Abstract
In the literature, several 3-valued logics can be found. They differ from a syntactic and proof-theoretic point of view as well as on the interpretation given to the third value, which, nevertheless, often assumes an epistemic flavor. This paper is a preliminary step in the attempt to clarify the situation of three-valued logics from a semantic point of view. Logical operations on three-valued functions are studied and their relationships put forward. They are also linked to existing logics, pointing out their usage and interpretation of the third value. In the long range, the idea is to be able to relate as many three-valued calculi as possible to classes of applications where the third truth-value is naturally interpreted, and the basic connectives make full sense.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Asenjo, F., Tamburino, J.: Logic of antinomies. Notre Dame Journal of Formal Logic 16, 17–44 (1975)
Avron, A.: On an implication connective of RM. Notre Dame Journal of Formal Logic 27, 201–209 (1986)
Baets, B.D., Fodor, J.C.: Residual operators of uninorms. Soft Computing 3, 89–100 (1999)
Bochvar, D.: On a three-valued logical calculus and its application to the analysis of the paradoxes of the classical extended functional calculus. History and Philosophy of Logic 2, 87–112 (1981)
D’Ottaviano, I., da Costa, N.: Sur un problème de Jaśkowski. Comptes Rendus de l’Académie des Science 270, 1349–1353 (1970)
Dubois, D.: On ignorance and contradiction considered as truth-values. Logic Journal of the IGPL 16, 195–216 (2008)
Dubois, D., Prade, H.: Fuzzy-set-theoretic differences and inclusions and their use in the analysis of fuzzy equations. Control and Cybernetics 13(3), 129–146 (1984)
Dubois, D., Prade, H.: An introduction to bipolar representations of information and preference. Int. J. Intelligent Systems 23(3), 866–877 (2008)
Dubois, D., Prade, H., Schockaert, S.: Stable models in generalized possibilistic logic. In: Proceedings KR 2012 (accepted, 2012)
Fodor, J.C.: On fuzzy implications operators. Fuzzy Sets and Systems 42, 293–300 (1991)
Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (1998)
Hardegree, G.M.: Material implication in orthomodular (and Boolean) lattices. Notre Dame Journal of Modal Logic 22, 163–182 (1981)
Kleene, S.C.: Introduction to metamathematics. North–Holland Pub. Co., Amsterdam (1952)
Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic, Dordrecht (2000)
Konikowska, B.: Mccarthy algebras: A model of mccarthy’s logical calculus. Fundamamenta Informaticae 26(2), 167–203 (1996)
Mas, M., Monserrat, M., Torrens, J.: s–implications and r–implications on a finite chain. Kibernetika 40, 3–20 (2004)
Mas, M., Monserrat, M., Torrens, J., Trillias, E.: A survey on fuzzy implication functions. IEEE Trans. on Fuzzy Systems 15, 1107–1121 (2007)
McCarthy: A basis for a mathematical theory of computation. In: Braort, P., Hirshberg, D. (eds.) Computer Programming and Formal Systems. North-Holland (1963)
Nelson, D.: Constructible falsity. J. of Symbolic Logic 14, 16–26 (1949)
Sabo, M.: On many valued implications. Tatra Mountain Mathematical Publication 14, 161–167 (1998)
Sette, A.: On propositional calculus p1. Mathematica Japonicae 16, 173–180 (1973)
Smets, P., Magrez, P.: Implication in fuzzy logic. Int. J. of Approximate Reasoning 1, 327–347 (1987)
Sobocinski, B.: Axiomatization of a partial system of three-value calculus of propositions. J. of Computing Systems 1, 23–55 (1952)
Vakarelov, D.: Notes on n-lattices and constructive logic with strong negation. Studia Logica 36, 109–125 (1977)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ciucci, D., Dubois, D. (2012). Relationships between Connectives in Three-Valued Logics. In: Greco, S., Bouchon-Meunier, B., Coletti, G., Fedrizzi, M., Matarazzo, B., Yager, R.R. (eds) Advances on Computational Intelligence. IPMU 2012. Communications in Computer and Information Science, vol 297. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31709-5_64
Download citation
DOI: https://doi.org/10.1007/978-3-642-31709-5_64
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-31708-8
Online ISBN: 978-3-642-31709-5
eBook Packages: Computer ScienceComputer Science (R0)