2019 International Workshop: Beyond Bell’s theorem

Bell inequalities based on the logic of plausible reasoning

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    Ilja Schmelzer

    In https://arxiv.org/abs/1712.04334 I show that the Bell inequalities follow as well from the logic of plausible reasoning also named the objective Bayesian interpretation of probability theory.

    Conceptually, in the objective Bayesian interpretation of probability, it is an extension of logic for the case of plausible reasoning. The axioms are, essentially, consistency requirements that different ways of argumentation should give the same results about the plausibility of some claim. So, the rules of probability theory obtain a quite different status, as rules of reasoning. In the frequency interpretation, they are some sort of frequencies we observe, and could be reasonably questioned, or empirically falsified or so, something which makes no sense for rules of reasoning.

    Technically, there was yet a minor technical difference between Kolmogorovian probability and the objective Bayesian interpretation, but it was easy to close, thus the logic of plausible reasoning now contains the full Kolmogorovian probability theory. But in this case, the key formula

    \[ E(AB|a,b) = \int A(a,b,l)B(a,b,l)\rho(a,b,l)dl \]

    which is considered to describe something nontrivial, like “realism”, becomes a formula of logic. It remains to get rid of a,b in \(\rho\) (superdeterminism), which appears a consequence of plausible reasoning too (logical independence: we have no information which suggests a dependence, thus, we have to assume independence) and the dependence of A on b resp. B of A, which is Einstein causality. So, the physically nontrivial assumption is only Einstein causality.

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