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A Refined Interpretation of Intuitionistic Logic by Means of Atomic Polymorphism
Studia Logica ( IF 0.7 ) Pub Date : 2019-03-27 , DOI: 10.1007/s11225-019-09858-1
José Espírito Santo , Gilda Ferreira

We study an alternative embedding of IPC into atomic system F whose translation of proofs is based, not on instantiation overflow, but instead on the admissibility of the elimination rules for disjunction and absurdity (where these connectives are defined according to the Russell–Prawitz translation). As compared to the embedding based on instantiation overflow, the alternative embedding works equally well at the levels of provability and preservation of proof identity, but it produces shorter derivations and shorter simulations of reduction sequences. Lambda-terms are employed in the technical development so that the algorithmic content is made explicit, both for the alternative and the original embeddings. The investigation of preservation of proof-reduction steps by the alternative embedding enables the analysis of generation of “administrative” redexes. These are the key, on the one hand, to understand the difference between the two embeddings; on the other hand, to understand whether the final word on the embedding of IPC into atomic system F has been said.

中文翻译:

原子多态对直觉逻辑的精解

我们研究了将 IPC 嵌入到原子系统 F 中的替代方法,其证明的翻译不是基于实例化溢出,而是基于析取和荒谬的消除规则的可接受性(其中这些连接词是根据 Russell-Prawitz 翻译定义的) . 与基于实例化溢出的嵌入相比,替代嵌入在证明身份的可证明性和保存级别上同样有效,但它产生更短的推导和更短的缩减序列模拟。在技​​术开发中使用了 Lambda 术语,以便使算法内容明确,无论是替代嵌入还是原始嵌入。通过替代嵌入对证明减少步骤的保存进行调查,可以分析“管理”redexes 的生成。一方面,这些是理解两种嵌入之间差异的关键;另一方面,了解是否已经说过将 IPC 嵌入原子系统 F 的最终决定。
更新日期:2019-03-27
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