This paper presents an algebraic framework for investigating proposed translations of classical logic into intuitionistic logic, such as the four negative translations introduced by Kolmogorov, Gödel, Gentzen and Glivenko. We view these as variant semantics and present a semantic formulation of Troelstra’s syntactic criteria for a satisfactory negative translation. We consider how each of the above-mentioned translation schemes behaves on two generalisations of Heyting algebras: bounded pocrims and bounded hoops. When a translation fails for a particular class of algebras, we demonstrate that failure via specific finite examples. Using these, we prove that the syntactic version of these translations will fail to satisfy Troelstra’s criteria in the corresponding substructural logical setting.

中文翻译：

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Heyting 代数推广的双重否定语义

本文提出了一个代数框架，用于研究从经典逻辑到直觉逻辑的拟议翻译，例如 Kolmogorov、Gödel、Gentzen 和 Glivenko 介绍的四种否定翻译。我们将这些视为变体语义，并提出了 Troelstra 句法标准的语义表述，以获得令人满意的否定翻译。我们考虑上述每种翻译方案在 Heyting 代数的两个推广上的表现：有界 pocrims 和有界箍。当特定类别的代数翻译失败时，我们通过特定的有限示例来证明该失败。使用这些，我们证明这些翻译的句法版本将无法在相应的子结构逻辑设置中满足 Troelstra 的标准。