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 F of n conditional events {E_1|H_1,ldots, E_n|H_n} we denote by C(F)=(E_1|H_1)& ... & (E_n|H_n) the conjunction of the conditional events in F. We introduce the iterated conditional C(F_2)|C(F_1), where F_1 and F_2 are two finite families of conditional events, by showing that the prevision of C(F_2)& C(F_1) is the product of the prevision of C(F_2)|C(F_1) and the prevision of C(F_1). Likewise the well known equality (A& H)|H=A|H, we show that (C(F_2)& C(F_1))|C(F_1)=C(F_2)|C(F_1). Then, we consider the case F_1=F_2=F and we verify for the prevision mu of C(F)|C(F) that the unique coherent assessment is mu=1 and, as a consequence, C(F)|C(F) coincides with the constant 1. Finally, by assuming F p-consistent, we deepen some previous characterizations of p-entailment by showing that 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})|C(F) is constant and equal to 1. We illustrate this characterization by an example related with weak transitivity.