The Liar Paradox

There are four main families of response alive today, plus a fifth that has been gaining ground. It is worth seeing each on its own terms, because each is coherent and each is expensive.

Tarski: stratify the language

Alfred Tarski (1933, and in accessible form 1944) diagnosed the problem as semantic closure: the Liar arises in any language that contains its own truth predicate, can refer to its own sentences, and obeys classical logic. Remove any one and the contradiction goes. Tarski removed the first.

On his account, a language cannot consistently contain "true-in-that-same- language". Truth for a language is defined in a stronger metalanguage that sits above it. There is no single predicate "true" but an unbounded hierarchy: true₀, true₁, true₂ … Every sentence lives at some level, and each level's truth predicate applies only to sentences below it. The Liar cannot be written, because "this sentence is false" would need a truth predicate of its own level.

Tarski was explicit that this makes natural language, which does contain "true" and does apply it to itself, inconsistent. He was untroubled by that: his target was formal languages, where the hierarchy is exactly right, and the resulting account of truth became the foundation of modern formal semantics.

Cost: it does not describe how anyone actually speaks. And Kripke's Nixon example shows the hierarchy cannot be imposed by inspecting sentences, since whether a sentence is safe depends on contingent facts about the world.

Kripke: one language, partial truth

Saul Kripke (1975) kept a single language containing its own truth predicate, and made the predicate partial — defined for some sentences and undefined for others.

Start with a truth predicate that applies to nothing. Now feed in every sentence whose truth you can settle without using the predicate: "snow is white" gets in. Then feed in everything settled by that: "'snow is white' is true" gets in. Keep going, transfinitely. The process only ever adds, never retracts, so it reaches a point where nothing further can be added — a fixed point.

Sentences that get a value along the way are called grounded. The Liar is never reached at any stage, so it stays undefined forever. It is not true, it is not false, it simply never enters the process. That is a principled account of the gap, rather than a stipulation, and it is why Kripke's construction is the single most influential technical contribution to the subject.

Cost: the strengthened Liar. Inside the fixed-point language you cannot say "the Liar sentence is not true", because "not true" in that language means "false" and the Liar is not false either. To say the thing Kripke's own theory says about the Liar, you have to step outside into a stronger language. Kripke saw this perfectly clearly and wrote that "the ghost of the Tarski hierarchy is still with us".

Priest and the dialetheists: some contradictions are true

Graham Priest's position, defended since the late 1970s, is the one most people find outrageous and hardest to refute. The Liar sentence is both true and false. It is a dialetheia, a true contradiction. There is no error in the reasoning; the reasoning is sound and its conclusion should be accepted.

The obvious objection is that from a contradiction everything follows, so the theory proves that the moon is a biscuit. Priest's answer is that this principle — ex contradictione quodlibet — is the thing that should go. In a paraconsistent logic such as his Logic of Paradox, a sentence can take the value "both", contradictions do not spread, and inference remains useful.

Cost: the logic is weaker everywhere, not just at the Liar. Disjunctive syllogism fails: from "P or Q" and "not P" you may no longer infer Q. That rule is used constantly in ordinary and mathematical reasoning, and giving it up in general to accommodate a handful of pathological sentences strikes most logicians as a bad trade. Priest replies that the rule is not truth-preserving in the presence of dialetheias, and that noticing this is progress.

Gupta, Herzberger and Belnap: truth as revision

A fourth approach denies that truth has a fixed extension at all. On the revision theory (Anil Gupta and Hans Herzberger, both 1982; developed with Nuel Belnap in 1993), "true" is a circular concept, like a definition that mentions itself. You cannot evaluate such a definition; you can only iterate it.

Start with any arbitrary guess about which sentences are true. Apply the truth rules once and you get a revised guess. Apply again, and again, transfinitely. Ordinary sentences settle down after a step or two and never move again — they are stable. The Liar flips at every single stage and never settles. Its instability is not a failure of the theory; it is the theory's account of what the Liar is.

Cost: you get a rich description of the Liar's behaviour, but "true" is no longer a property a sentence simply has, which many find too high a price for a concept this central.

Contextualists and paracompletes

Two further lines deserve naming. Contextualists — Charles Parsons, Tyler Burge, Jon Barwise and John Etchemendy (1987), Michael Glanzberg — argue that when we step back and pronounce the Liar "not true", we are speaking in a shifted context, about a different situation from the one the original sentence was about. The strengthened Liar is then not a contradiction but an equivocation across contexts. Paracomplete theorists, most prominently Hartry Field in Saving Truth from Paradox (2008), keep the full unrestricted truth principle and give up the law of excluded middle for pathological sentences, adding a new conditional carefully built to survive Curry.

Where this leaves things

There is no consensus. The nearest thing to agreement is negative and it is worth stating: no proposal is free. Every route pays in something — in expressive power, in logical strength, in the simplicity of the word "true", or in the claim that ordinary language is consistent. Judging the paradox now means judging which currency you would rather spend, and reasonable experts differ.

That is why this entry's status is debated rather than resolved, and why its nature is logical contradiction rather than apparent contradiction. There is no hidden equivocation waiting to be exposed. The premises are individually compelling and jointly inconsistent, and something genuinely has to go.

A note on the date. This entry is filed at 600 BCE because that is where tradition puts it, attached to Epimenides. The paradox proper is at least two centuries younger, and the tradition is wrong. Both facts are worth having in front of you.

The misattribution

Epimenides of Knossos was a Cretan seer and poet of the seventh or sixth century BCE, wrapped in legend — including a fifty-seven-year sleep in a cave. A line attributed to him says, in effect, "Cretans are always liars." It reaches most modern readers through the Epistle to Titus (1:12), whose author quotes it approvingly and adds "this testimony is true" — apparently without noticing anything odd. The same half-line appears in Callimachus' Hymn to Zeus.

Here is the thing everybody skips: that sentence is not paradoxical. Suppose a Cretan says all Cretans are always liars. If the sentence is true, then this Cretan is a liar, so the sentence is false — a contradiction, so it is not true. Therefore it is false. And "false" here means only that some Cretan, at some moment, said something true. That is not merely consistent, it is almost certainly the actual state of affairs. The argument terminates cleanly with a false statement and no contradiction. At most it is a sentence that cannot be true, which is a curiosity, not a paradox.

To get the paradox you need the negation to bite exactly, which requires a sentence about itself alone, and the Epimenides line is a sentence about all Cretans.

Why the misattribution stuck is not mysterious. It has a name, a date, a picturesque setting and a biblical citation, and the modern era passed it along: Russell listed "the Epimenides" in 1908 as the oldest contradiction of its kind, and Douglas Hofstadter's Gödel, Escher, Bach (1979) fixed the label for a generation of readers. Ancient sources do not connect Epimenides with the paradox at all.

The documented origin

The earliest surviving attribution of the paradox to a person is in Diogenes Laërtius (II.108), who credits it to Eubulides of Miletus, a Megarian logician and a contemporary and critic of Aristotle, working around the middle of the fourth century BCE. Diogenes lists it alongside several other puzzles credited to Eubulides — the Sorites or heap, the Hooded Man, the Horned Man, the Bald Man — which is a rather remarkable haul for one person, and suggests the Megarians were running a systematic programme of finding places where ordinary predicates break.

Whether Eubulides invented it or merely compiled it cannot be established. Diogenes wrote around 500 years later and is not a careful historian. What can be said is that the fourth-century BCE Megarian school is the earliest context in which anyone is recorded as having posed the paradox, and that the attribution to Epimenides has no ancient support whatever. The Kneales' Development of Logic (1962) is the standard reference for this correction.

Antiquity and the Middle Ages

The Stoics took it extremely seriously. Diogenes Laërtius' catalogue of the lost works of Chrysippus (VII.196–197) lists at least six titles on the Liar. None survive, so we know that the leading logician of the ancient world spent years on the problem and not what he concluded. Cicero refers to the puzzle in the Academica. A tradition reports that the poet-scholar Philetas of Cos wasted away from studying it, and that his epitaph said so; the story is charming and should be treated as a story.

Medieval logicians classified the Liar among the insolubilia and produced a genuinely sophisticated literature on it in the fourteenth century. Thomas Bradwardine argued that every sentence implicitly asserts its own truth, so the Liar asserts both its truth and its falsity and is simply false. Jean Buridan developed a related account and constructed a family of variants, including cases where paradox depends on contingent circumstance — anticipating Kripke's Nixon example by six hundred years. Paul of Venice surveyed fifteen distinct solutions. This work was largely forgotten and then rediscovered in the twentieth century, and modern commentators have not been kind to the assumption that nothing happened between Chrysippus and Russell.

The modern era

The paradox re-entered mathematics through the crisis in foundations. Russell discovered his own paradox in 1901 while examining Cantor's work, recognised the Liar as its semantic cousin, and grouped both under his theory of types in 1908. Gödel's incompleteness theorems (1931) turned the diagonal construction into a positive result. Tarski published the definition of truth and the undefinability theorem in Polish in 1933, in German in 1935, and in a non-technical English presentation in 1944, and for roughly forty years the hierarchy was the answer.

Kripke's 1975 paper broke that consensus and reopened the field. Priest put dialetheism on the table with "The Logic of Paradox" in 1979 and In Contradiction in 1987. Gupta and Herzberger independently published the revision theory in 1982. Barwise and Etchemendy's The Liar appeared in 1987, Yablo's sequence in 1993, Field's paracomplete theory in 2008. The literature has grown in every decade since Kripke's paper and shows no sign of converging.

It produced the modern theory of truth. Tarski's work on the Liar is the origin of formal semantics: the idea that you specify what a language's sentences mean by giving satisfaction conditions in a metalanguage. That method is now the standard apparatus of model theory, of the semantics of programming languages, and of formal linguistics. It was built to escape a paradox.

It produced a theorem about the limits of every formal system. Tarski's undefinability result — no sufficiently strong consistent language can define its own truth predicate — is not a piece of philosophy but a permanent constraint. It is a sibling of Gödel's incompleteness theorems and of Turing's undecidability of the halting problem, and all three run on the same diagonal argument that the Liar embodies. The Liar is the informal ancestor of the three most important limitative results of twentieth-century logic.

It showed that "true" is not innocent. Before the Liar was taken seriously it was natural to think of truth as a transparent device that adds nothing: saying a sentence is true is just saying it again. The paradox proves that this cannot be right without qualification, because the transparent reading is precisely what generates the contradiction. Anything you say about truth now has to specify how it avoids that.

It is the reason self-reference gets designed for rather than banned. The distinction between object language and metalanguage, the stratification of levels, and the careful controls on reflection in proof assistants and type theories all descend from the recognition that a system talking about itself is dangerous in a specific, characterisable way.

It is a working test case for non-classical logic. Paraconsistent and paracomplete logics are not curiosities; they have applications in databases, belief revision and automated reasoning. Much of their development was motivated by, and is still evaluated against, the Liar.

Incompleteness and undecidability. Gödel's theorems and Turing's halting result are the Liar with the semantic predicate replaced by a syntactic or computational one. This is not a loose analogy: all three use the diagonal lemma, and the proofs are close enough that each is routinely taught as a variant of the others. Every impossibility result in computability theory that proceeds by self-application traces to this argument.

Type theory and proof assistants. Curry's paradox is the direct reason that dependently typed languages must control recursion and universe levels. Girard showed in 1972 that allowing a type of all types makes the system inconsistent, and a Curry-style fixed point is the mechanism. Coq, Lean and Agda all maintain an unbounded hierarchy of universes for exactly the reason Tarski maintained an unbounded hierarchy of languages, and a paper proposing to collapse the hierarchy is a paper proposing to reinvent the Liar.

Reflection and self-modelling systems. Any system that reasons about its own outputs — a prover reasoning about its own provability, a planner evaluating its own plans, a model asked about its own reliability — is operating in the region the Liar maps. Löb's theorem, a direct descendant, says that a consistent system that can prove "if I can prove P, then P" can already prove P, which places real limits on how much a formal system can trust itself. Designers of self-referential architectures rediscover these constraints regularly.

Inconsistency-tolerant computing. Real databases and knowledge bases contain contradictory records, and classical logic's response to a contradiction is to license every conclusion, which is useless. Paraconsistent logics developed in the course of the Liar debate are used to reason usefully in the presence of inconsistency without repairing it first.

Everyday self-undermining claims. "This rule has no exceptions." "Never trust anything you read." "There are no absolute truths." Each is a small Liar, and the diagnosis in each case is the same one Tarski gave: the sentence is trying to apply a rule to itself from inside a level where it does not reach.

The derivation

Let $L$ be the language, and let $T$ be a one-place predicate of $L$ intended to mean "is true", applying to names of sentences of $L$. Write $\ulcorner \varphi \urcorner$ for the name of $\varphi$. The governing principle is the T-schema, Tarski's Convention T:

$$T(\ulcorner \varphi \urcorner) \leftrightarrow \varphi \qquad \text{for every sentence } \varphi \text{ of } L$$

Suppose $L$ contains a sentence $\lambda$ such that

$$\lambda \leftrightarrow \neg T(\ulcorner \lambda \urcorner)$$

Instantiating the T-schema at $\lambda$ and chaining gives

$$T(\ulcorner \lambda \urcorner) \leftrightarrow \lambda \leftrightarrow \neg T(\ulcorner \lambda \urcorner)$$

so $T(\ulcorner \lambda \urcorner) \leftrightarrow \neg T(\ulcorner \lambda \urcorner)$, and in classical logic every sentence follows.

The existence of $\lambda$ is not a hypothesis to be waved at. By the diagonal lemma, in any theory that can represent its own syntax — Robinson arithmetic suffices — for every formula $\varphi(x)$ there is a sentence $\sigma$ with

$$\vdash \sigma \leftrightarrow \varphi(\ulcorner \sigma \urcorner)$$

Take $\varphi(x) := \neg T(x)$ and $\lambda$ exists as a matter of arithmetic. No demonstrative pronoun, no cards, no pointing: self-reference is a theorem about any language rich enough to do elementary number theory.

The ingredients are therefore exactly four: (i) classical logic, (ii) the unrestricted T-schema, (iii) enough syntactic power for diagonalisation, and (iv) a truth predicate for the language within the language. Every known response gives up one of them.

Tarski's undefinability theorem

Tarski's positive result is the sharpest thing anyone has proved here.

Theorem (Tarski, 1933/1936). Let $\mathcal{T}$ be a consistent theory interpreting sufficient arithmetic, in a language $L$. Then there is no formula $\mathrm{Tr}(x)$ of $L$ such that for every sentence $\varphi$ of $L$, $\mathcal{T} \vdash \mathrm{Tr}(\ulcorner \varphi \urcorner) \leftrightarrow \varphi$.

Equivalently: the set of Gödel numbers of true sentences of arithmetic is not arithmetically definable. Truth for a language always requires a strictly stronger language.

The relationship to Gödel's first incompleteness theorem (1931) is exact and instructive. Gödel diagonalises on provability, not truth, obtaining $G$ with $G \leftrightarrow \neg \mathrm{Prov}(\ulcorner G \urcorner)$. Because provability is definable in the language — unlike truth — no contradiction follows. What follows instead is that $G$ is unprovable and true. The Liar's engine, redirected one notch, produces a theorem rather than a catastrophe. Gödel remarked on the analogy himself.

Kripke's construction

Kripke's fixed points are built as follows. Interpret $T$ partially: a partial interpretation is a pair $(S_1, S_2)$ of disjoint sets of sentences, the extension and the anti-extension. Evaluate compound sentences using the Strong Kleene scheme, in which a sentence is undefined when the classical rules give no answer.

Define the jump operator $\Phi$ that maps $(S_1, S_2)$ to the pair obtained by putting into the extension every sentence that comes out true under $(S_1, S_2)$, and into the anti-extension every sentence that comes out false. Strong Kleene is monotone: enlarging the interpretation never overturns a verdict already reached. By the Knaster–Tarski theorem, a monotone operator on this lattice has a least fixed point, reached by iterating $\Phi$ from $(\emptyset, \emptyset)$ through the ordinals:

$$(S_1^{0}, S_2^{0}) = (\emptyset,\emptyset), \qquad (S_1^{\alpha+1}, S_2^{\alpha+1}) = \Phi(S_1^{\alpha}, S_2^{\alpha}), \qquad \text{unions at limits}$$

Sentences entering at some stage are grounded. The Liar never enters: $\lambda \notin S_1^{\alpha} \cup S_2^{\alpha}$ for every $\alpha$. It is ungrounded, and undefined at the least fixed point.

There are many fixed points, not one. The least is distinguished by containing only grounded sentences; other fixed points can assign a value to, for instance, the truth-teller "this sentence is true", which is consistent either way and therefore genuinely underdetermined.

The residual problem. In the fixed-point language, $\neg T(\ulcorner \lambda \urcorner)$ is itself undefined, so the theory cannot assert its own central claim about $\lambda$. Kripke's own verdict on the situation is the sentence quoted earlier about the ghost of the hierarchy. The construction converts a contradiction into an expressive limitation, which is progress, but the limitation is exactly where the strengthened Liar lives.

Paraconsistency

Priest's Logic of Paradox (1979) takes three values ${t, b, f}$ — true, both, false — with ${t, b}$ designated. Negation maps $t \mapsto f$, $f \mapsto t$, and $b \mapsto b$. Conjunction and disjunction are the Kleene tables read on the ordering $f < b < t$.

Explosion fails: assign $b$ to $P$ and $f$ to $Q$, and $P \wedge \neg P$ is designated while $Q$ is not. So $\lambda$ can consistently — or rather, non-trivially — take the value $b$, satisfying the T-schema, without the theory collapsing.

The costs are structural rather than incidental. Disjunctive syllogism fails on the same valuation. The material conditional does not detach: modus ponens for $\supset$ defined as $\neg A \vee B$ fails when $A$ is $b$ and $B$ is $f$, which forces the addition of a primitive conditional. And that conditional must then be built so as not to validate contraction, $A \to (A \to B) \vdash A \to B$, or Curry's paradox reintroduces triviality without ever using negation. Curry is the binding constraint on this entire family of theories.

Revision

Gupta and Herzberger (1982) treat the T-schema not as a truth condition but as a rule of revision. Given a hypothesis $h$ about the extension of $T$, the rule delivers $\delta(h)$, the set of sentences true when $T$ is interpreted by $h$. Iterate through the ordinals with a limit rule for stages after $\omega$.

A sentence is stably true in a revision sequence if it belongs to every hypothesis beyond some stage, and categorically true if stably true in every revision sequence. Ordinary arithmetic sentences are categorical. The truth-teller is stable in some sequences and stably false in others, matching the intuition that it is arbitrary. And $\lambda$ is paradoxical: for every $h$, it is in $\delta(h)$ if and only if it is not in $h$, so it flips at every stage and is stable in no sequence at all.

The theory delivers a fine-grained taxonomy of pathology, at the price of abandoning the idea that $T$ has an extension.

What any solution must survive

Three benchmarks are standard, and a proposal that passes only the first is not taken seriously:

1. The Liar, $\lambda \leftrightarrow \neg T(\ulcorner \lambda \urcorner)$.
2. The strengthened Liar: the theory must be able to state its own verdict on $\lambda$ in its own language, or explain principledly why it cannot.
3. Curry: with $\kappa \leftrightarrow (T(\ulcorner \kappa \urcorner) \to \bot)$, the T-schema plus contraction and modus ponens yield $\bot$ with no negation used. Any restriction aimed at negation leaves this untouched.

Add a fourth if you take Yablo (1993) to be non-circular: the diagnosis cannot be that circularity is the culprit. Priest (1997) contests that reading, arguing the circularity has moved into the quantifier rather than disappeared, and the question remains open.