Let S be a set of states of a physical system and p(s) be the probability of the occurrence of an event when the system is in state s. A function p from S to [0,1] is called a numerical event or alternatively an S-probability. If a set P:={p(s)|s in S} is ordered by the order of real functions such that certain plausible requirements are fulfilled, P becomes an orthomodular poset in which properties can be described by the addition and comparison of functions. P is then called an algebra of S-probabilities or algebra of numerical events. We first answer the question under which circumstances it is possible to consider sets of empirically found numerical events as members of an algebra of S-probabilities. Then we discuss the problem to decide whether a given small set Pn of S-probabilities can be embedded into a Boolean subalgebra of an algebra P of S-probabilities, in which case we will call Pn Boolean embeddable. If Pn is not Boolean embeddable, then the physical system at hand will most likely be non-classical. In the case of a concrete logic P, that is a quantum logic which can be represented by sets, we derive criteria for the subset Pn of P to be Boolean embeddable which can be checked by very simple procedures, for arbitrary S-probabilities we provide sets of Bell-like inequalities which characterize the Boolean embeddability of Pn. Finally we will show how these Bell-like inequalities fit into a general framework of Bell inequalities by providing a method for generating Bell inequalities for S-probabilities from elementary Bell valuations.

中文翻译：

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数值事件的布尔性质和类钟不等式

设 S 是物理系统的一组状态，p(s) 是系统处于状态 s 时事件发生的概率。从 S 到 [0,1] 的函数 p 称为数值事件或 S 概率。如果一个集合 P:={p(s)|s in S} 是按照实函数的顺序排序的，以满足某些似是而非的要求，那么 P 就变成了一个正交模偏组，其中的属性可以通过函数的相加和比较来描述. P 被称为 S 概率的代数或数值事件的代数。我们首先回答在什么情况下可以将经验发现的数值事件集视为 S 概率代数的成员的问题。然后我们讨论决定一个给定的 S 概率的小集合 Pn 是否可以嵌入到 S 概率的代数 P 的布尔子代数中的问题，在这种情况下，我们将称 Pn 为布尔可嵌入的。如果 Pn 不是布尔可嵌入的，那​​么手头的物理系统很可能是非经典的。在具体逻辑 P 的情况下，即可以用集合表示的量子逻辑，我们推导出 P 的子集 Pn 是布尔可嵌入的标准，可以通过非常简单的程序进行检查，对于我们提供的任意 S 概率描述 Pn 的布尔可嵌入性的类 Bell 不等式的集合。最后，我们将通过提供一种从基本贝尔估值为 S 概率生成贝尔不等式的方法来展示这些类贝尔不等式如何适合贝尔不等式的一般框架。