A Generalized Notion of Conjunction for Two Conditional Events

Abstract

Traditionally the conjunction of conditional events has been defined as a three-valued object. However, in this way classical logical and probabilistic properties are not preserved. In recent literature, a notion of conjunction of two conditional events as a five-valued object satisfying classical probabilistic properties has been deepened in the setting of coherence. In this framework the conjunction of (A|H) \wedge (B|K) is defined as a conditional random quantity with set of possible values {1,0,x,y,z}, where x=P(A|H), y=P(B|K), and z is the prevision of (A|H) & (B|K). In this paper we propose a generalization of this object, denoted by (A|H) \wedge_{a,b} (B|K), where the values x and y are replaced by two arbitrary values a,b in [0,1]. Then, by means of a geometrical approach, we compute the set of all coherent assessments on the family {A|H,B|K,(A|H) &_{a,b} (B|K)}, by also showing that in the general case the Fréchet-Hoeffding bounds for the conjunction are not satisfied. We also analyze some particular cases. Finally, we study coherence in the imprecise case of an interval-valued probability assessment and we consider further aspects on (A|H) &_{a,b} (B|K).