On logical structures

Abstract

Universal logic, as described by Béziau, is a structural approach to logic which aims to unify the large kaleidoscopic variety of logics, while preserving their diversity. In order to do this, Béziau’s original proposal was to consider logics irrespective of how they are generated −beyond, e.g., the dichotomy of syntax/semantics − via logical structures (which, according to his initial proposal, are pairs of the form (L , |−) where L is a set and |− ⊆ P(L ) × L ).In this thesis, we investigate the fruitfulness of this idea in detail. The results we obtain can be divided into three classes. The first two chapters (after the introductory one) are concerned with the properties of some particular types of logical structures. The fourth one hovers around Suszko’s Thesis and many-valued logics. In this chapter, we provide a definition of many-valued logical structures taking into account the language/metalanguage hierarchy; a generalized version of Suszko’s Thesis is outlined as well. The fifth chapter deals with the question: is it possible to describe paraconsistency independently of the language or connectives? We show that the answer to this question is affirmative, and propose several syntax independent defnitions of paraconsistency. The sixth and the final one Is a contribution to proof theory. In this chapter, we investigate cut-elimination theorems in some detail. We provide two non-algorithmic proofs of the same for the propositional fragment of LK, followed by an addressal of the more general question of elimination of an arbitrary rule. This forces us to consider the notions of ‘rules’, ‘sequent systems’ and related notions in more detail. We generalise these concepts and manage to prove such‘ RULE-elimination theorems’ in these generalised settings. It is important to notice that in spite of the diversity of the areas to which these results are generally assumed to belong, the underlying unity rests entirely upon the very simple definition of logical structures. The establishment of this simple yet nontrivial fact, is perhaps one of the most important contributions of this thesis