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 suitable conditional random quantity, which satisfies classical probabilistic properties, has been deepened in the setting of coherence. In this framework the conjunction $(A|H) \wedge (B|K)$ of two conditional events $A|H$ and $B|K$ is defined as a five-valued object with set of possible values $\{1,0,x, y, z\}$, where $x=P(A|H), y=P(B|K)$, and $z=\mathbb{P}[(A|H) \wedge (B|K)]$. 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 $\{A|H,B|K,(A|H) \wedge_{a, b} (B|K)\}$, 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 $(A|H) \wedge_{a, b} (B|K)$.