Contradiction

Lore & Background

By creation of a paradox, Plato's Euthydemus dialogue demonstrates the need for the notion of contradiction. In the ensuing dialogue, Dionysodorus denies the existence of 'contradiction', all the while that Socrates is contradicting him. Socrates responds 'But how can I refute you, if, as you say, to tell a falsehood is impossible?'

Reader's Guide

In classical logic, a proposition φ is a contradiction if and only if φ ⊢ ⊥. Since for contradictory φ it is true that ⊢ φ → ψ for all ψ (because ⊥ ⊢ ψ), one may prove any proposition from a set of axioms which contains contradictions. This is called the 'principle of explosion', or 'ex falso quodlibet'. In a complete logic, a formula is contradictory if and only if it is unsatisfiable. For a set of consistent premises Σ and a proposition φ, it is true in classical logic that Σ ⊢ φ if and only if Σ ∪ {¬φ} ⊢ ⊥. This forms the basis of proof by contradiction, which mathematicians use extensively. Using minimal logic, various extensions yield intermediate logics: double-negation elimination yields classical logic; ex falso quodlibet yields intuitionistic logic; Peirce's rule captures proof by contradiction without explicitly referring to absurdity.

Did You Know?

• The law of noncontradiction states that 'It is impossible that the same thing can at the same time both belong and not belong to the same object and in the same respect.'
• In Plato's Euthydemus, Dionysodorus denies the existence of contradiction while Socrates contradicts him.
• The principle of explosion, or ex falso quodlibet, means 'from falsity, anything follows'.
• Proof by contradiction is used to prove that √2 is irrational.