Hostname: page-component-76fb5796d-dfsvx Total loading time: 0 Render date: 2024-04-25T07:55:09.742Z Has data issue: false hasContentIssue false
  • English
  • Français

SEQUENT CALCULI FOR SOME TRILATTICE LOGICS

Published online by Cambridge University Press:  09 July 2009

NORIHIRO KAMIDE  and
HEINRICH WANSING
NORIHIRO KAMIDE*
Affiliation:
Waseda Institute for Advanced Study, Waseda University
HEINRICH WANSING*
Affiliation:
Institute of Philosophy, Dresden University of Technology
*
*WASEDA INSTITUTE FOR ADVANCED STUDY, WASEDA UNIVERSITY, 1-6-1 NISHI WASEDA, SHINJUKU-KU, TOKYO 169-8050, JAPAN. E-mail:logician-kamide@aoni.waseda.jp
†INSTITUTE OF PHILOSOPHY, DRESDEN UNIVERSITY OF TECHNOLOGY, 01062 DRESDEN, GERMANY. E-mail:heinrich.wansing@tu-dresden.de
Get access Rights & Permissions [Opens in a new window]

Abstract

The trilattice SIXTEEN3 introduced in Shramko & Wansing (2005) is a natural generalization of the famous bilattice FOUR2. Some Hilbert-style proof systems for trilattice logics related to SIXTEEN3 have recently been studied (Odintsov, 2009; Shramko & Wansing, 2005). In this paper, three sequent calculi GB, FB, and QB are presented for Odintsov’s (2009) first-degree proof system ⊢B related to SIXTEEN3. The system GB is a standard Gentzen-type sequent calculus, FB is a four-place (horizontal) matrix sequent calculus, and QB is a quadruple (vertical) matrix sequent calculus. In contrast with GB, the calculus FB satisfies the subformula property, and the calculus QB reflects Odintsov’s coordinate valuations associated with valuations in SIXTEEN3. The equivalence between GB, FB, and QB, the cut-elimination theorems for these calculi, and the decidability of ⊢B are proved. In addition, it is shown how the sequent systems for ⊢B can be extended to cut-free sequent calculi for Odintsov’s LB, which is an extension of ⊢B by adding classical implication and negation connectives.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 2 , Issue 2 , June 2009 , pp. 374 - 395
Copyright
Copyright © Association for Symbolic Logic 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

Almukdad, A., & Nelson, D. (1984). Constructible falsity and inexact predicates. Journal of Symbolic Logic, 49, 231–233.Google Scholar
Anderson, A. R., & Belnap, N. D. (1975). Entailment: The Logic of Relevance and Necessity, Vol. I. Princeton, NJ: Princeton University Press.Google Scholar
Arieli, O., & Avron, A. (1996). Reasoning with logical bilattices. Journal of Logic, Language and Information, 5(1), 25–63.CrossRefGoogle Scholar
Belnap, N. D. (1977a). How computer should think. In Ryle, G., editor. Contemporary Aspects of Philosophy. Stocksfield, UK: Oriel Press, pp. 30–56.Google Scholar
Belnap, N. D. (1977b). A useful four-valued logic. In Dunn, J. M., and Epstein, G., editors. Modern Uses of Multiple-Valued Logic. Dordrecht, The Netherlands: Reidel, pp. 5–37.CrossRefGoogle Scholar
Chagrov, A., & Zakharyaschev, M. (1997). Modal Logic. Oxford, UK: Clarendon Press.CrossRefGoogle Scholar
Dunn, J. M. (1976). Intuitive semantics for first-degree entailment and ‘coupled trees’. Philosophical Studies, 29(3), 149–168.CrossRefGoogle Scholar
Gurevich, Y. (1977). Intuitionistic logic with strong negation. Studia Logica, 36, 49–59.Google Scholar
Kamide, N. (2002). Sequent calculi for intuitionistic linear logic with strong negation. Logic Journal of the IGPL, 10(6), 653–678.CrossRefGoogle Scholar
Kamide, N. (2005a). A cut-free system for 16-valued reasoning. Bulletin of the Section of Logic, 34(4), 213–226.Google Scholar
Kamide, N. (2005b). Gentzen-type methods for bilattice negation. Studia Logica, 80(2–3), 265–289.Google Scholar
Kamide, N., & Wansing, H. (2008). Alternative semantics for trilattice logics (manuscript).Google Scholar
Nelson, D. (1949). Constructible falsity. Journal of Symbolic Logic, 14, 16–26.CrossRefGoogle Scholar
Odintsov, S. P. (2009). On axiomatizing Shramko-Wansing’s logic. Studia Logica, 91(3), 407–428.CrossRefGoogle Scholar
Rautenberg, W. (1979). Klassische und nichtklassische Aussagenlogik. Braunschweig, Germany: Vieweg.CrossRefGoogle Scholar
Shramko, Y., & Wansing, H. (2005). Some useful 16-valued logics: How a computer network should think. Journal of Philosophical Logic, 34(2), 121–153.Google Scholar
Shramko, Y., & Wansing, H. (2006). Hyper-contradictions, generalized truth values and logics of truth and falsehood. Journal of Logic, Language and Information, 15(4), 403–424.Google Scholar
Wansing, H. (1999). Higher-arity Gentzen systems for Nelson’s logics. In Nida-Rümelin, J., editor. Proceedings of the 3rd International Congress of the Society for Analytical Philosophy (Rationality, Realism, Revision), 1997. Berlin: Walter de Gruyter, pp. 105–109.Google Scholar
Wansing, H., & Shramko, Y. (2008). An note on two ways of defining a many-valued logic. In Pelis, M., editor. Logica Yearbook 2007. Prague, Czech Republic: Filosofia, pp. 255–266.Google Scholar