Material implication and entailment (Q1119616)

The author argues that the so-called ``paradoxes'' of material implication are not really paradoxes, since the antecedent of a material conditional is in fact always relevant to its consequent. This is shown, he holds, by the two tautologies \(\sim p\to \{(p\to q)\leftrightarrow [(p\cdot q)\to q]\}\) and \(q\to \{(p\to q)\leftrightarrow [(p\cdot q)\to q]\}\). For the paradoxes arise when \(p\to q\) is inferred from \(\sim p\) of from q, and in both cases \(p\to q\) is equivalent to \((p\cdot q)\to q\). (The author says ``logically equivalent'', but that is idiosyncratic terminology.) Nevertheless, he does concede that ``we may regard a material implication which is true only because its antecedent is false as degenerate.'' Such a conditional, he suggests, ought to be considered together with its equally true contrary. The resulting composite conditional with a patently false consequent may be regarded as a negation of the antecedent, as has been suggested before. In conclusion, the author proposes to define \(p\to q\) as \(p\leftrightarrow (p\cdot q)\) and `p entails q' as the necessitation thereof, i.e. as strict implication. The trouble with the author's approach is that the very truth-functional considerations he adduces may be doubted to preserve relevance.