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Some first-order probability logics. (English) Zbl 0954.03024
Summary: We present some first-order probability logics. The logics allow making statements such as $$P_{\geq s}\alpha$$, with the intended meaning “the probability of truthfulness of $$\alpha$$ is greater than or equal to $$s$$”. We describe the corresponding probability models. We give a sound and complete infinitary axiomatic system for the most general of our logics, while for some restrictions of this logic we provide finitary axiomatic systems. We study the decidability of our logics. We discuss some related papers.

##### MSC:
 03B48 Probability and inductive logic
##### Keywords:
first-order logic; probability; possible worlds; completeness
Full Text:
##### References:
 [1] Abadi, M.; Halpern, J.Y., Decidability and expressiveness for first-order logics of probability, Inform. comput., 112, 1, 1-36, (1994) · Zbl 0799.03017 [2] Bacchus, F., $$Lp$$, A logic for representing and reasoning with statistical knowledge, Comput. intell., 6, 209-231, (1990) [3] Bhaskara Rao, K.P.S.; Bhaskara Rao, M., Theory of charges, (1983), Academic Press New York · Zbl 0516.28001 [4] Cross, C.B., From worlds to probabilities: a probabilistic semantics for modal logics, J. philos. logic, 22, 2, 169-192, (1993) · Zbl 0798.03014 [5] Fagin, R.; Halpern, J.Y.; Megiddo, N., A logic for reasoning about probabilities, Inform. comput., 87, 1/2, 78-128, (1990) · Zbl 0811.03014 [6] Fagin, R.; Halpern, J.Y., Reasoning about knowledge and probability, J. ACM, 41, 2, 340-367, (1994) · Zbl 0806.68098 [7] Garson, J.W., Quantification in modal logic, (), 249-307 · Zbl 0875.03050 [8] Gabbay, D.M.; Hodkinson, I.M., An axiomatization of the temporal logic with until and Since over real numbers, J. logic comput., 1, 2, 228-259, (1990) · Zbl 0744.03018 [9] Halpern, J.Y., An analysis of first-order logics of probability, Arti. intell., 46, 311-350, (1990) · Zbl 0723.03007 [10] J.Y. Halpern, Knowledge, belief and certainty, Ann. Math. Arti. Intell. (4) (1991) 301-322. · Zbl 0865.03016 [11] Hughes, G.E.; Cresswell, M.J., Modal logic, (1968), Methuen London · Zbl 0205.00503 [12] Hughes, G.E.; Cresswell, M.J., A companion to modal logic, (1984), Methuen London · Zbl 0625.03005 [13] Haddawy, P.; Frisch, A.M., Modal logics of higer-order probability, (), 133-148 [14] van der Hoeck, W., Some consideration on the logics $$PFD$$, J. appl. non-classical logics, 7, 3, 287-307, (1997) · Zbl 0885.03022 [15] Hoover, D.N., Probability logic, Ann. math. logic, 14, 287-313, (1978) · Zbl 0394.03033 [16] Kripke, S.A., The undecidability of monadic modal quantification theory, Z. mat. logik, 8, 113-116, (1962) · Zbl 0111.01101 [17] Ognjanović, Z.; Rašković, M., A logic with higher order probabilities, Pub. inst. math., (NS)-60, 74, 1-4, (1996) · Zbl 0884.03019 [18] Z. Ognjanović, M. Rašković, Some probability logics with new types of probability operators, J. Logic Comput., to be published. [19] Rašković, M., Classical logic with some probability operators, Publ. inst. math., (NS)-53, 67, 1-3, (1993) · Zbl 0799.03018 [20] Rašković, M.; Djordjević, R., Probability quantifiers and operators, (1996), Vesta Beograd · Zbl 0933.03044
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