(a) that Y 1 (1) = Y 2 (2) = 1 or (b) that Y 1 (1) = Y 2 (2) = − 1. Suppose that (a) holds: then M (0 , 2) = 0 and Y 2 (2) = 1 imply Y 1 (0) = − 1. But, M (1 , 0) = 0 and Y 1 (1) = 1 imply Y 2 (0) = − 1 and thus,by M (0 , 0) = 0, that Y 1 (0) = 1, a contradiction. Suppose instead that (b) holds. Then M (0 , 2) = 0 and Y 2 (2) = − 1 implies Y 1 (0) = 1. But M (1 , 0) = 0 and Y 1 (1) = − 1 implies Y 2 (0) = 1 and thus, by M (0 , 0) = 0, that Y 1 (0) = − 1, again a contradiction. Thus it cannot be the case that Y$_{i}$ ( x 1 ,x 2 ) = Y$_{i}$ ( x$_{i}$ ) ,i = 1 , 2 . The next result is given in VanderWeele (2010) in the context of testing for a causal interaction, sometimes referred to as ”epistasis” in genetics. It relates the empirical data E [ M ( x 1 ,x 2 )] to the existence of a unit satisfying M (1 , 2) = 1 ,M (0 , 2) = M (1 , 0) = M (0 , 0) = 0. Within the counterfactual framework, this would constitute a causal interaction for the variable M . Since the proof of the result relating observed data E [ M ( x 1 ,x 2 )] to units such that M (1 , 2) = 1 ,M (0 , 2) = M (1 , 0) = M (0 , 0) = 0 is essentially one line, we give it here also for completeness. Theorem 2 . If E [ M (1 , 2)] − E [ M (0 , 2)] − E [ M (1 , 0)] − E [ M (0 , 0)] > 0, then there must exist a unit with M (1 , 2) = 1 ,M (0 , 2) = M (1 , 0) = M (0 , 0) = 0. Proof. By contradiction. Suppose there were no unit with M (1 , 2) = 1 ,M (0 , 2) = M (1 , 0) = M (0 , 0) = 0. Then, for all units, M (1 , 2) − M (0 , 2) − M (1 , 0) − M (0 , 0) ≤ 0 which implies E [ M (1 , 2)] − E [ M (0 , 2)] − E [ M (1 , 0)] − E [ M (0 , 0)] ≤ 0, a contradiction. This corollary is referred to as Bell’s theorem in the physics literature. Its premise is referred to as Bell’s inequality. As noted above, from the quantum mechanics of the 2- particle system, and confirmed by experiment, E [ M ( x 1 = i,x 1 = j )] = sin $^{2}$(Δ$_{ij}$ / 2). Thus we have that: Corollary . If E [ M (1 , 2)] − E [ M (0 , 2)] − E [ M (1 , 0)] − E [ M (0 , 0)] > 0, then the the hypothesis of ’local hidden variables’ is false. An immediate corollary of Theorems 1 and 2 is then: 5