Valuations on Functionally Closed Sets of Quantum Mechanical Observables and Von Neumann’s ‘No-Hidden-Variables’ Theorem∗

We are not asserting that all realist interpretations of quantum mechanics must necessarily do this, nor are we asserting that doing this in itself counts as giving an interpretation. But certainly the central task of modal interpretations is to provide an ontology of the properties of quantum systems that circumvents the measurement problem, without falling prey to the various ‘no-hidden-variables’ theorems. So, to accomplish that task, modal interpretations must tell us which observables of a system we can and should be realists about. Morever, this must at least involve specifying which of a system’s discrete-valued observables can be said to possess definite values statistically distributed in conformity with the statistics prescribed by the density operator W of the system. Our main aim in this paper is to take a