Suszko's Thesis and Many-valued Logical Structures

TL;DR

This paper explores the concept of many-valued logical structures by analyzing Suszko's Thesis, proposing new semantics and entailment notions, and generalizing the thesis to connect graded consequence with many-valued logic.

Contribution

It introduces two semantics and three entailment notions to define and generalize many-valued logical structures and Suszko's Thesis.

Findings

Different semantics enable precise definitions of many-valued structures.

Generalizations of Suszko's Thesis relate graded consequence to many-valued logic.

Discarding bivalence in higher-order metalogics allows for new many-valued structures.

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.