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SEMANTICS: 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. EXAMPLE: 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
This implication is true for every combination of truth values for p and q. A logical consequence of predicate logic is the consequence of ThereIs(x) [ P(x) ] from P(c).
LIT. Gamut, L.T.F. (1991)