Equivalence is a 1. (material equivalence) the combination of two formulas with the connective <-> (if and only if, iff), which is only true if both formulas have the same truth value. phi <-> psi can also be defined as the conjunction of two implications: phi -> psi and psi -> phi. For this reason, the connective of material equivalence is sometimes called the biconditional. The truth table for material equivalence is as follows:
(i) phi psi phi <-> psi 1 1 1 1 0 0 0 1 0 0 0 1
See Connective. 2. (logical equivalence) a relation obtaining between two formulas phi and psi if their material equivalence phi <-> Psi is a tautology. In other words, two formulas which are logically equivalent have the same truth value for every possible model.
phi -> psi is logically equivalent with Neg [ phi & Neg psi ] in propositional logic and ThereIs(x) [ P(x) ] is equivalent with Neg All(x) [ Neg P(x) ] in predicate logic. When two expressions are logically equivalent, it is possible to substitute them for each other, without changing the truth values of the proposition they are contained in.
- Gamut, L.T.F. 1991. Logic, language, and meaning, Univ. of Chicago Press, Chicago.