Probabilistic entailment and iterated conditionals

In this chapter we exploit the notions of conjoined and iterated conditionals. These notions are defined, in the setting of coherence, by means of suitable conditional random quantities with values in the interval [0,1]. We examine the iterated conditional (B|K)|(A|H), by showing that A|H p-entails B|K if and only if (B|K)|(A|H) = 1. Then, we show that a (p-consistent) family F = {E1|H1, E2|H2} p-entails a conditional event E3|H3 if and only if E3|H3 = 1, or (E3|H3)|QC(S) = 1 for some nonempty subset S of F, where QC(S) is the quasi conjunction of the conditional events in S. We also examine the inference rules And, Cut, Cautious Monotonicity, and Or of System P, and other well-known inference rules (Modus Ponens, Modus Tollens, and Bayes). Furthermore, we show that QC(F)|C(F) = 1, where C(F) is the conjunction of the two conditional events in F. We characterize p-entailment by showing that F p-entails E3|H3 if and only if (E3|H3)|C(F) = 1. Finally, we examine Denial of the antecedent, Affirmation of the consequent, and Transitivity where the p-entailment of E3|H3 from F does not hold, so that (E3|H3)|C(F) is not equal to 1.