ar X iv : 0 90 7 . 07 67 v 1 [ qu an t - ph ] 4 J ul 2 00 9 epl draft Possible Experience : from Boole to Bell

Mainstream interpretations of quantum theory maintain that violations of the Bell inequalities deny at least either realism or Einstein locality. Here we investigate the premises of the Bell-type inequalities by returning to earlier inequalities presented by Boole and the findings of Vorob’ev as related to these inequalities. These findings together with a space-time generalization of Boole’s elements of logic lead us to a completely transparent Einstein local counterexample from everyday life that violates certain variations of the Bell inequalities. We show that the counterexample suggests an interpretation of the Born rule as a pre-measure of probability that can be transformed into a Kolmogorov probability measure by certain Einstein local space-time characterizations of the involved random variables. Introduction. – We discuss models of EinsteinPodolsky-Rosen-Bohm type [1, 2] of experiments as used by John Bell [3] when presenting his celebrated inequalities. These experiments result in outcomes of two spinvalues ±1 (in units of h̄/2) that in turn depend on certain magnet settings a,b, c... and have been linked to two-valued functions Aa(·), Ab(·), Ac(·) = ±1 by Bell and followers. Here (·) stands for the dependency on some element of a set of mathematical representations of elements of reality that do not depend on the magnet settings a,b, c.... This latter fact of independence on magnet settings was deduced by Bell from considerations of Einstein locality and the (physically unjustified) assumption that the elements of reality emanate exclusively from a distant source and not from the measurement equipment (including the magnets). There are numerous inequalities, delineated in the physics literature that are related to Bell’s functions Aa(·), . . .. These inequalities were first derived by Boole [4] in a much more general context. Here we discuss mainly a variation of the inequalities as published by Leggett and Garg [5], for which we also have developed a transparent counterexample. More complex counterex(a)E-mail: [email protected] amples have been developed in the past for the more elaborate inequalities [6] but have remained largely unappreciated because of their lack of transparency. Nevertheless, strong movements critical of Bell’s approach continue to grow as demonstrated in [7]. The Leggett-Garg inequality reads: Aa(·)Ab(·) +Aa(·)Ac(·) +Ab(·)Ac(·) ≥ −1. (1) Inserting all possible values of ±1 for the functions A(.) shows the correctness of this inequality. Because measurement outcomes of Einstein-Podolsky-Rosen (EPR) experiments [8] (that are closely related to such two-valued functions A(.)) do violate this inequality, it is commonly concluded that either (·) can not stand for any element of reality and one must therefore abandon realism or if it stands for an element of reality it must depend on the magnet settings and thus violate Einstein locality. There are, however, two important questions that have never been answered satisfactorily. If (·) stands for an element of reality, why does it have to appear identically for the three magnet setting pairs? If, on the other hand (·) is just seen as a random variable, why do the functions A not also depend on a measurement time label, as they are introduced in the theory of stochastic processes [9]?