Nelson's Negation on the Base of Weaker Versions of Intuitionistic Negation
- Dimiter Vakarelov
- … show all 1 hide
Purchase on Springer.com
$39.95 / €34.95 / £29.95*
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.
Constructive logic with Nelson negation is an extension of the intuitionistic logic with a special type of negation expressing some features of constructive falsity and refutation by counterexample. In this paper we generalize this logic weakening maximally the underlying intuitionistic negation. The resulting system, called subminimal logic with Nelson negation, is studied by means of a kind of algebras called generalized N-lattices. We show that generalized N-lattices admit representation formalizing the intuitive idea of refutation by means of counterexamples giving in this way a counterexample semantics of the logic in question and some of its natural extensions. Among the extensions which are near to the intuitionistic logic are the minimal logic with Nelson negation which is an extension of the Johansson's minimal logic with Nelson negation and its in a sense dual version — the co-minimal logic with Nelson negation. Among the extensions near to the classical logic are the well known 3-valued logic of Lukasiewicz, two 12-valued logics and one 48-valued logic. Standard questions for all these logics — decidability, Kripke-style semantics, complete axiomatizability, conservativeness are studied. At the end of the paper extensions based on a new connective of self-dual conjunction and an analog of the Lukasiewicz middle value ½ have also been considered.
- ARIELI, O., and A. AVRON, ‘Reasoning with logical bilattices’, Journal of Logic Language and Information, 5:25–63, 1996. CrossRef
- BIA LYNICKI-BIRULA, A., and H. RASIOWA, ‘On constructible falsity in constructive logic with strong negation’, Colloquium Mathematicum, 6:287–310, 1958.
- DUNN, J.M., ‘Generalized ortho negation’, in H. Wansing, Walter de Gruyer, (eds.), Negation, A notion in Focus, pp. 3–26, Berlin, New York, 1996.
- FIDEL, M.M., ‘An algebraic study of a propositional system of Nelson’, in Mathematical Logic, Proc. of the First Brasilian Conference, Campinas 1977, Lecture Notes in pure Appl. Math., 39, pp. 99–117, 1978.
- FITTING, M., ‘Bilattices and the semantics of logic programming’, Journal of logic programming, 11(2):91–116, 1991. CrossRef
- GARGOV, G., ‘Knowledge, uncertainty and ignorance in logic: bilattices and beyond’, Journal of Applied Non-Classical Logics, 9:195–283, 1999.
- GORANKO, V., ‘The Craig Interpolation Theorem for Propositional Logics with Strong Negation’, Studia Logica, 44:291–317, 1985. CrossRef
- KRACHT, M., ‘On extensions of intermediate logics by strong negation’, Journal of Philosophical Logic, 27:49–73, 1998. CrossRef
- LUKASIEWICZ, J., ‘O logice trójwartościowej’, Ruch Filozoficzny, 5:170–171, 1920.
- LUKASIEWICZ, J., ‘A system of Modal Logic’, The Journal of Computing Systems, 1:111–149 1953.
- LUKASIEWICZ, J., Aristotle's syllogistic from the standpoint of modern formal logic, Oxford, Clarendon Press, 1957.
- MARKOV, A. A., ‘Constructive Logic’, (in Russian), Uspekhi Matematicheskih Nauk 5:187–188, 1950.
- NELSON, D., ‘Constructible falsity’, Journal of Symbolic Logic, 14:16–26, 1949.
- ODINTSOV, S. P., ‘Algebraic semantics for paraconsistent Nelson's Logic’, Journal of Logic and Computation, 13(4):453–468, 2003. CrossRef
- ODINTSOV, S.P., ‘On the Representation of N4-Lattices’, Studia Logica, 76(3):385405, 2004. CrossRef
- PEARCE, D. and G. WAGNER, ‘Reasoning with negative information, I: Strong negation in logic programs’, Language, Knowledge and Intensionality (Acta Filosophica Fenica), Helsinki, 49:405–439, 1990.
- PEARCE, D., and G. WAGNER, ‘Logic programming with strong negation’, in P. Schroeder-Heister, (ed.), Extensions of Logic Programming, Lecture Notes in Artificial Intelligence, No. 475, pp. 311–326, Springer-Verlag, Berlin, 1991.
- PYNKO, A. P., ‘Functional completeness and axiomatizability within Belnap's four-valued logic and its expansions’, Journal of Applied Non-Classical Logics, 9:61–105, 1999.
- RASIOWA, H., ‘N-lattices and constructive logic with strong negation’, Fundamenta Mathematicae, 46:61–80, 1958.
- Rasiowa, H., An algebraic approach to non-classical logic, North-Holland Publishing Company, Amsterdam, London, 1974.
- SENDLEWSKI, A., ‘Some investigations of varieties of N-lattices’, Studia Logica, 43:257–280, 1984.
- SENDLEWSKI, A., ‘Nelson algebras through Heyting ones’, Studia Logica, 49:106–126, 1990. CrossRef
- VAKARELOV, D., ‘Ekstensionalnye Logiki’, (in Russian), Doklady BAN, 25:1609–1612, 1972.
- VAKARELOV, D., ‘Models for constructive logic with strong negation’, V Balkan Mathematical Congress, Abstracts, Beograd, 1974, 298.
- VAKARELOV, D., ‘Obobschennye reshetki Nelsona’, Chetvertaya Vsesoyuznaya Conferenciya po Matematicheskoy Logike, tezisy dokladov i soobschtenii, Kishinev, 1976.
- VAKARELOV, D., Theory of Negation in Certain Logical Systems. Algebraic and Semantical Approach, Ph.D. dissertation, University of Warsaw, 1976.
- VAKARELOV, D., ‘Notes on N-lattices and constructive logic with strong negation’, Studia Logica 36:109–125, 1977. CrossRef
- VAKARELOV, D., ‘Intuitive Semantics for Some Three-valued Logics Connected with Information, Contrariety and subcontrariety’, Studia Logica, 48(4):565–575, 1989. CrossRef
- VAKARELOV, D., ‘Consistency, Completeness and Negation’, in Gr. Priest, R. Routley and J. Norman, (eds.), Paraconsistent Logic. Essays on the Inconsistent, pp. 328–363, Analiytica, Philosophia Verlag, Munhen, 1989.
- VOROB'EV, N.N., ‘Constructive propositional calculus with strong negation’, (in Russian), Doklady Academii Nauk SSSR, 85:456–468, 1952.
- VOROB'EV, N.N., ‘The problem of provability in constructive propositional calculus with strong negation’, (in Russian), Doklady Academii Nauk SSSR, 85:689–692, 1952.
- VOROB'EV, N.N., ‘Constructive propositional calculus with strong negation’, wide(in Russian), Transactions of Steklov's institute, 72:195–227, 1964.
- Nelson's Negation on the Base of Weaker Versions of Intuitionistic Negation
Volume 80, Issue 2-3 , pp 393-430
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers
- Additional Links
- Nelson negation
- subminimal logic
- counterexample semantics
- many-valued logics
- Author Affiliations
- 1. Department of Mathematical Logic with Laboratory for Applied Logic Faculty of Mathematics and Computer Science, Sofia University, blvd James Bouchier 5, 1126, Sofia, Bulgaria