In this paper we consider the modal logic with both $$\Box $$ □ and $$\Diamond $$ ◊ arising from Kripke models with a crisp accessibility and whose propositions are valued over the standard Gödel algebra $$[0,1]_G$$ [ 0 , 1 ] G . We provide an axiomatic system extending the one from Caicedo and Rodriguez (J Logic Comput 25(1):37–55, 2015) for models with a valued accessibility with Dunn axiom from positive modal logics, and show it is strongly complete with respect to the intended semantics. The axiomatizations of the most usual frame restrictions are given too. We also prove that in the studied logic it is not possible to get $$\Diamond $$ ◊ as an abbreviation of $$\Box $$ □ , nor vice-versa, showing that indeed the axiomatic system we present does not coincide with any of the mono-modal fragments previously axiomatized in the literature.

中文翻译：

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Crisp Gödel 模态逻辑的公理化

在本文中，我们考虑了具有 $$\Box $$ □ 和 $$\Diamond $$ ◊ 的模态逻辑，这些模态逻辑源自具有清晰可访问性的 Kripke 模型，其命题的价值高于标准哥德尔代数 $$[0,1] _G$$ [ 0 , 1 ] G . 我们提供了一个公理系统，扩展了 Caicedo 和 Rodriguez (J Logic Comput 25(1):37–55, 2015) 的模型，该模型具有来自正模态逻辑的 Dunn 公理的有价值的可访问性，并表明它在预期的语义。也给出了最常见的框架限制的公理化。我们还证明，在所研究的逻辑中，不可能得到 $$\Diamond $$ ◊ 作为 $$\Box $$ □ 的缩写，反之亦然，表明我们提出的公理系统确实与以前在文献中公理化的任何单模态片段。