  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.
