Recherches sur la bivalence

Logic is traditionally defined according to underlying principles. Among them, three seem particularly important. The principle of bivalence says that there are exactly two truth values, usually called True and False. The principle of excluded middle states that a sentence has at least one truth value. The principle of non-contradiction states that a sentence has at most one truth value. A logic that satisfies the conjunction of these three principles is called classical. By contrast, a logic is called non-classical if it does not obey at least one of them.\nIn relation to these principles, three bivalent logics differ from classical logic insofar as they ignore the principle of excluded middle and/or the principle of non-contradiction: consistent logic satisfies the principle of non-contradiction, complete logic satisfies the principle of excluded middle and positive logic ignores both of these principles. In addition to classical logic, three bivalent (non-classical) logics can therefore be distinguished.\nI provide a unified framework for studying the semantic and syntactic relationships between these four bivalent logics. More specifically, my purpose is to characterize the notion of logical consequence within each of these logics. To do this, I propose a new definition of the notions of model and sequent which makes explicit these principles. For each of the logics mentioned above, I give a notion of validity and propose an associated sequent calculus.