Generalized conjunction and disjunction of two conditional events in the setting of conditional random quantities

In recent papers, notions of conjunction and disjunction of two conditional events as suitable conditional random quantities, which satisfy basic probabilistic properties, have been deepened in the setting of coherence. In this framework, the conjunction and the disjunction of two conditional events are defined as five-valued objects, among which are the values of the (subjectively) assigned probabilities of the two conditional events. In the present paper we propose a generalization of these structures, where these new objects, instead of depending on the probabilities of the two conditional events, depend on two arbitrary values $a,b$ in the unit interval. We show that they are connected by a generalized version of the De Morgan's law and, by means of a geometrical approach, we compute the lower and upper bounds on these new objects both in the precise and the imprecise case. Moreover, some particular cases, obtained for specific values of a and b or in case of some logical relations, are analyzed. The results of this paper lead to the conclusion that the only objects satisfying all the logical and the probabilistic properties already valid for the operations between events are the ones depending on the probabilities of the two conditional events.