Compound conditionals, Fréchet-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. In addition, we illustrate some details in the case of three 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 $(A|H)ee (B|K)$). Then, we analyze the case of logical dependence where $A=B$ and we obtain the set of coherent assessments on ${A|H,A|K,(A|H)wedge (A|K)}$; moreover we represent it in terms of the class of Frank t-norms $T_{lambda}$, with $lambdain[0,1]$. 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. In this case all the conjunctions coincide with the t-norms of the corresponding conditional events. We verify by a counterexample that, when the previsions of conjunctions are assessed by the Lukasiewicz t-norm, coherence is not assured. Then, the Lukasiewicz t-norm of conditional events may not be interpreted as their conjunction. Finally, we give two sufficient conditions for coherence and incoherence when using the Lukasiewicz t-norm.