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].
The original version of this chapter was revised: The coauthors’ names were added: The erratum to this chapter is available at: 10.1007/978-3-319-47012-2_8
An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-3-319-47012-2_8
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Bhaskara Rao, K.P.S., Bhaskara Rao, M.: Theory of Charges. Academic Press, Cambridge (1983)
Doder, D., Marinković, B., Maksimović, P., Perović, A.: A logic with conditional probability operators. Publications de L’Institut Mathematique (N.S.) 87(101), 85–96 (2010). http://elib.mi.sanu.ac.rs/files/journals/publ/107/n101p085.pdf
Doder, D., Ognjanović, Z.: Probabilistic logics with independence and probabilistic support. progic 2015: The Seventh Workshop on Combining Probability and Logic, 22–24 April 2015, University of Kent, Canterbury, 25–26 (2015). https://blogs.kent.ac.uk/jonw/files/2015/04/ProgicAbstractsBooklet2015.pdf
Dordević, R., Rašković, M., Ognjanović, Z.: Completeness theorem for propositional probabilistic models whose measures have only finite ranges. Arch. Math. Logic 43, 557–563 (2004)
Fagin, R., Halpern, J., Megiddo, N.: A logic for reasoning about probabilities. Inf. Comput. 87(1–2), 78–128 (1990)
Ghilezan, S., Ivetić, J., Ognjanovic, Z., Savić, N.: Towards probabilistic reasoning about lambda terms with intersection types. In: Ghilezan, S., Ivetic, J. (ed.), Proceedings of the 22nd International Conference on Types for Proofs and Programs, TYPES 2016, pp. 23–26. Novi Sad, Serbia (2016)
Ikodinović, N.: Some Probability and Topological Logics. Phd thesis (in Serbian). University of Kragujevac (2005). http://elibrary.matf.bg.ac.rs/bitstream/handle/123456789/194/phdNebojsaIkodinovic.pdf?sequence=1
Ikodinović, N., Ognjanović, Z.: A logic with coherent conditional probabilities. In: Godo, L. (ed.), Proceedings of the 8th European Conference Symbolic and Quantitative Approaches to Reasoning with Uncertainty, ECSQARU 2005, Barcelona, Spain, July 6–8, 2005. Lecture Notes in Computer Science, vol. 3571, pp. 726–736. Springer, Heidelberg (2005)
Ikodinović, N., Rašković, M., Marković, Z., Ognjanović, Z.: Logics with generalized measure operators. J Multiple Valued Logic Soft Comput. 20(5–6), 527–555 (2013)
Ikodinović, N., Ognjanović, Z., Perović, A., Rašković, M.: Hierarchies of probabilistic logics. Int. J. Approx. Reason. 55(9), 1830–1842 (2014)
Ilić-Stepić, A., Ognjanović, Z.: Complex valued probability logics. Publications de l’Institut Mathematique, N.s. tome 95(109), 73–86 (2014). http://elib.mi.sanu.ac.rs/files/journals/publ/115/n109p073.pdf
Ilić-Stepić, A., Ognjanović, Z.: Logics for reasoning about processes of thinking with information coded by p-adic numbers. Studia Logica 103, 145–174 (2015)
Ilić-Stepić, A., Ognjanović, Z., Ikodinović, N., Perović, A.: A p-adic probability logic. Math. Logic Q. 58(4–5), 63–280 (2012)
Ilić-Stepić, A., Ognjanović, Z., Ikodinović, N.: Conditional p-adic probability logic. Int. J. Approx. Reason. 55(9), 1843–1865 (2014)
Keisler, J.: Model Theory for Infinitary Logic. North-Holland, Amsterdam (1971)
Kokkinis, I., Maksimović, P., Ognjanović, Z., Studer, T.: First steps towards probabilistic justification logic. Logic J. IGPL 23(4), 662–687 (2015)
Kokkinis, I., Ognjanović, Z., Studer, T.: Probabilistic justification logic. In: Artemov, S., Nerode, A. (eds.), Proceedings of Logical Foundations of Computer Science International Symposium, LFCS 2016. Lecture Notes in Computer Science, January 4–7, 2016, vol. 9537, pp. 174–186. Deerfield Beach, FL, USA (2016)
Kripke, S.: Semantical analysis of intuitionistic logic I. In: Formal Systems and Recursive Functions, pp. 93–130. North-Holland, Amsterdam (1965)
Marković, Z., Ognjanović, Z., Rašković, M.: A probabilistic extension of intuitionistic logic. Math. Logic Q. 49, 415–424 (2003)
Marković, Z., Ognjanović, Z., Rašković, M.: An intuitionistic logic with probabilistic operators. Publications de L’Institute Matematique (N.S.) 73(87): 31–38 (2003). http://elib.mi.sanu.ac.rs/files/journals/publ/93/n087p031.pdf
Marković, Z., Ognjanović, Z., Rašković, M.: What is the proper propositional base for probabilistic logic? IPMU 2004, 443–450 (2004)
Milošević, M., Ognjanović, Z.: A first-order conditional probability logic. Logic J. Interest Gr. Pure Appl. Logics 20(1), 235–253 (2012)
Milošević, M., Ognjanović, Z.: A first-order conditional probability logic with iterations. Publications de L’Institute Matematique, N.S. 93(107), 19–27 (2013). http://elib.mi.sanu.ac.rs/files/journals/publ/113/n107p019.pdf
Ognjanović, Z., Ikodinović, N.: A logic with higher order conditional probabilities. Publications de L’Institute Matematique (Beograd), ns. 82(96), 141–154 (2007). http://elib.mi.sanu.ac.rs/files/journals/publ/102/n096p141.pdf
Ognjanović, Z., Rašković, M.: Some probability logics with new types of probability operators. J. Logic Comput. 9(2), 181–195 (1999)
Ognjanović, Z., Marković, Z., Rašković, M.: Completeness theorem for a Logic with imprecise and conditional probabilities. Publications de L’Institute Matematique Ns. 78(92), 35–49 (2005). http://elib.mi.sanu.ac.rs/files/journals/publ/98/n092p035.pdf
Ognjanović, Z., Perović, A., Rašković, M.: Logics with the qualitative probability operator. Logic J. IGPL 16(2), 105–120 (2008)
Perović, A.: Some applications of the formal method in set theory, model theory, probabilistic logics and fuzzy logics. Ph.d. thesis (in Serbian). University of Belgrade (2008). http://elibrary.matf.bg.ac.rs/bitstream/handle/123456789/100/phdAleksandarPerovic.pdf?sequence=1
Perović, A., Ognjanović, Z., Rašković, M., Marković, Z.: A probabilistic logic with polynomial weight formulas. In: Hartmann, S., Kern-Isberner, G, (eds.), 5th International Symposium on Foundations of Information and Knowledge Systems, FoIKS 2008, Proceedings. Lecture Notes in Computer Science, Pisa, Italy, February 11–15, 2008, vol. 4932, pp. 239–252. Springer, Heidelberg (2008)
Perović, A., Ognjanović, Z., Rašković, M., Marković, Z.: How to restore compactness into probabilistic logics? In: Hölldobler, S., Lutz, C., Wansing, H. (eds.), 11th European Conference on Logics in Artificial Intelligence, JELIA 2008, Proceedings. Lecture Notes in Computer Science, Dresden, Germany, September 28–October 1, 2008, vol. 5293, pp. 338–348. Springer, Heidelberg (2008)
Perović, A., Ognjanović, Z., Rašković, M., Radojević, D.: Finitely additive probability measures on classical propositional formulas definable by Godel’s t-norm and product t-norm. Fuzzy Sets Syst. 169, 65–90 (2011)
Rašković, M.: Completeness theorem for biprobability models. J. Symb. Logic 51(3), 586–590 (1986)
Rašković, M., Marković, Z., Ognjanović, Z.: A logic with approximate conditional probabilities that can model default reasoning. Int. J. Approx. Reason. 49(1), 52–66 (2008)
Savić, N., Doder, D., Ognjanović, Z.: A logic with upper and lower probability operators. In: Augustin, T., Doria, S., Miranda, E., Quaeghebeur, E. (eds.) Proceedings of the 9th International Symposium on Imprecise Probability: Theories and Applications, ISIPTA ’15, Pescara, Italy, 2015, pp. 267–276. SIPTA Society for Imprecise Probability, Theories and Applications (2015)
Tomović, S., Ognjanović, Z., Doder, D.: Probabilistic common knowledge among infinite number of agents. In: Destercke, S., Denoeux, T. (eds.), Proceedings of the 13th European Conference Symbolic and Quantitative Approaches to Reasoning with Uncertainty ECSQARU 2015. Lecture Notes in Artificial Intelligence, Compiegne, France, July 15–17, 2015, vol. 9161, pp.496–505 (2015)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2016 Springer International Publishing AG
About this chapter
Cite this chapter
Ognjanović, Z., Rašković, M., Marković, Z. (2016). Extensions of the Probability Logics LPP\(_2\) and LFOP\(_1\) . In: Probability Logics. Springer, Cham. https://doi.org/10.1007/978-3-319-47012-2_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-47012-2_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-47011-5
Online ISBN: 978-3-319-47012-2
eBook Packages: Computer ScienceComputer Science (R0)