implication a relation that holds between two statements when the truth of the first ensures the truth of the second. A number of statements together imply Q if their joint truth ensures the truth of Q. An argument is deductively valid exactly when its premises imply its conclusion. Expressions of the following forms are often interchanged one for the other: ‘P implies Q’, ‘Q follows from P’, and ‘P entails Q’. (‘Entailment’ also has a more restricted meaning.) In ordinary discourse, ‘implication’ has wider meanings that are important for understanding reasoning and communication of all kinds. The sentence ‘Last Tuesday, the editor remained sober throughout lunch’ does not imply that the editor is not always sober. But one who asserted the sentence typically would imply this. The theory of conversational implicature explains how speakers often imply more than their sentences imply. The term ‘implication’ also applies to conditional statements. A material implication of the form ‘if P, then Q’ (often symbolized ‘P P Q’ or ‘P / Q’) is true so long as either the if-clause P is false or the main clause Q is true; it is false only if P is true and Q is false. A strict implication of the form ‘if P, then Q’ (often symbolized ‘P Q’) is true exactly when the corresponding material implication is necessarily true; i.e., when it is impossible for P to be true when Q is false. The following valid forms of argument are called paradoxes of material implication: Q. Therefore, P / Q. Not-P. Therefore, P / Q. The appearance of paradox here is due to using ‘implication’ as a name both for a relation between statements and for statements of conditional form. A conditional statement can be true even though there is no relation between its components. Consider the following valid inference: Butter floats in milk. Therefore, fish sleep at night / butter floats in milk. Since the simple premise is true, the conditional conclusion is also true despite the fact that the nocturnal activities of fish and the comparative densities of milk and butter are completely unrelated. The statement ‘Fish sleep at night’ does not imply that butter floats in milk. It is better to call a conditional statement that is true just so long as it does not have a true if-clause and a false main clause a material conditional rather than a material implication.
Strict conditional is similarly preferable to ‘strict implication’. Respecting this distinction, however, does not dissolve all the puzzlement of the so-called paradoxes of strict implication:
Necessarily Q. Therefore, P Q.
Impossible that P. Therefore, P Q. Here is an example of the first pattern:
Necessarily, all rectangles are rectangles.
Therefore, fish sleep at night all rectangles are rectangles. ‘All rectangles are rectangles’ is an example of a vacuous truth, so called because it is devoid of content. ‘All squares are rectangles’ and ‘5 is greater than 3’ are not so obviously vacuous truths, although they are necessary truths. Vacuity is not a sharply defined notion.
Here is an example of the second pattern:
It is impossible that butter always floats in milk yet sometimes does not float in milk.
Therefore, butter always floats in milk yet sometimes does not float in milk fish sleep at night. Does the if-clause of the conclusion imply (or entail) the main clause? On one hand, what butter does in milk is, as before, irrelevant to whether fish sleep at night. On this ground, relevance logic denies there is a relation of implication or entailment. On the other hand, it is impossible for the if-clause to be true when the main clause is false, because it is impossible for the if-clause to be true in any circumstances whatever.
See also COUNTERFACTUALS , FORMAL LOGIC , IMPLICATURE , PRESUPPOSITION , RELE – VANCE LOGI. D.H.S.