The Liar Paradox

Take L: 'L is false.' Suppose L is true. Then things are as L says, so L is false—a contradiction. So L is not true; suppose instead it is simply false. But being false is exactly what L claims for itself, so L's claim holds, which makes L true. Either assumption refutes itself.

The strengthened liar closes the obvious exits: 'This sentence is not true.' Calling it merely false reasserts its content, and denying it a truth value while insisting it is not true lands back on the claim. The ancient versions were third-personal—Eubulides asked about a man saying he is lying, and the Cretan line quoted from Epimenides in Titus 1:12 becomes paradoxical only under extra assumptions—but modern treatments use direct self-reference because it is maximally sharp.

The liar matters because truth looks rule-bound: calling a sentence true seems equivalent to asserting it (the idea behind the T-schema). That equivalence plus unrestricted self-reference lets any language construct its own contradiction, which threatens the reliability of proof and assertion generally.

Alfred Tarski concluded that no consistent formal language may contain its own truth predicate: truth for a language must be defined in a richer metalanguage, producing an infinite hierarchy. The price is that natural languages, which do talk about themselves, are not formal languages in this sense—a result some accept calmly and others treat as the problem restated.

Saul Kripke's influential alternative lets a language build up truth inductively: start with sentences that contain no truth talk, extend truth step by step, and see where it stabilises. Liar sentences never stabilize—they come out 'ungrounded,' lacking a truth value at the least fixed point. The cost is reintroducing a strengthened liar at the meta-level, since saying the liar is 'not true' is itself meaningful talk.

Dialetheists such as Graham Priest take the opposite tack: the liar is both true and false, a genuine glut, and logic should be paraconsistent—that is, able to contain contradictions without entailing everything. Revision theorists (Gupta and Belnap) treat truth as a circular concept whose meaning lives in revision sequences rather than a definition. Each route preserves something classical logic must give up: bivalence, explosion, or definability. None is free.