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Kripke semantics interpret intuitionistic logic by a partially ordered set of worlds/situations. Consistent histories interpretation of QM elaborates the copenhagen interpretation where a consistent set of histories can be assigned probabilities.

I don't really understand either of these formalisms, but it does appear on the face of it that the formalisms should have something in common, do they?

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Check out the topos approaches to quantum mechanics, as initiated by Isham/Butterfield and Döring/Isham (the contravariant approach, see review by Flori: There's also a covariant approach, developed by Heunen, Landsman and Spitters (see Wolters' comparison of the approaches: That should get you started. – Ulrik Buchholtz Sep 22 '12 at 16:52
Isham addresses the topic specifically in 'Topos Theory and Consistent Histories' – David Corfield Sep 23 '12 at 14:34
great, thanks for both these references. – Mozibur Ullah Sep 23 '12 at 17:11

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