Yes, alternative "(A=B) OR (A=C) OR (B=C)" is always true (is tautology) for binary variables - what can be seen as example of Dirchlet's principle.
Or from probabilistic perspective, assume there is some probability distribution among 8 possibilities: sum_ABC pABC = 1
Pr(A=B) = p000 + p001 + p110 + p111
Pr(A=C) = p000 + p010 + p101 + p111
Pr(B=C) = p000 + p100 + p011 + p111
Pr(A=B) + Pr(A=C) + Pr(B=C) = 2p000 + 2p111 + sum_ABC pABC >= 1
Absolutely obvious, but violated e.g. by QM or MERW. No "local realism", but just standard probabilistics (
https://en.wikipedia.org/wiki/Probability_axioms ):
- first axiom: assumption of existence of probability distribution among all possibilities (that unknown = unmeasured), second axiom: Pr(Omega) = 1,
- third axiom - standard rule: "probability of alternative of disjoint events is sum of their probabilities" Pr(A=B) = p000 + p001 + p110 + p111
To violate it, we need to replace these two natural assumption with Born rule:
- instead of probabilities of events, there exist their amplitudes,
-
Born rule: "
probability of alternative of disjoint event is proportional to sum of squares of their probabilities": Pr_11 (A=B) ~ ((psi000 + psi001)^2 + (psi110 + psi111)^2)
Indeed formally there should be added indexes of which variables are measured - to distinguish just being unknown, from being unmeasured: the Born rule summation is over unmeasured variables.
Then we can violate inequalities derived using standard probability, like above.
for example assuming amplitude:
psi000 = psi111 = 0
psi001 = psi010 = psi011 = psi100 = psi101 = psi110 = 1
we get
Pr_11 (A=B) = ((psi000 + psi001)^2 + (psi110 + psi111)^2) /(sum_AB (psiAB0 + psiAB1)^2 ) = 1/5
Pr_11(A=B) + Pr_13(A=C) + Pr_23(B=C) = 3/5 violating the inequality.
Please write if I can elaborate on something.