tautology

Tautology is a property of a logical formula that is true in every possible interpretation of its basic components. In propositional logic, a tautology is a statement that remains true for all assignments of truth values to its propositional variables; equivalently, its truth table contains only true outcomes. Common examples include P ∨ ¬P (the law of the excluded middle) and P → P. More generally, a tautology expresses a truth that follows solely from the logical form, regardless of the content of the propositions involved.

In Boolean algebra and formal proof systems, tautologies correspond to expressions that simplify to the constant true (often denoted 1). A related notion is contradiction, a formula that is false under every valuation. Every tautology represents a valid form of inference: the implication from no premises to a tautologous conclusion is itself a tautology; conversely, a valid argument can be captured as a tautology of the form (P1 ∧ P2 ∧ ... ∧ Pn) → C.

In natural language, the term tautology can also refer to semantic redundancy or pleonasm—phrases that repeat the same idea (for example, "free gift" or "true truth")—though this linguistic use is distinct from the logical notion.

In non-classical logics, what counts as a tautology can differ; for example, certain formulas that are tautologies in classical logic may not hold in intuitionistic logic.