We take Carnap’s problem to be to what extent standard consequence relations in various formal languages fix the meaning of their logical vocabulary, alone or together with additional constraints on the form of the semantics. This paper studies Carnap’s problem for basic modal logic. Setting the stage, we show that neighborhood semantics is the most general form of compositional possible worlds semantics, and proceed to ask which standard modal logics (if any) constrain the box operator to be interpreted as in relational Kripke semantics. Except when restricted to finite domains, no modal logic characterizes exactly the Kripkean interpretations of $\Box $ . Moreover, we show that, in contrast with the case of first-order logic, the obvious requirement of permutation invariance is not adequate in the modal case. After pointing out some known facts about modal logics that nevertheless force the Kripkean interpretation, we focus on another feature often taken to embody the gist of modal logic: locality. We show that invariance under point-generated subframes (properly defined) does single out the Kripkean interpretations, but only among topological interpretations, not in general. Finally, we define a notion of bisimulation invariance—another aspect of locality—that, together with a reasonable closure condition, gives the desired general result. Along the way, we propose a new perspective on normal neighborhood frames as filter frames, consisting of a set of worlds equipped with an accessibility relation, and a free filter at every world.

我们将Carnap 的问题视为各种形式语言中的标准结果关系在多大程度上固定其逻辑词汇的含义，单独或与语义形式的附加约束一起。本文研究了基本模态逻辑的卡尔纳普问题。设置阶段，我们表明邻域语义是组合可能世界语义的最一般形式，并继续询问哪些标准模态逻辑（如果有的话）约束框运算符被解释为关系 Kripke 语义。 除非限制在有限域内，否则没有模态逻辑能准确描述$\Box$ 的 Kripkean 解释 . 此外，我们表明，与一阶逻辑的情况相比，置换不变性的明显要求在模态情况下是不够的。在指出一些关于模态逻辑的已知事实之后，这些事实仍然强制克里普克解释，我们关注另一个经常被用来体现模态逻辑要点的特征：局部性。我们表明，点生成子框架（正确定义）下的不变性确实挑出 Kripkean 解释，但仅在拓扑解释中，而不是一般情况下。最后，我们定义了互模拟不变性的概念——局部性的另一个方面——与合理的闭合条件一起，给出了期望的一般结果。在此过程中，我们提出了一种将普通邻域框架作为过滤器框架的新观点，