On infinite matrices and the paradoxes of material implication.
William Clyde Wilcox · Notre Dame Journal of Formal Logic · 1970
The purpose of this note is to lend some justification to Lukasiewicz's claim that, when one has gone beyond 3-valued logics, the next interesting case is infinitely many-valued logics. 1 One of the things that drives people into many-valued logics is the feeling of being 'cramped' in the classical, 2-valued logic.This feeling of being cramped particularly makes itself felt when we are working with the conditional, or implication, and to some extent furnishes an alibi for the paradoxes of material implication.We find ourselves saying something like this: 'Ordinarily we would not say that a conditional with a false antecedent (or whatever) was true, but it is in a 2-valued logic . . .."We feel forced to put different animals in the same cage.Propositions which strike us as being significantly different are assigned the same truth-value.This does not occur with alternation, conjunction, or any of the commutative functors-those in which the order of the arguments is irrelevant to the truth-value of the function of which they are arguments.As can be seen, we have as many values, in such cases, as there are ways in which the arguments can differ in value.Consider these cases 2 : 2-valued 3-valued n-valued