Compound conditionals, Fr\'echet-Hoeffding bounds, and Frank t-norms

In this paper we consider compound conditionals, Fr\'echet-Hoeffding bounds and the probabilistic interpretation of Frank t-norms. By studying the solvability of suitable linear systems, we show under logical independence the sharpness of the Fr\'echet-Hoeffding bounds for the prevision of conjunctions and disjunctions of $n$ conditional events. We study the set of all coherent prevision assessments on a family containing $n$ conditional events and their conjunction, by verifying that it is convex. We discuss the case where the prevision of conjunctions is assessed by Lukasiewicz t-norms and we give explicit solutions for the linear systems; then, we analyze a selected example. We obtain a probabilistic interpretation of Frank t-norms and t-conorms as prevision of conjunctions and disjunctions of conditional events, respectively. Then, we characterize the sets of coherent prevision assessments on a family containing $n$ conditional events and their conjunction, or their disjunction, by using Frank t-norms, or Frank t-conorms. By assuming logical independence, we show that any Frank t-norm (resp., t-conorm) of two conditional events $A|H$ and $B|K$, $T_{\lambda}(A|H,B|K)$ (resp., $S_{\lambda}(A|H,B|K)$), is a conjunction $(A|H)\wedge (B|K)$ (resp., a disjunction $(A|H)\vee (B|K)$). By considering a family $\mathcal{F}$ containing three conditional events, their conjunction, and all pairwise conjunctions we give some results on Frank t-norms and coherence of the prevision assessments on $\mathcal{F}$. By assuming logical independence, we show that it is coherent to assess the previsions of all the conjunctions by means of Minimum and Product t-norms. We verify by a counterexample that, when the previsions of conjunctions are assessed by the Lukasiewicz t-norm, coherence is not assured.