Suszko's Thesis and Many-valued Logical Structures

Sayantan Roy, Sankha S. Basu, Mihir K. Chakraborty
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Abstract

In this article, we try to formulate a definition of ''many-valued logical structure''. For this, we embark on a deeper study of Suszko's Thesis ($\mathbf{ST}$) and show that the truth or falsity of $\mathbf{ST}$ depends, at least, on the precise notion of semantics. We propose two different notions of semantics and three different notions of entailment. The first one helps us formulate a precise definition of inferentially many-valued logical structures. The second and the third help us to generalise Suszko Reduction and provide adequate bivalent semantics for monotonic and a couple of nonmonotonic logical structures. All these lead us to a closer examination of the played by language/metalanguage hierarchy vis-\'a-vis $\mathbf{ST}$. We conclude that many-valued logical structures can be obtained if the bivalence of all the higher-order metalogics of the logic under consideration is discarded, building formal bridges between the theory of graded consequence and the theory of many-valued logical structures, culminating in generalisations of Suszko's Thesis.
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苏兹科论文和多值逻辑结构
在本文中,我们试图为 "多值逻辑结构 "下一个定义。为此,我们开始深入研究苏斯科论题($\mathbf{ST}$),并证明$\mathbf{ST}$的真假至少取决于精确的语义学概念。我们提出了两种不同的语义概念和三种不同的蕴涵概念。第一个概念有助于我们为推论多值逻辑结构下一个精确的定义。第二个和第三个概念有助于我们对苏斯克还原进行广义化,并为单调逻辑结构和一些非单调逻辑结构提供适当的二价语义。所有这些都使我们对所扮演的语言/金属语言层次结构与$mathbf{ST}$的关系进行了更仔细的考察。我们的结论是,如果摒弃所考虑的逻辑的所有高阶金属语言的二价性,就可以得到多值逻辑结构,从而在分级后果理论与多值逻辑结构理论之间架起了正式的桥梁,并最终概括了苏兹科的论断。
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