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Iterated Conditionals, Trivalent Logics, and Conditional Random Quantities

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Scalable Uncertainty Management (SUM 2022)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 13562))

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Abstract

We consider some notions of iterated conditionals by checking the validity of some desirable basic logical and probabilistic properties, which are valid for simple conditionals. We consider de Finetti’s notion of conditional as a three-valued object and as a conditional random quantity in the betting framework. We recall the notions of conjunction and disjunction among conditionals in selected trivalent logics. Then, we analyze the two notions of iterated conditional introduced by Calabrese and de Finetti, respectively. We show that the compound probability theorem and other basic properties are not preserved by these objects, by also computing some probability propagation rules. Then, for each trivalent logic we introduce an iterated conditional as a suitable random quantity which satisfies the compound prevision theorem and some of the desirable properties. Finally, we remark that all the basic properties are satisfied by the iterated conditional mainly developed in recent papers by Gilio and Sanfilippo in the setting of conditonal random quantities.

L. Castronovo and G. Sanfilippo—Both authors contributed equally to the article and are listed alphabetically.

G. Sanfilippo—Supported by the FFR project of University of Palermo and by the INdAM-GNAMPA Research Group, Italy.

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Acknowledgements

We thank the three anonymous reviewers for their useful comments and suggestions. We also thank the Gugino Prize committee of University of Palermo.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Palermo, Palermo, Italy

    Lydia Castronovo & Giuseppe Sanfilippo

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  1. Lydia Castronovo
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  2. Giuseppe Sanfilippo
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Correspondence to Lydia Castronovo or Giuseppe Sanfilippo .

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Editors and Affiliations

  1. IRIT-Université Toulouse, Toulouse, France

    Florence Dupin de Saint-Cyr

  2. Université Paris-Dauphine - PSL, Paris, France

    Meltem Öztürk-Escoffier

  3. Imperial College London, London, UK

    Nico Potyka

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Castronovo, L., Sanfilippo, G. (2022). Iterated Conditionals, Trivalent Logics, and Conditional Random Quantities. In: Dupin de Saint-Cyr, F., Öztürk-Escoffier, M., Potyka, N. (eds) Scalable Uncertainty Management. SUM 2022. Lecture Notes in Computer Science(), vol 13562. Springer, Cham. https://doi.org/10.1007/978-3-031-18843-5_4

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  • DOI: https://doi.org/10.1007/978-3-031-18843-5_4

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