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

## 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 *A* ∧ *B* 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.

### REFERENCES

- 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 - Adams, E. W. (1975):
*The Logic of Conditionals*, D. Reidel, Dordrecht.Google Scholar - Adams, E.W. (1996): Four probability-preserving properties of inferences,
*J. Philos. Logic***25**: 1–24.Google Scholar - 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 - 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 - 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 - Bamber, D. (1994): Probabilistic entailment of conditionals by conditionals,
*IEEE Trans. Systems, Man, and Cybernetics***24**: 1714–1723.Google Scholar - 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 - 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 - Billingsley, P. (1971):
*Weak Convergence of Measures: Applications in Probability*, Society for Industrial and Applied Mathematics, Philadelphia.Google Scholar - 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 - Chernoff, H. (1956): Large sample theory: Parametric case,
*Ann. Math. Statist.***27**: 1–22.Google Scholar - 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 - 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 - 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 - Goldszmidt, M. and Pearl, J. (1996): Qualitative probabilities for default reasoning, belief revision, and causal modeling,
*Artif. Intell.***84**: 57–112.Google Scholar - 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
- 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 - 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
- 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
- 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 - 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 - Grove, A. J., Halpern, J. Y., and Koller, D. (1994): Random worlds and maximum entropy,
*J. Artif. Intell. Res.***2**: 33–88.Google Scholar - Grove, A. J., Halpern, J. Y., and Koller, D. (1996a): Asymptotic conditional probabilities: The unary case,
*SIAM J. Comput.***25**: 1–51.Google Scholar - 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 - Hailperin, T. (1996):
*Sentential Probability Logic: Origins, Development, Current Status, and Technical Applications*, Associated University Presses, Cranbury, NJ.Google Scholar - Hawthorne, J. (1996): On the logic of nonmonotonic conditionals and conditional probabilities,
*J. Philos. Logic***25**: 185–218.Google Scholar - Lehmann, D. and Magidor, M. (1992): What does a conditional knowledge base entail?
*Artif. Intell.***55**: 1–60.Google Scholar - Mann, H. B. and Wald, A. (1943): On stochastic limit and order relationships.
*Ann. Math. Statist.***14**: 217–226.Google Scholar - 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 - Paris, J. B. (1994):
*The Uncertain Reasoner's Companion*, Cambridge Univ. Press, Cambridge.Google Scholar - 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 - 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 - Pratt, J. W. (1959): On a general concept of “in probability”,
*Ann. Math. Statist.***30**: 549–558.Google Scholar - Rockafellar, R. T. (1970):
*Convex Analysis*, Princeton Univ. Press, Princeton.Google Scholar - 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 - 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 - Schurz, G. (1997b): Probabilistically reliable default reasoning, IPS Preprint 1997-2, Institut für Philosophie, Universität Salzburg, Salzburg, Austria.Google Scholar
- Schurz, G. (1998): Probabilistic semantics for Delgrande's conditional logic and a counterexample to his default logic,
*Artif. Intell.***102**: 81–95.Google Scholar - 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 - Webster, R. J. (1994):
*Convexity*, Oxford University Press, Oxford.Google Scholar