论文标题
复合条件，Fréchet-Hoeffding边界和Frank T-Norms
Compound conditionals, Fréchet-Hoeffding bounds, and Frank t-norms
论文作者
论文摘要
在本文中，我们考虑了复合条件，Fréchet-Hoeffding边界以及对Frank T-norms的概率解释。通过研究合适的线性系统的可溶性，我们在逻辑独立性下显示了Fréchet-Hoeffding边界的清晰度，以预防$ n $条件事件的连词和析取。我们通过验证它是凸的，研究了包含$ n $有条件事件的家庭的所有连贯预防评估及其连词的集合。我们讨论了通过Lukasiewicz T-Norms评估连词的预防并为线性系统提供明确的解决方案的情况。然后，我们分析了选定的示例。我们分别对有条件事件的结合和析出分别获得了对坦率的T-norms和T-conorms的概率解释。然后，我们通过使用Frank T-norms或Frank T-Conorms来表征包含$ n $有条件事件的家庭及其结合或脱节的家庭的连贯预防评估。通过假设逻辑独立性，我们表明，两个有条件事件的弗兰克t-norm（分别，t-conorm）$ a | h $ a | h $ a | h $ a | h $ a | h $，$t_λ（a | h，b | k）$（分别，$s_λ（a | h，b | k）是一个连接$（a | h） （b | k）$）。通过考虑一个家庭$ \ MATHCAL {F} $包含三个条件事件，它们的连词以及所有成对的连词，我们就可以对Frank T-norms和对$ \ Mathcal {f} $的预防评估进行一些结果。通过假设逻辑独立性，我们表明，通过最小值和产物T-norms评估所有连词的预防是一致的。我们通过反例验证，当通过Lukasiewicz t-norm评估连词的预防时，就不会确保连贯性。
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. 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 $B|K$, $T_λ(A|H,B|K)$ (resp., $S_λ(A|H,B|K)$), 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.
