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 $\{E_1|H_1,\dots ,E_n|H_n\}$ 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.

References in corpus (10)

• Conditional Random Quantities and Compounds of Conditionals
• Quasi Conjunction, Quasi Disjunction, T-norms and T-conorms: Probabilistic Aspects
• Probabilistic inferences from conjoined to iterated conditionals
• Probabilities of conditionals and previsions of iterated conditionals
• Generalized Logical Operations among Conditional Events
• Probabilistic entailment in the setting of coherence: The role of quasi conjunction and inclusion relation
• Boolean algebras of conditionals, probability and logic
• Compound conditionals, Fréchet-Hoeffding bounds, and Frank t-norms
• Algebraic aspects and coherence conditions for conjoined and disjoined conditionals
• Probabilistic entailment and iterated conditionals

Cited by in corpus (1)

• On Trivalent Logics, Compound Conditionals, and Probabilistic Deduction Theorems