Although a physicist probably would just shake his head and point to the usual Hilbert space framework, it is nonetheless tempting for anyone dabbling in Mathematical Logic to see if any of the existing structures out of Model Theory could be appropriate for quantum physics concepts. The closest I can find is some sort of possible world semantics, such as Kripke Frames, but I hit some snags very quickly. If we have a collection of possible worlds which are connected by an accessibility relation, then how is a superposition of different worlds represented? Are indeterminate states then part of the accessibility relation, or is a representation by truth values in a complex lattice sufficient? How are entanglement and the uncertainty relations handled? Perhaps I am not searching correctly, but I find no decent answers on the Internet. Can anyone point me in the right direction?