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Paraconsistent semantics for Pavelka style fuzzy sentential logic

机译：Pavelka风格模糊句子逻辑的超一致语义

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摘要

The root of this work is on the one hand in Belnap's four valued paraconsistent logic, and on the other hand on Pavelka's papers further developed by Turunen. We do not introduce a new non-classical logic but, based on a related study of Perny and Tsoukias, we introduce paraconsistent semantics of Pavelka style fuzzy sentential logic. Restricted to Lukasiewicz t-norm, our approach and the approach of Perny and Tsoukias partly overlap; the main difference lies in the interpretation of the logical connectives implication and negation. The essential mathematical tool proved in this paper is a one-one correspondence between evidence couples and evidence matrices that holds in all injective MV-algebras. Evidence couples associate to each atomic formula p two values a and b that can be interpreted as the degrees of pros and cons for p, respectively. Four values t, f, k, u, interpreted as the degrees of the truth, falsehood, contradiction and unknownness of p, respectively, can then be calculated by means of a and b and finally, the degrees of the truth, falsehood, contradiction and unknownness of any well formed formula a are available. The obtained logic is Pavelka style fuzzy sentential logic. In such an approach truth and falsehood are not each others complements. Moreover, we solve some open problems presented by Perny and Tsoukias.
机译：这项工作的根源一方面是贝尔纳普（Bernap）的四个有价值的超一致性逻辑，另一方面是图伦（Turunen）进一步开发的帕维尔卡（Pavelka）的论文。我们没有介绍新的非经典逻辑，但是，根据对Perny和Tsoukias的相关研究，我们介绍了Pavelka风格模糊句子逻辑的超一致语义。仅限于Lukasiewicz t范数，我们的方法与Perny和Tsoukias的方法部分重叠；主要区别在于对逻辑连接词的含义和否定的解释。本文证明的基本数学工具是证据对和证据矩阵之间的一一对应关系，该关系适用于所有内射MV代数。证据对与每个原子式p相关联，两个值a和b可以分别解释为p的优劣程度。然后可以通过a和b分别计算出四个值t，f，k，u，分别解释为p的真，假，矛盾和未知程度，最后得出真，假，矛盾的程度以及任何格式正确的公式a的未知性都是可用的。所获得的逻辑是Pavelka风格的模糊句子逻辑。在这种方法中，真与假不是彼此补充。此外，我们解决了Perny和Tsoukias提出的一些公开问题。

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