We define a physical magnitude as an equivalence class of measurement procedures and formulate sufficient restrictions on the equivalence relation to guarantee meaningful algebraic operations between magnitudes. These restrictions are not sufficient to let the Kochen and Specker argument go through. They are, however, stronger than mere statistical equivalence of measurement procedures and thus are relevant to the problem of the completeness of quantum mechanics. In fact, they give rise to a strong argument for the incompleteness of quantum mechanics.

This chapter focuses on alternative logics. It discusses a hierarchy of logical reform. It presents case studies that illustrate particular aspects of the logical revisionism discussed in the chapter. The first case study is of intuitionistic logic. The second case study turns to quantum logic, a system proposed on empirical grounds as a resolution of the antinomies of quantum mechanics. The third case study is concerned with systems of relevance logic, which have been the subject of an especially detailed reform (...) program. Finally, the fourth case study is paraconsistent logic, perhaps the most controversial of serious proposals. (shrink)

The standard axiomatization of quantum mechanics (QM) is not fully explicit about the role of the time-parameter. Especially, the time reference within the probability algorithm (the Born Rule, BR) is unclear. From a probability principle P1 and a second principle P2 affording a most natural way to make BR precise, a logical conflict with the standard expression for the completeness of QM can be derived. Rejecting P1 is implausible. Rejecting P2 leads to unphysical results and to a conflict with a (...) generalization of P2, a principle P3. All three principles are shown to be without alternative. It is thus shown that the standard expression of QM completeness must be revised. An absolutely explicit form of the axioms is provided, including a precise form of the projection postulate. An appropriate expression for QM completeness, reflecting the restrictions of the Gleason and Kochen-Specker theorems is proposed. (shrink)

Previous work has shown that the problem of measurement in quantum mechanics is not correctly seen as one of understanding some allegedly univocal process of measurement in nature which corresponds to the projection postulate. The present paper introduces a new perspective by showing that how we are to understand the nature of the change of quantum mechanical state on measurement depends very sensitively on the interpretation of the state function, and by showing how attention to this dependence can greatly sharpen (...) the problems and relations between them. In particular, the problems take a form resembling their traditional formulation only on an inexact value interpretation, according to which the state function attributes inexact values of quantities to systems. On other interpretations we can apply (with various drawbacks) the subensemble idea, according to which a discontinuous change of quantum mechanical description results on measurement simply because we need a new state function to describe a new object. (shrink)

Existing "no hidden variable proofs" for quantum mechanics deal exclusively with observables with discrete spectra. This note shows that similar results hold for observables with continuous spectra.

Fine has recently proved the surprising result that satisfaction of the Bell inequality in a Clauser-Horne experiment implies the existence of joint probabilities for pairs of noncommuting observables in the experiment. In this paper we show that if probabilities are interpreted in the von Mises-Church sense of relative frequencies on random sequences, a proof of the Bell inequality is nonetheless possible in which such joint probabilities are assumed not to exist. We also argue that Fine's theorem and related results do (...) not impugn the common view that local realists are committed to the Bell inequality. (shrink)

Recent discussions of experimental tests of the Sum Rule have been carried out in the context of the special circumstances attending the Cross-Ramsey experiment. A more general analysis of possible tests is presented. A technical mistake of Fine and Glymour concerned with a misunderstanding of the physics of the Cross-Ramsey experiment is explained and a detailed analysis of a thought experiment based on the Einstein-Podolsky-Rosen wave function is given. It is concluded, in agreement with Fine, that scattering experiments do not (...) test the Sum Rule as a principle which supplements standard quantum mechanics. (shrink)

In current philosophical debate Bell's theorem is often refered to as a proof of the impossibility of determinism in nature. It is argued here that this conclusion is wrong. The main consequence of the theorem is the non-local character of quantum theory itself and it is shown how this quality leads to a contradiction with the theory of relativity. If hidden variable theories are impossible, it is so because no empirically founded interpretation at all can be compatible with both quantum (...) mechanics and relativity. (shrink)

A new proof of the impossibility of reconciling realism and locality in quantum mechanics is given. Unlike proofs based on Bell's inequality, the present work makes minimal and transparent use of probability theory and proceeds by demonstrating a Kochen-Specker type of paradox based on the value assignments to the spin components of two spatially separated spin-1 systems in the singlet state of their total spin. An essential part of the argument is to distinguish carefully two commonly confused types of contextuality; (...) we call them ontological and environmental contextuality. These in turn are associated with two quite distinct senses of nonlocality. We indicate the relevance of our treatment to other related discussions in recent literature on the philosophy of quantum mechanics. (shrink)

The work of Gleason and of Kochen and Specker has been thought to refute a naïve realist approach to quantum mechanics. The argument of this paper substantially bears out this conclusion. The assumptions required by their work are not arbitrary, but have sound theoretical justification. Moreover, if they are false, there seems no reason why their falsity should not be demonstrable in some sufficiently ingenious experiment. Suitably interpreted, the work of Bell and Wigner may be seen to yield independent arguments (...) for the falsity of naïve realistic approach to quantum mechanics. Quantum mechanics is no more like classical statistical mechanics than its creators thought it was. (shrink)

Simon Kochen and Ernst Specker's well-known argument against hidden variable theories for quantum mechanics is also an argument against the possibility of quantum systems having, simultaneously, precise values for all of the dynamical quantities associated with such systems. Devices for defeating the argument were in the literature even before its publication, but recently Arthur Fine has raised a new difficulty. Fine points out that Kochen and Specker's argument requires the following principles:Sum Rule: At all times, in all states, the value (...) of the quantity with operator A + B is the sum of the values of the quantity with operator A and the quantity with operator B, provided A, B commute.Product Rule: At all times, in all states, the value of the quantity with operator AB is the product of the values of the quantities with operators A and B, provided A, B commute. (shrink)

This paper constructs two classes of models for the quantum correlation experiments used to test the Bell-type inequalities, synchronization models and prism models. Both classes employ deterministic hidden variables, satisfy the causal requirements of physical locality, and yield precisely the quantum mechanical statistics. In the synchronization models, the joint probabilities, for each emission, do not factor in the manner of stochastic independence, showing that such factorizability is not required for locality. In the prism models the observables are not random variables (...) over a common space; hence these models throw into question the entire random variables idiom of the literature. Both classes of models appear to be testable. (shrink)