Liar's Paradox — Definition, Formula & Examples

The Liar's Paradox is the self-referencing statement "This statement is false," which cannot consistently be assigned a truth value. If you assume it is true, then it must be false; if you assume it is false, then it must be true.

The Liar's Paradox arises from a sentence $L$ that asserts its own falsehood, i.e., $L \iff \neg L$ . Under classical two-valued logic, no truth-value assignment to $L$ is consistent, since both $L = T$ and $L = F$ lead to contradictions.

How It Works

Start by assuming the statement "This statement is false" is true. If it is true, then what it claims must hold — but it claims to be false, so it must be false. Now assume it is false. If it is false, then its claim (that it is false) does not hold, meaning it must be true. Either assumption collapses into its opposite, creating an endless loop with no stable truth value. This paradox reveals a fundamental limitation: not every grammatically correct sentence can be given a truth value in classical logic.

Example

Problem: Let L represent the sentence "L is false." Determine whether L can be true or false without contradiction.

Assume L is true: If L is true, then what L says is the case. L says "L is false." So L is false — contradicting our assumption.

L = T \implies L = F \quad \text{(contradiction)}

Assume L is false: If L is false, then what L says is not the case. L says "L is false," so the opposite holds: L is true — again a contradiction.

L = F \implies L = T \quad \text{(contradiction)}

Conclusion: Neither truth value can be assigned to L without generating a contradiction. L is therefore paradoxical under classical logic.

Answer: L cannot be consistently assigned true or false. It is a genuine paradox in two-valued logic.

Why It Matters

The Liar's Paradox motivates key developments in mathematical logic, including Tarski's undefinability theorem, which shows that a sufficiently powerful formal language cannot define its own truth predicate. In intro logic courses, it serves as a concrete example of why careful definitions and axiom systems matter when building proofs.

Common Mistakes

Mistake: Thinking the paradox can be resolved by simply declaring the statement "neither true nor false."

Correction: Adding a third value like "neither" leads to a strengthened liar: "This statement is not true." If it is neither true nor false, then it is indeed not true — which is exactly what it claims, making it true again. The paradox resurfaces at every level.