Time independence of Boole-Bell conditions of possible classical experience

– The theorem of Bell is a variant of Boole’s legendary consistency “conditions of possible experience.” Such an interpretation appears to be immune to arguments involving time dependencies put forward recently by Hess and Philipp [Europhysics Letters 57(6), 775-781, 2002], although experiments need not be. Boole’s “conditions of possible experience” and Pitowsky correlation polytopes. – In the middle of the 19th century the English mathematician George Boole formulated a theory of “conditions of possible experience” (COPE) [1–5]. These conditions subsume the consistency requirements satisfied by relative frequencies or probabilities of classical events. They are expressed by certain equations or inequalities. Here, the term “classical” refers to the fact that events can be joined and united by the usual rules of Boolean algebra. More recently, similar equations for a particular setup relevant in the quantum mechanical context have been discussed by Bell, Clauser&Horne and others [6–9]. Pitowsky has given a geometrical interpretation of COPE in terms of correlation polytopes [4, 5, 10, 11]. Thereby, the rows of the truth tables of events and their joints are interpreted as vectors in a real linear vector space. A correlation polytope is defined by taking all such vectors and interpreting them as the extreme points of the polytope. The Minkowski-Weyl representation theorem (e.g., [12, p. 29]) states that compact convex sets are “spanned” by their extreme points; and furthermore that the representation of this polytope by the inequalities corresponding to the planes of their faces is an equivalent one. Stated differently, every convex polytope has a dual description: either as the convex hull of its vertices (V-representation), or as the intersection of a finite number of half-spaces, each one given by a linear inequality (H-representation). The problem to obtain all inequalities from the vertices of a convex polytope is known as the hull problem. It is computationally hard [11] but recursively enumerable. One solution strategy is the Double Description Method [13]. The physical interpretation of the inequalities representing the boundaries of the Pitowsky correlation Polytope is this: Any face of the polytope has an “inside” and an “outside,” and corresponds to a Boole-Bell type inequality. It can be viewed as a sort of demarcation line, a maximal border, between the classically allowed probabilities and the ones (outside of the polytope) which are inconsistent with a classical description of events as a Boolean algebra and