Rational Acceptance and Conjunctive/Disjunctive Absorption

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.

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References

  1. Adams, E., 1975, The Logic of Conditionals, Dordrecht: Reidel.

    Google Scholar 

  2. Dekhtyar, A. and Subrahmanian, V.S., 2000, “Hybrid probabilistic programs,” Journal of Logic Programming 43(3), 187–250.

    Article  Google Scholar 

  3. Fagin, R., Halpern, J. and Megiddo, N., 1990, “A logic for reasoning about probabilities,” Information and Computation 87(1–2), 78–128.

    Article  Google Scholar 

  4. Fagin, R. and Halpern, J.Y., 1991, “Uncertainty, belief and probability”, Computational Intelligence 6, 160–173.

    Google Scholar 

  5. Halmos, P.R., 1950, Measure Theory, Berlin: Springer-Verlag.

    Google Scholar 

  6. Halpern, J.Y., 2003. Reasoning about Uncertainty, Cambridge, Mass: MIT Press.

    Google Scholar 

  7. Hawthorne, J. and Bovens, L., 1999, “The preface, the lottery, and the logic of belief”, Mind 108, 241–264.

    Article  Google Scholar 

  8. Kraus, S., Lehman, D., and Magidor, M., 1990, “Nonmonotonic reasoning, preferential models and cummulative logics,” Artificial Intelligence 44, 167–207.

    Article  Google Scholar 

  9. Kyburg, H.E. Jr., 1961, Probability and the Logic of Rational Belief, Middletown, CT: Wesleyan University Press

    Google Scholar 

  10. 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.

    Article  Google Scholar 

  11. Kyburg H.E. Jr., Teng, C.M., and Wheeler, G., “Conditionals and consequences,” Journal of Applied Logic, forthcoming.

  12. Lukasiewicz, T., 2002, “Probabilistic default reasoning with conditional constraints,” Annals of Mathematics and Artificial Intelligence 34, 35–88.

    Article  Google Scholar 

  13. Makinson, D.C., 1965, The paradox of the preface, Analysis, 25, 205–207.

    Article  Google Scholar 

  14. 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.

    Google Scholar 

  15. Ruspini, E.H., 1987 The logical foundations of evidential reasoning. Research note 408, SRI International, Stanford, CA.

  16. Walley, P., 1991, Statistical Reasoning with Imprecise Probabilities, London: Chapman and Hall.

    Google Scholar 

  17. Wheeler, G., 2004, A resource bounded default logic, Delgrande J. and Schaub, T., (eds). Proc. of NMR 2004, pp. 416–422.

  18. Wheeler, G., 2005, “The structural view of rational acceptance: Comments on Hawthorne and Bovens,” Synthese 144(2), 287–304.

    Article  Google Scholar 

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Correspondence to Gregory Wheeler.

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Wheeler, G. Rational Acceptance and Conjunctive/Disjunctive Absorption. JoLLI 15, 49–63 (2006). https://doi.org/10.1007/s10849-005-9006-6

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Keywords

  • probabilistic logic
  • rational acceptance
  • the lottery paradox
  • System P
  • bounded uncertain reasoning