Conclusion
Supervenience in most of its guises entails necessary coextension. Thus theoretical supervenience entails nomically necessary coextension. Kim's result, thus strengthened, has yet to hit home. I suspect that many supervenience enthusiasts would cool at necessary coextension: they didn't mean to be saying anything quite so strong. Furthermore, nomically necessary coextension can be a good reason for property identification, leading to reducibility in principle. This again is more than many supervenience theorists bargained for. They wanted supervenience without reducibility. It is not always available for this mediating role.
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Bacon, J. Supervenience, necessary coextension, and reducibility. Philosophical Studies 49, 163–176 (1986). https://doi.org/10.1007/BF00354332
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DOI: https://doi.org/10.1007/BF00354332