Implication is a binary propositional connective read most often as an "if-then" statement. An "If-then" statement is composed of an antecedent (what comes after the "if"), and a consequent (what comes after the "then"). Implication is defined as follows: the antecedent S implies the consequent P if and only if it is impossible for S to be true while P is false. For example, the proposition "If there are objects, then there is space" expresses an implication. Space is implied by objects. If it is true that there are objects, then it must also be true that there is space, since space is a necessary condition of the existence of objects. This type of implication is called material implication, and it can also be expressed as a necessary implication, as it is with respect to validity.

Implication is a form of inference closely related to the concept of validity. In an argument, a set of true propositions that implies or entails another true proposition with necessity is a valid argument. However, whereas validity is a feature of certain deductive arguments, implication expresses the relationship between propositions and sets of propositions.

Not all "if-then" propositions express material or necessary implication. For example, "If Lee Harvey Oswald had not killed Kennedy, then someone else would have killed Kennedy," expresses a counter-factual implication--that is, an "if-then" statement whose antecedent is false. The implication is only probable at best, because if it is false that Lee Harvey Oswald had killed Kennedy, then it is only probable that someone else would have killed Kennedy. There is no good reason to infer that someone necessarily would have killed Kennedy if Oswald had not.