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Soft Computing
Published in: Soft Computing 7/2019

02-01-2019 | Focus

Swap structures semantics for Ivlev-like modal logics

Authors: Marcelo E. Coniglio, Ana Claudia Golzio

Published in: Soft Computing | Issue 7/2019

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Abstract

In 1988, J. Ivlev proposed some (non-normal) modal systems which are semantically characterized by four-valued non-deterministic matrices in the sense of A. Avron and I. Lev. Swap structures are multialgebras (a.k.a. hyperalgebras) of a special kind, which were introduced in 2016 by W. Carnielli and M. Coniglio in order to give a non-deterministic semantical account for several paraconsistent logics known as logics of formal inconsistency, which are not algebraizable by means of the standard techniques. Each swap structure induces naturally a non-deterministic matrix. The aim of this paper is to obtain a swap structures semantics for some Ivlev-like modal systems proposed in 2015 by M. Coniglio, L. Fariñas del Cerro and N. Peron. Completeness results will be stated by means of the notion of Lindenbaum–Tarski swap structures, which constitute a natural generalization to multialgebras of the concept of Lindenbaum–Tarski algebras.
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Footnotes
1
This notation makes sense, provided that the set \(\varXi \) of generators will keep fixed along this paper.
 
2
Avron and his collaborators use in Avron and Lev (2001), Avron (2005) and Avron and Zamansky (2011) the term legal valuation to refer to valuations over an Nmatrix.
 
3
As observed in Sect. 1, Kearns restricts the valuations over the Nmatrices in order to deal with the necessitation rule.
 
4
This terminology is inspired by its use in computer science to refer to states.
 
5
Here \(\lnot \) represents the paraconsistent negation of a given LFI, while \({\circ }\) represents a consistency operator w.r.t. \(\lnot \). See, for instance, Carnielli and Coniglio 2016 for more information about LFIs.
 
6
Recall that, in any Boolean algebra, \(a \supset b = 1\) iff \(a \le b\).
 
Literature
Avron A (2005) Non-deterministic matrices and modular semantics of rules. In: Beziau J-Y (ed) Logica universalis: towards a general theory of logic. Birkhäuser, Basel, pp 149–167CrossRef
Avron A, Lev I (2001) Canonical propositional Gentzen-type systems. In: Goré R, Leitsch A, Nipkow T (eds) International joint conference on automated reasoning, IJCAR 2001. Lecture notes in computer science, vol 2083. Springer, Berlin, pp 529–544. https://​doi.​org/​10.​1007/​3-540-45744-5_​45
Avron A, Zamansky A (2011) Non-deterministic semantics for logical systems. In: Gabbay DM, Guenthner F (eds) Handbook of philosophical logic. volume 16 of handbook of philosophical logic. Springer, Dordrecht, pp 227–304
Carnielli W, Coniglio ME (2016) Paraconsistent logic: consistency, contradiction and negation. Logic, epistemology, and the unity of science, vol 40. Springer, ChamMATH
Coniglio ME, Fariñas del Cerro L, Peron NM (2015) Finite non-deterministic semantics for some modal systems. J Appl Non Class Log 25(1):20–45MathSciNetCrossRefMATH
Coniglio ME, Fariñas del Cerro L, Peron NM (2016) Errata and addenda to “Finite non-deterministic semantics for some modal systems”. J Appl Non Class Log 26(4):336–345MathSciNetCrossRefMATH
Marty F (1934) Sur une généralisation de la notion de groupe. In: Proceedings of 8th congress mathematiciens, Scandinaves, pp 45–49
Omori H, Skurt D (2016) More modal semantics without possible worlds. IfCoLog J Log Appl 3(5):815–846
Metadata
Title
Swap structures semantics for Ivlev-like modal logics
Authors
Marcelo E. Coniglio
Ana Claudia Golzio
Publication date
02-01-2019
Publisher
Springer Berlin Heidelberg
Published in
Soft Computing / Issue 7/2019
Print ISSN: 1432-7643
Electronic ISSN: 1433-7479
DOI
https://doi.org/10.1007/s00500-018-03707-4

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