Basic Logic and the Cube of its Extensions

Abstract

Up to the end of the last century, the only logic was classical logic (and possibly its extensions by modalities). Later some “weakenings” of classical logic were introduced, with the aim of expressing also at the level of logical propositions some distinctions which hold in a specific scientific context but are ignored by classical logic. The first example arises from intuitionism, which points out the distinction, when dealing with infinity, between constructive proofs and proofs based on reductio-adabsurdum; intuitionistic logic, by rejecting the law of double negation, allows to express such a distinction. In the thirties, it was realized that ortholattices (or orthomodular lattices), rather than boolean algebras, were the convenient algebraic structures to deal with quantum mechanics; thus in orthologic, as well as in ortholattices to which it corresponds, the classical equation given by distributivity of conjunction with disjunction fails Finally, various motivations lead to the third, more recent “weakening” of classical logic. The philosophical aim of overcoming paradoxes of classical implication produced relevant logics and, later, proof-theoretical motivations and the search for a logic well suited for theoretical computer science, produced linear logic; the common technical aspect is the rejection of one or more structural rules, which results for instance in the distinction made by linear logic between multiplicative and additive conjunction.