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L ogic In its ordinary sense, implication is a synonym of entailment , a logical relation between one or a set of premises and a consequence deduced from this premise or set of premises. It is most commonly expressed in sentences of the form “if p then q,” when p is the implying proposition (also called the antecedent or protasis), and q is the implied proposition (also called the consequent or apodasis). Russell and Whitehead used the term material implication to express the relation between the antecedent and consequent of a true conditional proposition, which is symbolized as p ⊃ q or as p → q. In order to avoid the so-called paradoxes of material implication, C. I. Lewis introduced a notion of strict implication , saying that p strictly implies q if and only if it was impossible that p should be true and q false. Other attempts to further clarify the meaning relation between antecedent and consequent include Carnap 's L-implication and the system of entailment. “In order to be able validly to infer the truth of a proposition, we must know that some other proposition is true, and that there is between the two a relation of the sort called ‘implication’, that is that (as we say) the premise ‘implies’ the conclusion.” Russell, Introduction to Mathematical Philosophy ... log in or subscribe to read full text