Generalized notions of conjunction and disjunction for two conditional events under coherence
Compound conditionals are largely used in natural language to describe decisions and inferences based on incomplete or uncertain information. Definitions of conjunction and disjunction of conditional events as three-valued objects are common in literature, although these do not preserve some classical logical and probabilistic properties. In recent papers, notions of conjunction and disjunction of two conditional events as suitable conditional random quantities, which satisfy classical probabilistic properties, have been deepened in the setting of coherence. In this framework the conjunction (A|H) \wedge (B|K) and the disjunction (A|H) \vee (B|K) of two conditional events A|H and B|K are defined as five-valued objects with sets of possible values {1,0,x,y,z} and {1,0,x,y,w}, respectively, where x=P(A|H), y=P(B|K), z=\mathbb{P}[(A|H) \wedge (B|K)] and w=\mathbb{P}[(A|H) \vee (B|K)]. In the present paper we propose a generalization of these structures, denoted by (A|H) \wedge_{a,b} (B|K) and (A|H) \vee_{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 sets of all coherent assessments on the families {A|H,B|K,(A|H) \wedge_{a,b} (B|K)} and {A|H,B|K,(A|H) \vee_{a,b} (B|K)}, by also showing that in the general case the Fréchet-Hoeffding bounds are not satisfied. We also analyze some particular cases, obtained for specific values of a and b or when some logical relations among the events A, B, H, K are taken into account. We give an interpretation of the generalized conjunction and disjunction within the context of a betting framework. Finally, we study coherence in the imprecise case of interval-valued assessments.
Area: CS56 - Conditional knowledge representation and information fusion under coherence (Lydia Castronovo and Giuseppe Sanfilippo)
Keywords: Coherence, Compound conditionals, Conditional prevision, Fréchet-Hoeffding bounds, Imprecise probability.
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