## Abstract

This paper aims to implement Bell’s notion of local causality into a framework, called local physical theory, which is general enough to integrate both probabilistic and spatiotemporal concepts and also classical and quantum theories. Bell’s original idea of local causality will then arise as the classical case of our definition. First, we investigate what is needed for a local physical theory to be locally causal. Then we compare local causality with Reichenbach’s common cause principle and relate both to the Bell inequalities. We find a nice parallelism: both local causality and the common cause principle are more general notions than captured by the Bell inequalities. Namely, Bell inequalities cannot be derived neither from local causality nor from a common cause unless the local physical theory is classical or the common cause is commuting, respectively.

## Notes

We note that our definition of a local physical theory does not embrace models beyond the Tsirelson bound. In order to incorporate also such models (Popescu–Rohrlich box) one should generalize the net of local algebras to a net of

*order-unit vector spaces*. See (Summers and Werner 1987a) and (Popescu and Rohrlich 1994).For the sake of uniformity we slightly changed Bell’s notation and figure.

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## Acknowledgments

This work has been supported by the Hungarian Scientific Research Fund OTKA K-100715 and K-108384, and the National Research, Development and Innovation Office, K-115593.

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Hofer-Szabó, G., Vecsernyés, P. A generalized definition of Bell’s local causality.
*Synthese* **193**, 3195–3207 (2016). https://doi.org/10.1007/s11229-015-0925-8

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DOI: https://doi.org/10.1007/s11229-015-0925-8