Contradiction

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

Ex: $p\wedge \neg p$

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 $\neg A$ must be false, so the compound statement $A\wedge (\neg A)$ is always false. The statement “$A\wedge (\neg A)$ is true” is called a contradiction.

Ex: Prove that $\sqrt{2}$ is irrational.

Personal Contradictions

See Paradox.