From Many-Valued Consequence to Many-Valued Connectives. Chemla, E. & Egré, P. Synthese, 198(22):5315-5352, 2021. ![From Many-Valued Consequence to Many-Valued Connectives [link]](https://bibbase.org/img/filetypes/link.svg "link")
Paper More doi abstract bibtex 20 downloads 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.
@article{ChemlaEgre-ManyValuedConsequenceAndConnectives,
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.},
author = {Emmanuel Chemla and Paul Egr\'e},
date-added = {2018-08-31 11:11:42 +0000},
date-modified = {2021-11-04 22:24:29 +0100},
doi = {10.1007/s11229-019-02344-0},
journal = {Synthese},
number = {22},
pages = {5315-5352},
title = {From Many-Valued Consequence to Many-Valued Connectives},
url = {https://arxiv.org/pdf/1809.01066},
url_more = {https://arxiv.org/abs/1809.01066v1/anc},
volume = {198},
year = {2021},
bdsk-url-1 = {https://www.dropbox.com/s/utmc0c09r59dk4j/Chemla-Egre-From%20Many-Valued%20Consequence%20to%20Many-Valued%20Connectives.pdf?dl=0}}
Downloads: 20
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