Abstract
There are several three-valued logical systems. They give the impression of a scattered landscape. The majority of the works on this subject gives the truth tables, sometimes an Hilbert style axiomatization in a basic propositional language and a completeness theorem with respect to those truth tables. We show that all the reasonable connectives in three-valued logics can be built starting from few of them. Nevertheless, the issue of the usefulness of each system in relation with the third truth value is often neglected. Here, we review the interpretations of the third truth value. Then, we focus on the unknown case, suggested by Kleene. We show that any formula in three-valued logics can be encoded as a fragment of an epistemic logic (formulae of modal depth 1, with modalities in front of literals), preserving all tautologies and inference rules. We study in particular, the translation of Kleene, Gödel, Łukasiewicz and Nelson logics. This work enables us to lay bare the limited expressive power of three-valued logics in uncertainty management.
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
Banerjee, M., Dubois, D.: A Simple Modal Logic for Reasoning about Revealed Beliefs. In: Sossai, C., Chemello, G. (eds.) ECSQARU 2009. LNCS (LNAI), vol. 5590, pp. 805–816. Springer, Heidelberg (2009)
Belnap, N.D.: A useful four-valued logic. In: Dunn, J.M., Epstein, G. (eds.) Modern Uses of Multiple-Valued Logic, pp. 8–37. D. Reidel (1977)
Bochvar, D.A.: 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)
Borowski, L. (ed.): Selected works of J. Łukasiewicz. North-Holland, Amsterdam (1970)
Ciucci, D., Dubois, D.: Relationships between Connectives in Three-Valued Logics. In: Greco, S., Bouchon-Meunier, B., Coletti, G., Fedrizzi, M., Matarazzo, B., Yager, R.R. (eds.) IPMU 2012, Part I. CCIS, vol. 297, pp. 633–642. Springer, Heidelberg (2012)
Dubois, D.: On ignorance and contradiction considered as truth-values. Logic Journal of the IGPL 16, 195–216 (2008)
Dubois, D., Prade, H.: Possibility theory, probability theory and multiple-valued logics: A clarification. Ann. Math. and AI 32, 35–66 (2001)
Dubois, D., Prade, H., Schockaert, S.: Stable models in generalized possibilistic logic. In: Proceedings KR 2012, Roma, pp. 519–529 (2012)
Dubois, D., Prade, H.: Generalized Possibilistic Logic. In: Benferhat, S., Grant, J. (eds.) SUM 2011. LNCS, vol. 6929, pp. 428–432. Springer, Heidelberg (2011)
Fox, J.: Motivation and demotivation of a four-valued logic. Notre Dame Journal of Formal Logic 31(1), 76–80 (1990)
Gaines, B.R.: Foundations of fuzzy reasoning. Int. J. of Man-Machine Studies 6, 623–668 (1976)
Gödel, K.: Zum intuitionistischen aussagenkalkül. Anzeiger Akademie der Wissenschaften Wien 69, 65–66 (1932)
Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (1998)
Hosoi, T.: The axiomatization of the intermediate propositional systems sn of gödel. J. Coll. Sci., Imp. Univ. Tokyo 13, 183–187 (1996)
Jaśkowski, S.: Propositional calculus for contradictory deductive systems. Studia Logica 24, 143–160 (1969)
Kleene, S.C.: Introduction to metamathematics. North–Holland Pub. Co., Amsterdam (1952)
Nelson, D.: Constructible falsity. J. of Symbolic Logic 14, 16–26 (1949)
Pearce, D.: Equilibrium logic. Annals of Mathematics and Artificial Intelligence 47, 3–41 (2006)
Sette, A.M.: On propositional calculus p1. Math. Japon 16, 173–180 (1973)
Sobociński, B.: Axiomatization of a partial system of three-value calculus of propositions. J. of Computing Systems 1, 23–55 (1952)
Surma, S.: Logical Works. Polish Academy of Sciences, Wroclaw (1977)
Urquhart, A.: Many-valued logic. In: Gabbay, D.M., Guenthner, F. (eds.) Handbook of Philosophical Logic: vol. III, Alternatives to Classical Logic, Springer (1986)
Vakarelov, D.: Notes on n-lattices and constructive logic with strong negation. Studia Logica 36, 109–125 (1977)
Wajsberg, M.: Aksjomatyzacja trówartościowego rachunkuzdań (Axiomatization of the three-valued propositional calculus). Comptes Rendus des Séances de la Societé des Sciences et des Lettres de Varsovie 24, 126–148 (1931); English Translation in [21]
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). Three-Valued Logics for Incomplete Information and Epistemic Logic. In: del Cerro, L.F., Herzig, A., Mengin, J. (eds) Logics in Artificial Intelligence. JELIA 2012. Lecture Notes in Computer Science(), vol 7519. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33353-8_12
Download citation
DOI: https://doi.org/10.1007/978-3-642-33353-8_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-33352-1
Online ISBN: 978-3-642-33353-8
eBook Packages: Computer ScienceComputer Science (R0)