A UNIFORM APPROACH TO AXIOMATIZING BUNDLED OPERATORS

Abstract Recent years have witnessed extensive logical studies of bundled operators. A known difficulty of such studies is how to axiomatize bundled operators. In this paper, we propose a uniform approach to axiomatizing such operators, which we call ‘the method based on almost-definability schemas’. The approach is useful in finding axioms and inference rules, and in defining a suitable canonical relation, thus in the completeness proof of logical systems. This is, hopefully, good news for modal logicians who are interested in axiomatizing bundled operators. To explicate this approach, we choose four bundled operators—the operator N of purely physical necessity, the operator $N'$ called ‘All $_1$ and Only $_2$ ’, the operator $N"$ , and the operator $N"'$ , in the literature, where $N\varphi := \Box _1\varphi \land \neg \Box _2\varphi $ , $N'\varphi := \Box _1\varphi \land \Box _2\neg \varphi $ , $N"\varphi := \Box _1\varphi \vee \neg \Box _2\varphi $ , $N"'\varphi := \Box _1\varphi \vee \Box _2\neg \varphi $ . This approach can uniformly deal with axiomatizing these bundled operators. Among other contributions, we also answer several open questions, and obtain alternative axiomatizations which are deductively equivalent to the existing ones in the literature.