The different modal interpretations all advance a core property ascription {〈P j , C α j 〉} j . This chapter is about how this core property ascription fixes the full property ascription. I start with some logic and algebra. I then present two existing proposals for determining the full property ascriptions as well as four conditions one can impose on them. Lastly I give my own proposal and end by discussing how the full property ascription leads to a value assignment to magnitudes. Some logic and algebra To prepare the ground for discussing the full property ascription, I firstly define the logical connectives (negation, conjunction and disjunction) for properties. Then because, as discussed in Section 4.1, all the modal interpretations which I consider give up on the idea that thefull property ascription assigns definite values to all properties, I secondly define two types of subsets of properties: Boolean algebras and faux-Boolean algebras. I have already assumed that every property Q α pertaining to a system α is represented by one and only one projection Q α defined on the Hilbert space ℋα associated with α. Each projection Q α in its turn corresponds one-to-one to the subspace of ℋα, denoted by 2α, onto which it projects. One thus has a bijective mapping from a property Q α to a projection Q α to a subspace 2α. For the set of subspaces of a Hilbert space one can in a natural way define an orthocomplement, a meet and a join. I now choose 45 to define the logical connectives for properties pertaining to a system by means of this orthocomplement, meet and join for subspaces and the bijective mapping between properties and subspaces (by means of an isomorphism, more shortly).
The different modal interpretations all advance a core property ascription {〈P j , C α j 〉} j . This chapter is about how this core property ascription fixes the full property ascription. I start with some logic and algebra. I then present two existing proposals for determining the full property ascriptions as well as four conditions one can impose on them. Lastly I give my own proposal and end by discussing how the full property ascription leads to a value assignment to magnitudes. Some logic and algebra To prepare the ground for discussing the full property ascription, I firstly define the logical connectives (negation, conjunction and disjunction) for properties. Then because, as discussed in Section 4.1, all the modal interpretations which I consider give up on the idea that thefull property ascription assigns definite values to all properties, I secondly define two types of subsets of properties: Boolean algebras and faux-Boolean algebras. I have already assumed that every property Q α pertaining to a system α is represented by one and only one projection Q α defined on the Hilbert space ℋα associated with α. Each projection Q α in its turn corresponds one-to-one to the subspace of ℋα, denoted by 2α, onto which it projects. One thus has a bijective mapping from a property Q α to a projection Q α to a subspace 2α. For the set of subspaces of a Hilbert space one can in a natural way define an orthocomplement, a meet and a join. I now choose 45 to define the logical connectives for properties pertaining to a system by means of this orthocomplement, meet and join for subspaces and the bijective mapping between properties and subspaces (by means of an isomorphism, more shortly).
In this chapter I introduce the different versions of the modal interpretation, including the three on which I focus in this book. The property ascription of these versions is characterised by the map W α ↦ {〈p j , C α j 〉} j which I call the core property ascription. In the next chapter I discuss how this core property ascription determines the full property ascription W α ↦ The best modal interpretation The best imaginable modal interpretation is, I guess, an interpretation which (A) ascribes at all times all the properties to a system which pertain to that system, and (B) ascribes these properties such that the classical logical relations between the negation, conjunction and disjunction of properties are satisfied. The content of the first requirement (A) is clear: assuming that every projection onto a subspace of a Hilbert space ℋ α represents one and only one property of α, it follows that all sets of definite-valued projections should contain all the projections in ℋ α and that all the maps {. j } j should be maps from all the projections in ℋ α to the values {0,1}. The content of requirement (B) is, however, less clear because there is consensus neither about how to define the negation, conjunction and disjunction of properties in quantum mechanics nor about how to impose the logical relations. In Section 5.1 I present my choice for the definitions of the negation, conjunction and disjunction.
In Chapter 4 the different modal interpretations are introduced. Their property ascription to a system α is characterised by a core property ascription {〈P j ,C α J 〉} j . Chapter 5 treats the question of how the core property ascription to a system determines the full property ascription to the system, and of how that full property ascription induces an assignment of values to the magnitudes of the system. In Chapter 6 it is determined whether the properties that modal interpretations simultaneously ascribe to different systems can be correlated. A no-go theorem is derived which restricts the possibility of giving such correlations. In Chapter 7 it is shown that the set of properties, which a system possibly possesses, evolves in a number of undesirable ways. This evolution is, for instance, discontinuous and unstable. Chapter 8 is concerned with the evolution of the actually possessed properties of a system. It is proved for the case of freely evolving systems that this evolution is deterministic and for the case of interacting systems it is argued that this evolution cannot be uniquely fixed. In Chapter 9 it is proved that the evolution of the actually possessed properties of systems violates a number of the Dynamical Autonomy assumptions presented in Section 3.3. It is shown that this allows the descriptions of reality by modal interpretations to be non-local in a quite explicit way.
As a last step towards general conclusions, I collect and assess the more important results obtained in the previous chapters. In the next and final chapter I then consider the question of whether the modal descriptions of reality meet the demands of being well developed, empirically adequate and metaphysically tenable. Indefinite properties and inexact magnitudes The bi, spectral and atomic modal interpretations do manage to interpret quantum mechanics by providing a description of what reality could be like in terms of systems possessing properties. Moreover, if one accepts the full property ascription (5.12) as developed in Section 5.4, this description of reality satisfies a number of desirable conditions. However, these modal interpretations do not simultaneously ascribe all the possible properties to a system. Instead they select at each instant a specific subset of the properties pertaining to a system and ascribe only these. And the properties which are not selected are taken to be indefinite. This possibility that properties of systems are sometimes not definite is in itself not that problematic for in some cases it is perfectly understandable what this means. Consider again the example of a tossing coin. During the toss the properties ‘heads’ and ‘tails’ can be said to be indefinite and this can be understood as meaning that these two properties are applicable only to coins which lie on surfaces. However, in other cases it is less clear what is going on.
Let's return to the three demands I have imposed on the bi, spectral and atomic modal interpretations. The first was the demand that these interpretations should provide well-developed descriptions of reality. And a first conclusion is that the bi, spectral and atomic modal interpretations indeed provide the necessary starting points to develop them. That is, they all describe reality by means of well-defined property ascriptions: for all it is clear on which points these descriptions of reality need improvement, and for all one has the means to make a start with these improvements. A second conclusion is, however, that the success with which the bi, spectral and atomic modal interpretations can be developed varies. The results with regard to the full property ascription to a single system are in my opinion satisfactory for all three interpretations (Chapter 5). But with regard to the correlations between the properties ascribed to different systems the results start to diverge. For the atomic modal interpretation such correlations can be given (Section 6.4). But for the spectral modal interpretation it is proved that such correlations do not always exist for the properties ascribed to non-disjoint systems (Section 6.3). And for the bi modal interpretation correlations are unknown or, when one accepts Instantaneous Autonomy, Dynamical Autonomy for measurements and Empirical Adequacy (Section 3.3), do not exist as well (Section 6.3). Only if one adopts perspectivalism, that is, if one assumes that one can simultaneously only consider the properties of two disjoint systems in the bi modal interpretation and of n disjoint systems in the spectral modal interpretation, these two interpretations give all the defined correlations.
The thesis investigates the possibility of a classical semantics for quantum propositional structures. A classical semantics is defined as a set of mappings each of which is (i) bivalent, i.e., the value 1 (true) or 0 (false) is assigned to each proposition, and (ii) truth-functional, i.e., the logical operations are preserved. In addition, this set must be "full", i.e., any pair of distinct propositions is assigned different values by some mapping in the set. When the propositions make assertions about the properties of classical or of quantum systems, the mappings should also be (iii) "state-induced", i.e., values assigned by the semantics should accord with values assigned by classical or by quantum mechanics. In classical propositional logic, (equivalence classes of) propositions form a Boolean algebra, and each semantic mapping assigns the value 1 to the members of a certain subset of the algebra, namely, an ultrafilter, and assigns 0 to the members of the dual ultraideal, where the union of these two subsets is the entire algebra. The propositional structures of classical mechanics are likewise Boolean algebras, so one can straightforwardly provide a classical semantics, which also satisfies (iii). However, quantum propositional structures are non-Boolean, so it is an open question whether a semantics satisfying (i), (ii) and (iii) can be provided. Von Neumann first proposed (1932) that the algebraic structures of the subspaces (or projectors) of Hilbert space be regarded as the pro-positional structures PQM of quantum mechanics. These structures have been formalized in two ways: as orthomodular lattices which have the binary operations "and", "or", defined among all elements, compatible [symbol omitted] and incompatible [symbol omitted]; and as partial-Boolean algebras which have the binary operations defined among only compatible elements. In the thesis, two basic senses in which these structures are non-Boolean are discriminated. And two notions of truth-functionality are distinguished: truth-functionality [symbols omitted] applicable to the PQM lattices; and truth-functionality [symbol omitted] applicable to both the lattices and partial-Boolean algebras. Then it is shown how the lattice definitions of "and", "or", among incompatibles rule out a bivalent, truth-functional [symbols omitted] semantics for any lattice containing incompatible, elements. In contrast, the Gleason and Kochen-Specker proofs of the impossibility of hidden-variables for quantum mechanics show the impossibility of a bivalent, truth-functional [symbol omitted] semantics for three-or-higher dimensional Hilbert space structures; and the presence of incompatible elements is necessary but is not sufficient to rule out such a semantics. As for (iii), each quantum state-induced expectation-function on a PQM truth-functionally assigns 1 and 0 values to the elements in a ultrafilter and dual ultraideal of PQM³ where, in general the union of an ultrafilter and its dual ultraideal is smaller than the entire structure. Thus it is argued that each expectation-function is the quantum analog of a classical semantic mapping, even though the domain where each expectation-function is bivalent and truth-functional is usually a non-Boolean substructure of PQM. The final portion of the thesis surveys proposals for the introduction of hidden variables into quantum mechanics, proofs of the impossibility of such hidden-variable proposals, and criticisms of these impossibility proofs. And arguments in favour of the partial-Boolean algebra, rather than the orthomodular lattice, formalization of the quantum propositional structures are reviewed.
In this chapter I begin the discussion of the question of how the core properties of systems evolve in time. This question breaks up in two separate subquestions. Firstly, there is the question of how the set of all possible core properties of a system evolves. Secondly, one has the question of how, given this dynamics of the set of possible core properties, the actually possessed core property of that system evolves. Here, I consider only the first question of the dynamics of the whole set of core properties. In the next chapter I focus on the evolution of the actually possessed core property. The modal interpretations that I consider ascribe core properties to systems by means of the spectral resolutions of states. The dynamics of the set of core properties is thus determined by the dynamical behaviour of these spectral resolutions. Study of the dynamics of the spectral resolutions now reveals a number of problematic features of the modal property ascription. Firstly, the set of the eigenprojections of a state W α( t ) can evolve discontinuously. Secondly, this set can evolve highly unstably. Finally, the eigenprojections of a state can evolve rather deviantly when compared with the evolution of the state itself. As a result of all this, the evolution of the set of core properties of a system can be discontinuous, unstable and deviant as well. Discontinuities Up to now I have only considered the property ascription to systems at single instants. It is now time to set things in motion.