Extensions of the Probability Logics LPP\(_2\) and LFOP\(_1\)

Abstract

We describe various extensions of logics introduced in the Chaps.  3 and  4 that concern introduction of new types of probability operators, and various ranges of probability functions (finite ranges, non-Archimedean ranges, and unordered ranges). We outline the general features of the corresponding completeness-proof techniques. We present finitary probability logics for reasoning about probability measures with fixed finite ranges, and an infinitary logic with probability functions with arbitrary (not fixed) finite ranges. We introduce logics with the additional probability operators of the form \(Q_F\). The intended meaning of \(Q_F\alpha \) is that the probability of \(\alpha \) is in F. A characterization of the hierarchy of logics with \(Q_F\)-operators is provided. We give strongly complete axiomatization for a logic with the qualitative probability operator \(\preceq \). A probability extension of the intuitionistic logic is presented. Logics that correspond to Kolmogorov’s and de Finetti’s notions of conditional probabilities, and a logic with \([0,1]_{\mathbb {Q}(\varepsilon )}\)-valued probability functions with binary operators for conditional and approximate probabilities are presented. We describe strongly complete propositional axiomatizations for logics with linear and polynomial weight formulas. We consider axiomatization of probability functions with unordered ranges, and illustrate that using p-adic valued probabilities. This Chapter covers some results from Doder et al., Publications de L’Institut Mathematique (N.S.), 87(101), 85–96 (2010), [2], Doder and Ognjanović, Probabilistic logics with independence and probabilistic support, (2015), [3], Dordević et al. Arch. Math. Logic, 43, 557–563 (2004), [4], Ghilezan et al. Proceedings of the 22nd international conference on types for proofs and programs, TYPES (2016), [6], Ikodinović, Some Probability and Topological Logics (2005), [7], Ikodinović and Ognjanović, Proceedings of the 8th European Conference Symbolic and Quantitative Approaches to Reasoning with Uncertainty, ECSQARU (2005), [8], Ikodinović, J Multiple Valued Logic Soft Comput., 20(5–6), 527–555 (2013), [9], Ikodinović, Int. J. Approx. Reason. 55(9), 1830–1842, 2014), [13] (Ilić-Stepić, Math. Logic Q. 58(4–5), 63–280 (2012), [10], Ilić-Stepić, Int. J. Approx. Reason., 55(9), 1843–1865 (2014), [14], Ilić-Stepić, Ognjanović, Publications de l’Institut Mathematique, N.s. tome, 95(109), 73–86 (2014), [11], Ilić-Stepić and Ognjanović, Studia Logica, 103, 145–174 (2015), [12], Kokkinis et al. Logic J. IGPL, 23(4), 662–687 (2015), [16], Kokkinis et al. roceedings of Logical Foundations of Computer Science International Symposium, LFCS, (2016), [17], Marković et al. Math. Logic Q. 49, 415–424 (2003), [19], Marković et al. Publications de L’Institute Matematique (N.S.), 73(87), 31–38 (2003), [20], Marković et al. IPMU 2004, 443–450 (2004), [21], Milošević and Ognjanović, Logic J. Interest Gr. Pure Appl. Logics, 20(1), 235–25 (2012), [22], Milošević and Ognjanović, Publications de L’Institute Matematique, N.S., 93(107), 19–27 (2013), [23], Ognjanović and Rašković, J. Logic Comput., 9(2), 181–195 (1999), [25], Ognjanović, Publications de L’Institute Matematique Ns., 78(92), 35–49 (2005), [26], Ognjanović and Ikodinović, Publications de L’Institute Matematique (Beograd), ns., 82(96), 141–154 (2007), [24], Ognjanović et al. Logic J. IGPL, 16(2), 105–120 (2008), [27], Perović, Some applications of the formal method in set theory, model theory, probabilistic logics and fuzzy logics, (2008), [28], Perović et al. 5th International Symposium on Foundations of Information and Knowledge Systems, FoIKS 2008, Proceedings, (2008), [29], Perović, et al. 11th European Conference on Logics in Artificial Intelligence, JELIA 2008, Proceedings, (2008), [30], Perović, et al. Fuzzy Sets Syst., 169, 65–90 (2011), [31], Rašković, J. Symb. Logic 51(3), 586–590 (1986), [32], Rašković et al. Int. J. Approx. Reason., 49(1), 52–66 (2008), [33], Savić et al. Proceedings of the 9th International Symposium on Imprecise Probability: Theories and Applications, ISIPTA, (2015), [34], Tomović, Proceedings of the 13th European Conference Symbolic and Quantitative Approaches to Reasoning with Uncertainty ECSQARU (2015), [35].