Implication

Definition
Implication is a 1. (material implication) the combination in propositional logic of two formulae with the connective -&gt; (if ... then ...), also called conditional. The implication of phi and psi, phi -&gt; psi, is only false if phi (which is called the antecedent) is true while psi (the consequent) is false:

(i)	phi		psi	phi -&gt; 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 -&gt; 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) -&gt; 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).

Link

 * Utrecht Lexicon of Linguistics