A Propositional Logic With 4 Values: True, False, Divergent and Meaningless

In this paper, we survey 3-valued logics and their complete axiomatizations, one of which is new. We then propose a 4-valued, functionally complete logic that incorporates these 3-valued systems and provide notations for interesting operators and subsystems. Key Words & Phrases: 3-valued logic, 4-valued logic, axiomatics of non-classical theories. 1991 Mathematics Subject Classification: 03B50, 03B60, 03G10. Contents 1 Introduction 1 2 Three-valued propositional calculi 1 3 Incorporating three-valued logic into four-valued logic 8 1 Introduction Three-valued logic is relevant from different points of view and has been studied intensively by many authors including D.A. Bochvar, S.C. Kleene and J. McCarthy. On the basis of an inspection of their systems, we adopt a couple of classical principles which exclude some prominent options such as the systems of A. Heyting, J. / Lukasiewicz and N.D. Belnap. We thus obtain a survey of 3-valued logics with complete axiom systems, one of which ..