The logics CS4 and IS4 are intuitionistic variants of the modal logic S4. Whether the finite model property holds for each of these logics has been a long-standing open problem. In this paper we introduce two logics closely related to IS4: GS4, obtained by adding the Godel-Dummett axiom to IS4, and S4I, obtained by reversing the roles of the modal and intuitionistic relations. We then prove that CS4, GS4, and S4I all enjoy the finite model property.

中文翻译：

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具有有限模型属性的S4的一些构造性变体

逻辑CS4和IS4是模态逻辑S4的直觉变体。有限模型属性是否适用于这些逻辑中的每一个，一直是一个长期存在的问题。在本文中，我们介绍了与IS4密切相关的两种逻辑：通过将Godel-Dummett公理添加到IS4获得的GS4，以及通过逆转模态和直觉关系的作用获得的S4I。然后，我们证明CS4，GS4和S4I都具有有限模型属性。