The Liar and Semantic Paradoxes

'This sentence is false' cannot be consistently assigned a truth value, and the resulting Liar paradox has driven the modern theory of truth.

Definition

A semantic paradox arises when a language containing its own truth predicate permits a self-referential sentence (such as one saying of itself that it is false) whose evaluation leads to contradiction.

Scope

This topic covers the Liar paradox and the family of semantic paradoxes (Curry's, the Strengthened Liar, the truth-teller) that arise from combining a truth predicate with self-reference. It treats the main responses: Tarski's hierarchy of object- and metalanguages, Kripke's fixed-point theory with truth-value gaps, paraconsistent (dialetheic) and revision-theoretic approaches, and the persistent problem of revenge, whereby strengthened versions reinstate paradox against proposed solutions.

Core questions

• How can a sentence consistently say of itself that it is false?
• Should we restrict the truth predicate (Tarski) or allow truth-value gaps (Kripke)?
• Can any solution avoid 'revenge' paradoxes that target it directly?
• Do the paradoxes show that some contradictions are true?

Key concepts

• self-reference
• the T-schema
• truth-value gaps and gluts
• Tarskian hierarchy
• groundedness
• revenge paradoxes

Key theories

Tarskian hierarchy

Tarski blocks the Liar by denying any language a univocal self-applicable truth predicate, stratifying truth into a hierarchy of object- and metalanguages so that 'true' always applies from a higher level.

Fixed-point (gap) theory

Kripke allows a single self-applicable truth predicate but uses a fixed-point construction in which paradoxical sentences are ungrounded and fall into a truth-value gap, avoiding contradiction without a hierarchy.

History

The Liar dates to antiquity (Eubulides). Tarski's 1930s-40s work diagnosed it via the undefinability of truth and proposed the hierarchy. Kripke's 1975 fixed-point theory revived a single truth predicate with gaps, after which revision theory (Gupta-Belnap), paraconsistent approaches (Priest), and Field's gap-based logic of 2008 sought to overcome revenge.

Debates

Can the Liar be solved without revenge?

Whether any consistent account of truth can handle the Liar without a strengthened, 'revenge' version reinstating paradox using the very notions the solution introduces (such as 'not true' or 'gappy'), or whether dialetheism's acceptance of true contradictions is the only stable option.

Key figures

• Alfred Tarski
• Saul Kripke
• Hartry Field
• Graham Priest
• Anil Gupta

Seminal works

• tarski1944
• kripke1975
• field2008

Frequently asked questions

What is a revenge paradox?

A revenge paradox is a strengthened Liar built using the very concepts a proposed solution relies on. If you solve the Liar by saying it is 'neither true nor false', the sentence 'This sentence is not true' uses 'not true' to reignite the contradiction. Revenge is the central obstacle facing any theory of the Liar.