A Note on Disjunction and Existence Properties in Predicate Extensions of Intuitionistic Logic?An Application of Jankov Formulas to Predicate Logics
A Note on Disjunction and Existence Properties in Predicate Extensions of Intuitionistic Logic?An Application of Jankov Formulas to Predicate Logics
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直觉逻辑谓词扩展中析取和存在性的注解？Jankov公式在谓词逻辑中的应用
DOI：
10.1007/978-3-031-06843-0_9
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发表时间：
2022
期刊：
影响因子：
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通讯作者：
Suzuki Nobu-Yuki
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文献类型：
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作者：
小林誉明（大森佐和・西村幹子編）;西田充;西平等;Kyoko Hatakeyama;Suzuki Nobu-Yuki
直觉逻辑的谓词扩展（PEI）是具有与直觉逻辑相同的命题部分的中间谓词逻辑。直觉上，PEI必须类似于直觉主义逻辑。我们从分离性（DP）和存在性（EP）的角度讨论PEI。注意，DP和EP被认为是直觉主义逻辑的构造性的“标志”。然而，有无数的PEI同时具有DP和EP。此外，存在两种PEI的连续体：（1）每一种都缺乏DP和EP两者，以及（2）每一种都具有EP但缺乏DP。现在，一个自然的问题出现了：是否存在无数个PEI，每个都有DP而没有EP？我们肯定地回答这个问题。具体来说，我们通过使用修改后的Jankov公式构建了无数个这样的PEI。这一结果表明，尽管PEI的生活接近直觉逻辑，但其性质的多样性似乎很丰富。换句话说，PEI中的逻辑从逻辑的角度来看是迷人的，还有待探索。
Predicate extensions of intuitionistic logic (PEI’s) are intermediate predicate logics having the same propositional part as intuitionistic logic. Intuitively, PEI’s must resemble intuitionistic logic. We discuss PEI’s from the viewpoint of disjunction property (DP) and existence property (EP). Note that DP and EP are regarded as “hallmarks” of constructivity of intuitionistic logic. There are, however, uncountably many PEI’s having both of DP and EP. Moreover, there are two continua of PEI’s: (1) each of which lacks both of DP and EP, and (2) each of which has EP but lacks DP. Now, a natural question arises:Do there exist uncountably many PEI’s each of which has DP and lacks EP?We answer this question affirmatively. Specifically, we construct uncountably many such PEI’s by making use of modified Jankov formulas. This result suggests that although PEI’s are living near to intuitionistic logic, the diversity of their nature seems rich. In other words, logics among PEI’s are fascinating from the logical point of view and yet to be explored.