# Contradiction

Definition. A propositional formula $A$ is a contradiction (or falsehood) if $[A]=F$ for all Boolean valuations.

Ex: $p∧¬p$

The opposite of a contradiction is a Tautology. Something that is neither a contradiction nor a tautology is a Tautology. Something that is neither a contradiction nor a tautology is a Contingent Formula.

### Proof by Contradiction

Let $A$ be a statement. Note that either $A$ or $¬A$ must be false, so the compound statement $A∧(¬A)$ is always false. The statement “$A∧(¬A)$ is true” is called a **contradiction**.

Ex: Prove that $2 $ is irrational.

### Personal Contradictions

See Paradox.