A comparison between monoidal and substructural logics. Issue 2 (2nd April 2016)

• Record Type: Journal Article
• Authors: Peterson, Clayton
• Is Part Of: Journal of applied non-classical logics. Volume 26:Issue 2(2016)
• Page Start: 126
• Page End: 159
• Publication Date: 2016-04-02
• Subjects: Non-classical logics -- linear logic -- display logic -- relevance logic -- classical systems -- De Morgan dualities; Nonclassical mathematical logic -- Periodicals; Wiskundige logica; Logic; Periodicals; 511.3105
• DOI: 10.1080/11663081.2016.1179528
• Languages: English
• ISSNs: 1166-3081

Abstract

Abstract : Monoidal logics were introduced as a foundational framework to analyse the proof theory of deontic logic. Building on Lambek's work in categorical logic, logical systems are defined as deductive systems, that is, as collections of equivalence classes of proofs satisfying specific rules and axiom schemata. This approach enables the classification of deductive systems with respect to their categorical structure. When looking at their proof theory, however, one can see that there are similarities between monoidal and substructural logics. The purpose of the present paper is to address this issue and highlight the differences between these two approaches. We argue that monoidal logics provide a more flexible foundational framework that enables a finer analysis of the relationship between negation(s) and other logical connectives. We show that the elimination of double negation(s) is independent from the de Morgan dualities, that monoidal deductive systems are not necessarily weakly distributive and that deductive systems satisfying the elimination of double negation(s) and the law of excluded middle are not necessarily classical.