I’ve been seeing this on the web lately:
"This sentence is false."
Answer by Craig Skinner
This is the shortest version of the liar paradox, so called because of the ancient tale of the Cretan who said ‘All Cretans are liars’ (Epimenides poem of c.a. 600 BCE actually says ‘The Cretans, always liars…’, but no matter).
As with the Cretan’s utterance, it’s paradoxical because, although it seems to make sense, we can’t say that it is true (as opposed to false) or false (as opposed to true).
Assume the sentence is true. Then what it says is correct. But it says it’s false. So it’s false. So, if it’s true, it’s false.
Assume, on the other hand, it’s false. Then what it says is incorrect. But it says it’s false. So this is incorrect. So, it’s true. So, if it’s false, it’s true.
The sentence is self contradictory.
Ways of dealing with this.
1. Ban self-referring statements, and say that comment about a sentence must be in a higher-level metalanguage. This is like Russell’s Theory of Types as a ‘solution’ to the set-of-sets-which-are-not-members-of-themself paradox. But It seems to dodge the issue. In any case you can avoid the self-referring sentence as follows:
* The sentence below is true.
* The sentence above is false.
2. Abandon true/ false bivalence, admit a third truth value of neither-true-nor-false, or both-true-and-false, or a null value (truth gap).
3. Accept that some contradictions exist. The sentence IS true, and the sentence IS not-true. There are true contradictions.
My preference is for 3. Some people go wild at this, saying that if we accept a single contradiction, then ANYTHING can be proved (the explosion problem). But I doubt this. If you’re interested, read Graham Priest In Contradiction 2nd ed Oxford University Press 2006.
Hegel held that there are true contradictions, but based this on acceptance of Kant’s antinomies (which are fallacious) and on arguments of his own which are incomprehensible (to me at any rate). But I think he was right.
Logic including true contradictions is called dialetheic logic or paraconsistent logic, and its proponents say that it is to classical logic as Einstein’s theory of gravity is to Newton’s — both get it right in ordinary circumstances, but the newer view is more correct and also gets it right in extreme circumstances.