Abstract

We show that classical two-valued logic is included in weak extensions of normal three-valued logics and also that normal three-valued logics are best viewed not as deviant logics but instead as strong extensions of classical two-valued logic obtained by adding a modal operator and the right axioms. This article develops a general method for formulating the right axioms to construct a two-valued system with theorems that correspond to all of the logical truths of any normal three-valued logic. The extended classical system can then express anything that can be expressed in the three-valued logic, so there can be no reason to abandon two-valued logic in favor of three-valued logic. Moreover, the two-valued modal system is preferable, because it enables us to study interactions of different operators with different rationales. It also makes it easier to introduce quantifiers and iteration. Nothing is lost and much is gained by choosing the extended two-valued approach over normal three-valued logics