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Conditional logic and the Principle of Entropy. (English) Zbl 0939.68854
Summary: The conditional three-valued logic of Calabrese is applied to the language $$L^{*}$$ of conditionals on propositional variables with finite domain. The conditionals in $$L^{*}$$ serve as a means for the construction and manipulation of probability distributions respecting the principle of maximum entropy and of minimum relative entropy. This principle allows a sound inference even in the presence of uncertain evidence. The inference is directed, it respects a probabilistic version of Modus Ponens – not of Modus Tollens –, it permits transitive chaining and supports a cautious monotony. Conjunctive, conditional and material deduction are manageable in this probabilistic logic, too. The concept is not merely theoretical, but enables large-scale applications in the expert system-shell SPIRIT.

##### MSC:
 68T35 Theory of languages and software systems (knowledge-based systems, expert systems, etc.) for artificial intelligence 68T27 Logic in artificial intelligence
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##### References:
 [1] Calabrese, P.G., Deduction and inference using conditional logic and probability, (), 71-100 [2] Calabrese, P.G., Theory of conditional information with applications, IEEE trans. systems man cybernet. (special issue on conditional event algebra), 24, 12, 1676-1684, (1994) · Zbl 1371.03030 [3] Calabrese, P.G., Conditional events: doing for logic and probability what fractions do for integer arithmetic, (), 175-212 [4] Csiszàr, I., I-divergence geometry of probability distributions and minimisation problems, Ann. probab., 3, 1, 146-158, (1975) · Zbl 0318.60013 [5] Csiszàr, I., Why least squares and maximum entropy? an axiomatic approach to inference for linear inverse problems, Ann. statist., 19, 4, 2032-2066, (1991) · Zbl 0753.62003 [6] de Finetti, B., Induction and statistics, (1972), Wiley New York · Zbl 0275.60001 [7] Dubois, D.; Prade, H., The logical view of conditioning and its application to possibility and evidence theories, Internat. J. approx. reason., 4, 23-46, (1990) · Zbl 0696.03006 [8] Dubois, D.; Prade, H., Conditioning, non-monotonic logic, and non-standard uncertainty models, () [9] Kern-Isberner, G., Characterizing the principle of minimum cross-entropy within a conditional-logical framework, Artificial intelligence, 98, 169-208, (1998) · Zbl 0903.68181 [10] Kern-Isberner, G., A logically sound method for uncertain reasoning with quantified conditionals, (), 365-379 [11] Lauritzen, S.L.; Spiegelhalter, D.J., Local computations with probabilities in graphical structures and their applications to expert systems, J. roy. statist. soc., 13, 2, 415-448, (1988) · Zbl 0684.68106 [12] Sombé, Lea, Schließen bei unsicherem wissen in der künstlichen intelligenz, (1996), Vieweg Braunschweig, Wiesbaden · Zbl 0755.68128 [13] Meyer, C.-H., Korrektes schließen bei unvollständiger information, dissertation, (1997), Fern Universität Hagen [14] Paris, J.B.; Vencovská, A., A note on the inevitability of maximum entropy, Internat. J. approx. reason., 14, 183-223, (1990) · Zbl 0697.68089 [15] Pearl, J., Probabilistic reasoning in intelligent systems, (1988), Morgan Kaufmann San Mateo, CA [16] Rescher, N., Many-valued logics, (1969), McGraw-Hill New York [17] Rödder, W.; Kern-Isberner, G., Representation and extraction of information by probabilistic logic, Information systems, 21, 8, 637-652, (1996) · Zbl 0869.68100 [18] Rödder, W.; Meyer, C.-H., Coherent knowledge processing at maximum entropy by SPIRIT, (), 470-476 [19] Rödder, W.; Xu, L., Entropy-driven inference and inconsistency, (), 272-277 [20] Shore, J.E.; Johnson, R.W., Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross entropy, IEEE trans. inform. theory, 26, 1, 26-37, (1980) · Zbl 0429.94011 [21] Shore, J.E., Relative entropy, probabilistic inference, and AI, (), 211-215 [22] Suppes, P., Probabilistic inference and the concept of total evidence, (), 49-65 · Zbl 0202.29603 [23] Tarski, A., Wahrscheinlichkeitslehre und mehrwertige logik, Erkenntnis, 5, 174-175, (193536) [24] Voorbraak, F., Probabilistic belief expansion and conditioning, research report LP-96-07, (1996), Institute for Logic, Language and Computation (ILLC), University of Amsterdam [25] Williams, P.M., Bayesian conditionalisation and the principle of minimum information, Brit. J. phil. sci., 131-144, (1980) [26] Zadeh, L.A., Fuzzy sets, Inform. control, 8, 338-353, (1965) · Zbl 0139.24606
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