2018/09/04 by Emmanuel Chemla, Chemla, Emmanuel, Paul Egré +1
Mathematics · #03B50 #03B53 #FOS: Mathematics #Logic (math.LO) #math.LO #msc:03B50 #msc:03B53
paper · pdf · doi:10.48550/arxiv.1809.01066
Updated version [corrections of an incorrect claim in first version; two bib entries added]
arxiv created 2019/07/25 · arxiv updated 2019/07/26
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.