In a recent article https://doi.org/10.1007/s10699-020-09666-0) Marek Czachor claims that the Bell inequality cannot be proved because variables of complementary measurements cannot be added or multiplied. Even though he has correctly identified the problems existing with the orthodox interpretation of the Bell inequality and dealt with them in an original way, the interpretation he addresses do not pertain to the original formulation given by John Stewart Bell.

Counterfactual definiteness is supposed to underlie the Bell theorem. An old controversy exists among those who reject the theorem implications by rejecting counterfactual definiteness and those who claim that, since it is a direct consequence of locality, it cannot be independently rejected. We propose a different approach for solving this contentious issue by realizing that counterfactual definiteness is an unnecessary and inconsistent assumption. Counterfactual definiteness is not equivalent to realism or determinism neither it follows from locality. It merely reduces to (...) an incongruent application of counterfactual reasoning. Being incompatible with falsifiability, it constitutes an unjustified assumption that goes against the scientific method rigor. Correct formulations of the Bell theorem’s bases show it is absent either as a fundamental hypothesis or as a consequence of something else. Most importantly, we present a coherent Bell inequality derivation carefully devised to show explicitly and convincingly the absence of incompatible experiments or counterfactual reasoning. Thus, even admitting that counterfactual definiteness could be a consistent assumption, the necessary conclusion is that it is irrelevant for the inequality formulation and can be safely ignored when discussing Bell’s inequality philosophical and physical implications. (shrink)

Local hidden-variable model of singlet-state correlations discussed in Czachor is shown to be a particular case of an infinite hierarchy of local hidden-variable models based on an infinite hierarchy of calculi. Violation of Bell-type inequalities can be interpreted as a ‘confusion of languages’ problem, a result of mixing different but neighboring levels of the hierarchy. Mixing of non-neighboring levels results in violations beyond the Tsirelson bounds.