In Part one the description of reality by modal interpretations has been developed as far as possible. In this part I determine whether modal interpretations are empirically adequate by considering their descriptions of measurements.

In Chapter 10 I focus on the measurement problem. After a measurement,according to our observations, the measurement device displays a definite outcome. Such an outcome is traditionally called a pointer reading and the question is whether modal interpretations manage to ascribe such pointer readings.

In the standard formulation of quantum mechanics one predicts by means the Born rule the probabilities and the correlations with which measurements have outcomes. In Chapter 11 I discuss whether modal interpretations can reproduce these empirically correct predictions.

In this chapter I prove that the transition probabilities, which describe the dynamics of the actually possessed properties of systems, violate the assumption of Dynamical Autonomy for composite systems. And, related to this violation, it is also proved that the dynamics of the actually possessed properties is non-local in a quite explicit way.

Introduction

When I introduced the assumptions of Instantaneous and Dynamical Autonomy in Section 3.3, I motivated them by making an appeal to locality. In the case of Instantaneous Autonomy, I argued that a state of a system α should codify all the information about the property ascription to that system. For if a state does not codify all the information, it may happen that a change in the state of some distant system σ (I used the example of a far-away asteroid) means that the properties of α change as well. And this would make modal property ascriptions undesirably non-local. In the case of Dynamical Autonomy one can give a similar motivation. The state of a system α, as it evolves during a time interval i, should codify all the information about the simultaneous and sequential correlations between the properties ascribed to α in that interval i. For if the evolving state does not do so, it may happen that a change in a distant system changes the correlations between the properties of α as well.

Modal interpretations satisfy a number of the Autonomy assumptions. All modal interpretations satisfy Instantaneous Autonomy by construction. The transition probabilities derived in Section 8.2 prove that the bi and spectral modal interpretations satisfy Dynamical Autonomy for whole systems and that the atomic modal interpretation satisfies Dynamical Autonomy for atomic systems.

In quantum mechanics the properties of a composite system can be divided into properties which are reducible to the properties of the subsystems of the composite and properties which are irreducible to those subsystem properties. The irreducible properties are called holistic and emerge when one describes the system as a whole.

In this chapter I show that the bi and spectral modal interpretations can ascribe the holistic properties to composites but, in general, fail to reproduce the relations between the reducible properties of a composite and the properties of the subsystems. The atomic modal interpretation, on the other hand, fails to ascribe the holistic properties but is much more successful in reproducing the relations between the reducible properties and the subsystem properties. I discuss whether these failures harm the empirical adequacy and the metaphysical tenability of modal interpretations.

Holistic properties of composite systems

The basic principle of reductionism, that all the properties of composite systems are amalgamates of the properties of the subsystems of these composites, has triumphed so much in classical physics that I find it difficult to come up with a clear example of a classical property that cannot be taken as a complex of the subsystem properties. So, in order to make a case for the idea that composites can also possess, in addition to reducible properties, so-called holistic properties which cannot be reduced to the properties of the subsystems, one has to move outside physics or to draw upon some counterfactual development of physics.

When I decided to enter research on modal interpretations of quantum mechanics, I barely knew what it was about. I had attended a talk on the subject and read bits about them, but the ideas behind these interpretations didn't stick in my mind. Modal interpretations were at that time (1993) not widely known, and their approach to quantum mechanics was not common knowledge in the philosophy of physics. So my decision was a step in the dark. But what I did know was that I was beginning research on one of the most irritating and challenging problems of contemporary physics. Namely, the problem that quantum theories, unlike all other fundamental theories in physics, cannot be understood as descriptions of an outside world consisting of systems with definite physical properties.

Your decision to read this book may be a step in the dark as well, because modal interpretations are presently, especially among physicists, still rather unknown. The reason for this may lie in their somewhat isolated and slow development. The first modal interpretation was formulated in 1972 by Van Fraassen. Then, in the 1980s, Kochen, Dieks and Healey put forward similar proposals which, later on, were united under Van Fraassen's heading as modal interpretations. But these proposals were not immediately developed to fully elaborated accounts of quantum mechanics. Moreover, modal interpretations were proposed and discussed in journals and at conferences which were mainly directed towards philosophers of physics, rather than towards general physicists. Modal interpretations are in that sense true philosophers' understandings of quantum mechanics.

In Chapter 4 the different modal interpretations are introduced. Their property ascription to a system α is characterised by a core property ascription {〈Pj,CαJ〉}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.

Imagine this strange island you have just set foot on. The travel agencies had advertised it as the latest and most exciting place to visit, an absolute must for those who still want to explore the unknown. So, of course, you decided to visit this island and booked with your friends a three-week stay. And now you've arrived and are sitting in a cab taking you from the airport into town. The landscape looks beautiful but strange. For some reason you can't take it in at one glance. You clearly see the part right in front of you, but, possibly because of the tiring flight, everything in the corners of your eyes appears more blurred than usual.

In town you buy a map. They don't sell one single map of the island but offer instead a booklet containing on each page a little map which covers only a small patch of the town or of the surrounding countryside. ‘How convenient,’ your friends say and off they go to explore this new and exciting place. But you approach things differently. You want to figure out where the places of interest are. So you buy the booklet, seek the nearest café, take out the pages and try to join the little maps together to make a single big one. Unfortunately you don't succeed; the little maps seem not to match at the edges.

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 Wα ↦ {〈pj, Cαj〉}j which I call the core property ascription. In the next chapter I discuss how this core property ascription determines the full property ascription Wα ↦

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 ℋα represents one and only one property of α, it follows that all sets of definite-valued projections should contain all the projections in ℋα and that all the maps {.j}j should be maps from all the projections in ℋα 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.

The different modal interpretations all advance a core property ascription {〈Pj,Cαj〉}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 Qα pertaining to a system α is represented by one and only one projection Qα defined on the Hilbert space ℋα associated with α. Each projection Qα 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 Qα to a projection Qα 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 choose45 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).

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.

Before considering the particulars of the different versions of the modal interpretation in the next chapter, I first present their common characteristics. Then I list the starting points from which I develop these modal interpretations to fully-fledged descriptions of reality. Finally, I give the criteria I think interpretations should meet and present a number of desiderata I hope they meet.

General characteristics

The name ‘modal interpretation’ originates with Van Fraassen (1972) who, in order to interpret quantum mechanics, transposed the semantic analysis of modal logics to the analysis of quantum logic. The resulting interpretation was for obvious reasons called the modal interpretation of quantum logic. Since then, the term modal interpretation has acquired a much more general meaning and lost its direct kinship with modal logics. In particular new interpretations of quantum mechanics developed in the 1980s by Kochen (1985), Krips (1987), Dieks (1988), Healey (1989) and Bub (1992) became known as modal interpretations and also older traditions like Bohmian mechanics (Bohm 1952; Bohm and Hiley 1993) were identified as modal ones. But why are all these interpretations still called modal? And what is the present-day meaning of this term?

I think part of the answer to the first question has to do with public relations. The name ‘modal’ is short, sounds nice and is rather intriguing. Furthermore, I guess that also Van Fraassen's prestige as a philosopher of science adds a special gloss to the term. But, apart from all this, I believe the name ‘modal interpretation’ is quite suited.

Starting with a more philosophical review, I discuss how modal interpretations describe reality. Firstly, it is shown that they describe states of affairs which need not be observable. Secondly, I underpin my position that in order to be metaphysically tenable, interpretations need only to satisfy the criteria of Consistency and Internal Completeness. Thirdly, I analyse how properties, states and measurement outcomes are related to one another within the modal descriptions of reality. Finally, I show how modal interpretations recover the standard formulation of quantum mechanics if one defines, in addition to the true states, so-called effective states of systems.

Noumenal states of affairs

The aim of an interpretation of quantum mechanics is, as I said in the introduction, to provide a description of what reality would be like if quantum mechanics were true. This formulation underlines two aspects of interpretations. Firstly, interpretations intend to construe quantum mechanics in terms of a description of reality and not merely in terms of the outcomes of measurements. Secondly, this description need not be correct. The objective is only to prove that there exists a construal of quantum mechanics which yields an acceptable description of reality. And because it is already difficult enough to give such a proof of existence, it is not (yet) the aim of interpretations to provide the one and only correct description of reality.

The above formulation, however, disregards another aspect of interpretations of quantum mechanics, namely that they describe states of affairs which need not always be observable.