Abstract
Possibilistic logic is an important uncertainty reasoning mechanism based on Zadeh's possibility theory and classical logic. Its inference rules are derived from the classical resolution rule by attaching possibility or necessity weights to ordinary clauses. However, since not all possibility-valued formulae can be converted into equivalent possibilistic clauses, these inference rules are somewhat restricted. In this paper, we develop Gentzen sequent calculus for possibilistic reasoning to lift this restriction. This is done by first formulating possibilistic reasoning ⇒ a kind of modal logic. Then the Gentzen method for modal logics generalized to cover possibilistic logic. Finally, some properties of possibilistic logic, such as Craig's interpolation lemma and Beth's definability theorem are discussed in the context of Gentzen methods.
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References
Beth, E. W.: “On Padoa's method in the theory of definition,” in: Indag. Math. 15 (1953) 330–339.
Craig, W.: “Linear reasoning. A new form of the Herbrand-Gentzen theorem,” in: Journal of Symbol Logic 22 (1957a) 250–268.
Craig, W.: “Three uses of the Herbrand-Gentzen theorem in relating model theory to proof theory,” in: Journal of Symbol Logic 22 (1957b) 269–285.
Dubois, D., J. Lang, and H. Prade: “Fuzzy sets in approximate reasoning. Part 2: Logical approache,” in: Fuzzy Sets and Systems, 25th Anniversary volume, 1990.
Dubois, D., and H. Prade: “An introduction to possibilistic and fuzzy logics,” in: P. Smets et al. (eds.), Non-standard Logics for Automated Reasoning, Academic Press (1988) 287–325.
Dubois, D., H. Prade, and C. Testmale: “In search of a modal system for possibility theory,” in: Proc. of ECAI-88 (1988) 501–506.
Farinas del Cerro, L., and A. Herzig: “A modal analysis of possibility theory,” in: Kruse and Siegel (eds.), Proc. of ECSQAU. Springer Verlag, LNAI 548 (1991) 58–62.
Fitting, M.C.: Proof Methods for Modal and Intuitionistic Logics. Vol. 169 of Synthese Library, D. Reidel Publishing Company, 1983.
Gaifman, H.: “A theory of higher order probabilities,” in: J. Y. Halpern (ed.), Proceeding of 1st Conference on Theoretical Aspects of Reasoning about Knowledge. Morgan Kaufmann, (1986) 275–292.
Gentzen, G.: “Investigation into logical deduction,” in: M. E. Szabo (ed.), The Collected Papers of Gerhard Gentzen Amsterdam: North-Holland (1969) 68–131.
Liau, C. J., and I. P. Lin: “Quantitative modal logic and possibilistic reasoning,” in: Proc. of ECAI 92, 1992.
Robinson, J. A.: “A machine oriented logic based on the resolution principle,” in: JACM 12 (1965) 23–41.
Wallen, L.: “Matrix proofs for modal logics,” in: Proc. of IJCAI (1987) 917–923.
Zadeh, L. A.: “Fuzzy sets as a basis for a theory of possibility,” in: Fuzzy Sets and Systems, 1 (1978) 3–28.
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© 1994 Springer-Verlag Berlin Heidelberg
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Liau, C.J., Lin, B.I.P. (1994). Gentzen sequent calculus for possibilistic 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_3
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DOI: https://doi.org/10.1007/3-540-58095-6_3
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