A modal logic for uncertainty: a completeness theorem

Esther Anna Corsi, Tommaso Flaminio, Lluı́s Godo, Hykel Hosni
Proceedings of the Thirteenth International Symposium on Imprecise Probability: Theories and Applications, PMLR 215:119-129, 2023.

Abstract

In the present paper, we axiomatize a logic that allows a general approach for reasoning about probability functions, belief functions, lower probabilities and their corresponding duals. The formal setting we consider arises from combining a modal S5 necessity operator $\Box$ that applies to the formulas of the infinite-valued {Ł}ukasiewicz logic with the unary modality $P$ that describes the behaviour of probability functions. The modality $P$ together with an S5 {modality} $\Box$ provides a language rich enough to characterise probability, belief and lower probability theories. For this logic, we provide an axiomatization and we prove that, once we restrict to suitable sublanguages, it turns out to be sound and complete with respect to belief functions and lower probability models.

Cite this Paper

BibTeX
@InProceedings{pmlr-v215-corsi23a, title = {A modal logic for uncertainty: a completeness theorem}, author = {Corsi, Esther Anna and Flaminio, Tommaso and Godo, Llu{\'\i}s and Hosni, Hykel}, booktitle = {Proceedings of the Thirteenth International Symposium on Imprecise Probability: Theories and Applications}, pages = {119--129}, year = {2023}, editor = {Miranda, Enrique and Montes, Ignacio and Quaeghebeur, Erik and Vantaggi, Barbara}, volume = {215}, series = {Proceedings of Machine Learning Research}, month = {11--14 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v215/corsi23a/corsi23a.pdf}, url = {https://proceedings.mlr.press/v215/corsi23a.html}, abstract = {In the present paper, we axiomatize a logic that allows a general approach for reasoning about probability functions, belief functions, lower probabilities and their corresponding duals. The formal setting we consider arises from combining a modal S5 necessity operator $\Box$ that applies to the formulas of the infinite-valued {Ł}ukasiewicz logic with the unary modality $P$ that describes the behaviour of probability functions. The modality $P$ together with an S5 {modality} $\Box$ provides a language rich enough to characterise probability, belief and lower probability theories. For this logic, we provide an axiomatization and we prove that, once we restrict to suitable sublanguages, it turns out to be sound and complete with respect to belief functions and lower probability models.} }
Endnote
%0 Conference Paper %T A modal logic for uncertainty: a completeness theorem %A Esther Anna Corsi %A Tommaso Flaminio %A Lluı́s Godo %A Hykel Hosni %B Proceedings of the Thirteenth International Symposium on Imprecise Probability: Theories and Applications %C Proceedings of Machine Learning Research %D 2023 %E Enrique Miranda %E Ignacio Montes %E Erik Quaeghebeur %E Barbara Vantaggi %F pmlr-v215-corsi23a %I PMLR %P 119--129 %U https://proceedings.mlr.press/v215/corsi23a.html %V 215 %X In the present paper, we axiomatize a logic that allows a general approach for reasoning about probability functions, belief functions, lower probabilities and their corresponding duals. The formal setting we consider arises from combining a modal S5 necessity operator $\Box$ that applies to the formulas of the infinite-valued {Ł}ukasiewicz logic with the unary modality $P$ that describes the behaviour of probability functions. The modality $P$ together with an S5 {modality} $\Box$ provides a language rich enough to characterise probability, belief and lower probability theories. For this logic, we provide an axiomatization and we prove that, once we restrict to suitable sublanguages, it turns out to be sound and complete with respect to belief functions and lower probability models.
APA
Corsi, E.A., Flaminio, T., Godo, L. & Hosni, H.. (2023). A modal logic for uncertainty: a completeness theorem. Proceedings of the Thirteenth International Symposium on Imprecise Probability: Theories and Applications, in Proceedings of Machine Learning Research 215:119-129 Available from https://proceedings.mlr.press/v215/corsi23a.html.

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