In this paper we consider compound conditionals, Fréchet-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 Fréchet-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 $(A|H)\vee (B|K)$). 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 $\lambda \in [0,1]$. 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.

在本文中，我们考虑复合条件、Fréchet-Hoeffding 边界和 Frank t-范数的概率解释。通过研究合适的线性系统的可解性，我们在逻辑独立性下展示了 Fréchet-Hoeffding 边界的锐度，用于预测n 个条件事件的连接和分离。此外，我们在三个条件事件的情况下说明了一些细节。我们研究包含n的家庭的所有连贯的预测评估的集合条件事件及其联合，通过验证它是凸的。我们讨论了通过 Lukasiewicz t 范数评估连接的预判的情况，并给出了线性系统的明确解；然后，我们分析一个选定的例子。我们将 Frank t-norms 和 t-conorms 的概率解释分别作为条件事件的连接和分离的预测。然后，我们通过使用 Frank t-norms 或 Frank t-conorms 来表征包含n 个条件事件及其结合或分离的家庭的连贯预测评估集。通过假设逻辑独立，我们证明两个条件事件的任何 Frank t-norm (resp., t-conorm)$\mathrm{一种}|H$ 和 $乙|钾$, ${吨}_{\lambda }(\mathrm{一种}|H,乙|钾)$ （分别， ${秒}_{\lambda }(\mathrm{一种}|H,乙|钾)$)，是连词 $(\mathrm{一种}|H)\wedge (乙|钾)$ （分别是分离 $(\mathrm{一种}|H)\vee (乙|钾)$）。然后，我们分析逻辑依赖的情况，其中$\mathrm{一种}=乙$ 我们获得了一套连贯的评估 $\mathrm{一种}|H,\mathrm{一种}|钾,(\mathrm{一种}|H)\wedge (\mathrm{一种}|钾)$; 此外，我们用 Frank t-norms 的类来表示它${吨}_{\lambda }$， 和 $\lambda \in [0,1]$. 通过考虑家庭$\mathcal{F}$ 包含三个条件事件、它们的连词和所有成对连词，我们给出了一些关于弗兰克 t 范数和预测评估的连贯性的结果 $\mathcal{F}$. 通过假设逻辑独立性，我们表明通过最小值和乘积 t 范数评估所有连词的前提是一致的。在这种情况下，所有的连词都与相应条件事件的 t 范数一致。我们通过反例验证，当用 Lukasiewicz t 范数评估连词的前提时，不能保证连贯性。那么，条件事件的 Lukasiewicz t 范数可能不会被解释为它们的合取。最后，我们给出了使用 Lukasiewicz t 范数时相干和非相干的两个充分条件。