We establish completeness and the finite model property for logics featuring the pooling modalities that were introduced in Van De Putte and Klein (Pooling modalities and pointwise intersection: semantics, expressivity, and applications). The definition of our canonical models combines standard techniques with a so-called “puzzle piece construction”, which we first illustrate informally. After that, we apply it to the weakest classical logics with pooling modalities and investigate the technique’s potential for the axiomatization of stronger logics, obtained by imposing well-known frame conditions on the models.

中文翻译：

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池化模态和逐点交集：公理化和可判定性

我们为具有 Van De Putte 和 Klein 中引入的池化模态（池化模态和逐点交集：语义、表达性和应用）的逻辑建立完整性和有限模型属性。我们的规范模型的定义将标准技术与所谓的“拼图结构”相结合，我们首先非正式地说明了这一点。之后，我们将其应用于具有池化模态的最弱经典逻辑，并通过在模型上强加众所周知的框架条件来研究该技术对更强逻辑公理化的潜力。