From Many-Valued Consequence to Many-Valued Connectives

TL;DR

This paper investigates which connectives, including conditionals, negation, conjunction, and disjunction, can be defined within many-valued logics, especially intersective mixed consequence relations, using computer-aided analysis for 3- and 4-valued cases.

Contribution

It provides necessary and sufficient conditions for the existence of Gentzen-regular connectives in multi-valued logics, extending understanding of connectives in many-valued consequence relations.

Findings

Mixed consequence relations admit all classical connectives.

Pure consequence relations admit no other Gentzen-regular connectives.

Conditionals exist for broader classes of intersective mixed relations, excluding order-theoretic ones.

Abstract

Given a consequence relation in many-valued logic, what connectives can be defined? For instance, does there always exist a conditional operator internalizing the consequence relation, and which form should it take? In this paper, we pose this question in a multi-premise multi-conclusion setting for the class of so-called intersective mixed consequence relations, which extends the class of Tarskian relations. Using computer-aided methods, we answer extensively for 3-valued and 4-valued logics, focusing not only on conditional operators, but on what we call Gentzen-regular connectives (including negation, conjunction, and disjunction). For arbitrary N-valued logics, we state necessary and sufficient conditions for the existence of such connectives in a multi-premise multi-conclusion setting. The results show that mixed consequence relations admit all classical connectives, and among them pure consequence relations are those that admit no other Gentzen-regular connectives. Conditionals can also be found for a broader class of intersective mixed consequence relations, but with the exclusion of order-theoretic consequence relations.