ABSTRACT
Cf. also Paton’s commentary, The Categorical Imperative, 1947.) B.A.O.Williams and P.T.Geach, ‘Imperative inference’, Analysis, 1963 (supplement).
(Can one infer one imperative from another?)
‘Implication’ is the most general name for those relations between propositions or statements in virtue of which we can infer the truth of a proposition or statement from something else. A minimum condition for such a relation to hold (except contextual implication: see below) is that if one proposition, p, implies another, q, it is not the case that p is true and q is false. Whenever this condition is fulfilled, and provided p and q are each either true or false, we say that p materially implies q. Hence a false proposition
materially implies any proposition (for if p is false it is not the case that p is true and q false), and any proposition materially implies a true proposition; these facts are called the ‘paradoxes’ of material implication, though they are only paradoxical in the sense of sounding odd because ‘implies’ in ordinary speech suggests a stronger relation. Material implication is usually symbolized by ‘⊃’, which is specific to it, or ‘→’, which can also stand for other relations, including entailment.
