Abstract
This paper, presents, results, of, the, application, to epistemic, logic structures of the method proposed by Carnap for the development of logical foundations of probability theory. These results, which provide firm conceptual bases for the Dempster-Shafer calculus of evidence, are derived by exclusively using basic concepts from probability and modal logic theories, without resorting to any other theoretical notions or structures.
A form of epistemic logic (equivalent in power to the modal system S5), is used to define a space of possible worlds or states of affairs. This space, called the epistemic universe, consists of all possible combined descriptions of the state of the real world and of the state of knowledge that certain rational agents have about it. These representations generalize those derived by Carnap, which were confined exclusively to descriptions of possible states of the real world.
Probabilities defined on certain classes of sets of this universe, representing different states of knowledge about the world, have the properties of the major functions of the Dempster-Shafer calculus of evidence: belief functions and mass assignments. The importance of these epistemic probabilities lies in their ability to represent the effect of uncertain evidence in the states of knowledge of rational agents. Furthermore, if an epistemic probability is extended to a probability function defined over subsets of the epistemic universe that represent true states of the real world, then any such extension must satisfy the well-known interval bounds derived from the Dempster-Shafer theory.
Application of this logic-based approach to problems of knowledge integration results in a general expression, called the additive combination formula, which can be applied to a wide variety of problems of integration of dependent and independent knowledge. Under assumptions of probabilistic independence this formula is equivalent to Dempster’s rule of combination.
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References
R. Carnap, Meaning and necessity, University of Chicago Press, Chicago, Illinois, 1956.
R. Carnap, Logical foundations of probability, University of Chicago Press, Chicago, Illinois, 1962.
A. P. Dempster, Upper and lower probabilities induced by a multivalued mapping, Annals of Mathematical Statistics 38 (1967), 325–339.
M. Jr Hall, Combinatorial theory, second edition, John Wiley and Sons, New York, 1986.
J. Y. Halpern and D. A. McAllester, Likelihood, probability, and knowledge, Proceedings, 1984 AAAI Conference (AAAI, Menlo Park, California), 1984, pp. 137–141.
J. Hintikka, Knowledge and belief, Cornell University Press, Ithaca, New York, 1962.
G. E. Hughes and M. E. Creswell, An introduction to modal logic, Methuen, London, 1968.
R. Moore, Reasoning about knowledge and action, Tech. report, SRI International, Menlon Park, California, 1980.
J. Neveu, Bases mathematiques du calcul des probabilities, Masson, Paris, France, 1964.
N. Nilsson, Probabilistic logic, Artificial Intelligence 28 (1986), 71–88.
S. J. Rosenschein and L. P. Kaelbling, The synthesis of digital machine with provable epistemic properties, Proceedings of 1986 Conference on Theoretical Aspects of Reasoning about Knowledge (Los Altos, California), Kaufmann, 1986, pp. 83–98.
E. H. Ruspini, Approximate deduction in single evidential bodies, Proceedings of 1986 AAAI Uncertainty Workshop (AAAI-86) (Menlo Park, California), 1986.
E. H. Ruspini, The logical foundations of evidential reasoning, Tech. report, SRI International, Menlo Park, California, 1986.
G. Shafer, A mathematical theory of evidence, Princeton University Press, Princeton, NJ, 1976.
P. Suppes, The measurement of belief, Journal of Royal Statistical Society Series B 36 (1974), 160–175.
P. M. Williams,P. M. Williams, Indeterminate probabilities, Formal Methods in the Methodology of Empirical Sciences (K. Szaniawski M. Przelecki and R. Wojciki, eds.), Ossolineum and D. Reidel, 1976.
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Ruspini, E.H. (2008). Epistemic Logics, Probability, and the Calculus of Evidence. In: Yager, R.R., Liu, L. (eds) Classic Works of the Dempster-Shafer Theory of Belief Functions. Studies in Fuzziness and Soft Computing, vol 219. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-44792-4_17
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DOI: https://doi.org/10.1007/978-3-540-44792-4_17
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