Patrick Fraser

It has been argued that the quantity valuation scheme invoked by spontaneous collapse interpretations of quantum mechanics has the unacceptable consequence of rendering violations of arithmetic, and of conjunction introduction more generally, physically possible. We argue that what is ultimately at issue in this argument and the debate that followed has nothing to do with spontaneous collapse theories, their quantity valuation schemes, nor arithmetic. Instead, we show that the argument in question hinges on an implausible assumption about the relation between the semantics of conjunction and the representation of quantum-mechanical quantities. We argue that this episode holds important lessons for how to treat the semantics of Hilbert space theories in particular, and the semantics of physical theories in general.

We use the framework of determinable and determinate properties to provide a novel account of the constraints imposed by the Kochen-Specker theorem. Specifically, we show that the result may be understood as a prohibition on the possibility of providing an account of quantum properties, as they are usually represented using Hilbert space operators, according to which quantum systems are fully determinate. That is, the Kochen-Specker theorem entails that any account of quantum properties compatible with the predictions of quantum mechanics must exhibit determinable-based metaphysical indeterminacy.

Physical theories assign values of quantities to physical systems, and they assign probabilities to the possible outcomes of measurements of those quantities. In quantum theory, probabilities are assigned using the Born rule. Here, I show that the Born rule is uniquely determined by the conditions under which quantum systems instantiate values of quantities as they are specified by the eigenstate-eigenvalue link, together with minimal theory-independent assumptions about how probabilities are sensitive to the structure of physical quantities.

We articulate a collection of desiderata for an account of the dynamical quantities of a physical theory, and we present a theory that meets these desiderata in the case of quantum mechanics. Our theory retains a distinction between the values of dynamical quantities and the truth values of sentences asserting that a system has a particular value of a particular quantity. This allows our theory to incorporate the phenomenon of quantum indeterminacy as a pattern in the properties instantiated by a system in a particular quantum state, while also retaining the semantics of classical logic. We contrast our theory with quantum logic, which flattens the distinction between quantity values and truth values, and we address a series of objections that have been raised to previous attempts to reconcile quantum indeterminacy with classical logic.

Del Santo and Gisin have recently argued that classical mechanics exhibits indeterminacy and that by treating the observables of classical mechanics with real number precision we introduce hidden variables that restore determinacy. In this article we introduce the conceptual machinery required to critically evaluate these claims. We present a characterization of indeterminacy which can capture both quantum indeterminacy and the classical indeterminacy Del Santo and Gisin propose. This allows us to show that there is an important difference between the two: their classical indeterminacy can be resolved with hidden variables in a manner which is not possible for quantum indeterminacy.

Del Santo and Gisin have recently argued that classical mechanics exhibits a form of indeterminacy and that by treating the observables of classical mechanics with real number precision we introduce hidden variables that restore determinacy. In this article we introduce the conceptual machinery required to critically evaluate these claims. We present a characterization of indeterminacy which can capture both quantum indeterminacy and the classical indeterminacy of Del Santo and Gisin. This allows us to show that there is an important difference in kind between the two: their classical indeterminacy can be resolved with hidden variables in a manner which is not possible for quantum indeterminacy.

In 1957, Feyerabend delivered a paper titled ‘On the Quantum-Theory of Measurement’ at the Colston Research Symposium in Bristol to sketch a completion of von Neumann's measurement scheme without collapse, using only unitary quantum dynamics and well-motivated statistical assumptions about macroscopic quantum systems. Feyerabend's paper has been recognised as an early contribution to quantum measurement, anticipating certain aspects of decoherence. Our paper reassesses the physical and philosophical content of Feyerabend's contribution, detailing the technical steps as well as its overall philosophical motivations and consequences. Summarising our results, Feyerabend interpreted collapse as a positivist assumption in quantum mechanics leading to a strict distinction between the uninterpreted formalism of unitary evolution in quantum mechanics and the classically interpreted observational language describing post-measurement outcomes. Thus Feyerabend took the no-collapse completion of the von Neumann measurement scheme to show the dispensability of the positivist assumption, leading the way to a realistic interpretation of quantum theory. We note, however, that there are substantial problems with his account of measurement that bring into question its viability as a legitimate foil to the orthodox view. We further argue that his dissatisfaction with the von Neumann measurement scheme is indicative of early views on theoretical pluralism.

Constructivist epistemology posits that all truths are knowable. One might ask to what extent constructivism is compatible with naturalized epistemology and knowledge obtained from inference-making using successful scientific theories. If quantum theory correctly describes the structure of the physical world, and if quantum theoretic inferences about which measurement outcomes will be observed with unit probability count as knowledge, we demonstrate that constructivism cannot be upheld. Our derivation is compatible with both intuitionistic and quantum propositional logic. This result is implied by the Frauchiger-Renner theorem, though it is of independent importance as well.