This sentence is false

Question

"This sentence is false". Is this sentence true or false?

My attempt:

If this sentence were true, then what it says would be the truth , it implies that it is false which is a contradiction. if it were false then it is false that this sentence is false, hence it is true which is also a contradiction. So this sentence can neither be true be true nor false. Is my reasoning correct?

PS: I would like to see more of such statements.

@Demosthene this sentence is neither true nor false, hence its not false, wouldn't that imply that its true?? — hashir, May 13 '15 at 10:22
As @Adayah pointed out, it is a version of the well-known liar paradox. — Demosthene, May 13 '15 at 10:36
Hint: First, consider a slight variation: "This sentence is true." Though grammatically correct, it is obviously meaningless nonsense. Can changing "true" to "false" suddenly imbue it with meaning? — Dan Christensen, May 14 '15 at 03:29

score 6 · Answer 1 · answered May 13 '15 at 15:00

Great question. This is an example of the liar paradox and is studied in the scholastic logic of insolubles. What you did was prove using the self-principle of the excluded middle (every proposition is either true or false) that this sentence (this sentence is false) can neither have truth or falsity because truth implies falsity and falsity implies truth. Consequently, since the definition of a proposition is that of a declarative sentence having truth or falsity, this sentence (this sentence is false) can not even be a proposition. So it is called a self-contradictory non-propositional sentence. This is different from p:(no proposition is true). Supposing p to be true, no proposition is true. Therefore, the proposition p can not be true. Therefore, p is false. But if p is false so that (some proposition is true) there is no implication of the truth of p nor any contradiction. Therefore, p is a self-contradictory proposition as distinguished from (this sentence is false).

Therefore, (this sentence is false) is in fact not a statement, i.e. proposition, but a self-contradictory non-propositional sentence. — Eric Brown, May 13 '15 at 15:01
(This sentence is false) is also called a self-contradictory insoluble sentence. — Eric Brown, May 13 '15 at 15:02

score 2 · Answer 2 · edited Apr 13 '17 at 12:20

I know this is an old question, but in fact such paradoxes are one of the motivations for inventing systems that are not purely classical but allow something called a truth-value gap, namely that although there are no contradictions there could be statements that are neither true nor false, which are said to fall into that gap. One such system is Kleene's 3-valued logic, which is fully compatible with this analysis of various well-known paradoxes. The apparent disadvantage that there are no tautologies in Kleene's 3-valued logic is easily circumvented by overlaying some type theory that allows one to quantify over propositions with classical truth values, which would result in classical logic being embedded into the new system.

This sentence is false

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