english), in Liber Amicorum Pascal Engel, J. Dutant, G. Fassio & A. Meylan (éd.), Université de Genève, 2014, pp. 581-608.
The Sorite paradox (or heap paradox) is one of the oldest and most important paradoxes known. Its origin is attributed to Eubulides of Miletus, the ancient Greek philosopher to whom we also owe the Liar’s paradox. The paradox can be informally described as follows. First of all, it is commonly accepted that a collection of 100,000 grains of sand is a heap. Furthermore, it appears that if a set with a given number of grains of sand is a heap, then a set with one less grain of sand is also a heap. Given these premises, it follows that a set with one grain of sand is also a heap. Indeed, if a set with 100,000 grains of sand is a heap, it follows that a set with 99999 grains of sand is a heap; and the same applies to a set with 99998 grains of sand, then 99997, 99996, 99995, …, and so on, down to a single grain of sand. The paradox arises from the fact that the corresponding reasoning appears to be perfectly valid, while the conclusion that follows is unacceptable.
The different steps leading to the Sorite paradox can be detailed as follows:
(1) a set with 100,000 grains of sand is a heap (2) if a set with n grains of sand is a heap, then a set with n – 1 grains of sand is a heap (3) if a set with 100000 grains of sand is a heap, then a set with 99999 grains of sand is a heap (4) ∴ a set with 99999 grains of sand is a heap (5) if a set with 99999 grains of sand is a heap, then a set with 99998 grains of sand is a heap (6) ∴ a set with 99998 grains of sand is a heap (7) if a set with 99998 grains of sand is a heap, then a set with 99997 grains of sand is a heap (8) ∴ a set with 99997 grains of sand is a heap (9) … (10) ∴ a set with 1 grain of sand is a heap
The Liar’s paradox is one of the oldest and most profound paradoxes known. It is attributed to the Greek philosopher Eubulides of Miletus, who lived in the 4th century BC. The Liar’s paradox can be expressed very simply, as it arises directly from the consideration of the following statement: “This sentence is false”. The paradox arises from the fact that if the latter sentence is true, then it follows that it is false; but if the same sentence is false, then it is false that it is false and therefore 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 this last conclusion is paradoxical. We often denote “This sentence is false” by (λ). It is useful at this point, to describe in detail the different steps of reasoning that lead to the Liar’s paradox (the symbol ∴ denotes the conclusion here):
(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 [of (λ),(2)]
(4) then (λ) is false [from (3) (5) if (λ) is false assumption 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) here is paradoxical, since it follows that (λ) is neither true nor false, in contradiction with the principle (1) of bivalence. The problem raised by the Liar is thus the following: what is the truth value of the proposition (λ), given that it cannot be assigned, without contradiction, the truth value true or false?
The doctrine of the mean (sometimes termed ‘doctrine of the golden mean’) is a principle formulated by Aristotle in the Nicomachean Ethics, according to which a virtue is found in a position that occupies a median location between two extremes associated with it, one erring by excess, and the other by defect.
Applying the doctrine of the mean to the notion of courage, Aristotle arrives at the following definition: courage stands in a middle position between the two corresponding extremes of recklessness and cowardice.
Ross, W.D. and Brown, L. (2009) Aristotle The Nicomachean Ethics. Oxford University Press, Oxford.
The doctrine of the mean (sometimes termed ‘doctrine of the golden mean’) is a principle formulated by Aristotle in the Nicomachean Ethics, according to which a virtue is found in a position that occupies a median location between two extremes associated with it, one erring by excess, and the other by defect.
For example, courage stands in a middle position between the two corresponding extremes of recklessness and cowardice.
Ross, W.D. and Brown, L. (2009) Aristotle The Nicomachean Ethics. Oxford University Press, Oxford.
The classical version of the surprise examination paradox goes as follows: a teacher tells his students that an examination will take place on the next week, but they will not know in advance the precise date on which the examination will occur. The examination will thus occur surprisingly. The students reason then as follows. The examination cannot take place on Saturday, they think, otherwise they would know in advance that the examination would take place on Saturday and therefore it could not occur surprisingly. Thus, Saturday is eliminated. In addition, the examination can not take place on Friday, otherwise the students would know in advance that the examination would take place on Friday and so it could not occur surprisingly. Thus, Friday is also ruled out. By a similar reasoning, the students eliminate successively Thursday, Wednesday, Tuesday and Monday. Finally, every day of the week is eliminated. However, this does not preclude the examination of finally occurring by surprise, say on Wednesday. Thus, the reasoning of the students proved to be fallacious. However, such reasoning seems intuitively valid. The paradox lies here in the fact the students’ reasoning is apparently valid, whereas it finally proves inconsistent with the facts, i.e. that the examination can truly occur by surprise, as initially announced by the professor.
In order to introduce the dichotomous analysis (Franceschi 2005) that can be applied to the surprise examination paradox, it is worth considering first two variations of the paradox that turn out to be structurally different. The first variation is associated with the solution to the paradox proposed by Quine (1953). Quine considers then the student’s final conclusion that the examination can not take place surprisingly on any day of the week. According to Quine, the student’s error lies in the fact of not having envisaged from the beginning that the examination could take place on the last day. Because the fact of considering precisely that the examination will not take place on the last day finally allows the examination to occur by surprise on the last day. If the student had also considered this possibility from the beginning, he would not have been committed to the false conclusion that the examination can not occur surprisingly.
The second variation of the paradox that proves interesting in this context is the one associated with the remark made by several authors (Hall 1999, p. 661, Williamson 2000), according to which the paradox emerges clearly when the number n of units is large. Such a number is usually associated with a number n of days, but we may as well use hours, minutes, seconds, etc.. An interesting feature of the paradox is indeed that it emerges intuitively more significantly when large values of n are involved. A striking illustration of this phenomenon is thus provided by the variation of the paradox that corresponds to the following situation, described by Timothy Williamson (2000, p 139).
Advance knowledge that there will be a test, fire drill, or the like of which one will not know the time in advance is an everyday fact of social life, but one denied by a surprising proportion of early work on the Surprise Examination. Who has not waited for the telephone to ring, knowing that it will do so within a week and that one will not know a second before it rings that it will ring a second later?
The variation described by Williamson corresponds to the announcement made to someone that he/she will receive a phone call during the week, but without being able to determine in advance at what exact second the latter event will occur. This variation highlights how surprise may occur, in a quite plausible way, when the value of n is high. The unit of time considered here by Williamson is the second, in relation with a time duration that corresponds to one week. The corresponding value of n here is very high and equal to 604800 (60 x 60 x 24 x 7) seconds. However, it is not necessary to take into account a value as large of n, and a value of n equal to 365, for example, should also be well-suited.
The fact that two versions of the paradox that seem a priori quite different coexist suggests that two structurally different versions of the paradox could be inextricably intertwined within the surprise examination paradox. In fact, if we analyse the version of the paradox that leads to Quine’s solution, we find that it has a peculiarity: it is likely to occur for a value of n equal to 1. The corresponding version of the professor’s announcement is then as follows: “An examination will take place tomorrow, but you will not know in advance that this will happen and therefore it will occur surprisingly.” Quine’s analysis applies directly to this version of the paradox for which n = 1. In this case, the student’s error resides, according to Quine, in the fact of having only considered the hypothesis: (i) “the examination will take place tomorrow and I predict that it will take place.” In fact, the student should also have considered three cases: (ii) “the examination will not take place tomorrow, and I predict that it will take place” (iii) “the examination will not take place tomorrow and I do not predict that it will take place” (iv) “the examination will take place tomorrow and I do not predict that it will take place.” And the fact of having envisaged hypothesis (i), but also hypothesis (iv) which is compatible with the professor’s announcement would have prevented the student to conclude that the examination would not finally take place. Therefore, as Quine stresses, it is the fact of having only taken into account the hypothesis (i) that can be identified as the cause of the fallacious reasoning.
As we can see it, the very structure of the version of the paradox on which Quine’s solution is based has the following features: first, the non-surprise may actually occur on the last day, and second, the examination may also occur surprisingly on the last day. The same goes for the version of the paradox where n = 1: the non-surprise and the surprise may occur on day n. This allows to represent such structure of the paradox with the following matrix S[k, s] (where k denotes the day on which the examination takes place and S[k, s] denotes whether the corresponding case of non-surprise (s = 0) or surprise (s = 1) is possible (in this case, S[k, i] = 1) or not (in this case, S[k, i] = 0)):
day
non-surprise
surprise
1
1
1
2
1
1
3
1
1
4
1
1
5
1
1
6
1
1
7
1
1
Matrix structure of the version of the paradox corresponding to Quine’s solution for n = 7 (one week)
day
non-surprise
surprise
1
1
1
Matrix structure of the version of the paradox corresponding to Quine’s solution for n = 1 (one day)
Given the structure of the corresponding matrix which includes values that are equal to 1 in both cases of non-surprise and of surprise, for a given day, we shall term joint such a matrix structure.
If we examine the above-mentioned variation of the paradox set by Williamson, it presents the particularity, in contrast to the previous variation, of emerging neatly when n is large. In this context, the professor’s announcement corresponding for example to a value of n equal to 365, is the following: “An examination will take place in the coming year but the date of the examination will be a surprise.” If such a variation is analysed in terms of the matrix of non-surprise and of surprise, it turns out that this version of the paradox has the following properties: the non-surprise cannot occur on the first day while the surprise is possible on this very first day; however, on the last day, the non-surprise is possible whereas the surprise is not possible.
day
non-surprise
surprise
1
0
1
…
…
…
365
1
0
Matrix structure of the version of the paradox corresponding to Williamson’s variation for n = 365 (one year)
The foregoing allows now to identify precisely what is at fault in the student’s reasoning, when applied to this particular version of the paradox. Under these circumstances, the student would then have reasoned as follows. The surprise cannot occur on the last day but it can occur on day 1, and the non-surprise can occur on the last day, but cannot occur on the first day. These are proper instances of non-surprise and of surprise, which prove to be disjoint. However, the notion of surprise is not captured exhaustively by the extension and the anti-extension of the surprise. But such a definition is consistent with the definition of a vague predicate, which is characterized by an extension and an anti-extension which are mutually exclusive and non-exhaustive. Thus, the notion of surprise associated with a disjoint structure is that of a vague notion. Thus, the student’s error of reasoning at the origin of the fallacy lies in not having taken into account the fact that the surprise is in the case of a disjoint structure, a vague concept and includes therefore the presence of a penumbra corresponding to borderline cases between non-surprise and surprise. Hence, the mere consideration of the fact that the surprise notion is here a vague notion would have prohibited the student to conclude that S[k, 1] = 0, for all values of k, that is to say that the examination can not occur surprisingly on any day of the period.
Finally, it turns out that the analysis leads to distinguish between two independent variations with regard to the surprise examination paradox. The matrix definition of the cases of non-surprise and of surprise leads to two variations of the paradox, according to the joint/disjoint duality. In the first case, the paradox is based on a joint definition of the cases of non-surprise and of surprise. In the second case, the paradox is grounded on a disjoint definition. Both of these variations lead to a structurally different variation of the paradox and to an independent solution. When the variation of the paradox is based on a joint definition, the solution put forth by Quine applies. However, when the variation of the paradox is based on a disjoint definition, the solution is based on the prior recognition of the vague nature of the concept of surprise associated with this variation of the paradox.
As we finally see it, the dichotomous analysis of the surprise examination paradox leads to consider the class of the matrices associated with the very definition of the paradox and to distinguish whether their structure is joint or disjoint. Therefore, it follows an independent solution for each of the resulting two structurally different versions of the paradox.
Franceschi, P., “Éléments d’un contextualisme dialectique” (in english), in Liber Amicorum Pascal Engel, J. Dutant, G. Fassio & A. Meylan (éd.), Université de Genève, 2014, pp. 581-608.
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 turns out to be paradoxical.
We shall endeavour now to detail the dichotomous analysis on which is based the solution proposed in Franceschi (1999). The corresponding approach is based on finding a reference class associated with the statement of the paradox, which may be defined with the help of an A/Ā duality. If we scrutinise the concepts and categories that underlie propositions (H) and (H*), we first note that there are four categories:
ravens
black objects
non-black objects
non-black objects.
It turns out that three of the four classes do not pose any particular problem. 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.
A blue blackbird
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 vanishes.
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.
Franceschi, P., « Comment l’urne de Carter et Leslie se déverse dans celle de Carter », Canadian Journal of Philosophy, vol. 29, Mars 1999, pages 139-156, The Doomsday Argument and Hempel’s problem (English translation)
Franceschi, P., “Éléments d’un contextualisme dialectique” (in english), in Liber Amicorum Pascal Engel, J. Dutant, G. Fassio & A. Meylan (éd.), Université de Genève, 2014, p. 581-608.
A two-sided viewpoint is a viewpoint applied to an object o which takes into account both poles of a given duality. A two-sided viewpoint is opposed to a one-sided viewpoint, where only one pole of a duality is considered. On the other hand, in the two-sided viewpoint, both poles of a duality are taken into account and applied to a given object o. Thus, if the poles A and Ā of a matrix of concepts are taken into account, it will be a two-sided viewpoint.
Franceschi, P., “Éléments d’un contextualisme dialectique” (in english), in Liber Amicorum Pascal Engel, J. Dutant, G. Fassio & A. Meylan (éd.), Université de Genève, 2014, p. 581-608.
The matrix of concepts is a structure that includes six concepts, which is suitable for modeling many common concepts, such as: courage, recklessness, irresolution, eclecticism, superficiality, clemency, instability, selfishness, objectivity, frankness, brusqueness, altruism, etc. Of the six concepts in the matrix:
two are neutral: A0 and Ā0
two are positive: A+ and Ā+
two are negative: A– and Ā–
These six concepts constitute the canonical poles of the matrix.
The six concepts of the matrix are in particular relationships with each other. Thus:
the neutral concepts A0 and Ā0 are dual
the positive concept A+ and the negative concept Ā– are opposite (or contrary); similarly, the negative concept A– and the positive concept Ā+ are opposite (or contrary)
the propensity to take risks and the propensity to avoid risks are dual
audacity and cowardice are contrary, opposite; in the same way, temerity and prudence are contrary, opposite
audacity and prudence are complementary
temerity and cowardice are extreme opposites
Moreover, the three concepts located on the left of the matrix constitute a half-matrix: it is the half-matrix associated with the pole A. In the same way, the three concepts located on the right of the matrix constitute the half-matrix associated to the pole Ā.
Dichotomous analysis is a technique of analysis that tends to consider an object o from the point of view of a given duality, and to distinguish two different situations, each of which corresponds to one of the two poles of the duality considered. Examples of dualities are: Quantitative/Qualitative, Absolute/Relative, Static/Dynamic, Diachronic/Synchronic, Extension/Restriction.
The dichotomous analysis is directly linked to the very statement of the principle of dialectical indifference. Its use responds to the concern not to be vulnerable to one-sidedness bias and to demonstrate, on the contrary, a two-sided viewpoint.
Franceschi, P., “Éléments d’un contextualisme dialectique” (in english), in Liber Amicorum Pascal Engel, J. Dutant, G. Fassio & A. Meylan (éd.), Université de Genève, 2014, p. 581-608.
Dialectical monism is a philosophical doctrine that tends to consider that objects present an intrinsic unity, which results from the union of the dual poles that characterize them. The difference with monism in general lies in the fact that unity results in dialectical monism, from the union of the dual or opposite poles which are inherent to the objects.
For example, arbitrary focus is a general distortion. And mental filter is a specific cognitive distortion that consists in an instance of arbitrary focus that applies to the positive/negative duality.
In the same manner, the disqualification of one pole is the general cognitive distortion which leads to give an arbitrary priority to one of the poles of a given duality. And the disqualification of the positive is then a specific cognitive distortion that consists in the disqualification of one of the poles applied to the positive/negative duality.
The disqualification of the positive is a specific instance of the disqualification of one pole, which applies to the Positive/Negative duality. The subject grants an arbitrary priority in the negative pole of the positive/negative duality, in order to qualify the elements of a given reference class. It consists then in the fact of attributing more importance to the negative pole rather than to the positive one, in the lack of objective motivation.
For example, a specific instance of disqualification of the positive applies to the reference class including the facts and events of the subject’s life. The subject tends then to ignore positive events, by considering that they do not count, for any reason.
Mental filter (or selective abstraction) is a specific instance of arbitrary focus, which relates to the reference class of the facts of the subject’s life, and to the Positive/Negative duality. It consists in the fact of choosing one detail with a negative connotation and to focalise on it. Suchlike, the subject sees only this detail, and his/her view of reality is darkened because it is entirely tinted with this particular event.
Selective abstraction (or mental filter) is a specific instance of arbitrary focus, which relates to the reference class of the facts of the subject’s life, and to the Positive/Negative duality. It consists in the fact of choosing one detail with a negative connotation and to focalise on it. Suchlike, the subject sees only this detail, and his/her view of reality is darkened because it is entirely tinted with this particular event.
Arbitrary focus is a type of general cognitive distortion that consists in arbitrary focusing on a modality of a given duality. In the present context, this type of general cognitive distortion leads to favour one taxon in the subject’s system of taxa, by ignoring all the others. In arbitrary focusing, the taxon being discussed is present in the subject’s system of taxa, and is affected to an unique element of the reference class. There is eclipsing (in general temporary) of others taxa and other elements of the reference class, so that the subject is haunted by this specific element.
A particular instance of this type of general cognitive distortion, relates to the reference class of the facts of the subject’s life, and to the Positive/Negative duality. It is a specific cognitive distortion, which consists in focusing on a negative event of the subject’s life. It is then one of the classical cognitive distortions, defined as selective abstraction (Mental filter), which consists in the fact of choosing one detail with a negative connotation and to focalise on it. Suchlike, the subject sees only this detail, and his/her view of reality is darkened because it is entirely tinted with this particular event.
The disqualification of one pole is the general cognitive distortion which leads to grant an arbitrary priority in one of the poles of a given duality, in order to qualify the elements of a given reference class. It consists then in the fact of attributing more importance to one of the poles rather than to the other one, in the lack of objective motivation. The taxa corresponding to one of the poles of a given duality are lacking in the patient’s system of taxa. So, the subject sees things only through the prism of pole A (respectively Ā), by ignoring completely the viewpoint of the opposed pole Ā (respectively A).
Formally, the disqualification of one of the poles leads to consider only the Ei such that d[Ei] ≤ (respectivelyd[Ei] ≥ 0), by ignoring any events such thatd[Ei] > 0 (respectivelyd[Ei] < 0).
An instance of the disqualification of one of the poles consists in the disqualification of the positive. The latter can be analysed as a specific instance of the disqualification of one of the poles, which applies to the Positive/Negative duality and to the reference class including the facts and events of the subject’s life. The subject tends then to ignore positive events, by considering that they do not count, for any reason.
Another instance of the disqualification of one pole also applies to the Positive/Negative duality and to the reference class which comprises the character’s traits of the subject. This one completely ignores his/her positive character’s traits (qualities) and only directs his/her attention to his/her negative character’s traits (defects). This encourages then him/her to conclude that he/she “is worth nothing”, that he/she is “a failure”.
Dichotomous reasoning (or all-or-nothing thinking) can be defined as a general cognitive distortion which leads the subject to consider a given reference class only according to the two extreme taxa which relate to every pole of a given duality. With this type of reasoning, the subject ignores completely the presence of degrees or of intermediate steps. In his/her system of taxa, the subject has as well the two extreme taxa corresponding to poles A and Ā. The defect in that way of considering things is that facts or objects corresponding to intermediary taxa are not taken into account. So it results from it a reasoning without nuances nor gradation, which proves to be maladapted to properly apprehend the diversity of human situations. Formally, dichotomous reasoning consists in taking into account only the elements of the reference class such as|d[Ei]| = 1, ord[E1] = 1 ord[E11] = -1, by ignoring all the others.
