Logical difficulty from combining counterfactuals in the GHZ-Bell theorems

In eliminating the fair sampling assumption, the Greenberger, Horne, Zeilinger (GHZ) theorem is believed to confirm Bell's historic conclusion that local hidden variables are inconsistent with the results of quantum mechanics. The GHZ theorem depends on predicting the results of sets of measurements of which only one may be performed. In the present paper, the non-commutative aspects of these unperformed measurements are critically examined. Classical examples and the logic of the GHZ construction are analyzed to demonstrate that combined counterfactual results of non-commuting operations are in general logically inconsistent with performed measurement sequences whose results depend on non-commutation. The Bell theorem is also revisited in the light of this result. It is concluded that negative conclusions regarding local hidden variables do not follow from the GHZ and Bell theorems as historically reasoned.