What Is a Tautology? Statements That Cannot Be False and Say Nothing

In propositional logic, a tautology is a formula that comes out true under every possible assignment of truth values to its components. The simplest is the statement that either something is the case or it is not, which holds whatever the something is. Others include the assertion that if a thing is true then it is true, and more complex formulas whose tautological status is not obvious and must be checked, which is what truth tables are for. The property matters because the valid inference rules of a logical system correspond to tautologies, so establishing that a formula is one establishes that the corresponding argument pattern never leads from truth to falsehood. The cost of that guarantee is informational emptiness: a statement that is true no matter how the world is arranged excludes no possibility, and excluding possibilities is what carrying information amounts to. That trade is exact rather than metaphorical and it is the basis of a substantial claim about the nature of logic.

Several distinctions in the neighbourhood are frequently conflated:

• •Contradiction, the mirror image, false under every assignment, which also carries no information because it excludes everything
• •Contingent statements, which are true under some assignments and false under others, and which are therefore the ones that say something
• •Analytic truths, true in virtue of the meanings of their terms, as with the claim that all bachelors are unmarried, which is a broader category than logical tautology
• •Necessary truths, true in all possible circumstances, which may include mathematical and metaphysical claims that are not obviously true by meaning alone
• •Definitions, which are stipulations rather than discoveries and which are sometimes disguised as substantive claims
• •Circular arguments, where a premise assumes the conclusion, which is a defect in an argument rather than a property of a statement

Wittgenstein argued in the Tractatus that all the propositions of logic are tautologies and therefore say nothing, which was intended not as a dismissal but as an account of what logic is, namely something that shows the structure of language rather than describing facts. The logical positivists built on that, dividing meaningful statements into analytic ones, true by meaning and empty of factual content, and synthetic ones, verifiable by observation, and concluding that anything falling into neither category, including much of traditional metaphysics and ethics, was literally meaningless. That programme collapsed for several reasons, including that its own central principle fits neither category, and Quine attacked the analytic and synthetic distinction itself in a paper of 1951 arguing that no clear line separates truths held by meaning from truths held by fact, since both are revisable in the face of experience. The debate is not closed and the vocabulary from it is now standard.

Ordinary usage calls a statement tautological when it repeats itself or when it is trivially true in a way that pretends to be informative, as with a manager observing that the team needs to score more goals than the opposition. Sports commentary is the standard source of examples and a considerable body of humour depends on it. There is also a linguistic category, sometimes called pleonasm, covering redundant phrases such as free gift, added bonus and advance planning, and the objection to these is stylistic rather than logical, since redundancy in speech serves real functions including emphasis, clarity in noisy conditions and politeness. A third use appears in argument, where a claim is made to seem substantive by leaving the definition of a key term unstated, so that the claim turns out true by definition once the term is pinned down. That last case is worth spotting, since it is a way of appearing to assert something while risking nothing.

A formula true under every possible assignment cannot be false and therefore excludes no possibility, which is exactly why it carries no information, and that trade is the basis of Wittgenstein's claim that the propositions of logic say nothing. The positivist division into analytic and synthetic statements was attacked by Quine in 1951. Ordinary usage covers trivial repetition, and the useful version is spotting a claim that is true only by definition.