Boolean Properties and Bell-Like Inequalities of Numerical Events

Abstract

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 ∈ S. A function p : S → [0, 1] is called a numerical event or alternatively an S-probability. If a set P := {p(s) | s ∈ 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 P n of S-probabilities can be embedded into a Boolean subalgebra of an algebra P of S-probabilities, in which case we will call P n Boolean embeddable. If P n 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 P n ⊆ 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 P n . 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.

On Boolean posets of numerical events

“…, p n are contained, as it is well known for every subset of mutually orthogonal elements of an orthomodular poset (cf. [8]), and as proved in [13] every algebra of S-probabilities is orthomodular, and by Theorem 3.4 also structured sets of S-probabilities have this property. So let us suppose that {p 1 , .…”

“…Assume n = 2. Then according to Theorem 3.4 in[8] {p 1 , p 2 } is contained in a Boolean subalgebra of P if and only if p 1 ⊼ p 2 (:= min(p 1 , p 2 )) ∈ P which in our notion means that p 1 • p 2 ∈ P . In this theorem it is also stated that p 1 ⊼ p 2 (= p 1 • p 2 ) ∈ P is equivalent to p 1 C p 2 .…”

On Boolean posets of numerical events

“…, p n are contained, as it is well known for every subset of mutually orthogonal elements of an orthomodular poset (cf. Dorninger et al (2020)), and as proved in Maczyński and Traczyk (1973), every algebra of S-probabilities is orthomodular. So, let us suppose that { p 1 , .…”

“…Then according to Theorem 3.4, in Dorninger et al (2020), { p 1 , p 2 } is contained in a Boolean subalgebra of P if and only if p 1 p 2 (:= min( p 1 , p 2 )) ∈ P which in our notion means that p 1 • p 2 ∈ P. In this theorem, it is also stated that p 1 p 2 (= p 1 • p 2 ) ∈ P is equivalent to p 1 C p 2 .…”