Tautology (logic) - Tautological Implication

Tautological Implication

A formula R is said to tautologically imply a formula S if every valuation that causes R to be true also causes S to be true. This situation is denoted . It is equivalent to the formula being a tautology (Kleene 1967 p. 27).

For example, let S be . Then S is not a tautology, because any valuation that makes A false will make S false. But any valuation that makes A true will make S true, because is a tautology. Let R be the formula . Then, because any valuation satisfying R makes A true and thus makes S true.

It follows from the definition that if a formula R is a contradiction then R tautologically implies every formula, because there is no truth valuation that causes R to be true and so the definition of tautological implication is trivially satisfied. Similarly, if S is a tautology then S is tautologically implied by every formula.