Abstract
We present a logic for reasoning about higher-order upper and lower probabilities of justification formulas. We provide sound and strongly complete axiomatization for the logic. Furthermore, we show that the introduced logic generalizes the existing probabilistic justification logic \(\mathsf {PPJ}\).
This work was supported by the SNSF project 200021\(\_\)165549 Justifications and non-classical reasoning and by the Serbian Ministry of Education and Science through Mathematical Institute of Serbian Academy of Sciences and Arts.
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Notes
- 1.
\(\mathsf {I}\) stands for iterations, \(\mathsf {LUP}\) for lower and upper probabilities and \(\mathsf {J}\) for the justification logic \(\mathsf {J}\).
- 2.
\([A]_{M,w}\) represents the set of all worlds from W(w) in a model M where A holds and will be defined later.
- 3.
When M is clear from the context we will write \([A]_w.\)
References
Anger, B., Lembcke, J.: Infinitely subadditive capacities as upper envelopes of measures. Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete 68, 403–414 (1985)
Artemov, S., Fitting, S.: Justification Logic: Reasoning with Reasons, Cambridge University Press, New York, June 2019
Artemov, S.N.: Explicit provability and constructive semantics. Bull. Symbol. Logic 7(1), 1–36 (2001)
Artemov, S.: On aggregating probabilistic evidence. In: Artemov, S., Nerode, A. (eds.) LFCS 2016. LNCS, vol. 9537, pp. 27–42. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-27683-0_3
Doder, D., Marinković, B., Maksimović, P., Perović, A.: A logic with conditional probability operators. Publications de l’Institut Mathématique 87(101) (2010)
Doder, D., Savić, N., Ognjanović, Z.: Multi-agent logics for reasoning about higher-order upper and lower probabilities. J. Logic Lang. Inf. 1–31 (2019)
Fitting, M.: The logic of proofs, semantically. Ann. Pure Appl. Logic 132(1), 1–25 (2005)
Halpern, J.Y., Pucella, R.: A logic for reasoning about upper probabilities. J. Artif. Intell. Res. 17, 57–81 (2002)
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.) LFCS 2016. LNCS, vol. 9537, pp. 174–186. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-27683-0_13
Kuznets, R., Studer, T.: Weak arithmetical interpretations for the logic of proofs. Logic J. IGP 24(3), 424–440 (2016)
Kuznets, R., Studer, F.: Logics of Proofs and Justifications. College Publications, London (2019)
Marinkovic, B., Glavan, P., Ognjanovic, Z., Studer, T.: A temporal epistemic logic with a non-rigid set of agents for analyzing the blockchain protocol. J. Log. Comput. 29(5), 803–830 (2019)
Marinkovic, B., Ognjanovic, Z., Doder, D., Perovic, A.: A propositional linear time logic with time flow isomorphic to \(\omega \)\({}^{\text{2 }}\). J. Appl. Logic 12(2), 208–229 (2014)
Milnikel, R.S.: The logic of uncertain justifications. Ann. Pure Appl. Logic 165(1), 305–315 (2014)
Ognjanovic, Z., Raskovic, M., Markovic, Z.: Probability Logics - Probability-Based Formalization of Uncertain Reasoning. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-47012-2
Ognjanović, Z., Savić, N., Studer, T.: Justification logic with approximate conditional probabilities. In: Baltag, A., Seligman, J., Yamada, T. (eds.) LORI 2017. LNCS, vol. 10455, pp. 681–686. Springer, Heidelberg (2017). https://doi.org/10.1007/978-3-662-55665-8_52
Savić, N., Doder, D., Ognjanović, Z.: Logics with lower and upper probability operators. Int. J. Approx. Reason. 88, 148–168 (2017)
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We would like to thank the anonymous reviewers whose comments helped to improve the paper substantially.
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Doder, D., Ognjanović, Z., Savić, N., Studer, T. (2022). Incomplete Information and Justifications. In: Özgün, A., Zinova, Y. (eds) Language, Logic, and Computation. TbiLLC 2019. Lecture Notes in Computer Science, vol 13206. Springer, Cham. https://doi.org/10.1007/978-3-030-98479-3_13
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