In proof-theoretic semantics, meaning is based on inference. It may seen as the mathematical expression of the inferentialist interpretation of logic. Much recent work has focused on base-extension semantics, in which the validity of formulas is given by an inductive definition generated by provability in a ‘base’ of atomic rules. Base-extension semantics for classical and intuitionistic propositional logic have been explored by several authors. In this paper, we develop base-extension semantics for the classical propositional modal systems $K$, $KT$, $K4$ and $S4$, with $\square $ as the primary modal operator. We establish appropriate soundness and completeness theorems and establish the duality between $\square $ and a natural presentation of $\lozenge $. We also show that our semantics is in its current form not complete with respect to euclidean modal logics. Our formulation makes essential use of relational structures on bases.

中文翻译：

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模态逻辑的基础扩展语义

在证明理论语义学中，意义基于推理。它可以被视为逻辑推理主义解释的数学表达。最近的许多工作都集中在基础扩展语义上，其中公式的有效性是由原子规则“基础”中的可证明性生成的归纳定义给出的。几位作者已经探索了经典命题逻辑和直觉命题逻辑的基础扩展语义。在本文中，我们为经典命题模态系统 $K$、$KT$、$K4$ 和 $S4$ 开发了基础扩展语义，其中 $\square $ 作为主要模态运算符。我们建立适当的健全性和完整性定理，并建立 $\square $ 和 $\lozenge $ 的自然表示之间的对偶性。我们还表明，我们的语义目前的形式相对于欧几里得模态逻辑来说并不完整。我们的公式在基础上充分利用了关系结构。