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

We consider compound conditionals, Frechet-Hoeffding bounds and the probabilistic interpretation of Frank t-norms. We show under logical independence the sharpness of the Frechet-Hoeffding bounds for the prevision of conjunctions and disjunctions of $n$ conditional events. 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 $(A|H)\vee (B|K)$). Then, we analyze the case of logical dependence where $A=B$. We give some results on Frank t-norms and coherence of the prevision assessments on a family of three conditional events. 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 using the Lukasiewicz t-norm to assess the previsions of conjunctions, coherence is not assured; thus, 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.