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In this paper, by adopting a coherence-based probabilistic approach todefault reasoning, we focus the study on the logical operation of quasiconjunction and the Goodman-Nguyen inclusion relation for conditional events.We recall that quasi conjunction is a basic notion for defining consistency ofconditional knowledge bases. By deepening some results given in a previouspaper we show that, given any finite family of conditional events F and anynonempty subset S of F, the family F p-entails the quasi conjunction C(S);then, given any conditional event E|H, we analyze the equivalence betweenp-entailment of E|H from F and p-entailment of E|H from C(S), where S is somenonempty subset of F. We also illustrate some alternative theorems related withp-consistency and p-entailment. Finally, we deepen the study of the connectionsbetween the notions of p-entailment and inclusion relation by introducing for apair (F,E|H) the (possibly empty) class K of the subsets S of F such that C(S)implies E|H. We show that the class K satisfies many properties; in particularK is additive and has a greatest element which can be determined by applying asuitable algorithm.

Probabilistic entailment in the setting of coherence: The role of quasi conjunction and inclusion relation