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AN ABSTRACT APPROACH TO CONSEQUENCE RELATIONS

Published online by Cambridge University Press:  15 February 2019

PETR CINTULA ,
JOSÉ GIL-FÉREZ ,
TOMMASO MORASCHINI  and
FRANCESCO PAOLI
PETR CINTULA*
Affiliation:
Institute of Computer Science of the Czech Academy of Sciences
JOSÉ GIL-FÉREZ*
Affiliation:
University of Bern
TOMMASO MORASCHINI*
Affiliation:
Institute of Computer Science of the Czech Academy of Sciences
FRANCESCO PAOLI*
Affiliation:
University of Cagliari
*
*INSTITUTE OF COMPUTER SCIENCE OF THE CZECH ACADEMY OF SCIENCES POD VODÁRENSKOU VĚŽÍ 271/2 182 07 PRAGUE 8, CZECH REPUBLIC E-mail: cintula@cs.cas.czE-mail: moraschini@cs.cas.cz
†MATHEMATHICAL INSTITUTE UNIVERSITY OF BERN SIDLERSTRASSE 5 3012 BERN, SWITZERLAND E-mail: gilferez@gmail.com
*INSTITUTE OF COMPUTER SCIENCE OF THE CZECH ACADEMY OF SCIENCES POD VODÁRENSKOU VĚŽÍ 271/2 182 07 PRAGUE 8, CZECH REPUBLIC E-mail: cintula@cs.cas.czE-mail: moraschini@cs.cas.cz
‡DEPARTMENT OF PEDAGOGY, PSYCHOLOGY, PHILOSOPHY UNIVERSITY OF CAGLIARI VIA IS MIRRIONIS, 1 09123 CAGLIARI, ITALY E-mail: paoli@unica.it
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Abstract

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 nonidempotent 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.

Keywords

Primary 03G27Secondary 06F05consequence relationmultiset consequenceGentzen systemM-sethypermatrixAbstract Algebraic Logic
Type
Research Article
Information
The Review of Symbolic Logic , Volume 12 , Issue 2 , June 2019 , pp. 331 - 371
Copyright
Copyright © Association for Symbolic Logic 2019 

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References

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