## Abstract

“Bell’s theorem” can refer to two different theorems that John Bell proved, the first in 1964 and the second in 1976. His 1964 theorem is the incompatibility of quantum phenomena with the joint assumptions of locality and predetermination. His 1976 theorem is their incompatibility with the single property of local causality. This is contrary to Bell’s own later assertions, that his 1964 theorem began with the assumption of local causality, even if not by that name. Although the two Bell’s theorems are logically equivalent, their assumptions are not. Hence, the earlier and later theorems suggest quite different conclusions, embraced by operationalists and realists, respectively. The key issue is whether locality or local causality is the appropriate notion emanating from relativistic causality, and this rests on one’s basic notion of causation. For operationalists the appropriate notion is what is here called the Principle of agent-causation, while for realists it is Reichenbach’s Principle of common cause. By breaking down the latter into even more basic Postulates, it is possible to obtain a version of Bell’s theorem in which each camp could reject one assumption, happy that the remaining assumptions reflect its *weltanschauung*. Formulating Bell’s theorem in terms of causation is fruitful not just for attempting to reconcile the two camps, but also for better describing the ontology of different quantum interpretations and for more deeply understanding the implications of Bell’s marvellous work.

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## Notes

- 1.
This first section is written in first person by one of us (Wiseman), who spoke at the Quantum [Un]speakables conference. The other of us (Cavalcanti) has been a long-time co-worker and correspondent with Wiseman on Bell’s theorem. In particular, since the conference, their discussions have convinced Wiseman of a better way to formulate the causal assumptions in Bell’s theorem, and this is reflected in the latter parts of the paper, and in its authorship.

- 2.
It was the year he wrote his review of hidden variables (HVs) [2], by misfortune not published until 1966, in which he dismissed von Neumann’s anti-HV proof, gave the first proof of the necessity of contextuality for deterministic HV theories, and raised the question of the necessity of nonlocality, which he immediately answered in the positive in his 1964 paper [3].

- 3.
In earlier, albeit recent, publications [4, 5] I used the term ‘determinism’ for Bell’s second assumption in 1964. However, Bell did not actually use this word in 1964, and it is useful to reserve it for a different notion [10]. Note the use here of small-capitals for terms with a precise technical meaning, even when the relevant definition is given only at a later point in the chapter.

- 4.
Note that here we are not making a commitment to frequentism, but rather simply recognising that real experimental data are relative frequencies.

- 5.
More generally [12], one could sandwich these variables between two suitable SLHs, but the above formulation will suffice.

- 6.
Given footnote 4, this equation is not to be read as a strict equality, but as carrying the same meaning as that of any probabilistic prediction.

- 7.
He introduces it in the first paragraph of the paper proper, which serves as motivation for the formulation of the assumptions he will use. There, Bell unfortunately misapplies his notion, in attempting to derive predetermination via EPR-Bohm-correlations and locality. See Refs. [4, 11] for discussions of the irrelevance of this flawed paragraph to Bell’s 1964 theorem (i.e. the “result to be proved”, as he calls it, which he does indeed prove).

- 8.
Indeed, by 1981 [22] he was implying that by ‘locality’ he had always meant local causality. This historical revisionism is perfectly understandable, and probably unconscious—a plausible unfolding of the localistic intuition Bell had in 1964 is local causality, since this would have worked, where locality failed, in Bell’s attempt to reproduce the EPR argument (see footnote 7).

- 9.
- 10.
*Les passions de l’âme*(1649),*De natura*(415), and*De natura et gratia*(415). - 11.
The alert reader will have noticed some sleight of hand. In Bell’s 1976 paper, where he introduced local causality, \(\lambda \) denoted all events in the intersection of the past light-cones. But in Ref. [12] Bell took the \(\lambda \)s to be defined between two SLHs, and the limit when these become one corresponds to the situation he considered in 1964, where the \(\lambda \)s were “initial values of the [relevant] variables at some suitable instant.” As in Fig. 6.2, that suitable

*instant*(SLH) may not even cross the intersection of the past light-cones. The resolution is that since, by assumption, the variables \(\left\{ {\lambda } \right\} \) are the only relevant ones, they must carry the information that was present in the common causes \(\mathcal{C}\) in the common Minkowski past of*A*and*B*. This they can without falling foul of local causality (in the sense of Principle 2) because there is a part of the SLH that is in the future light-cone of the events in \(\mathcal{C}\), but in the past light-cone of*A*, and likewise for*B*. - 12.
Bell was immediately criticised for the vagueness of this statement (and for what followed, some of which was not sufficient for his purpose) by Shimony, Horne and Clauser [39]. The immediacy was, according to Clauser [40], because the latter two authors had originally drafted their 1974 paper [21] using the above Principle of local causality (or something like it), but Shimony pointed out to them that this was not sufficient to derive Bell-local ity without an extra assumption relating to free choice. As a consequence they retreated from such a principled formulation of local causality to the more specific Definition 5, which they said characterized “objective local theories” [21], enabling a “less general and more plausible” [39] assumption (than local agency, for example) relating to free choice. Clauser and Horne [21] deserve credit for first (as far as we are aware) discussing, in their footnote 13, the need for independence of the hidden variables \(\lambda \) from the free choices

*a*and*b*which is implicit in Eq. (2). They note that to justify that assumption one has to rule out the “possibility” that “Systems originate within the intersection of the backward light cones of both analyzers and the source ...[which] effect [*sic.*] both the experimenters’ selections of analyzer orientations and the emissions from the source.” - 13.
- 14.
This may sound like a strong statement, and the reader may feel tempted to follow neither the operationalist nor the realist camp, but rather to reject Postulate 1 from the list of assumptions in Theorem 8. This temptation should vanish if the reader thinks through what it would actually mean to explain away Bell-correlations through the real (not just in-principle) failure of free choice. There is no general theory that does this. If such a theory did exist, it would require a grand conspiracy of causal relationships leading to results in precise agreement with quantum mechanics, even though the theory itself would bear no resemblance to quantum mechanics. Moreover, it is hard to imagine why it should only be in Bell experiments that free choices would be significantly influenced by causes relevant also to the observed outcomes; rather, every conclusion based upon observed correlations, scientific or casual, would be meaningless because the observers’s method would always be suspect. It seems to us that any such theory would be about as plausible, and appealing, as, belief in ubiquitous alien mind-control.

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## Acknowledgments

This research was supported by the Australian Research Council (ARC) Discovery Project DP140100648, the ARC DECRA project DE120100559, and a grant (FQXi-RFP-1504) from the Foundational Questions Institute Fund (fqxi.org) at the Silicon Valley Community Foundation. We thank Eleanor Rieffel for helpful comments.

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Wiseman, H.M., Cavalcanti, E.G. (2017).
*Causarum Investigatio* and the Two Bell’s Theorems of John Bell.
In: Bertlmann, R., Zeilinger, A. (eds) Quantum [Un]Speakables II. The Frontiers Collection. Springer, Cham. https://doi.org/10.1007/978-3-319-38987-5_6

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