Bell's Theorem requires any theory which obeys the technical definitions of Free Choice and Local Causality to satisfy the Bell inequality. Invariant set theory is a finite theory of quantum physics which violates the Bell inequality exactly as does quantum theory: in it neither Free Choice nor Local Causality hold, consistent with Bell's Theorem. However, within the proposed theory, the mathematical expressions of both Free Choice and Local Causality involve states which, for number-theoretic reasons, cannot be ontic (cannot lie on the theory's fractal-like invariant set
IU in state space). Weaker versions of Free Choice and Local Causality are proposed involving only the theory's ontic states. Conventional hidden-variable theories satisfying only these weaker definitions still obey the Bell inequality. However, invariant set theory, which violates the Bell inequality, satisfies these weaker definitions. It is argued that the weaker definitions are consistent with the physical meaning behind free choice and local causality as defined in space-time, and hence that Free Choice and Local Causality are physically too strong. It is concluded that the experimental violation of the Bell inequality may have less to do with free choice or local causality \emph{per se}, and more to do with the presence of a holistic but causal state-space geometry onto which quantum ontic states are constrained.
It is generally believed that Bell's inequality holds for the case of entangled states, including two correlated particles or special states of a single particle. Here, we derive a single-particle Bell's inequality for two correlated spin states at two successive times, appealing to the statistical independence condition in an ideal experiment, for a locally causal hidden variables theory. We show that regardless of the locality assumption, the inequality can be violated by some quantum predictions.
It is widely accepted that the violation of Bell inequalities excludes local theories of the quantum realm. This paper presents a stronger Bell argument which even forbids certain non-local theories. Among these excluded non-local theories are those whose only non-local connection is a probabilistic (or functional) dependence between the space-like separated measurement outcomes of EPR/B experiments (a subset of outcome dependent theories). In this way, the new argument shows that the result of the received Bell argument, which requires just any kind of non-locality, is inappropriately weak. Positively, the remaining non-local theories, which can violate Bell inequalities (among them quantum theory), are characterized by the fact that at least one of the measurement outcomes in some sense probabilistically depends both on its local as well as on its distant measurement setting (probabilistic Bell contextuality). Whether an additional dependence between the outcomes holds, is irrelevant for the question whether a certain theory can violate Bell inequalities. This new concept of quantum non-locality is considerably tighter and more informative than the one following from the usual Bell argument. We prove that (given usual background assumptions) the result of the stronger Bell argument presented here is the strongest possible consequence from the violation of Bell inequalities on a qualitative probabilistic level.