Sayantan Roy, Sankha S. Basu, Mihir K. Chakraborty
{"title":"Suszko's Thesis and Many-valued Logical Structures","authors":"Sayantan Roy, Sankha S. Basu, Mihir K. Chakraborty","doi":"arxiv-2408.13769","DOIUrl":null,"url":null,"abstract":"In this article, we try to formulate a definition of ''many-valued logical\nstructure''. For this, we embark on a deeper study of Suszko's Thesis\n($\\mathbf{ST}$) and show that the truth or falsity of $\\mathbf{ST}$ depends, at\nleast, on the precise notion of semantics. We propose two different notions of\nsemantics and three different notions of entailment. The first one helps us\nformulate a precise definition of inferentially many-valued logical structures.\nThe second and the third help us to generalise Suszko Reduction and provide\nadequate bivalent semantics for monotonic and a couple of nonmonotonic logical\nstructures. All these lead us to a closer examination of the played by\nlanguage/metalanguage hierarchy vis-\\'a-vis $\\mathbf{ST}$. We conclude that\nmany-valued logical structures can be obtained if the bivalence of all the\nhigher-order metalogics of the logic under consideration is discarded, building\nformal bridges between the theory of graded consequence and the theory of\nmany-valued logical structures, culminating in generalisations of Suszko's\nThesis.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Logic","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.13769","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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.