Outline of a Theory of Truth

The central claim is that a formal language can consistently contain its own truth predicate if truth is treated as a partially defined predicate rather than a total one. Kripke constructs a minimal fixed point in which sentences that are grounded receive classical truth values while ungrounded sentences—including the liar sentence—remain undefined. The theory rejects Tarski's impossibility theorem, which held that any language rich enough for arithmetic and containing its own truth predicate must be inconsistent. Kripke's alternative accepts truth-value gaps rather than a hierarchy of metalanguages, showing that self-referential truth need not lead to contradiction if the semantics permits partiality.

Groundedness is Kripke's diagnostic notion for distinguishing sentences that can receive a determinate truth value from those that cannot. A sentence is grounded if its truth value is determined by facts that do not depend on the truth value of the sentence itself—ultimately, by empirical facts and the base language. A sentence is ungrounded if its semantic dependencies form a loop or an infinite regress that never reaches a non-semantic ground. The liar sentence ('this sentence is false') is the paradigm of ungroundedness. Kripke's fixed-point construction formalizes this intuition: grounded sentences stabilize at classical values across all fixed points, while ungrounded sentences remain undefined in the minimal fixed point.

Tarski's approach prohibits a language from containing its own truth predicate, requiring a hierarchy of metalanguages in which each level's truth predicate is defined in a language one level higher. This avoids paradox but cannot accommodate natural language, which contains its own truth predicate. Kripke's theory permits a single language to contain its own truth predicate by abandoning the requirement that truth be a total predicate. Where Tarski demands that every sentence be either true or false, Kripke allows truth-value gaps. The result is a language that can express its own truth without inconsistency, though at the cost of accepting that some sentences (the ungrounded ones) are neither true nor false.

The strengthened liar paradox is the objection that Kripke's theory cannot express its own diagnosis of ungroundedness. If the liar sentence is 'neither true nor false,' then a sentence asserting 'this sentence is not true' (where 'not true' includes being undefined) generates a new paradox. Kripke's theory, the objection goes, cannot consistently claim that the liar is ungrounded because the language lacks the expressive resources to state that fact without reintroducing contradiction. Kripke acknowledges this limitation: his object language cannot express the notion of ungroundedness that his metalanguage uses to analyze it. Later work by Vann McGee and others has attempted to address this expressive gap, with varying degrees of success.

The essay is widely regarded as one of the most influential post-Tarskian contributions to the theory of truth. Google Scholar records over 2,000 citations, and it is a standard reference in debates about semantic paradox, truth-value gaps, and the expressive capacity of formal languages. Its fixed-point construction has been adapted and criticized by philosophers including Vann McGee, Hartry Field, and Tyler Burge. McGee's work on vague truth predicates and Field's revisionary approach to truth both engage Kripke's framework, either extending it or arguing that its gaps are inadequate. The essay also influenced work on the semantics of natural language, where partiality and context-dependence are pervasive.

A common misreading treats Kripke's theory as a solution to the liar paradox in the sense of providing a determinate truth value for the liar sentence. It does not. Kripke does not claim that the liar is true or false; he claims that it is ungrounded and therefore undefined. Another misreading assumes the theory is intended as a description of natural language truth. Kripke explicitly presents it as a formal model and cautions against drawing direct conclusions about ordinary usage. The essay's title—'Outline'—signals its provisional character, and Kripke himself notes that the construction is one of many possible formal approaches, not a definitive account of truth.