Tautology

https://en.wikipedia.org/wiki/Tautology_(logic)

In logic, a tautology is a formula or assertion that is true in every possible interpretation. ταυτολογία

The philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921, borrowing from rhetoric, where a tautology is a repetitive statement.

In logic, a formula is satisfiable if it is true under at least one interpretation, and thus a tautology is a formula whose negation is unsatisfiable.

Unsatisfiable statements, both through negation and affirmation, are known formally as contradictions.

A formula that is neither a tautology nor a contradiction is said to be logically contingent. Such a formula can be made either true or false based on the values assigned to its propositional variables.

The double turnstile notation ⊨ S is used to indicate that S is a tautology. Tautology is sometimes symbolized by "Vpq", and contradiction by "Opq".

The ⊤ symbol is sometimes used to denote an arbitrary tautology, with the dual symbol ⊥ (falsum) representing an arbitrary contradiction. In any symbolism, a tautology may be substituted for the truth value "true".

Tautologies are a key concept in propositional logic, where a tautology is defined as a propositional formula that is true under any possible Boolean valuation of its propositional variables. A key property of tautologies in propositional logic is that an effective method exists for testing whether a given formula is always satisfied; or equivalentlly, whether its negation is unsatisfiable.

The definition of tautology can be extended to sentences in predicate logic, which may contain quantifiers. In propositional logic, there is no distinction between a tautology and a logically valid formula. In the context of predicate logic, many authors define a tautology to be a sentence that can be obtained by taking a tautology of propositional logic, and uniformly replacing each propositional variable by a first-order formula (one formula per propositional variable). The set of such formulas is a proper subset of the set of logically valid sentences of predicate logic (that is, sentences that are true in every model).