Partial and paraconsistent three-valued logics
DOI:
https://doi.org/10.12775/LLP.2016.003Keywords
four-valued logic, three-valued logic, partial logic, paraconsistent logic, sequent calculus, functional completeness, cut redundancy, proof-search procedureAbstract
On the sidelines of classical logic, many partial and paraconsistent three-valued logics have been developed. Most of them differ in the notion of logical consequence or in the definition of logical connectives. This article aims, firstly, to provide both a model-theoretic and a proof-theoretic unified framework for these logics and, secondly, to apply these general frameworks to several well-known three-valued logics. The proof-theoretic approach to which we give preference is sequent calculus. In this perspective, several results concerning the properties of functional completeness, cut redundancy, and proof-search procedure are shown. We also provide a general proof for the soundness and the completeness of the three sequent calculi discussed.References
A.R. Anderson and N.D. Belnap. Entailment: The Logic of Relevance and Necessity. Volume I. Princeton University Press, Princeton, 1975.
A. Avron. On the expressive power of three-valued and four-valued languages. Journal of Logic and Computation, 9(6):977–994, 1999. 10.1093/logcom/9.6.977
M. Baaz, C.G. Fermüller, G. Salzer, and R. Zach. Labeled calculi and finite-valued logics. Studia Logica, 61(1):7–33, 1998. 10.1023/A:1005022012721
M. Baaz, C.G. Fermüller, and G. Salzer. Automated deduction for many-valued logics. In A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning. Volume II, pages 1355–1402. Elsevier Science Publishers B.V., Amsterdam, 2001. 10.1016/B978-044450813-3/50022-9
N.D. Belnap. A useful four-valued logic. In J.M. Dunn and G. Epstein, editors, Modern Uses of Multiple-Valued Logic, pages 8–37. Reidel Publishing Company, Dordrecht, 1977. 10.1007/978-94-010-1161-7
A. Bochman. Biconsequence relations: a four-valued formalism of reasoning with inconsistency and incompleteness. Notre Dame Journal of Formal Logic, 39(1):47–73, 1998. 10.1305/ndjfl/1039293020
R.T. Brady. Completeness proofs for the systems RM3 and BN4. Logique et Analyse, 25(97):9–32, 1982.
A.B.M. Brunner and W.A. Carnielli. Anti-intuitionism and paraconsistency. Journal of Applied Logic, 3(1):161–184, 2005. 1016/j.jal.2004.07.016
N. Chomsky. Aspects of the Theory of Syntax. The M.I.T. Press, Cambridge, Massachusetts, 1965.
V. Degauquier. Cuts, gluts and gaps. Logique et Analyse, 55(218):229–240, 2012.
V. Degauquier. Normal forms and functional completeness for four-valued languages. Bulletin of the Belgian Mathematical Society – Simon Stevin, 22(4):579–588, 2015.
I.M.L. D’Ottaviano and N.C.A. da Costa. Sur un problème de Jaśkowski. Comptes Rendus de l’Académie de Sciences de Paris, 270:1349–1353, 1970.
J.M. Dunn. Intuitive semantics for first-degree entailments and ‘coupled trees’. Philosophical Studies, 29(3):149–168, 1976. http://dx.doi.org/10.1007/BF00373152
J.M. Dunn. Partiality and its dual. Studia Logica, 66(1):5–40, 2000. 10.1023/A:1026740726955
M. Fitting. Bilattices and the semantics of logic programming. The Journal of Logic Programming, 11(2):91–116, 1991. 10.1016/0743-1066(91)90014-G
G. Gentzen. Untersuchungen über das logische Schließen. I. Mathematische Zeitschrift, 39(1):176–210, 1935. 10.1007/BF01201353
J.-Y. Girard. Three-valued logic and cut-elimination: the actual meaning of Takeuti’s conjecture. Dissertationes Mathematicae (Rozprawy Matematyczne), 136:1–49, 1976.
K. Gödel. On the intuitionistic propositional calculus. In S. Feferman, editor, Collected Works. Volume I. Publications 1929–1936, pages 222–225. Oxford University Press, New York, 1986.
S.C. Kleene. Introduction to Metamathematics. North-Holland Publishing Company, Amsterdam, 1952.
J. Łukasiewicz. Philosophical remarks on many-valued systems of propositional logic. In L. Borkowski, editor, Selected Works, pages 153–178. North-Holland Publishing Company, Amsterdam, 1970.
J. Łukasiewicz. On three-valued logic. In L. Borkowski, editor, Selected Works, pp. 87–88. North-Holland Publishing Company, Amsterdam, 1970.
R. Muskens. Meaning and partiality. Studies in Logic, Language and Information Publications, Stanford, 1995.
R. Muskens. On partial and paraconsistent logics. Notre Dame Journal of Formal Logic, 40(3):352–374, 1999. 10.1305/ndjfl/1022615616
G. Priest. In Contradiction. Oxford University Press, New York, 2006. 10.1093/acprof:oso/9780199263301.001.0001
A.M. Sette. On the propositional calculus P1. Mathematica Japonicae, 18(13):173–180, 1973.
A.M. Sette and W.A. Carnielli. Maximal weakly-intuitionisticlogics. Studia Logica, 55(1):181–203, 1995. 10.1007/BF01053037
A.S. Troelstra and H. Schwichtenberg. Basic Proof Theory. Cambridge University Press, Cambridge, 2000. 10.1017/CBO9781139168717
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