EPR-Bell realism as a part of logic

We argue that for the proof of Bell’s theorem no assumptions about realism or free will are necessary. The key formula $$E(AB\mathrm{\mid }a,b)=\int A(a,b,\lambda )B(a,b,\lambda )\rho (\lambda )d\lambda$$ follows from the logic of plausible reasoning (the objective Bayesian interpretation of probability theory) taken alone, without any further assumptions about realism. The space $\mathrm{\Lambda }$, usually interpreted as some space of hidden variables', can be constructed for an arbitrary field of discourse’ using Stone’s theorem. The rejection of superdeterminism follows from logical independence – the non-existence of information which suggest a dependence – of the free decisions of the experimenters from everything else. To prove the Bell inequality it is, then, sufficient to reduce this to $$E(AB\mathrm{\mid }a,b)=\int A(a,\lambda )B(b,\lambda )\rho (\lambda )d\lambda \mathrm{.}$$ This follows for space-like separated measurements from Einstein causality alone. So, the consequence of the violation of Bell’s inequality is that Einstein causality has been empirically falsified, without any loopholes left. We consider and reject the idea of weakening the logic of plausible reasoning could be used to create a loophole. Finally, we discuss what can be used to replace Einstein causality. While weak (signal) Einstein causality holds, it follows either from a notion of strong causality or from human inability to prepare states which violate quantum uncertainty, which makes it uninteresting for fundamental considerations. A notion of strong causality has either unacceptable causal loops or requires a hidden preferred frame. Given that a hidden preferred frame is completely compatible with modern physics, it is argued that this choice is preferable.