迭代条件句与P-蕴涵的刻画
Iterated Conditionals and Characterization of P-entailment
摘要：
在本文中，我们深化，在设置的连贯性，在最近的文件中所获得的一些结果的概念的p-蕴涵的亚当斯和它的关系，连接和迭代条件。我们记得，联合和迭代条件是适当的条件随机量的框架中定义的。给定$n$条件事件$\{E_1|H_1,\ldots, E_n|H_n\}$的族$\mathcal{F}$，我们用$\mathscr{C}(\mathcal{F})=(E_1|H_1)\wedge \cdots \wedge (E_n|H_n)$表示$\mathcal{F}$中的条件事件的合取。通过证明$\mathscr{C}(\mathcal{F}_2)\wedge \mathscr{C}(\mathcal{F}_1)$的预见是$\mathscr{C}(\mathcal{F}_2)|\mathscr{C}(\mathcal{F}_1)$的预见和$\mathscr{C}(\mathcal{F}_1)$的预见的乘积，我们引入了迭代条件$\mathscr{C}(\mathcal{F}_2)|\mathscr{C}(\mathcal{F}_1)$，其中$\mathcal{F}_1$和$\mathcal{F}_2$是两个有限的条件事件族.同样，众所周知的平等$(A\wedge H)|H=A|H$，我们表明，$ (\mathscr{C}(\mathcal{F}_2)\wedge \mathscr{C}(\mathcal{F}_1))|\mathscr{C}(\mathcal{F}_1)= \mathscr{C}(\mathcal{F}_2)|\mathscr{C}(\mathcal{F}_1)$。然后，我们考虑的情况下$\mathcal{F}_1=\mathcal{F}_2=\mathcal{F}$和我们验证的预见$μ$的$\mathscr{C}(\mathcal{F})|\mathscr{C}(\mathcal{F})$，唯一的连贯评估是$μ=1$，因此，$\mathscr{C}(\mathcal{F})|\mathscr{C}(\mathcal{F})$符合常数1。最后，通过假设$\mathcal{F}$ p-一致的，我们深化了一些以前的特征的p-蕴涵表明，$\mathcal{F}$ p-蕴涵一个条件事件$E_{n+1}|H_{n+1}$当且仅当迭代条件$(E_{n+1}|H_{n+1})\,|\,\mathscr{C}(\mathcal{F})$是常数，等于1。我们通过一个与弱传递性相关的例子来说明这一特征。
英文摘要：
In this paper we deepen, in the setting of coherence, some results obtained in recent papers on the notion of p-entailment of Adams and its relationship with conjoined and iterated conditionals. We recall that conjoined and iterated conditionals are suitably defined in the framework of conditional random quantities. Given a family $\mathcal{F}$ of $n$ conditional events $\{E_1|H_1,\ldots, E_n|H_n\}$ we denote by $\mathscr{C}(\mathcal{F})=(E_1|H_1)\wedge \cdots \wedge (E_n|H_n)$ the conjunction of the conditional events in $\mathcal{F}$. We introduce the iterated conditional $\mathscr{C}(\mathcal{F}_2)|\mathscr{C}(\mathcal{F}_1)$, where $\mathcal{F}_1$ and $\mathcal{F}_2$ are two finite families of conditional events, by showing that the prevision of $\mathscr{C}(\mathcal{F}_2)\wedge \mathscr{C}(\mathcal{F}_1)$ is the product of the prevision of $\mathscr{C}(\mathcal{F}_2)|\mathscr{C}(\mathcal{F}_1)$ and the prevision of $\mathscr{C}(\mathcal{F}_1)$. Likewise the well known equality $(A\wedge H)|H=A|H$, we show that $ (\mathscr{C}(\mathcal{F}_2)\wedge \mathscr{C}(\mathcal{F}_1))|\mathscr{C}(\mathcal{F}_1)= \mathscr{C}(\mathcal{F}_2)|\mathscr{C}(\mathcal{F}_1)$. Then, we consider the case $\mathcal{F}_1=\mathcal{F}_2=\mathcal{F}$ and we verify for the prevision $μ$ of $\mathscr{C}(\mathcal{F})|\mathscr{C}(\mathcal{F})$ that the unique coherent assessment is $μ=1$ and, as a consequence, $\mathscr{C}(\mathcal{F})|\mathscr{C}(\mathcal{F})$ coincides with the constant 1. Finally, by assuming $\mathcal{F}$ p-consistent, we deepen some previous characterizations of p-entailment by showing that $\mathcal{F}$ p-entails a conditional event $E_{n+1}|H_{n+1}$ if and only if the iterated conditional $(E_{n+1}|H_{n+1})\,|\,\mathscr{C}(\mathcal{F})$ is constant and equal to 1. We illustrate this characterization by an example related with weak transitivity.
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