Monthly Archives: September 2022

Doomsday argument

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

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

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

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

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

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

Ambiguous images ▣ Arbitrary focus ▣ Bistable perception ▣ Complementarity relationship ▣ Conflict resolution ▣ Conflict resolution with matrices of concepts ▣ Conflict types relating to matrices of concepts ▣ Contrary relationship ▣ Courage ▣ Dialectical contextualism ▣ Dialectical monism ▣ Dialectical monism in Aztec philosophy ▣ Dialectical monism in Heraclitus ▣ Dichotomic analysis ▣ Dichotomic analysis applied to paradox resolution ▣ Dichotomous reasoning ▣ Disqualification of one pole ▣ Disqualification of the positive ▣ Doctrine of the mean ▣ Doomsday argument ▣ Dualities ▣ Dual poles ▣ Extreme opposition ▣ General cognitive distortions ▣ Instance of one-sidedness bias ▣ Liar paradox ▣ Matrix of concepts ▣ Maximization ▣ Mental filter ▣ Minimization ▣ Bistable cognition ▣ Omission of the neutral ▣ One-sidedness bias ▣ One-sided viewpoint ▣ Opposition relationship ▣ Principle of dialectical indifference ▣ Requalification into the other pole ▣ Reference class ▣ Reference class problem ▣ Reference class problem in philosophical paradoxes ▣ Reference class problem in the Doomsday argument ▣ Reference class problem in Hempel’s paradox ▣ Reference class problem in the surprise examination paradox ▣ Selective abstraction ▣ Sorites paradox ▣ Specific cognitive distortions ▣ Surprise examination paradox ▣ System of taxa ▣ Two-sided viewpoint ▣ Viewpoint of a duality ▣ Viewpoint of a pole

Surprise examination paradox

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., Une analyse dichotomique du paradoxe de l’examen surprise, Philosophiques, vol. 32-2, 2005, pp. 399-421, A dichotomic analysis of the surprise examination paradox (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, pp. 581-608.

Ambiguous images ▣ Arbitrary focus ▣ Bistable perception ▣ Complementarity relationship ▣ Conflict resolution ▣ Conflict resolution with matrices of concepts ▣ Conflict types relating to matrices of concepts ▣ Contrary relationship ▣ Courage ▣ Dialectical contextualism ▣ Dialectical monism ▣ Dialectical monism in Aztec philosophy ▣ Dialectical monism in Heraclitus ▣ Dichotomic analysis ▣ Dichotomic analysis applied to paradox resolution ▣ Dichotomous reasoning ▣ Disqualification of one pole ▣ Disqualification of the positive ▣ Doctrine of the mean ▣ Doomsday argument ▣ Dualities ▣ Dual poles ▣ Extreme opposition ▣ General cognitive distortions ▣ Instance of one-sidedness bias ▣ Liar paradox ▣ Matrix of concepts ▣ Maximization ▣ Mental filter ▣ Minimization ▣ Bistable cognition ▣ Omission of the neutral ▣ One-sidedness bias ▣ One-sided viewpoint ▣ Opposition relationship ▣ Principle of dialectical indifference ▣ Requalification into the other pole ▣ Reference class ▣ Reference class problem ▣ Reference class problem in philosophical paradoxes ▣ Reference class problem in the Doomsday argument ▣ Reference class problem in Hempel’s paradox ▣ Reference class problem in the surprise examination paradox ▣ Selective abstraction ▣ Sorites paradox ▣ Specific cognitive distortions ▣ Surprise examination paradox ▣ System of taxa ▣ Two-sided viewpoint ▣ Viewpoint of a duality ▣ Viewpoint of a pole

Sorites paradox

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

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

Ambiguous images ▣ Arbitrary focus ▣ Bistable perception ▣ Complementarity relationship ▣ Conflict resolution ▣ Conflict resolution with matrices of concepts ▣ Conflict types relating to matrices of concepts ▣ Contrary relationship ▣ Courage ▣ Dialectical contextualism ▣ Dialectical monism ▣ Dialectical monism in Aztec philosophy ▣ Dialectical monism in Heraclitus ▣ Dichotomic analysis ▣ Dichotomic analysis applied to paradox resolution ▣ Dichotomous reasoning ▣ Disqualification of one pole ▣ Disqualification of the positive ▣ Doctrine of the mean ▣ Doomsday argument ▣ Dualities ▣ Dual poles ▣ Extreme opposition ▣ General cognitive distortions ▣ Instance of one-sidedness bias ▣ Liar paradox ▣ Matrix of concepts ▣ Maximization ▣ Mental filter ▣ Minimization ▣ Bistable cognition ▣ Omission of the neutral ▣ One-sidedness bias ▣ One-sided viewpoint ▣ Opposition relationship ▣ Principle of dialectical indifference ▣ Requalification into the other pole ▣ Reference class ▣ Reference class problem ▣ Reference class problem in philosophical paradoxes ▣ Reference class problem in the Doomsday argument ▣ Reference class problem in Hempel’s paradox ▣ Reference class problem in the surprise examination paradox ▣ Selective abstraction ▣ Sorites paradox ▣ Specific cognitive distortions ▣ Surprise examination paradox ▣ System of taxa ▣ Two-sided viewpoint ▣ Viewpoint of a duality ▣ Viewpoint of a pole

Liar paradox

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?

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

Courage

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.

Ambiguous images ▣ Arbitrary focus ▣ Bistable perception ▣ Complementarity relationship ▣ Conflict resolution ▣ Conflict resolution with matrices of concepts ▣ Conflict types relating to matrices of concepts ▣ Contrary relationship ▣ Courage ▣ Dialectical contextualism ▣ Dialectical monism ▣ Dialectical monism in Aztec philosophy ▣ Dialectical monism in Heraclitus ▣ Dichotomic analysis ▣ Dichotomic analysis applied to paradox resolution ▣ Dichotomous reasoning ▣ Disqualification of one pole ▣ Disqualification of the positive ▣ Doctrine of the mean ▣ Doomsday argument ▣ Dualities ▣ Dual poles ▣ Extreme opposition ▣ General cognitive distortions ▣ Instance of one-sidedness bias ▣ Liar paradox ▣ Matrix of concepts ▣ Maximization ▣ Mental filter ▣ Minimization ▣ Bistable cognition ▣ Omission of the neutral ▣ One-sidedness bias ▣ One-sided viewpoint ▣ Opposition relationship ▣ Principle of dialectical indifference ▣ Requalification into the other pole ▣ Reference class ▣ Reference class problem ▣ Reference class problem in philosophical paradoxes ▣ Reference class problem in the Doomsday argument ▣ Reference class problem in Hempel’s paradox ▣ Reference class problem in the surprise examination paradox ▣ Selective abstraction ▣ Sorites paradox ▣ Specific cognitive distortions ▣ Surprise examination paradox ▣ System of taxa ▣ Two-sided viewpoint ▣ Viewpoint of a duality ▣ Viewpoint of a pole

The doctrine of the mean

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.

Ambiguous images ▣ Arbitrary focus ▣ Bistable perception ▣ Complementarity relationship ▣ Conflict resolution ▣ Conflict resolution with matrices of concepts ▣ Conflict types relating to matrices of concepts ▣ Contrary relationship ▣ Courage ▣ Dialectical contextualism ▣ Dialectical monism ▣ Dialectical monism in Aztec philosophy ▣ Dialectical monism in Heraclitus ▣ Dichotomic analysis ▣ Dichotomic analysis applied to paradox resolution ▣ Dichotomous reasoning ▣ Disqualification of one pole ▣ Disqualification of the positive ▣ Doctrine of the mean ▣ Doomsday argument ▣ Dualities ▣ Dual poles ▣ Extreme opposition ▣ General cognitive distortions ▣ Instance of one-sidedness bias ▣ Liar paradox ▣ Matrix of concepts ▣ Maximization ▣ Mental filter ▣ Minimization ▣ Bistable cognition ▣ Omission of the neutral ▣ One-sidedness bias ▣ One-sided viewpoint ▣ Opposition relationship ▣ Principle of dialectical indifference ▣ Requalification into the other pole ▣ Reference class ▣ Reference class problem ▣ Reference class problem in philosophical paradoxes ▣ Reference class problem in the Doomsday argument ▣ Reference class problem in Hempel’s paradox ▣ Reference class problem in the surprise examination paradox ▣ Selective abstraction ▣ Sorites paradox ▣ Specific cognitive distortions ▣ Surprise examination paradox ▣ System of taxa ▣ Two-sided viewpoint ▣ Viewpoint of a duality ▣ Viewpoint of a pole