Probability Logics with Iterations of Probability Operators

Abstract

The first order probability logic \({ LFOP}_{1}\) is introduced. The logic allows formulas about higher-order probabilities. To give semantics to the logic we introduced first order probability models with constant domains and rigid terms. An infinitary axiom system for \({ LFOP}_{1}\) is presented. The logic \({ LFOP}_{1}\) inherits the main properties of \(LPP_2\), so, in spite of all differences, we use the techniques from Chap. 3 to prove strong completeness. The same technique can be also applied to other kinds of probability logics: the first order \({ LFOP}_2\), and propositional \({ LPP}_1\), etc. In \({ LFOP}_{1}\)-formulas probabilistic operators and the classical quantifiers can be mixed and nested, so the logic is related to modal logics and we discuss how (translations of) some properties of modal logics (e.g., Barcan formula) behave in the probabilistic settings. Since \({ LFOP}_{1}\) extends classical first order logic, it is undecidable, but we show that the monadic fragment of \({ LFOP}_{1}\) without iterations of probability operators is decidable. The same hold for the propositional \({ LPP}_1\). Finally, the logic \({ LPP}_1^{\mathrm {LTL}}\) suitable for a combination of probability and temporal reasoning is presented. This chapter covers some results from Ikodinovic, Proceedings of the 8th European conference symbolic and quantitative approaches to reasoning with uncertainty ECSQARU 2005, vol 3571, pp 726–736, 2005, [10], Markovic et al. Int J Approx Reason, 49(1): 52–66, 2008, [11], Milosevic, Ognjanovic, Logic J Interes Group Pure Appl Logics, 20(1): 235–253, 2012, [12], Milosevic, Ognjanovic, Publications de L’InstituteMatematique, ns, 93(107): 19–27, 2013, [8], Ognjanovic, J Logic Comput 16(2): 257–285, 2006, [15], Ognjanovic, Publications de L’Institute Matematique (Beograd), ns, 82(96): 141–154, 2007, [16], Ognjanovic, Raskovic, Publications de L’Institute Matematique (Beograd), ns, 60(74): 1–4, 1996, [13], Ognjanovic, Raskovic, Theor Comput Sci, 247(1–2): 191–212, 2000, [14], Ognjanovic et al. Fifth international conference on scalable uncertainty management SUM-2011, vol 6929, pp 219–232, 2011, [18], Ognjanovic et al. Zbornik radova. Subseries logic in computer science, vol 12(20), pp 35-111, [17], Perovic et al. Proceedings of the 5th international symposium foundations of information and knowledge systems FoIKS 2008, vol 4932, pp 239–252, 2008, [21], Raskovic, Ognjanovic, Proceedings of the Kurepa’s symposium 1996, pp 83–90, 1996, [19].