Comparing Lemmon's Essay-Faithful And Fruitful Logic

Faithful and Fruitful Logic

Appropriate for a conference relating philosophy and education, we seek ways more faithful than the truth-functional (TF) hook to understand and represent that ordinary-language conditional which we use in, e.g., modus ponens, and that conditional’s remote and counterfactual counterparts, and also the proper negations of all three. Such a logic might obviate the paradoxes caused by T-F representation, and be educationally fruitful. William and Martha Kneale and Gilbert Ryle assist us: "In the hypothetical case in which p, it is inferable, on the basis that p and at least in the given context, that q." "Inferable" is explained. This paraphrase is the foundation of the logic of hypothetical inferability ("HI logic").

–q’ comes out even more clearly in relation to Lemmon’s claim (Beginning Logic, p.61) that it is a logical truth that either if it is raining then it is snowing or if it is snowing then it is raining. Again the negation is crucial; deny the symbolic representation of that supposed logical truth and you doubly contradict yourself, with p . –q . q . –p. But will any educated person (not already imbued with symbolic logic) who is aware how few of the infinite range of imaginable conditionals are true accept that, as a matter of logic, at least one of any pair of statements representable by ‘If p then q’ and ‘If q then p’ must be true? No: for such persons will agree that conditionals are not properly asserted merely on the strength of knowledge, or of belief, that –p, or that q, nor properly denied by categorical statements of the form ‘p .

Consider the simplest case of hypothetical syllogism. Let us number the premises after a capital P, write ‘Con’ and ‘NCon’ for the conclusion and its negation, C for ‘Compatible with’ and I for ‘Incompatible with’. Let us also write S for ‘Subjunct’ for anything we can join on underneath because we could infer it if we supposed to be true both the NCon and a premise, or both that conjunction and a further premise, and so on (we can number subjuncts if we have more than one). Thus we