A tautology can also refer to a redundant or repetitive phrase in ordinary language.
Definition
Within propositional logic, a tautology is a propositional formula that evaluates to true for every possible assignment of truth values to its propositional variables. It is commonly represented by the symbol — a mark that is also associated with the truth value "true". The English term comes from the Greek ταυτολογία and was introduced into the study of logical propositions by Ludwig Wittgenstein in the early twentieth century (1921).
One standard way to show that a formula is a tautology is to verify that every row of its truth table yields true. Typical logical tautologies include laws such as the law of excluded middle (for any proposition P, "P or not P" is always true) and laws of identity and noncontradiction expressed in propositional form.
Relation to other logical classes
A formula that is false under every valuation is called a contradiction (sometimes labelled as identically false). A statement is described as contingent when it is neither a tautology nor a contradiction — that is, it is true for some assignments and false for others.
Examples in everyday language
- "I know this is Wikipedia because I already know it is Wikipedia."
- "I am the club president because I am the president of this club."
- "The first rule of the tautology club: the first rule of the tautology club."
These sentences share the same logical form: "A is true because A is true." Their truth does not depend on any additional information; they are vacuously or trivially true in the sense that their asserted reason is identical to the claim itself.