Jan Łukasiewicz

"When I became aware of the incompatibility of traditional theorems of modal propositions in 1920, I was in the process of establishing a normal bivalent propositional calculus based on the matrix method. At that time I was convinced that it was possible to demonstrate all the thesis of the ordinary propositional calculus assuming that propositional variables could take on only two values, "0" (false), and "1" (true). This assumption corresponds to the basic theorem that every proposition is either true or false. For brevity's sake, I will refer to it as the law of bivalence. Although it is sometimes referred to as the law of excluded middle, I prefer to restrict this latter term to the well known principle of classical logic which states that two contradictory propositions cannot both be false at the same time.

Our whole system of logic is based on the law of bivalence, even though it has been fiercely disputed since ancient times. Aristotle knew this law, but he questioned its validity as it referred to future contingent propositions. The law of bivalence was flatly rejected by the Epicureans. Chrysippus and the Stoics were the first ones to develop it fully and incorporate it as a principle of their dialectic, the equivalent of modern day propositional calculus. The arguments regarding the law of bivalence have metaphysical overtones: its supporters are resolute determinists; whereas its opponents generally have an indeterministic Weltanschauung. Thus, we are once again in the area of concepts of possibility and necessity."