The paradoxes of material implication

Abstract

'Paradoxes of material implication' is a significant topic in modern symbolic and mathematical logic. Various attempts have been taken to resolve these paradoxes. Thus, a number of schools of logic have been developed in this regard. In our present paper we examine three of the main schools of modern logic which deal with these paradoxes: many valued logic, modal logic and relevance logic. Three-valued logic, which is a kind of many-valued logic, fails to show any promise in resolving these paradoxes as it adopts the entire truth table based on traditional bivalence. Five-valued logic, another kind of many-valued logic, shows some promises in resolving these paradoxes. But it destroys the system of propositional calculus and the process of judging the validity/invalidity of arguments. So it is difficult to accept this solution with such a price. Modal logic, the second approach discussed in this paper, resolves these paradoxes by introducing the device of strict implication. But the problem is that it creates some new paradoxes, namely paradoxes of strict implication, which are analogous to the paradoxes of material implication. Hence modal logic is also not adequate. Relevance logic, however, resolves these paradoxes without creating any new paradox. It does not destroy the system of propositional calculus or the process of judging the validity/invalidity of arguments. Hence, relevance logic solves the problems of the logic of implication. Although there are some minor difficulties in relevance logic, we hope that more work in this area will resolve these problems soon, so that relevance logic will provide a fully adequate logic of implication.