EPR discussed such systems as a way of critiquing what they called the completeness (as opposed to the correctness) of the quantum description. In 1964 John Bell used David Bohm's version of EPR's argument to ironically rule against them: EPR's assumption of locality is found to be false, releasing quantum theory from any claim of incompleteness or inconsistency.

The status of nonlocality in quantum mechanics is still being debated.

EPR's starting point and central idea is that we can measure some aspect of one of the pair and subsequently predict outcomes of measuring corresponding aspects of the second one.

EPR said that since this property of the second particle was predictable after the first measurement, it was therefore "real." (EPR suggested that the particle's properties were like the inevitably mismatched socks of Bell's colleague Bertlemann: once you see one of Dr. Bertlemann's socks you could make predictions about the other consistent with the fact that the socks are real.)

The counterfactual idea: But for that matter, EPR argued, we might just as well instead measure some other, perhaps contradictory (complimentary) property of the first particle, and in this case we would know some corresponding other fact about the second particle in the pair: this 'other' property of the second particle would be "real."

The assumption of locality: We at first go nowhere near the second particle (which might be at the other end of the galaxy by this time): we are only measuring the first particle to start with - EPR said that surely we could not alter "the real situation" of the second, distant particle, by measuring the first...

EPR said, since

1. we could make either measurement of the first particle, and
2. since (by locality) this cannot itself influence the second particle,

3. therefore the second particle must have pre-prepared "real" values for both (all) possible properties corresponding to measurements on the first particle.

Yet in the case of complementary variables quantum theory denies that both properties can be simultaneously present.

EPR concluded that since there were "real" properties of the world not even defineable in quantum theory, quantum theory is "incomplete."

The line of argument initiated by EPR was formally updated in 1964 when John Bell showed that the assumption of locality made by Einstein, Podolsky and Rosen was itself actually in contradiction with facts predicted by quantum mechanics.

The quantum properties are not like Bertlemann's socks after all.

(Eventually these nonlocal "predictions" were experimentally verified.)

Bell summed it up this way in 1964,

The paradox of Einstein, Podolsky, and Rosen was advanced as an argument that quantum mechanics could not be a complete theory but should be supplemented by additional variables. These additional variables were to restore to the theory causality and locality... That idea will be formulated mathematically and shown to be incompatible with the statistical predictions of quantum mechanics.