Abstract
A bounded formula is a pair consisting of a propositional formula φ in the first coordinate and a real number within the unit interval in the second coordinate, interpreted to express the lower-bound probability of φ. Converting conjunctive/disjunctive combinations of bounded formulas to a single bounded formula consisting of the conjunction/disjunction of the propositions occurring in the collection along with a newly calculated lower probability is called absorption. This paper introduces two inference rules for effecting conjunctive and disjunctive absorption and compares the resulting logical system, called System Y, to axiom System P. Finally, we demonstrate how absorption resolves the lottery paradox and the paradox of the preference.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adams, E., 1975, The Logic of Conditionals, Dordrecht: Reidel.
Dekhtyar, A. and Subrahmanian, V.S., 2000, “Hybrid probabilistic programs,” Journal of Logic Programming 43(3), 187–250.
Fagin, R., Halpern, J. and Megiddo, N., 1990, “A logic for reasoning about probabilities,” Information and Computation 87(1–2), 78–128.
Fagin, R. and Halpern, J.Y., 1991, “Uncertainty, belief and probability”, Computational Intelligence 6, 160–173.
Halmos, P.R., 1950, Measure Theory, Berlin: Springer-Verlag.
Halpern, J.Y., 2003. Reasoning about Uncertainty, Cambridge, Mass: MIT Press.
Hawthorne, J. and Bovens, L., 1999, “The preface, the lottery, and the logic of belief”, Mind 108, 241–264.
Kraus, S., Lehman, D., and Magidor, M., 1990, “Nonmonotonic reasoning, preferential models and cummulative logics,” Artificial Intelligence 44, 167–207.
Kyburg, H.E. Jr., 1961, Probability and the Logic of Rational Belief, Middletown, CT: Wesleyan University Press
Kyburg, H.E. Jr., and Teng, C.M., 1999, “Statistical inference as default logic,” International Journal of Pattern Recognition and Artificial Intelligence 13(2), 267–283.
Kyburg H.E. Jr., Teng, C.M., and Wheeler, G., “Conditionals and consequences,” Journal of Applied Logic, forthcoming.
Lukasiewicz, T., 2002, “Probabilistic default reasoning with conditional constraints,” Annals of Mathematics and Artificial Intelligence 34, 35–88.
Makinson, D.C., 1965, The paradox of the preface, Analysis, 25, 205–207.
Pearl, J., 1990, System Z: A natural ordering of defaults with tractable applications to default reasoning. In Theoretical Aspects of Reasoning about Knowledge, Morgan Kaufman: San Francisco, pp. 121–135.
Ruspini, E.H., 1987 The logical foundations of evidential reasoning. Research note 408, SRI International, Stanford, CA.
Walley, P., 1991, Statistical Reasoning with Imprecise Probabilities, London: Chapman and Hall.
Wheeler, G., 2004, A resource bounded default logic, Delgrande J. and Schaub, T., (eds). Proc. of NMR 2004, pp. 416–422.
Wheeler, G., 2005, “The structural view of rational acceptance: Comments on Hawthorne and Bovens,” Synthese 144(2), 287–304.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wheeler, G. Rational Acceptance and Conjunctive/Disjunctive Absorption. JoLLI 15, 49–63 (2006). https://doi.org/10.1007/s10849-005-9006-6
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10849-005-9006-6