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Soft Computing
Erschienen in: Soft Computing 7/2019

29.10.2018 | Focus

Combining fragments of classical logic: When are interaction principles needed?

verfasst von: Carlos Caleiro, Sérgio Marcelino, João Marcos

Erschienen in: Soft Computing | Ausgabe 7/2019

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Abstract

We investigate the combination of fragments of classical logic as a way of conservatively extending a given Boolean logic by the addition of new connectives, and we precisely characterize the circumstances in which such a combination produces the corresponding fragment of classical logic over the signature containing connectives from both fragments given as input. If the thereby produced combined fragment is only incompletely characterized by the components given as input, this means that connectives from one component need to interact with connectives from the other component, giving rise to interaction principles. The main contributions strongly rely on the (well-known) description of the 2-valued clones made by Post, on the (not so well-known) axiomatization procedures for 2-valued matrices laid out by Rautenberg, and on Avron’s non-deterministic matrices, which have (recently) been used to produce a significant advance on the understanding of the semantics of fibring.
Vorheriger Artikel Pseudomonadic BL-algebras: an algebraic approach to possibilistic BL-logic
Nächster Artikel Toward a general frame semantics for modal many-valued logics

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Fußnoten
1
\({\langle V,\cdot _{\mathbb {M}}\rangle }\) is a multi-algebra, see Grätzer (1962) and Corsini and Leoreanu (2009).
 
2
The problem is that, while it is true that any logical matrix \(\mathbb {M}\) and any translation \({\mathbf {t}}:\Xi \longrightarrow L_\Sigma (P)\) allow one to interpret unambiguously any derived connective, so that \(\text {Val}_P(\mathbb {M}^{\mathbf {t}})=\{v\circ {\mathbf {t}}:v\in \text {Val}_P(\mathbb {M})\}\), the latter equality is not true, in general, if \(\mathbb {M}\) is an Nmatrix. For an example of that, suppose the 1-place derived connective \({\sim }\in \Xi \) is introduced through \({\mathbf {t}}\) as https://static-content.springer.com/image/art%3A10.1007%2Fs00500-018-3584-0/MediaObjects/500_2018_3584_IEq1562_HTML.gif , where  https://static-content.springer.com/image/art%3A10.1007%2Fs00500-018-3584-0/MediaObjects/500_2018_3584_IEq1563_HTML.gif is the Boolean implication and \(\bot \top \) is an unrestrained 0-place connective. In that case, the induced Nmatrix \(\mathbb {M}^{\mathbf {t}}\) would take \({\sim }_{\mathbb {M}^{\mathbf {t}}}=\{0,1\}\), while the original Nmatrix \(\mathbb {M}\) can only allow \({\sim }_{\mathbb {M}}\) to be affirmation connective or the negation connective.
 
Literatur
Avron A, Zohar Y (2017) Non-deterministic matrices in action: expansions, refinements, and rexpansions. In: 2017 IEEE 47th international symposium on multiple-valued logic (ISMVL), pp 118–123
Caleiro C, Ramos J (2007) Combining classical and intuitionistic implications. In: Konev B, Wolter F (eds) FroCoS 2007: frontiers of combining systems, LNAI. Springer, New York, pp 118–132
Caleiro C, Carnielli W, Rasga J, Sernadas C (2005) Fibring of logics as a universal construction. In: Gabbay D, Guenthner F (eds) Handbook of philosophical logic, vol 13, 2nd edn. Kluwer, Dordrecht, pp 123–187
Caleiro C, Marcelino S, Marcos J (2017) Merging fragments of classical logic. In: Dixon C, Finger M (eds) FroCoS 2017: frontiers of combining systems, 11th international symposium, Brasília, volume 10483 of LNCS. Springer, pp 298–315
Carnielli W, Coniglio M, Gabbay D, Gouveia P, Sernadas C (2008) Analysis and synthesis of logics: how to cut and paste reasoning systems, vol 35. Applied logic series. Springer, New York
Corsini P, Leoreanu V (2009) Applications of hyperstructure theory (advances in mathematics). Springer, New YorkMATH
del Cerro LF, Herzig A (1996) Combining classical and intuitionistic logic. Or: intuitionistic implication as a conditional. In: FroCoS 1996: frontiers of combining systems. Kluwer, pp 93–102
Gabbay D (1996) Fibred semantics and the weaving of logics: part 1. J Symb Logic 61:1057–1120CrossRefMATH
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Humberstone L (2015) Béziau on And and Or. In: Koslow A, Buchsbaum A (eds) The road to universal logic: Festschrift for 50th birthday of Jean-Yves Béziau, vol I. Springer, New York, pp 283–307
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Marcelino S, Caleiro C (2017) On the characterization of fibred logics, with applications to conservativity and finite-valuedness. J Logic Comput 27(7):2063–2088MathSciNetMATH
Marcelino S, Caleiro C (2017) Disjoint fibring of non-deterministic matrices. In: Logic, language, information, and computation: 24th international workshop, WoLLIC 2017 proceedings, London, UK, July 18–21, pp 242–255
Post EL (1941) On the two-valued iterative systems of mathematical logic. Princeton University Press, PrincetonMATH
Shoesmith DJ, Smiley TJ (1978) Multiple-conclusion logic. Cambridge University Press, CambridgeCrossRefMATH
Metadaten
Titel
Combining fragments of classical logic: When are interaction principles needed?
verfasst von
Carlos Caleiro
Sérgio Marcelino
João Marcos
Publikationsdatum
29.10.2018
Verlag
Springer Berlin Heidelberg
Erschienen in
Soft Computing / Ausgabe 7/2019
Print ISSN: 1432-7643
Elektronische ISSN: 1433-7479
DOI
https://doi.org/10.1007/s00500-018-3584-0

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