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.

翻译：在证明论语义学中，意义基于推理，可视为逻辑推理主义解释的数学表达。近期大量研究聚焦于基础扩展语义学——该理论通过原子规则"基础"中的可证性生成归纳定义，从而确定公式的有效性。多位学者已探索了经典命题逻辑和直觉主义命题逻辑的基础扩展语义。本文针对以□为主要模态算子的经典命题模态系统 K、KT、K4 和 S4 发展基础扩展语义，建立了相应的可靠性与完备性定理，并揭示了□与其自然呈现的◇之间的对偶性。同时证明该语义在当前形式下对欧几里得模态逻辑不完全成立。我们的表述本质上依赖于基础之上的关系结构。