6533b7d0fe1ef96bd125ad5c

RESEARCH PRODUCT

Iterated Conditionals and Characterization of P-Entailment

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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.