Joseph S. Fulda
Consider first the denied conditional: It is not the case that if I win the lottery, I will travel the world. I might know this to be true because I know myself and because when I won the lottery two years back, I was not taken with a desire to travel the world. Yet, the truth conditions of this denied conditional are met only if I win the lottery and do not travel the world. From this, in turn, it follows that if the proposition is true, as I claim it is, I will win the lottery! The deduction is immediate: ~(L->T) |- L.
Consider, as a second example, the denied biconditional: It is not the case that I am eligible for Social Security if and only if I am of retirement age. This denial is true: I might be disabled. However, in classical logic, we have: ~(E<->R)<->((~E)<->R). Yet this is surely false, for it claims, contrary-to-fact, that I am ineligible for Social Security if and only if I am of retirement age! All this arises, it would seem, from the traditional definition of material implication.
There is a way, however, of saving denied conditionals within classical logic, and it is quite simple. What we must do is move from the propositional calculus to the predicate calculus and quantify over cases in the latter. Thus, the claim about Social Security becomes ~(Vx)(Ex<->Rx), from which (Ex)(Ex&~Rx) follows, but (Vx)(~Ex<->Rx) most assuredly does not. Likewise, the claim about the lottery-winner becomes ~(Vx)(Lx->Tx), from which the truth of Lx is not known a priori but depends on the case, x, in question.
Acknowledgments
The author wishes to acknowledge the very helpful comments of Professors Fred Johnson, Peter Milne, and David Sherry and to lovingly dedicate this article to Lycette Mizrahi.
Joseph S Fulda, CSE, PhD
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