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Iterated Conditionals and Characterization of P-Entailment

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Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU 2021)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 12897))

Abstract

In this paper we deepen, in the setting of coherence, some results obtained in recent papers on the notion of p-entailment of Adams and its relationship with conjoined and iterated conditionals. We recall that conjoined and iterated conditionals are suitably defined in the framework of conditional random quantities. Given a family \(\mathcal {F}\) of n conditional events \(\{E_{1}|H_{1},\ldots , E_{n}|H_{n}\}\) we denote by \(\mathcal {C}(\mathcal {F})=(E_{1}|H_{1})\wedge \cdots \wedge (E_{n}|H_{n})\) the conjunction of the conditional events in \(\mathcal F\). We introduce the iterated conditional \(\mathcal {C}(\mathcal {F}_{2})|\mathcal {C}(\mathcal {F}_{1})\), where \(\mathcal {F}_{1}\) and \(\mathcal {F}_{2}\) are two finite families of conditional events, by showing that the prevision of \(\mathcal {C}(\mathcal {F}_{2})\wedge \mathcal {C}(\mathcal {F}_{1})\) is the product of the prevision of \(\mathcal {C}(\mathcal {F}_{2})|\mathcal {C}(\mathcal {F}_{1})\) and the prevision of \(\mathcal {C}(\mathcal {F}_{1})\). Likewise the well known equality \((A\wedge H)|H=A|H\), we show that \( (\mathcal {C}(\mathcal {F}_{2})\wedge \mathcal {C}(\mathcal {F}_{1}))|\mathcal {C}(\mathcal {F}_{1})= \mathcal {C}(\mathcal {F}_{2})|\mathcal {C}(\mathcal {F}_{1})\). Then, we consider the case \(\mathcal {F}_{1}=\mathcal {F}_{2}=\mathcal {F}\) and we verify for the prevision \(\mu \) of \(\mathcal {C}(\mathcal F)|\mathcal {C}(\mathcal {F})\) that the unique coherent assessment is \(\mu =1\) and, as a consequence, \(\mathcal {C}(\mathcal {F})|\mathcal {C}(\mathcal {F})\) coincides with the constant 1. Finally, by assuming \(\mathcal {F}\) p-consistent, we deepen some previous characterizations of p-entailment by showing that \(\mathcal {F}\) p-entails a conditional event \(E_{n+1}|H_{n+1}\) if and only if the iterated conditional \((E_{n+1}|H_{n+1})\,|\,\mathcal {C}(\mathcal {F})\) is constant and equal to 1. We illustrate this characterization by an example related with weak transitivity.

A. Gilio and G. Sanfilippo—Both authors contributed equally to the article and are listed alphabetically.

A. Gilio—Retired.

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Acknowledgements

We thank the four anonymous reviewers for their useful comments and suggestions. G. Sanfilippo has been partially supported by the INdAM–GNAMPA Project 2020 Grant U-UFMBAZ-2020-000819.

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Authors and Affiliations

  1. Department SBAI, University of Rome “La Sapienza”, Rome, Italy

    Angelo Gilio

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

    Giuseppe Sanfilippo

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Correspondence to Giuseppe Sanfilippo .

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  1. Institute of Information Theory and Automation, Prague, Czech Republic

    Jiřina Vejnarová

  2. Insight Centre for Data Analytics, School of Computer Science and IT University College Cork, Cork, Ireland

    Nic Wilson

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Gilio, A., Sanfilippo, G. (2021). Iterated Conditionals and Characterization of P-Entailment. In: Vejnarová, J., Wilson, N. (eds) Symbolic and Quantitative Approaches to Reasoning with Uncertainty. ECSQARU 2021. Lecture Notes in Computer Science(), vol 12897. Springer, Cham. https://doi.org/10.1007/978-3-030-86772-0_45

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