In logic, a tautology is a statement that simply cannot be false
Either it is raining right now or it is not. Whatever the weather, the sentence holds. Logicians call such statements tautologies: formulas true under every possible assignment of true and false to their parts. The word once insulted needless repetition before logic gave it a precise, respectable job.
Ancient Greeks used tautology for a claim that was supposedly true only because it said the same thing twice, a sneer that survives in rhetoric. Between 1800 and 1940 it gained a technical meaning. Kant described some analytic judgments as tautological, and Frege distinguished truths resting on meaning from statements empty of content. Ludwig Wittgenstein, in his 1921 Tractatus, first applied the word to logic, arguing that anything derivable by pure deduction says nothing about the world. Bertrand Russell initially disagreed but came round by 1918.
Formally, a statement that can be true under at least one assignment is satisfiable. A tautology is one whose negation is never satisfiable, a contradiction is never true, and anything in between is contingent, true or false depending on the facts. In propositional logic there is a mechanical method to test whether any given formula is a tautology.
Many familiar reasoning rules turn out to be tautologies. The law of excluded middle says A or not A. Contraposition says that whenever A leads to B, the absence of B rules out A: if everything bound is a book, anything that is not a book is not bound. De Morgan's law says that not both A and B means not A or not B. Proof by cases, hypothetical syllogism and reductio ad absurdum all fit the same mould.
Usage has narrowed over time. Early 20th-century logicians including Tarski and Gödel called any universally valid formula a tautology. Modern textbooks usually reserve the word for valid statements of propositional logic, or predicate-logic statements reducible to them by substitution. There are infinitely many.
Source: Tautology (logic)
