Iterated Conditionals and Characterization of P-entailment

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 $\mathscr{C}(\mathcal{F})=(E_1|H_1)\wedge \cdots \wedge (E_n|H_n)$ the conjunction of the conditional events in $\mathcal{F}$. We introduce the iterated conditional $\mathscr{C}(\mathcal{F}_2)|\mathscr{C}(\mathcal{F}_1)$, where $\mathcal{F}_1$ and $\mathcal{F}_2$ are two finite families of conditional events, by showing that the prevision of $\mathscr{C}(\mathcal{F}_2)\wedge \mathscr{C}(\mathcal{F}_1)$ is the product of the prevision of $\mathscr{C}(\mathcal{F}_2)|\mathscr{C}(\mathcal{F}_1)$ and the prevision of $\mathscr{C}(\mathcal{F}_1)$. Likewise the well known equality $(A\wedge H)|H=A|H$, we show that $ (\mathscr{C}(\mathcal{F}_2)\wedge \mathscr{C}(\mathcal{F}_1))|\mathscr{C}(\mathcal{F}_1)= \mathscr{C}(\mathcal{F}_2)|\mathscr{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 $\mathscr{C}(\mathcal{F})|\mathscr{C}(\mathcal{F})$ that the unique coherent assessment is $\mu=1$ and, as a consequence, $\mathscr{C}(\mathcal{F})|\mathscr{C}(\mathcal{F})$ 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 $(E_{n+1}|H_{n+1})\,|\,\mathscr{C}(\mathcal{F})$ is constant and equal to 1. We illustrate this characterization by an example related with weak transitivity.