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Does determinism (or even the incompleteness of quantum mechanics) follow from locality and perfect correlations? In a 1964 paper John Bell gave the first demonstration that quantum mechanics is incompatible with local hidden variables. Since then a vigorous debate has rung out over whether he relied on an assumption of determinism or instead, as he later claimed in a 1981 paper, derived determinism from assumptions of locality and perfect correlation. This paper aims to bring clarity to the debate via simple examples and rigorous results. It is first recalled, via quantum and classical counterexamples, that the weakest statistical form of locality consistent with Bell's 1964 paper (parameter independence) is insufficient for the derivation of determinism. Attention is then turned to critically assess Bell's appealing to the Einstein-Rosen-Podolsky (EPR) incompleteness argument to support his claim. It is shown this argument is itself incomplete, via counterexamples that expose two logical gaps. Closing these gaps via a strong "counterfactual" reality criterion enables a rigorous derivation of both determinism and parameter independence, and in this sense justifies Bell's claim. Conversely, however, it is noted that whereas the EPR argument requires a weaker "measurement choice" assumption than Bell's demonstration, it nevertheless leads to a similar incompatibility with quantum predictions rather than to quantum incompleteness., Comment: v2: Significant update. New section 5 compares the weaker "accessible choice" assumption required for the EPR argument vs the "measurement independence" assumption required for Bell inequalities, and discusses a relevant recent experiment. 4 puns, no figures

A simple minimalist argument is given for why some correlations between quantum systems boggle our classical intuition. The argument relies on two elementary physical assumptions, and recovers the standard experimentally-testable Bell inequality in a form that applies equally well to correlations between six-sided dice and between photon polarizations. The first assumption, that measurement selection in a first lab leaves the measurement statistics in a remote lab invariant (no-signaling), has been empirically verified, and is shown to be equivalent to the existence of a corresponding joint probability distribution for quantities measured in the first lab. The observed violation of the Bell inequality is then equivalent to the failure of a second assumption, that measurement selection in the remote lab leaves such a joint distribution invariant. Indeed, the degree of violation lower-bounds the variation of the joint distribution. It directly follows there are just three possible physical mechanisms underlying such violations -- action-at-a-distance (superluminality), unavoidable common factors linking measurement choice and distant properties (conspiracy), and intrinsically incompatible physical quantities (complementarity). The argument extends to all Bell inequalities, and is briefly compared with other derivations., Comment: 10 pages, 1 figure (accepted version, with minor clarifications)