Tautology

Definition

Let $\varphi$ be a formula of propositional logic with the propositional variables $p_1, \dots, p_n$. The formula $\varphi$ is a tautology when $\varphi$ is true for each of the $2^n$ assignments of truth values to $p_1, \dots, p_n$.

Remarks

Examples are $p \lor \neg p$ and $p \to p$. A formula that is false for each assignment is a contradiction. A truth table decides in a finite number of steps whether a formula is a tautology.

Explanations

L1 human

A tautology is true only because of its form. The sentence "it rains or it does not rain" is true whatever the weather. Its form $p \lor \neg p$ is a tautology.