H. P. Grice, "Implicata for the Two Alleged Paradoxes of 'If'"

material implication, paradoxes of: the unpalatable consequences arising from the definition of material implication. A false proposition, merely because it is false, implies every proposition; and a true proposition, merely because it is true, is implied by every proposition. Put it in another way, whenever P is false, P ⊃ Q is true; whenever Q is true, P ⊃ Q is true. The problem arises because implication is ordinarily used for a relation between two propositions, while a statement of material implication can be true even if (as is usually the case with H. P. Grice, when playing bridge) there is NO RELATION at all between its component propositions. Material implication does not concern the subject-matter or content of its components. To avoid these slightly ‘paradoxical’ consequences, it is suggested that we speak of the material ‘conditional,’ or just ‘the ‘if’’ instead of material implication. “Russell’s definition of ‘p implies q’ as synonymous with ‘either not p or q’ solicited the justified objection that according to it (1) a true proposition is implied by any proposition and (2) a false proposition implies any proposition (the two paradoxes of material implication – Johnson recognised only one).” Pap, Elements of Analytic Philosophy