The Liar Paradox

The Puzzle

Consider the sentence:

$\lambda$: "This sentence is false."

Suppose $\lambda$ is true. Then what $\lambda$ says must be the case: $\lambda$ is false. So $\lambda$ is true and false. Contradiction.

Suppose $\lambda$ is false. Then what $\lambda$ says is not the case: it is not true that $\lambda$ is false, i.e., $\lambda$ is true. So $\lambda$ is false and true. Contradiction.

The sentence cannot be consistently assigned either truth-value. There is no consistent valuation of $\lambda$.

This is the Liar Paradox. Variants include the Strengthened Liar ("this sentence is not true") and the Liar Cycle ("the next sentence is true; the previous sentence is false"). The paradox is ancient (Eubulides of Miletus, fourth century BCE) and is the motivating example for most of twentieth-century work on truth, self-reference, and the foundations of formal semantics.

This page assumes What Is Logic? and Syntax vs Semantics in Formal Systems. The latter is especially important: the Liar lives at the intersection of self-reference and truth, exactly where Tarski's distinction between object language and metalanguage was designed to do work.

Reconstructed Formally

What makes the Liar a paradox rather than a curiosity is that the contradiction follows from premises that all look mandatory. To make the dependency precise, we need three ingredients.

Ingredient 1: Tarski's T-schema. For any sentence $\varphi$ of a language and its name $\langle \varphi \rangle$ in the language's truth predicate, the schema says:

In words: the predicate $T$ applied to the name of $\varphi$ is true iff $\varphi$ itself is true. The T-schema looks indispensable. It is the formal expression of what we mean by "true."

Ingredient 2: classical bivalent logic. Every sentence is either true or false; the law of excluded middle ($\varphi \lor \neg \varphi$) holds without restriction; the law of non-contradiction ($\neg (\varphi \land \neg \varphi)$) holds without restriction. These are the standard axioms of classical logic. (See Propositional Logic.)

Ingredient 3: self-reference. The language can construct a sentence $\lambda$ that says of itself that it is not true. Formally, $\lambda$ is a sentence such that

This is not a logical axiom; it is an empirical claim about what the language can express. In natural language and in any formal language rich enough to encode arithmetic via Goedel-numbering, self-reference of this kind is constructible.

The derivation. From the three ingredients, the contradiction is mechanical.

By the T-schema applied to $\lambda$:

By the construction of $\lambda$:

Substituting:

This is a contradiction: a sentence equivalent to its own negation. By classical logic, this entails $\bot$ (any false statement). By the principle of explosion in classical logic ($\bot \vdash \psi$ for any $\psi$), the entire system becomes inconsistent: anything is provable.

The Liar Paradox is therefore a system-breaker. A formal system in which all three ingredients hold proves everything, which means it proves nothing.

What the Paradox Forces

The derivation is valid. The contradiction is real. To preserve a coherent theory of truth, at least one of the three ingredients must give:

• Reject the T-schema (or restrict it). This is Tarski's solution.
• Reject classical bivalent logic. This is Kripke's solution and the dialetheist solution, in different ways.
• Reject self-reference. This is unattractive: any formal system with arithmetic has self-reference via Goedel-numbering. Forbidding self-reference at the language level is too strong.

Each surviving solution family corresponds to a different choice of which ingredient to restrict. The choice is non-trivial: each option carries philosophical and technical costs.

Solution Family 1: Tarski's Hierarchy

Alfred Tarski (1933, 1936) gave the first technically successful solution.¹ His diagnosis: a single language cannot contain a complete truth predicate for itself. Truth must be relativized.

The construction. We have an object language $L_0$. Truth-of-$L_0$ is not expressible in $L_0$; it is expressible only in a metalanguage $L_1$ that contains $L_0$ as a sublanguage. In $L_1$, we can write $T_0(\langle \varphi \rangle)$ for sentences $\varphi$ of $L_0$, but not for sentences of $L_1$ itself. Truth-of-$L_1$ is expressible only in $L_2$. And so on, generating a hierarchy:

with $T_n$ definable in $L_{n+1}$.

How this dissolves the Liar. The Liar sentence requires self-reference within a single language using that language's own truth predicate. In Tarski's hierarchy, this is forbidden: $\lambda$ would have to say $\lambda$ is not $T_n$-true for some $n$, but $\lambda$ itself lives in some $L_m$ where $T_n$ may or may not be expressible. The cross-level construction never gets off the ground in the same form.

Cost of the solution. We lose the apparent universality of "truth." Natural-language truth seems unstratified: we say things like "everything Plato said is false" without choosing a level. Tarski's hierarchy makes this kind of statement formally awkward. For working formal languages (mathematics, programming, formal verification), the hierarchy is well-behaved and is the standard solution; the philosophy of language has continued to grapple with whether natural-language truth is stratified or whether something else is going on.

The formal-systems use of Tarski's hierarchy is what makes Syntax vs Semantics in Formal Systems work cleanly: the truth predicate for a formal system always lives in a richer metalanguage, never in the system itself.

Solution Family 2: Kripke's Fixed-Point Semantics

Saul Kripke (1975) proposed an alternative.² Keep a single unstratified language; abandon the assumption that every sentence is either true or false. Allow sentences to be undefined or gappy.

The construction starts with a partial valuation: each sentence is initially marked True, False, or Undefined. The truth predicate $T$ is then defined by a monotonic operator: at each stage, $T(\langle \varphi \rangle)$ takes the value of $\varphi$ if $\varphi$ already has a defined value at the previous stage; otherwise $T(\langle \varphi \rangle)$ remains Undefined. The construction iterates and reaches a fixed point: a stage where iterating the operator does not change the assignment.

At the fixed point, the Liar sentence $\lambda$ has the value Undefined. It does not get assigned True (which would commit to its negation also being True) or False (which would commit to its truth). It simply has no classical value. The classical contradiction does not arise because the law of excluded middle no longer applies unrestrictedly.

Cost of the solution. Kripke's theory is mathematically beautiful and avoids hierarchy. But it relies on partial logic (Kleene's three-valued logic, often), so the surrounding logical apparatus must be modified. Many natural classical reasoning steps are no longer valid. There is also the Strengthened Liar: the sentence "this sentence is not true" (where "not true" includes "false or gappy"). At Kripke's fixed point, this strengthened Liar still cannot be consistently assigned; the theory has to declare it gappy at a level the original Liar already addressed, leading to a regress that mirrors Tarski's hierarchy in a different form. The theory is technically clean but does not fully escape the relativization Tarski accepted upfront.

Solution Family 3: Dialetheism and Paraconsistent Logic

Graham Priest (1979 and after) proposes the most radical move.³ Accept that the Liar is both true and false, and adopt a logic in which contradictions do not entail everything.

The view, called dialetheism: there are some true contradictions. The Liar is one. We do not have to declare it true or false, gappy, or stratified; we can let it have both truth values. The only thing that prevents us from doing this in classical logic is the principle of explosion ($\bot \vdash \psi$, "from a contradiction, anything follows"), which makes the whole system collapse. So the move is: replace classical logic with a paraconsistent logic in which explosion is invalid.

In a paraconsistent logic, contradictions can be local: $\lambda \land \neg \lambda$ holds, but it does not propagate to make everything else true. The system remains useful; it just contains some specific true contradictions.

Cost of the solution. Most logicians and mathematicians find dialetheism deeply counterintuitive. Accepting that some sentences are both true and false runs against the law of non-contradiction, one of the most cherished principles in the western logical tradition (Aristotle, Metaphysics IV.3-6). Defenders argue the cost is worth paying because the Liar shows classical logic was already committed to true contradictions in disguise; we should make the commitment explicit and adopt a logic that handles it cleanly. Critics argue the position confuses the meta-level acknowledgment of the paradox with an object-level claim about truth. The debate is alive.

For technical work, paraconsistent logics have niche but genuine applications: reasoning about inconsistent legal codes, inconsistent databases, inconsistent historical sources where explosion would be catastrophic.

Solution Family 4: Revision Theory

Anil Gupta and Nuel Belnap (1993) propose treating truth as a circular concept.⁴ Instead of looking for a fixed-point or stratified definition, give a revision rule: at each stage, the truth values of sentences are reassessed based on the previous stage's assignment.

The construction. Start with an arbitrary hypothesis $H_0$ about which sentences are true. Apply a revision operator: at stage $n+1$, a sentence $\varphi$ counts as true iff what $\varphi$ says is the case under hypothesis $H_n$. Iterate.

For the Liar, the revision sequence oscillates: at any stage where $\lambda$ is hypothesized true, the next stage hypothesizes it false; at any stage where $\lambda$ is hypothesized false, the next stage hypothesizes it true. The Liar has no stable value, but it has a perfectly determinate behavior: it oscillates with period 2 across stages.

The thesis is then that truth is a circular concept, like other circular definitions in mathematics, and that its semantics is correctly given by the long-run behavior of revision sequences rather than by a fixed point.

Cost of the solution. The mathematics is more involved than Kripke's fixed-point construction. The view also faces criticism that it does not really explain the paradox; it only describes the oscillation. Revision theory remains a serious option in the technical literature but has fewer adherents than Tarski stratification or Kripke fixed points.

Where the Solutions Stand Today

The four solution families are not mutually exclusive in practice. Working logicians and philosophers commonly pick a solution per problem.

For formal systems (the use most relevant to the rest of PhilosophyPath and TheoremPath), Tarski's hierarchy is the working consensus. It is what is built into modern model theory, type theory, and the foundations of programming-language semantics. Goedel's first incompleteness theorem (covered in Syntax vs Semantics) constructs a sentence that is like the Liar in form but says "I am not provable" rather than "I am not true." The crucial difference: provability is a syntactic relation (decidable from the proof system), while truth in the standard model is a semantic relation (not decidable from inside arithmetic). Goedel's sentence is true but unprovable, not paradoxical.

Why the Liar Matters

Three reasons the Liar continues to motivate research.

It exposes the structure of self-reference. Any formal system rich enough to encode arithmetic admits self-reference via Goedel-numbering. The Liar shows that self-reference plus a complete truth predicate is incompatible with classical logic. This is not a quirk; it is a structural feature of formal systems with sufficient expressive power. Anyone designing such a system (a logic, a programming language, a proof assistant) must take a position on the four solution families, even if the position is implicit.

It motivated modern formal semantics. Tarski's response to the Liar gave us the model-theoretic definition of truth that is now standard in mathematical logic. Without the Liar, the field probably would not have developed the syntax-semantics distinction with the technical sharpness it has. The paradox was the forcing function.

It surfaces in machine-checkable formal systems. Type theories used in proof assistants (Coq, Lean, Agda) deliberately restrict self-reference to avoid Liar-style paradoxes that would make the whole system inconsistent. The hierarchy of universes in Martin-Loef type theory is a Tarski-style stratification at the type level, designed precisely so that types do not contain themselves. Anyone who has written "Type : Type" in a type theory and triggered Girard's paradox has met the Liar in computational dress.

Common Confusions

Confusion 1: the Liar is "just a sentence." The Liar can look like a wordplay until the formal derivation is laid out. The contradiction follows from the T-schema, classical logic, and self-reference, all of which seem mandatory. Calling the sentence "ill-formed" or "meaningless" is a position (a particular form of solution) and needs argument; it is not a default response.

Confusion 2: the Liar refutes classical logic. The Liar shows classical logic plus the T-schema plus self-reference is inconsistent. It does not directly refute classical logic alone. The inconsistency can be removed by restricting any of the three ingredients. Most logicians prefer to restrict the T-schema (Tarski) rather than abandon classical logic, but this is a choice, not a forced move.

Confusion 3: the Liar and Goedel are the same. They share the technical tool of self-reference but make different claims. The Liar uses self-reference plus a complete truth predicate to produce a contradiction. Goedel's theorem uses self-reference plus a provability predicate to produce a sentence that is true but unprovable. The Liar is paradoxical; Goedel's sentence is consistent and informative. The difference is exactly the difference between truth (semantic) and provability (syntactic), as established in Syntax vs Semantics.

Confusion 4: dialetheism is irrationalism. Dialetheism is a technical position that requires a paraconsistent logic to be coherent. It does not abandon logic; it changes which logical principles are taken as fundamental. Whether the change is acceptable is contested, but the position is internally rigorous, not a license for arbitrary contradiction.

Two Exercises

Exercise 1. Construct a Liar Cycle: two sentences $\alpha$ and $\beta$ such that $\alpha$ says "$\beta$ is true" and $\beta$ says "$\alpha$ is false." Show that no consistent assignment of truth values to $\alpha$ and $\beta$ exists. Then explain how each of the four solution families would handle the cycle.

Exercise 2. The Curry Paradox uses an apparently innocuous sentence:

$\kappa$: "If $\kappa$ is true, then 0 = 1."

Show that classical logic plus the T-schema plus self-reference (no negation needed) entails $0 = 1$. Then explain why the Liar paradox is sometimes called the negation version and Curry the implication version.

Sketch of answers

Answer 1. Suppose $\alpha$ true. Then "$\beta$ is true" is true, so $\beta$ true. But $\beta$ says "$\alpha$ is false," so $\alpha$ false. Contradiction.

Suppose $\alpha$ false. Then "$\beta$ is true" is false, so $\beta$ false. But $\beta$ false means "$\alpha$ is false" is false, so $\alpha$ true. Contradiction.

No consistent assignment.

How each solution handles it: Tarski stratification would say $\alpha$ and $\beta$ live at different metalanguage levels, so the cyclic reference is malformed. Kripke would assign both Undefined at the relevant fixed point. Dialetheism would accept some truth-value assignment that is inconsistent and confine the contradiction. Revision theory would describe the oscillation pattern of $\alpha$ and $\beta$ as the revision rule iterates; each oscillates with period 2 in opposite phase.

Answer 2. From the T-schema: $T(\langle \kappa \rangle) \leftrightarrow \kappa$. From the construction: $\kappa = (T(\langle \kappa \rangle) \to (0 = 1))$. So $T(\langle \kappa \rangle) \leftrightarrow (T(\langle \kappa \rangle) \to (0 = 1))$.

Now: assume $T(\langle \kappa \rangle)$. Then by the right-to-left direction of the biconditional, $T(\langle \kappa \rangle) \to (0 = 1)$. By modus ponens with our assumption, $0 = 1$. So we have shown $T(\langle \kappa \rangle) \to (0 = 1)$, by conditional proof. Now the left-to-right direction of the biconditional gives $T(\langle \kappa \rangle)$. By modus ponens, $0 = 1$.

The Curry paradox derives an arbitrary conclusion (here $0 = 1$, but it could be any sentence) without using negation. This is why it is called the implication version of the Liar: it shows that contraction (the inference rule $\varphi \to (\varphi \to \psi) \vdash \varphi \to \psi$) plus the T-schema plus self-reference entail collapse, even without negation as in the standard Liar.

Curry is one reason some logicians replace classical logic with substructural logics (linear logic, relevance logic) that drop contraction. Whether this fixes Curry without losing too much expressive power is an active question.

Prerequisites and Next Pages

• Prerequisites: What Is Logic?, Syntax vs Semantics in Formal Systems, Propositional Logic.
• Next: Russell's Paradox, the set-theoretic counterpart that drove the development of modern axiomatic set theory.
• Related: What Is a Symbolic System?, where the discipline of stratified type-universes that prevents Liar-style contradictions in modern proof assistants is mentioned.

References

Primary texts:

• Tarski, Alfred. "The Concept of Truth in Formalized Languages." 1933 (Polish), 1956 English in Logic, Semantics, Metamathematics, Oxford. The foundational technical paper on the hierarchy solution.
• Kripke, Saul A. "Outline of a Theory of Truth." Journal of Philosophy 72 (1975): 690-716. Fixed-point semantics with truth-value gaps.
• Priest, Graham. "The Logic of Paradox." Journal of Philosophical Logic 8 (1979): 219-241. The original dialetheist paper.
• Gupta, Anil, and Nuel Belnap. The Revision Theory of Truth. MIT Press, 1993. The standard revision-theory reference.

Modern reference:

• Beall, J. C., ed. Liars and Heaps: New Essays on Paradox. Oxford, 2003. Edited collection covering all four solution families.
• Field, Hartry. Saving Truth from Paradox. Oxford, 2008. A book-length defense of a particular hybrid solution.
• Priest, Graham. In Contradiction: A Study of the Transconsistent. Oxford, 2nd ed. 2006. The dialetheist position fully developed.
• Simmons, Keith. Universality and the Liar. Cambridge, 1993. The contextualist solution.

Stanford Encyclopedia entries (link, do not paraphrase):

• "The Liar Paradox" by Bradley Dowden.
• "Tarski's Truth Definitions."
• "Self-Reference."
• "Dialetheism."
• "Paraconsistent Logic."