Philosophy:Yablo's paradox

Yablo's paradox is a logical paradox published by Stephen Yablo in 1985. It is similar to the liar paradox. Unlike the liar paradox, which uses a single sentence, this paradox uses an infinite list of sentences, each referring to sentences occurring further down the list. Analysis of the list shows that there is no consistent way to assign truth values to any of its members. Since everything on the list refers only to later sentences, Yablo claims that his paradox is "not in any way circular". However, Graham Priest disputes this.

Statement

Consider the following infinite set of sentences:

S₁: For each i > 1, S_i is not true.

S₂: For each i > 2, S_i is not true.

S₃: For each i > 3, S_i is not true.

...

Analysis

Assume that there is an n such that S_n is true. Then S_{n + 1} is not true, so there is some k > n + 1 such that S_k is true. But S_k is not true because S_n is true and k > n. Assuming S_n to be true implies a contradiction: some later S_k is both true and not true. So our assumption is absurd, and we must conclude that for each i, the sentence S_i is not true. But if each S_i is not true, then given that each attributes untruth to later sentences, they are all true. So we have the paradox that each sentence in Yablo's list is true and not true.

External links

• "Liar Paradox". Internet Encyclopedia of Philosophy. http://www.iep.utm.edu/par-liar. 
• "Yablo's Paradox". Internet Encyclopedia of Philosophy. http://www.iep.utm.edu/yablo-pa.