## Abstract

Contemporary debate over laws of nature centers around Humean supervenience, the thesis that everything supervenes on the distribution of non-nomic facts. The key ingredient of this thesis is the idea that nomic-like concepts—law, chance, causation, etc.—are expressible in terms of the *regularities* of non-nomic facts. Inherent to this idea is the tacit conviction that regularities, “constant conjunctions” of non-nomic facts do supervene on the distribution of non-nomic facts. This paper raises a challenge for this conviction. It will be pointed out that the notion of regularity, understood as statistical correlation, has a necessary conceptual component not clearly identified before—I shall call this the “conjunctive relation” of the correlated events. On the other hand, it will be argued that there exists no unambiguous, non-circular way in which this relation could be determined. In this regard, the notion of correlation is similar to that of distant simultaneity where the necessary conceptual component is the one-way speed of light, whose value doesn’t seem to be determined by matters of (non-nomic) facts.

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## Notes

*X*is sometimes called “sample space.” Note that the elements of the sample space should not be confused with the elements of the statistical ensemble. The elements of the sample space are event*types*, while the elements of the statistical ensemble are*token*events. In the two coin example, the sample space we consider is \(\left\{ \left( H,H\right) ,\left( H,T\right) ,\left( T,H\right) ,\left( T,H\right) \right\}\); while the statistical ensemble consists of particular instances of double tosses, different elements of which may realize one and the same element of the sample space.Conditions (1)–(3), defining the notions of independence and correlation, are not expected to hold exactly in general, but only with some finite precision depending on the size of the statistical ensemble. In this sense there is no sharp boundary between cases of statistical independence and correlation, and the question of independence versus correlation should be seen as a matter of degree rather than a matter of kind. Note however that the choice of the conjunctive relation may alter whether a pattern of data is seen as evidence of independence or

*maximal*correlation (with some finite precision), which are the two*extremes*on the correlation scale.This condition may or may not be specifiable merely in terms of \(\mathscr {F}\) and \(\mathscr {F}'\).

Alternatively, one can take a finite partition of the unit square into small cells of equal size, and identify the ensembles with the finite collections of these cells.

It must be noted that the Clauser–Horne inequalities are only sufficient in conjunction with the following supplementary inequalities (as also noted by [7, p. 293]):

$$\begin{aligned} \begin{array}{ccc} p\left( A_{i}\wedge B_{j}|a_{i}\wedge b_{j}\right) \le p\left( A_{i}|a_{i}\right) \\ p\left( A_{i}\wedge B_{j}|a_{i}\wedge b_{j}\right) \le p\left( B_{j}|b_{j}\right) \\ p\left( A_{i}|a_{i}\right) +p\left( B_{j}|b_{j}\right) -p\left( A_{i}\wedge B_{j}|a_{i}\wedge b_{j}\right) \le 1 \end{array} \end{aligned}$$These however follow from (12)–(16) and thus are respected by all three conjunctive products.

A local hidden variable model provides a common causal explanation. However, after modifying the conjunctive relation the pairs of events in the two wings matched together may no longer be spatially separated and so explaining the statistics relative to the modified ensemble may also invoke direct causal connection between the two wings. Nevertheless, what we have demonstrated here is that an explanation purely in terms of common causes is also available.

There do exist experiments in which particle pairs can be identified independently of their detections. However, in none of these experiments they succeeded to ensure spatial separation of the two wings [19, Sect. 3].

The thought that the particles’ time of arrival at the detectors may depend on the hidden variable is not new. This is the main idea of hidden variable models based on the so-called coincidence loophole [6, 8, 20, 21]. In these models one builds on the fact that in real EPR-type experiments pairs of detection events in the two wings are counted as “coincident” if they happen within a given time window; if the retardation of one detection with respect to the other is larger then the prescribed amount the data is simply ignored and will not contribute to the statistics. If, as it is assumed in the models in question, there is a systematic connection between the value of the hidden variable and the detections that are counted as coincident, then the statistics over the subensemble of coincident detections may exhibit violations of Bell’s inequalities even though the statistics over the original ensemble, being produced by a fully local mechanism, adhere to them.

There is another type of “loophole” in Bell’s theorem related to the fact that the statistical data is given as a temporal series of outcomes. In the original Bell analysis events in different runs of the experiment are treated as independent. However, since different runs of an EPR-type experiment are timelike separated, earlier values of outcomes, settings and the hidden variable can in principle influence the later ones. For example, if the consecutive measurement choices in one wing are correlated, one can imagine a mechanism whereby the local measurement apparatus predicts the current remote measurement choice by registering the earlier ones. This way the remote measurement setting could influence the local outcome, resulting in a violation of Bell’s inequalities, by means of a fully local mechanism. This problem is sometimes referred to as the memory loophole [2, 9,10,11, 14].

While both the coincidence and memory loopholes are related with the role of time in the EPR–Bell problem, and thereby they have some similarities with the idea of the time-delayed models discussed here, it must be emphasized that the main thought behind the time-delayed models is different from the problems discussed in the literature. The reason is that all existing hidden variable models assume that it is the statistics of the

*simultaneous*measurements that is to be explained by a local mechanism, while the essence of the time-delayed model idea is to*question*that the simultaneous statistics exhibit real correlations in need of explanation.Here we assume that a token event can be individuated without referring to the event types it falls under. Such an identification seems possible, for example, by referring to the spatiotemporal location of the token events.

It must be emphasized that the physical realizability of the simultaneous pairing has nothing to do with the issue of whether simultaneity is factual or conventional, in the sense as it arises with regard to the problem of distant clock synchronization. What gives physical objectivity to the simultaneous pairing is not that the local clocks in question are synchronized or that they measure the “same” (temporal) order—in fact no such an assumption has to be made. What is important here is that the counters that we call clocks—synchronized or not—are identically constructed and therefore there is a physical correspondence between their readings, based on their similarity. This physical correspondence is what grounds the reality of the simultaneous pairing.

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## Acknowledgements

This work has grown out from discussions with László E. Szabó on his physicalist account of scientific theories [27], in which the semantics of a theory is analyzed in terms of the “distant correlation” of facts of the world and “facts” of the theory. I wish to thank László for these discussions. I am also grateful to Gábor Hofer-Szabó and Balázs Gyenis for valuable comments on earlier versions of the paper. The research was supported by the (Hungarian) National Research, Development and Innovation Office (NKFIH), No. K115593

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Gömöri, M. On the Very Idea of Distant Correlations.
*Found Phys* **50**, 530–554 (2020). https://doi.org/10.1007/s10701-020-00332-w

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DOI: https://doi.org/10.1007/s10701-020-00332-w