Implication is a 1. (material implication) the combination in propositional logic of two formulae with the connective -> (if ... then ...), also called conditional. The implication of phi and psi, phi -> psi, is only false if phi (which is called the antecedent) is true while psi (the consequent) is false:
(i) phi psi phi -> psi 1 1 1 1 0 0 0 1 1 0 0 1
2. (logical implication) the relation that exists between two sentences phi and psi if phi -> psi is a tautology. In other words, psi is the logical implication or logical consequence of phi if psi is true in every model in which phi is true.
That q is a logical implication of (p V q) can be demonstrated by merely setting up the truth table for the formula in (ii):
(ii) (p V q) -> q
- Gamut, L.T.F. 1991. Logic, language, and meaning, Univ. of Chicago Press, Chicago.