Conditional Statement and Material Implication (1956)

Abstract

From a general Iecture course on formal logic students expect to be familiarized with the inferential schemata which underlie arguments in everyday life and in science. They expect that the inference forms presented in the lectures on logic will be applicable to their own thinking. Classical logic did not disappoint such expectations. Categorical syllogisms and so- called hypothetical syllogisms of classical logic were formulated in terms taken from ordinary language without altering the usual meaning of those terms; hence its inferential schemas were indeed ‘forms’ of actual inferences found in actual thinking. Contemporary mathematical logic seems not to live up to these expectations. The terms which occur in its theorems and inference schemas (so-called logical constants) seem to have meaning which none of the terms of ordinary language have, hence the theorems and inference schemas of contemporary logic appear to be useless for actual thinking in everyday- life and in science. This applies above all to the symbol of material implication. It seems that there is no term in ordinary language which would have a meaning identical with that of material implication. The conditional connective ‘if…, then,…’ used in ordinary language has only a similar but not identical meaning with that assigned by contemporary logic to the sign of material implication. This is why, though some theorems and inference schemas of mathematical logic formulated with the help of the sign of implication remain valid when that sign is replaced by the conditional connective ‘if…, then…’ taken in its usual sense (e.g. modus ponens. modus tollens, the law of transposition, etc.), others seem to become invalid when the implication sign is replaced in them by the conditional connective ‘if…’. then This applies to those theorems which depend on the fact that implication (i.e. ‘p→q’) is true whenever its antecedent is false or its consequent is true.