Bell inequalities are usually derived by assuming locality and realism, and therefore violations of the Bell-CHSH inequality are usually taken to imply violations of either locality or realism, or both. But, after reviewing an oversight by Bell, in the Corollary below we derive the Bell-CHSH inequality by assuming only that Bob can measure along vectors b and b' simultaneously while Alice measures along either a or a', and likewise Alice can measure along vectors a and a' simultaneously while Bob measures (...) along either b or b', without assuming locality. The violations of the Bell-CHSH inequality therefore only mean impossibility of measuring along b and b' simultaneously. (shrink)

We show that the respective oversights in the von Neumann's general theorem against all hidden variable theories and Bell's theorem against their local-realistic counterparts are homologous. When latter oversight is rectified, the bounds on the CHSH correlator work out to be ±2√2 instead of ±2.

I demonstrate that Bell's theorem is based on circular reasoning and thus a fundamentally flawed argument. It unjustifiably assumes the additivity of expectation values for dispersion-free states of contextual hidden variable theories for non-commuting observables involved in Bell-test experiments, which is tautologous to assuming the bounds of ±2 on the Bell-CHSH sum of expectation values. Its premises thus assume in a different guise the bounds of ±2 it sets out to prove. Once this oversight is ameliorated from Bell's argument by (...) identifying the impediment that leads to it and local realism is implemented correctly, the bounds on the Bell-CHSH sum of expectation values work out to be ±2√2 instead of ±2, thereby mitigating the conclusion of Bell's theorem. Consequently, what is ruled out by any of the Bell-test experiments is not local realism but the linear additivity of expectation values, which does not hold for non-commuting observables in any hidden variable theories to begin with. (shrink)