Journal of Philosophical Logic

, Volume 29, Issue 1, pp 1–74

# Entailment with Near Surety of Scaled Assertions of High Conditional Probability

• Donald Bamber

## Abstract

An assertion of high conditional probability or, more briefly, an HCP assertion is a statement of the type: The conditional probability of B given A is close to one. The goal of this paper is to construct logics of HCP assertions whose conclusions are highly likely to be correct rather than certain to be correct. Such logics would allow useful conclusions to be drawn when the premises are not strong enough to allow conclusions to be reached with certainty. This goal is achieved by taking Adams" (1966) logic, changing its intended application from conditionals to HCP assertions, and then weakening its criterion for entailment. According to the weakened entailment criterion, called the Criterion of Near Surety and which may be loosely interpreted as a Bayesian criterion, a conclusion is entailed if and only if nearly every model of the premises is a model of the conclusion. The resulting logic, called NSL, is nonmonotonic. Entailment in this logic, although not as strict as entailment in Adams" logic, is more strict than entailment in the propositional logic of material conditionals. Next, NSL was modified by requiring that each HCP assertion be scaled; this means that to each HCP assertion was associated a bound on the deviation from 1 of the conditional probability that is the subject of the assertion. Scaling of HCP assertions is useful for breaking entailment deadlocks. For example, it it is known that the conditional probabilities of C given A and of ¬ C given B are both close to one but the bound on the former"s deviation from 1 is much smaller than the latter"s, then it may be concluded that in all likelihood the conditional probability of C given AB is close to one. The resulting logic, called NSL-S, is also nonmonotonic. Despite great differences in their definitions of entailment, entailment in NSL is equivalent to Lehmann and Magidor"s rational closure and, disregarding minor differences concerning which premise sets are considered consistent, entailment in NSL-S is equivalent to entailment in Goldszmidt and Pearl"s System-Z+. Bacchus, Grove, Halpern, and Koller proposed two methods of developing a predicate calculus based on the Criterion of Near Surety. In their random-structures method, which assumed a prior distribution similar to that of NSL, it appears possible to define an entailment relation equivalent to that of NSL. In their random-worlds method, which assumed a prior distribution dramatically different from that of NSL, it is known that the entailment relation is different from that of NSL.

nonmonotonic logic nonmonotonic reasoning entailment conditional probability second-order probability Bayesian inference conditionals rule-based systems exceptions

### 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, pp. 265–316.Google Scholar
2. Adams, E. W. (1975): The Logic of Conditionals, D. Reidel, Dordrecht.Google Scholar
3. Adams, E. W. (1986): On the logic of high probability, J. Philos. Logic 15: 255–279.Google Scholar
4. Adams, E.W. (1996): Four probability-preserving properties of inferences, J. Philos. Logic 25: 1–24.Google Scholar
5. Bacchus, F., Grove, A. J., Halpern, J. Y., and Koller, D. (1992): From statistics to belief, in Proceedings of the Tenth National Conference on Artificial Intelligence (AAAI-92), pp. 602–608.Google Scholar
6. Bacchus, F., Grove, A. J., Halpern, J. Y., and Koller, D. (1993): Statistical foundations for default reasoning, in Proceedings of the 13th International Joint Conference on Artificial Intelligence (IJCAI-93), Vol. 1, pp. 563–569.Google Scholar
7. Bacchus, F., Grove, A. J., Halpern, J. Y., and Koller, D. (1996): From statistical knowledge bases to degrees of belief, Artif. Intell. 87: 75–143.Google Scholar
8. Bamber, D. (1994): Probabilistic entailment of conditionals by conditionals, IEEE Trans. Systems, Man, and Cybernetics 24: 1714–1723.Google Scholar
9. Bamber, D. (1996): Entailment in probability of thresholded generalizations, in Proceedings of the Twelfth Conference on Uncertainty in Artificial Intelligence (UAI-96), pp. 57–64.Google Scholar
10. Bamber, D. (1998): How probability theory can help us design rule-based systems, in Proceedings of the 1998 Command and Control Research and Technology Symposium, Monterey, CA, June 29–July 1, 1998, pp. 441–451.Google Scholar
11. Billingsley, P. (1971): Weak Convergence of Measures: Applications in Probability, Society for Industrial and Applied Mathematics, Philadelphia.Google Scholar
12. Bishop, Y. M. M., Fienberg, S. E., and Holland, P. W. (1975): Discrete Multivariate Analysis: Theory and Practice, M.I.T. Press, Cambridge, MA.Google Scholar
13. Chernoff, H. (1956): Large sample theory: Parametric case, Ann. Math. Statist. 27: 1–22.Google Scholar
14. Goldszmidt, M., Morris, P. and Pearl, J. (1993): A maximum entropy approach to nonmonotonic reasoning, IEEE Trans. on Pattern Analysis and Machine Intelligence 15: 220–232.Google Scholar
15. Goldszmidt, M. and Pearl, J. (1991): System-Z +: A formalism for reasoning with variable strength defaults, in Proceedings of the Ninth National Conference on Artificial Intelligence (AAAI-91), Vol. 1, pp. 399–404.Google Scholar
16. Goldszmidt, M. and Pearl, J. (1992): Reasoning with qualitative probabilities can be tractable, in Proceedings of the Eighth Conference on Uncertainty in Artificial Intelligence (UAI-92), pp. 112–120.Google Scholar
17. Goldszmidt, M. and Pearl, J. (1996): Qualitative probabilities for default reasoning, belief revision, and causal modeling, Artif. Intell. 84: 57–112.Google Scholar
18. Goodman, I. R. (1998): New perspectives on deduction in data fusion, Paper presented at the 1998 Multi-Sensor Data Fusion Open Symposium, Marietta, GA, March 30–31, 1998.Google Scholar
19. Goodman, I. R. and Nguyen, H. T. (1998): Adams' high probability deduction and combination of information in the context of product probability conditional event algebra, in Proceedings of the 1998 International Conference on Multisource-Multisensor Data Fusion (Fusion '98). Google Scholar
20. Goodman, I. R. and Nguyen, H. T. (1999): A new application of Bayesian analysis to the resolution of anomalies between commonsense reasoning, classical logic, and high probability logic, Paper presented at the 37th Annual Bayesian Research Conference, Los Angeles, CA, February 18–19, 1999.Google Scholar
21. Goodman, I. R. and Nguyen, H. T. (in preparation): Product space conditional event algebra as a basis for Adams' high probability logic and its extensions to second order probability logics, Manuscript to be submitted for publication.Google Scholar
22. Grabisch, M., Nguyen, H. T., and Walker, E. A. (1995): Fundamentals of Uncertainty Calculi with Applications to Fuzzy Inference, Kluwer Acad. Publ., Dordrecht.Google Scholar
23. Gritzmann, P. and Klee, V. (1994): On the complexity of some basic problems in computational convexity: II. Volume and mixed volumes, in T. Bisztriczky, P. McMullen, R. Schneider, and A. I. Weiss (eds.), Polytopes: Abstract, Convex and Computational, Kluwer Acad. Publ., Dordrecht, pp. 373–466.Google Scholar
24. Grove, A. J., Halpern, J. Y., and Koller, D. (1994): Random worlds and maximum entropy, J. Artif. Intell. Res. 2: 33–88.Google Scholar
25. Grove, A. J., Halpern, J. Y., and Koller, D. (1996a): Asymptotic conditional probabilities: The unary case, SIAM J. Comput. 25: 1–51.Google Scholar
26. Grove, A. J., Halpern, J. Y., and Koller, D. (1996b): Asymptotic conditional probabilities: The non-unary case, J. Symbolic Logic 61: 250–276.Google Scholar
27. Hailperin, T. (1996): Sentential Probability Logic: Origins, Development, Current Status, and Technical Applications, Associated University Presses, Cranbury, NJ.Google Scholar
28. Hawthorne, J. (1996): On the logic of nonmonotonic conditionals and conditional probabilities, J. Philos. Logic 25: 185–218.Google Scholar
29. Lehmann, D. and Magidor, M. (1992): What does a conditional knowledge base entail? Artif. Intell. 55: 1–60.Google Scholar
30. Mann, H. B. and Wald, A. (1943): On stochastic limit and order relationships. Ann. Math. Statist. 14: 217–226.Google Scholar
31. McGee, V. (1994): Learning the impossible, in E. Eells and B. Skyrms (eds.), Probability and Conditionals: Belief Revision and Rational Decision, Cambridge Univ. Press, Cambridge, pp. 179–199.Google Scholar
32. Nilsson, N. J. (1986): Probabilistic logic, Artif. Intell. 28: 71–87.Google Scholar
33. Paris, J. B. (1994): The Uncertain Reasoner's Companion, Cambridge Univ. Press, Cambridge.Google Scholar
34. Pearl, J. (1990): System Z: A natural ordering of defaults with tractable applications to nonmonotonic reasoning, in R. Parikh (ed.), Theoretical Aspects of Reasoning about Knowledge. Proceedings of the Third Conference (TARK 1990), Morgan Kaufmann, San Mateo, CA, pp. 121–135.Google Scholar
35. Pearl, J. (1994): From Adams' conditionals to default expressions, causal conditionals, and counterfactuals, in E. Eells and B. Skyrms (eds.), Probability and Conditionals: Belief Revision and Rational Decision, Cambridge Univ. Press, Cambridge, pp. 47–74.Google Scholar
36. Pratt, J. W. (1959): On a general concept of “in probability”, Ann. Math. Statist. 30: 549–558.Google Scholar
37. Rockafellar, R. T. (1970): Convex Analysis, Princeton Univ. Press, Princeton.Google Scholar
38. Schurz, G. (1994): Probabilistic justification of default reasoning, in B. Knebel and L. D. Dreschler-Fischer (eds.), KI-94: Advances in Artificial Intelligence. Proceedings 18th German Annual Conference on Artificial Intelligence, Springer-Verlag, Berlin, pp. 248–259.Google Scholar
39. Schurz, G. (1997a): Probabilistic default logic based on irrelevance and relevance assumptions, in D. Gabbay, et al. (eds.), Qualitative and Quantitative Practical Reasoning, Lecture Notes in Artif. Intell. 1244, Springer-Verlag, Berlin.Google Scholar
40. Schurz, G. (1997b): Probabilistically reliable default reasoning, IPS Preprint 1997-2, Institut für Philosophie, Universität Salzburg, Salzburg, Austria.Google Scholar
41. Schurz, G. (1998): Probabilistic semantics for Delgrande's conditional logic and a counterexample to his default logic, Artif. Intell. 102: 81–95.Google Scholar
42. Spohn, W. (1988): Ordinal conditional functions: A dynamic theory of epistemic states, in W. L. Harper and B. Skyrms (eds.), Causation in Decision, Belief Change, and Statistics. Proceedings of the Irvine Conference on Probability and Causation, Vol. II, Kluwer Acad. Publ., Dordrecht, pp. 105–134.Google Scholar
43. Webster, R. J. (1994): Convexity, Oxford University Press, Oxford.Google Scholar