Abstract
A logical analysis of reasoning under conditions of uncertainty and vagueness is presented, using many valued and modal logics and their generalizations. The emphasis is on the distinction between degrees of belief and degrees of truth. The paper is mainly a survey of the relevant results.
Preview
Unable to display preview. Download preview PDF.
References
Adams, J.B.: “A Probability Model of Medical Reasoning and the MYCIN Model,” in: Mathematical Biosciences, Vol.32 (1976) 177–186.
Bendová, K., and P. Hájek: “Possibilistic logic as tense logic,” in: Piera Carreté, N. et al. (eds.), Qualitative Reasoning and Decision Technologies, CIMNE Barcelona: Proceedings QUARDET'93 (1993) 441–450.
Bonissone, P.P.: “Summarizing and Propagating Uncertain Information with Triangular Norms,” International Journal Approximate Reasoning 1 (1987) 71–101.
Boutilier, C.: “Modal logics for qualitative possibility and beliefs,” in: Dubois, D. et al. (eds.), Uncertainty in Artificial Intelligence VIII, San Mateo, CA: Morgan-Kaufmann Publishers (1992) 17–24.
Buchanan, B.G., and E.H. Shortliffe: “Rule Based Expert Systems,” in: MYCIN Experiments of the Stanford Heuristic Programming Project, Reading, MA: Addison-Wesley Pub. Comp. (1984) 739.
Burghess, J.P.: “Basic tense logics,” in: Gabbay, D. and F. Guenthner (eds.), Handbook of Philosophical Logic, Vol.II. Reidel, 1984.
Di Nola, A., S. Sessa, W. Pedrycz, and E. Sanchez: Fuzzy Relation Equations and Their Applications to Knowledge Engineering. Dordrecht: Kluwer Academic Publishers, 1989.
Dubois, D., and H. Prade: “An Introduction to Possibilistic and Fuzzy Logics,” in: Smets et al., Non-Standard Logics for Automated Reasoning. London: Academic Press (1988), p.287–326.
Dubois, D., and H. Prade: “Resolution Principles in Possibilistic Logic,” in: International Journal of Approximate Reasoning, Vol.4. No.1 (1990) 1–21.
Dubois, D., and H. Prade: “Fuzzy sets in approximate reasoning, Part 1: Inference with possibility distribution,” Fuzzy Sets and Systems, No.40 (1991a) 143–202.
Dubois, D., and H. Prade: “Fuzzy sets in approximate reasoning, Part 2: Logical approaches,” in: Fuzzy Sets and Systems, No.40 (1991b) 203–244.
Duda, R.O., P.E. Hart, and N.J. Nilsson: “Subjective Bayesian Methods for Rule-Based Inference Systems,” in: Proceedings Nat. Comp. Conf., AFIPS, (1976) 1075–1082.
Esteva, F., P. García, and L. Godo: “On the relationship between preference and similarity based approaches to possibilistic reasoning,” in: Proceedings FUZZ-IEEE'93, 1993.
Fariñas del Cerro, L., and A. Herzig: “A modal analysis of possibilistic logic,” in: Kruse et al. (eds.), Symbolic and Quantitative Approaches to Uncertainty. LNCS 548, Springer-Verlag (1991) 58 ff. Also in: Jorrand and Kelemen (eds.), Fundamentals of AI research, Lecture Notes in AI 535, Springer-Verlag (1991) 11 ff.
Fitting, M.: “Many-valued modal logics I,” in: Fundamenta Informatical 15 (1992a) 235–254.
Fitting, M.: “Many-valued modal logics II,” in: Fundamenta Informatical 17 (1992) 55–73.
Gabbay, D.M.: “Tense logics with discrete moments of time I,” in: Journal of Philosophical Logic 1 (1972) 35 ff.
Gödel, K.: Zum intuitionistischen Aussugenkalkül, Ergebnisse eines Math. Colloquium 4 (1993) 40 ff. (see also Gödel's collected works Vol.1).
Godo L., and R. Lopez de Mantaras: “Fuzzy logic,” in: Encyclopedia of Computer Science, to appear.
Gottwald, S.: Mehrwertige Logik. Berlin: Akademie-Verlag, 1988.
Hájek, P.: “Decision Problems of Some Statistically Motivated Monadic Modal Calculi,” in: International Journal Man-Machine Studies, Vol.15 (1981) 351–358.
Hájek, P.: “Towards a Probabilistic Analysis of MYCIN-like Expert Systems,” in: COMPSTAT'88 Copenhagen. Heidelberg: Physica-Verlag, 1988.
Hájek, P.: “Towards a probabilistic analysis of MYCIN-like Systems II,” in: Plander (ed.), Artificial Intelligence and Information-Control Systems of Robots. Amsterdam: North Holland, 1989.
Hájek, P., and D. Harmancová: “A comparative fuzzy modal logic,” in: Klement, Slany (eds.), Fuzzy Logic in Artificial Intelligence. Lecture Notes in AI 695, Springer-Verlag (1993) 27–34.
Hájek, P., and D. Harmanec: “An exercise in Dempster-Shafer theory,” in: International Journal General Systems 20 (1992) 137–140.
Hájek, P., and T. Havránek: Mechanizing Hypothesis Formation: Mathematical foundations for a general theory. Berlin, Heidelberg: Springer-Verlag (1978) 396.
Hájek, P., T. Havránek, and R. Jiroušek: Uncertain Information Processing in Expert Systems. Boca Raton: CRC Press (1992) 285.
Hájek, P., and J. Valdes: “Algebraic Foundations of Uncertainty Processing in Rule-based Expert Systems (group-theoretical approach),” in: Computers and Artificial Intelligence Vol.9 (1990) 325–344.
Hájek, P., and J. Valdes: “Generalized Algebraic Foundations of Uncertainty Processing in Rule-based Expert Systems (dempsteroids),” in: Computers and Artificial Intelligence Vol.10 (1991) 29–42.
Hughess, C.E., and M.J. Cresswell: An Introduction to Modal Logic. London: Methuen, 1968.
Johnson, R.W.: “Independence and Bayesian Updating Methods,” in: Uncertainty in AI, Amsterdam: North Holland (1986) 197.
Kleene, S.C.: Introduction to Metamathematics. Amsterdam: North Holland, 1952.
Lukasiewicz, J.: Selected Works. Amsterdam: North Holland, 1970.
Mukaidono, M.:” Fuzzy inference in resolution style,” in: Yager (ed.), Fuzzy sets and possibility theory-recent developments. Peyamon Press (1982) 224–231.
Nilsson, N.J.: “Probabilistic logic,” in: Artificial Intelligence 28 (1986) 71–87.
Pavelka, J.: “On fuzzy logic I — Many valued rules of inference,” in: Zeitschr. f. Math. Logic und Grundlagen d. Math. 25 (1979a) 45–52.
Pavelka, J.: “On fuzzy logic II — Enriched residuated lattices and semantics of propositional calculi,” in: Zeitschrift f. Math. Logik und Grundlagen d. Math. 25 (1979b) 119–134.
Pavelka, J.: “On fuzzy logic III — Semantical completeness of some many-valued propositional calculi,” in: Zeitschrift f. Math. Logic und Grundlagen d. Math. 25 (1979c) 446–464.
Resconi, G., G.J. Klir, U. St.Clair, and D. Harmanec: “On the integration of uncertainty theories,” in: International J. of Uncertainty, Fuzziness, and Knowledge-Based Systems 1, 1993.
Resconi, G., G.J. Klir, and U. St.Clair: “Hierarchical uncertainty metatheory based upon modal logic,” in: Int. J. General Systems 21 (1992) 23–50.
Rose, A., and J.B. Rosser: “Fragments of many-valued statement calculi,” in: Trans. Amer. Math. Soc. 87 (1958) 1–53.
Rosser, J.B., and A.R. Turquette: Many-valued logic. Amsterdam: North Holland, 1952.
Ruspini, E.H.: The logical foundations of evidential reasoning. Techn. Note 408, AI Center. Menlo Park CA: SRI International, 1986 (revised 1987).
Ruspini, E.H.: “On the semantics of fuzzy logic,” in: Int. J. Approach Reasoning 5 (1991) 45–88.
Schweitzer, B., and A. Sklar: Probability metric spaces. New York: North Holland, 1983.
Shafer, G.: ‘A Mathematical Theory of Evidence. Princenton: Princenton University Press, 1976.
Shortliffe, E.H., and B.G. Buchanan: “A Model of Inexact Reasoning in Medicine,” in: Mathematical Biosciences Vol.23 (1975) 351–379.
Smets, P., A. Mamdani, D. Dubois, and H. Prade (eds.): Non-standard Logics for Automated Reasoning. London: Academic Press, 1988.
Smoryński, C.: Self-reference and Modal Logic. Springer Verlag, 1985.
Takeuti, G., and S. Titani: “Fuzzy logic and fuzzy set theory,” in: Archive for Math. Logic, to appear.
Van Benthem, J.: The Logic of Time. Dordrecht: Kluwer Academic Publisher, 1991.
Visser, A.: “Interpretability Logic,” in: Mathematical Logic, Proceedings of Heyting Conference Bulgaria 1988. Plenum Press, 1990.
Wajsberg, M.: “Axiomatization of the three-valued propositional calculus,” (Polish with German summary) in: C.R. Soc. Sci. Lett Varsovie, Cl. 3, Vol. 24 (1931) 269–283. (Translation in: M. Wajsberg, Logical Works. Warsaw: Polish Academy of Sciences, 1977.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1994 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hájek, P. (1994). On logics of approximate reasoning. In: Masuch, M., Pólos, L. (eds) Knowledge Representation and Reasoning Under Uncertainty. Logic at Work 1992. Lecture Notes in Computer Science, vol 808. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58095-6_2
Download citation
DOI: https://doi.org/10.1007/3-540-58095-6_2
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-58095-9
Online ISBN: 978-3-540-48451-6
eBook Packages: Springer Book Archive