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Van Fraassen's Modal Model of Quantum Mechanics

Bas van Fraassen in [4] has recently tried to use modal logic to solve the measurement problem of quantum mechanics. His model is based on a method of expressing quantum states developed by Hugh Everett [1] called the "relative state formulation." Unfortunately, Everett's mathematics cannot be generalized as van Fraassen requires. The difficulty itself is elementary enough. But a revision of van Fraassen's postulates (such as the one he adopted on p. 341 after he had been told of the problem) can save the mathematics only on pain of making the whole study irrelevant to the physics. I shall explain the dilemma in this note. In a later paper [5], van Fraassen avoids this dilemma by developing an abstract modal interpretation which does not rely on any expression for quantum states. In van Fraassen's view, the problem of measurement is to reconcile the contradiction between the predictions of the Schrodinger equation for the final state of the combined system, object plus measuring apparatus, with the predictions provided by the "naive" account of measurement described by von Neumann, [6]. For an object initially in the state I Cr,r, the naive account tells us that if the pointer states are {i}, and the eigenstates of the measured observable are {}J, then the final state of the composite will be the mixture