Tautology (logic)
A formula true under all interpretations of its terms.
In mathematical logic, a tautology is a formula that is true under every possible interpretation of its component terms, with only the logical constants having a fixed meaning. It is a logical truth, such as 'the ball is green or the ball is not green,' which remains true regardless of what a ball is or its color. Tautologies are a key concept in propositional logic, where an effective method exists for testing whether a given formula is always satisfied.
- field
- Mathematical logic, propositional logic, predicate logic
- known_for
- A formula that is true under all interpretations; a logical truth; a key concept in propositional logic
- first_applied_by
- Ludwig Wittgenstein in 1921
- earlier_use_by
- Immanuel Kant in 1800, Gottlob Frege in 1884
- symbolism
- Double turnstile ⊨ S, tee symbol ⊤ for arbitrary tautology, 'Vpq' sometimes used
Lore & Background
The word tautology was used by ancient Greeks to describe a statement true merely by saying the same thing twice, a pejorative meaning still used for rhetorical tautologies. Between 1800 and 1940, the word gained new meaning in logic. In 1800, Immanuel Kant wrote in his book Logic that analytic propositions are tautological when the identity of concepts is explicit. In 1884, Gottlob Frege proposed that a truth is analytic if it can be derived using logic, but he maintained a distinction between analytic truths and tautologies (statements devoid of content).
Reader's Guide
Tautologies are fundamental to propositional logic, defined as propositional formulas true under any possible Boolean valuation of their variables. A key property is that an effective method exists for testing whether a given formula is always satisfied (or whether its negation is unsatisfiable). The definition extends to predicate logic, where many authors define a tautology as a sentence obtained by uniformly replacing each propositional variable of a propositional tautology with a first-order formula. This set is a proper subset of all logically valid sentences of predicate logic. The double turnstile notation ⊨ S indicates that S is a tautology, and the tee symbol ⊤ denotes an arbitrary tautology, with ⊥ representing an arbitrary contradiction. Tautologies are distinct from contradictions (unsatisfiable statements) and logically contingent formulas (neither tautology nor contradiction). Modern textbooks commonly restrict 'tautology' to valid sentences of propositional logic or valid predicate sentences reducible to propositional tautologies by substitution.
Did You Know?
- The philosopher Ludwig Wittgenstein first applied the term 'tautology' to redundancies of propositional logic in 1921, borrowing from rhetoric.
- A tautology is a formula whose negation is unsatisfiable; it cannot be false.
- The double turnstile notation ⊨ S is used to indicate that S is a tautology.
- Immanuel Kant wrote in 1800 that analytic propositions are tautological when the identity of concepts is explicit.
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