Tautology (from Greek tautología, same-saying; to auto + logos) — In propositional logic, a tautology is a formula that is true under every possible assignment of truth values (true/false) to its propositional variables. A tautology represents the particular and maximal case of logical validity in the propositional calculus: a logically valid formula is true in every possible interpretation.
The canonical example is the law of excluded middle: p ∨ ¬p (“either p is true or p is false”) — this formula is true regardless of what p designates. Equally tautological are p → p (“if p, then p”) and (p → q) → (¬q → ¬p) (the contrapositive form of the conditional). The verification method is the truth table: all possible combinations of truth values for the variables are systematically constructed, and the formula is checked to see whether it results in truth in every case.
The opposite of a tautology is a contradiction (or unsatisfiable formula): a formula that is false under every assignment, such as p ∧ ¬p. A formula that is neither a tautology nor a contradiction is contingent: it is true under some assignments and false under others, and its truth value depends on facts about the world.
Wittgenstein and the Tractatus Logico-Philosophicus (1921): Ludwig Wittgenstein introduced a profound philosophical interpretation of tautologies that goes beyond their formal characterisation. In the Tractatus, he maintains that meaningful (sinnvoll) propositions are those that represent possible states of affairs in the world — they can be true or false. Tautologies, by contrast, are true regardless of any state of affairs: they say nothing about the world; they are senseless (sinnlos, to be distinguished from unsinnig, nonsensical). The propositions of logic are all tautological: “The propositions of logic say nothing” (Tractatus, 6.11). They show the formal structure of language and the world, but they convey no empirical content. Hence Wittgenstein’s idea that logic is transcendental — it is the condition of possibility of every meaningful proposition, yet itself adds no fact to the world.
This view profoundly influenced the Vienna Circle: for logical positivists such as Rudolf Carnap, analytic propositions (including tautologies) lack empirical content, while synthetic propositions (about the world) possess it. The analytic/synthetic distinction became central to logical empiricism, but was subsequently attacked by W.V.O. Quine in “Two Dogmas of Empiricism” (1951).
In classical first-order logic, the notion of tautology generalises to logical validity: a formula is logically valid if it is true in every possible structure (model). Gödel’s Completeness Theorem (1930) demonstrated that every valid formula of first-order logic is formally derivable — that is, the set of first-order tautologies coincides with the set of theorems of the predicate calculus.
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