Daily Archives: September 16, 2022

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