An Abstract Approach to Consequence Relations

We generalise the Blok-Jónsson account of structural consequence relations, later developed by Galatos, Tsinakis and other authors, in such a way as to naturally accommodate multiset consequence. While Blok and Jónsson admit, in place of sheer formulas, a wider range of syntactic units to be manipul...

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Bibliographic Details
Published in:arXiv.org 2019-03
Main Authors: Cintula, Petr, José Gil Férez, Moraschini, Tommaso, Paoli, Francesco
Format: Article
Language:English
Subjects:
Fuzzy logic
Fuzzy sets
Monoids
Semantics
Online Access:Get full text
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Summary:We generalise the Blok-Jónsson account of structural consequence relations, later developed by Galatos, Tsinakis and other authors, in such a way as to naturally accommodate multiset consequence. While Blok and Jónsson admit, in place of sheer formulas, a wider range of syntactic units to be manipulated in deductions (including sequents or equations), these objects are invariably aggregated via set-theoretical union. Our approach is more general in that non-idempotent forms of premiss and conclusion aggregation, including multiset sum and fuzzy set union, are considered. In their abstract form, thus, deductive relations are defined as additional compatible preorderings over certain partially ordered monoids. We investigate these relations using categorical methods, and provide analogues of the main results obtained in the general theory of consequence relations. Then we focus on the driving example of multiset deductive relations, providing variations of the methods of matrix semantics and Hilbert systems in Abstract Algebraic Logic.
ISSN:2331-8422
DOI:10.48550/arxiv.1710.00220