Journal of Philosophical Logic

, Volume 25, Issue 1, pp 1–24 | Cite as

Four probability-preserving properties of inferences

  • Ernest W. Adams
Article

Abstract

Different inferences in probabilistic logics of conditionals ‘preserve’ the probabilities of their premisses to different degrees. Some preserve certainty, some high probability, some positive probability, and some minimum probability. In the first case conclusions must have probability 1 when premisses have probability 1, though they might have probability 0 when their premisses have any lower probability. In the second case, roughly speaking, if premisses are highly probable though not certain then conclusions must also be highly probable. In the third case conclusions must have positive probability when premisses do, and in the last case conclusions must be at least as probable as their least probable premisses. Precise definitions and well known examples are given for each of these properties, characteristic principles are shown to be valid and complete for deriving conclusions of each of these kinds, and simple trivalent truthtable ‘tests’ are described for determining which properties are possessed by any given inference. Brief comments are made on the application of these results to certain modal inferences such as “Jones may own a car, and if he does he will have a driver's license. Therefore, he may have a driver's license.”

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References

  1. Adams, E.W., 1966. “Probability and the logic of conditionals”, in J. Hintikka and P. Suppes (eds), Aspects of Inductive Logic, North-Holland, Amsterdam, 265–316.Google Scholar
  2. Adams, E.W., 1975. The Logic of Conditionals; an Application of Probability to Deductive Logic, D. Reidel, Dordrecht.Google Scholar
  3. Adams, E.W., 1986. “On the logic of high probability”, Journal of Philosophical Logic 15, 255–279.Google Scholar
  4. Adams, E.W., 1993. “On the rightness of certain counterfactuals”, Pacific Philosophical Quarterly 74, (1), 1–10.Google Scholar
  5. Adams, E.W., 1995, “Remarks on a theorem of McGee”, Journal of Philosophical Logic 24, 343–348.Google Scholar
  6. Adams, E.W. and Levine, H.P., 1975. “On the uncertainties transmitted from premises to conclusions in deductive inferences”, Synthese 30, nos. 3/4, 429–460.Google Scholar
  7. Calabrese, P., 1990. “Reasoning with uncertainty using conditional logic and probability”, in I.R. Goodman, M.M. Gupta, H.T. Nguyen and G.S. Rogers (eds), Conditional Logic in Expert Systems, North-Holland, Amsterdam, 71–100.Google Scholar
  8. Dubois, D. and Prade, H., 1993. “Conditional objects: a three-valued scmantics for nonmontonic inference”, in Conditionals in Knowledge Representation, a workshop held in conjunction with IJCAI-93, C. Boutelier and J. Delgrande, organizers, IJCAI-93, 81–87.Google Scholar
  9. Gilio, Angelo, to appear. “On the logic of conditionals and coherence principles”.Google Scholar
  10. Goodman, I.R., Nguyen, H.T. and Walker, E.A., 1991. Conditional Inference and Logic for Intelligent Systems. North-Holland, Amsterdam.Google Scholar
  11. Kyburg, H., 1990. “Probability, rationality, and the rule of detachment”, in Y. Bar-Hillel (ed.), Logic, Methodology, and Philosophy of Science, North-Holland, Amsterdam, 301–310.Google Scholar
  12. Lewis, D., 1974. Counterfactuals, Harvard University Press.Google Scholar
  13. McGee, V., 1981. “Finite Matrices and the Logic of Conditionals”, Journal of Philosophical Logic 10, 349–351.Google Scholar
  14. McGee, V., 1994. “Learning the impossible”, in B. Skyrms and E. Eells (eds), Probability and Conditionals, Belief Revision and Rational Decision, Cambridge University Press, Cambridge, 179–199.Google Scholar
  15. Popper, K.R., 1959. The Logic of Scientific Discovery, Hutchinson of London, London.Google Scholar
  16. Stalnaker, R.C., 1969. “A theory of conditionals”, in N. Rescher (ed.), Studies in Logical Theory, Blackwell.Google Scholar
  17. Suppes, P., 1966. “Probabilistic inference and the principle of total evidence”, in J. Hintikka and P. Suppes (eds), Aspects of Inductive Logic, North-Holland, Amsterdam, 49–65.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Ernest W. Adams
    • 1
  1. 1.University of CaliforniaBerkeleyUSA

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