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Philosophical Issues raised by Quantum
Theory and its Interpretations
Wayne C. Myrvold
Department of Philosophy
The University of Western Ontario
wmyrvold@uwo.ca
In Olival Friere, Jr., ed., The Oxford Handbook of the History
of Quantum Interpretations, Oxford University Press, 2022.
Abstract This chapter serves as an introduction to the philosophical issues raised by quantum theory. It begins with a brief overview of the for- malism of quantum theory. The so-called “measurement problem” is introduced, and the main approaches to it surveyed. We then discuss the implications of quantum theory for metaphysics. One question concerns the implications of quantum nonlocality for our understand- ing of spacetime and of causality. Another has to do with the ontology of quantum states. Should these be regarded as physically real, and, if so, what sort of reality should be ascribed to them?
1 Introduction
The philosophical questions surrounding quantum theory revolve around the question: What, if anything, does the empirical success of quan- tum theory tell us about the physical world? Since the key papers formulating what we now call quantum mechanics were published in the years 1925–27, we are only a few years away from the centennial of the theory’s inception. As we approach the centennial, there is more intense discussion than ever about its import.
Why is this? At the heart of the discussions is the following sit- uation. We have in our textbooks, and teach to our students, what amounts to an operational recipe sufficiently precise for most appli- cations. We learn to associate quantum states with various physical situations, and use them to calculate probabilities of outcomes of ex- periments. This is enormously important, as it is the basis both for the experimental testing of the theory and for its application. The success of the theory in these contexts is the reason we are taking it seriously at all. But the operational recipe does not, without further ado, yield anything like a clear description of what the physical sys- tems to which it is applied are like, or what they are doing in between experiments. The various approaches to the question of description of physi- cal systems and processes (among which are those that hold that we should refrain from describing physical systems) are sometimes re- ferred to as interpretations of quantum mechanics. This terminology is potentially misleading, for two reasons. The first is that it might suggest that the task of interpreting quantum theory is akin to sup- plying a model for an uninterpreted formal system. This is nothing at all like the task at hand. We don’t have an uninterpreted formal sys- tem, or a mathematical theory devoid of physical significance. What we have is a formalism with an agreed-upon operational significance (or, at least, sufficiently close to agreed-upon for most applications). The question is what, if anything, is to be added to this operational core. The second reason that the phrase “interpretations of quantum mechanics” is potentially misleading is that some of the avenues of approach involve formulation of a physical theory distinct from stan- dard quantum theory, in some cases differing in empirical content. These are not merely different interpretations of a common theory, but alternate physical theories.
2 What is a quantum theory?
In this section quantum theories are briefly described, with an empha- sis on the agreed-upon operational core, which every interpretational project must take into account. Quantum theories can be expressed in a number of different mathematical forms that are equivalent as far as the operational core is concerned. To avoid the pitfall of tying
commutator, [A, B] = AB − BA. (1)
When AB is equal to BA, A and B are said to commute. The distinctive quantum relations are the canonical commutation relations. These specify commutators for the operators {(Qi, Pi)} that correspond to canonical variables {(qi, pi)}. The rules are,
- Operators corresponding to different degrees of freedom com- mute. This means that, for distinct i, j, Qi and Pi commute with Qj and Pj.
- [Qi, Pi] = iℏ 1 , where 1 is the identity operator and ℏ = h/ 2 π, where h is Planck’s constant. The special operator 1 is the mul- tiplicative identity; the result of multiplying any operator A by 1 is just A itself. For any operator A, there is an operator A†, called the adjoint of A. These satisfy,
- (A†)†^ = A.
- For any operators A, B, and any complex numbers a, b, i). (a A + b B)†^ = a∗A†^ + b∗B†. ii). (AB)†^ = B†A†.
An operator that is its own adjoint is said to be self-adjoint. We can associate with any operator a set of real or complex numbers called its spectrum. If the operator is self-adjoint, its spectrum consists of real numbers only. We associate with any experiment a self-adjoint operator whose spectrum is the set of possible values of the outcome variable. For an experiment that, classically, would be regarded as a measurement of a given quantity that is a function of the canonical variables {(qi, pi)} , the associated operator is the corresponding function of the operators {(Qi, Pi)} (this does not yield a unique prescription, but this is not a matter we will go into in this chapter). We associate quantum states with preparation procedures that the system can be subjected to. A quantum state is an assignment ρ of numbers to operators, required to satisfy the conditions,
- Positivity. For any A, ρ(A†A) is a non-negative real number.
- Normalization. ρ( 1 ) = 1.
- Linearity. For any complex numbers a, b,
ρ(a A + b B) = a ρ(A) + b ρ(B).
For self-adjoint A, the value of ρ(A) is to be interpreted as the expec- tation value, in state ρ, of the outcome of an experiment with which is associated the operator A. The positivity condition ensures that self-adjoint operators are assigned real numbers. This condition, to- gether with the normalization condition, ensures that, if the spectrum of a self-adjoint operator A is bounded, ρ(A) is not above or below the bounds of the spectrum of A. This is required in order for these numbers to be interpreted as expectation values for the outcomes of experiments that yield results in the spectrum of A. The linearity condition is a non-trivial constraint, as it relates expectation values assigned to outcomes of incompatible experiments. It is a central prin- ciple of quantum mechanics, but is not something that is dictated by the operational significance of ρ(A) as an expectation value of the outcomes of an experiment (one could imagine other, non-quantum theories that violate it). For any state ρ, and any self-adjoint operator A, let a be the expectation value of A in state ρ, ρ(A). We define the variance of A in state ρ as,
Varρ(A) = ρ((A − a 1 )^2 ) = ρ(A^2 ) − ρ(A)^2. (2)
This is one way to quantify the spread in the probability distribution of outcomes of an A-experiment. It is small if the distribution is tightly focussed near the expectation value ρ(A). It is zero only when there is a single outcome that will be obtained with probability one. If this is the case—that is, if Varρ(A) is equal to zero—then ρ is said to be an eigenstate of A, with eigenvalue ρ(A). For such a state, one in which there is a definite value of the observable corresponding to A that will with certainty be obtained as the outcome of an appropriate experiment, it is usual to ascribe the property of possessing this value to the system. This is known as the eigenstate-eigenvalue link. For example, a state that is an eigenstate of the operator corresponding to energy, with eigenvalue E, is taken to be a state in which the system has energy E. This has been, since the early days of quantum mechanics, a central interpretational principle of quantum theories. Given any two states ρ 1 , ρ 2 , and any two positive numbers p 1 and p 2 that sum to one, we can always form the corresponding mixture of
A wave function representing the state is not unique; any two func- tions that differ only on a set of measure zero represent the same state, and multiplying any wave function by any complex number yields an- other function that represents the same state. A wave function yields probabilities for outcomes of detection experiments as follows: the probability of finding particle 1 in a set ∆ 1 , particle 2 in ∆ 2 , etc., is given by the integral of |ψ|^2 over the region of configuration space with x 1 in ∆ 1 and x 2 in ∆ 2 , etc., divided by the integral of |ψ|^2 over all of configuration space. For a system consisting of n particles with spin, the total spin state of the system can be represented by a vector in a finite-dimensional Hilbert space HS. A wave function for such a sys- tem is an assignment of a vector in HS to each point in configuration space.
2.2 Entangled and unentangled states
Consider a quantum theory of two non-overlapping systems. The al- gebra of observables AQ has two commuting subalgebras, AA and AB , corresponding to the observables of the two subsystems. A state ρ is a product state if any only if
ρ(AB) = ρ(A)ρ(B) (5)
for all A in AA and B in AB. A state that is either a product state or mixture of product states is called a separable state. A state (pure or mixed) that is not a separable state is an entangled state. We can also characterize pure entangled states more directly. A pure state ρ of AQ is a product state if the restriction of ρ to AA is a pure state of AA; it is an entangled state if the restriction of ρ to AA is a mixed state of AA. For any state that is not a product state, the state of a composite system is not uniquely determined by the states of the components, even if the state of the composite is pure. This is a striking differ- ence between quantum and classical theories. For a classical theory, the restriction of any pure state—that is, a maximally specific state description—of a composite to one of its components is a pure state of the component, and specification of the states of the components uniquely determines the state of the composite. Following Howard (1985), this feature of classical theories has come to be known as sep- arability, and the fact that it is not satisfied by quantum theories, as nonseparability.
2.3 Temporal evolution: Schr¨odinger and Heisen-
berg pictures
Suppose that we have a system whose dynamical variables are {(qi(t), pi(t))}. To construct a quantum theory of the system, we require operators {(Qi(t), Pi(t))} to represent the dynamical variables. The dynamical laws of our quantum theory specify how expecta- tion values of variables at different times are related to each other. The basic equation of evolution is,
iℏ d dt
ρ(A(t)) = ρ(A(t)H − HA(t)), (6)
where H is the operator corresponding to the system’s Hamiltonian H. Suppose, now, we want to construct a Hilbert space representation of our theory. This means assigning, to each operator A(t), a Hilbert space operator Aˆ(t), and choosing, for each time t, a density operator ρˆ(t) to represent the state, in such a way that
ρ(A(t)) = Tr[ˆρ(t) Aˆ(t)]. (7)
As we have to specify both Aˆ(t) and ˆρ(t), this gives us some lee-way. One way to do this is to choose the same Hilbert space operators ( Qˆi, Pˆi) to represent (qi(t), pi(t)) at all times. Then the density oper- ators ˆρ(t) will have to satisfy,
iℏ d dt
ρˆ(t) = Hˆ ρˆ(t) − ρˆ(t) H.ˆ (8)
For a pure state represented by a state vector |ψ(t)⟩, we have,
iℏ d dt
|ψ(t)⟩ = Hˆ|ψ(t)⟩. (9)
This is the Schr¨odinger equation, and the choice of Hilbert space rep- resentation on which the operators representing (qi(t), pi(t)) are time- independent, is called the Schr¨odinger picture. Another choice is to choose a fixed density operator ˆρ to represent the state at any time. This requires the operators Aˆ(t) to satisfy
iℏ d dt
Aˆ(t) = Aˆ(t) Hˆ − Hˆ Aˆ(t). (10)
discontinuous change of the quantum state, sometimes referred to as collapse of the state vector, or state vector reduction. There are two interpretations of the postulate about collapse, corresponding to two different conceptions of quantum states. If a quantum state represents nothing more than our knowledge about the system, then the collapse of the state to one corresponding to the observed result can be thought of as representing nothing more than an updating of knowledge. If, however, quantum states represent physical reality, in such a way that distinct pure states always represent distinct physical states of affairs, then the collapse postulate entails an abrupt, perhaps discontinuous, change of the physical state of the system. Considerable confusion can arise if the two interpretations are conflated. The collapse postulate is found already in Heisenberg’s The Phys- ical Principles of the Quantum Theory, based on lectures presented in 1929 (Heisenberg 1930a, 27; 1930b, 36). Von Neumann, in his refor- mulation of quantum theory a few years later, distinguished between two types of processes: Process 1., which occurs upon performance of an experiment, and Process 2., the unitary evolution that takes place as long as no measurement is made (von Neumann 1932; 1955, §V.I). He does not take this distinction to be a difference between two physically distinct processes. Rather, the invocation of one process or the other depends on a somewhat arbitrary division of the world into an observing part and an observed part (see von Neumann 1932, 224; 1955, 420). There is a persistent misconception that, for von Neumann, col- lapse is to be invoked only when a conscious observer becomes aware of the result. This is the opposite of his attitude; for him it is essential that the location of the boundary between the observed part of the world and the observing part is somewhat arbitrary. It may be placed between the system under the study and the experimental apparatus. On the other hand, we could include the experimental apparatus in the quantum description, and place the cut at the moment when light indicating the result hits the observer’s retina. Or we could go further, and include the retina and relevant parts of the observer’s nervous sys- tem in the quantum system. That the cut may be pushed arbitrarily far into the perceptual apparatus of the observer is required, according to von Neumann, by the principle of psycho-physical parallelism. The collapse postulate does not appear in the first edition (1930) of Dirac’s Principles of Quantum Mechanics; it is introduced in the second edition (1935), which appeared subsequent to von Neumann’s
treatment. Dirac, in contrast to Heisenberg and von Neumann, ap- pears to take the distinction between unitary and collapse evolution to be a distinction between two physical processes. Also, for Dirac it is an act of measurement, not observation, that causes a system to “jump” into an eigenstate of the observable being measured (Dirac, 1935, 26). According to Dirac, this jump is caused by the interaction of the system with the experimental apparatus. A formulation of a version of the collapse postulate according to which a measurement is not completed until the result is observed is found in London and Bauer (1939). For them, as for Heisenberg, this is a matter of an increase of knowledge on the part of the observer. Wigner (1961) combined elements of the two interpretations. Like those who take the collapse to be a matter of updating of belief in light of information newly acquired by an observer, he takes collapse to take place when a conscious observer becomes aware of an experimental result. However, like Dirac, he takes it to be a real physical process. His conclusion is that consciousness has an influence on the physical world not captured by the laws of quantum mechanics. This involves a rejection of von Neumann’s principle of psycho-physical parallelism, according to which it must be possible to treat the process of subjective perception as if it were a physical process like any other.^2
4 The so-called “measurement prob-
lem”
If it is possible for any quantum theory to be a comprehensive physical theory, it must be capable of treating of our experimental apparatus, and, indeed, everything else. Suppose, then, we analyze an experi- mental set-up quantum-mechanically. Let S be the system to be experimented on, which we call the studied system. We suppose that it has at least two distinguishable states, |+⟩S and |−⟩S , and that an apparatus, A, can be devised that distinguishes these states. This means that there are distinguishable sets A+^ and A−^ of states of the apparatus, which we will call indicator states, such that the apparatus can be coupled to the system S in such a way that, if the apparatus is started out in a ready state and the (^2) Despite this, Wigner’s proposal is sometimes wrongly attributed to von Neumann,
and is sometimes called the “von Neumann-Wigner interpretation.”
would be like to find the apparatus in such a state, then it makes no sense either to affirm or to deny that we have ever found the apparatus in such a state. Though we have chosen an experimental set-up as an illustra- tion, situations in which linear evolution would yield superpositions of macroscopically distinct states are ubiquitous. Nonetheless, the prob- lem of what to make of this fact—that applying linear evolution to quantum states involving macroscopic objects will lead to superposi- tions of macroscopically distinct states—has come to be known as the measurement problem. If there is a unique outcome of the experiment, and if (12) is the correct quantum state, then the outcome fails to be represented by the quantum state, which must be supplemented by something that does indicate the outcome. On the other hand, it might be that neither Schr¨odinger evolution nor any other linear evolution applies to situa- tions like the one envisaged, and that the correct evolution leads to a state that we can take to be indicating a determinate outcome. These two options were summarized by J.S. Bell in his remark, “Either the wavefunction, as given by the Schr¨odinger equation, is not everything, or it is not right” (Bell 1987a, 41; 1987b and 2004, 201). This gives us a (prima facie) neat way of classifying approaches to the so-called “measurement problem.”
- There are approaches that involve a denial that a quantum wave function (or any other way of representing a quantum state) yields a complete description of a physical system.
- There are approaches that involve modification of the dynam- ics to produce a collapse of the quantum state in appropriate circumstances.
- There are approaches that reject both horns of Bell’s dilemma, and hold that quantum states undergo unitary evolution at all times and that there is no more to be said about the physical state of a system than can be represented by a quantum state.
We include in the first category approaches that deny that a quantum state should be thought of as representing anything in physical reality at all. If quantum states do not represent anything, and if there is something rather than nothing, then quantum states do not represent everything. In this category are Bohrian approaches, according to which there are principled reasons not to seek a complete description, and Einsteinian approaches, according to which seeking a theory that
need not leave anything out in its descriptions is a project worthy of pursuit. Also included in the first category are approaches that take quan- tum states to represent something, but not everything, in physical reality. These include “hidden-variables” theories, and modal inter- pretations (see Lombardi and Dieks 2017). The best-known and most thoroughly worked-out theory of this sort is the de Broglie-Bohm pilot wave theory, which takes particles with definite trajectories as the ba- sic ontology. The role of the wave function is to provide dynamics for the particles. See Bacciagaluppi and Valentini (2009) for a historical introduction, and D¨urr et al. (1992) and Pearle and Valentini (2006) for current perspectives. The second category embraces the dynamical collapse theory pro- gramme, which seeks a modified dynamics that approximates unitary evolution in the domains in which we have good evidence for its cor- rectness, and approximates collapse in other situations, including, but not limited to, experimental set-ups. The best-known version of this is the Ghirardi-Rimini-Weber (GRW) theory (Ghirardi, Rimini, and Weber 1986), referred to by its creators as Quantum Mechanics with Spontaneous Localization (QMSL). On this theory, Schr¨odinger evo- lution of the quantum state is punctuated by discontinuous jumps. The GRW theory has the defect that it does not respect the sym- metrization/antisymmetrization requirements for states of a system containing identical particles. This is remedied in a successor theory, the Continuous Spontaneous Localization (CSL) theory (Pearle 1989; Ghirardi, Pearle, and Rimini 1990). Approaches that reject both horns of Bell’s dilemma are typified by Everettian, or “many-worlds” interpretations. The basic idea is denial that there is a unique experimental outcome; rather, there is a splitting, and different results obtain on different branches of the multiverse. These have their roots in the work of Hugh Everett III (see Barrett and Byrne 2012). See Saunders et al. (2010), Wallace (2012), and Carroll and Singh (2019) for some recent approaches along these lines. An approach that does not fit neatly into these categories is the relational interpretation advocated by Carlo Rovelli. It is akin in some ways to Everett’s original conception, which he called the relative-state interpretation. It differs from it in not taking quantum states to be representational. For more on this, see Rovelli’s contribution to this volume, and also Laudisa and Rovelli (2019), and references therein.
Violation of Bell inequalities has been abundantly confirmed by experiment. What does this tell us? It is sometimes said that violation of Bell inequalities straightfor- wardly entails violation of relativistic causality. Things are not so simple, as there is no interpretation-independent answer to the ques- tion of compatibility with relativity. The question is most straightforward in connection with hidden- variables theories such as the de Broglie-Bohm theory. Any deter- ministic theory that violates Bell inequalities must violate parameter independence, and thus must have cause-effect dependencies between spacelike separated events. Because, in a multi-particle system, the velocity of each particle may depend on the positions of all the others, the de Broglie-Bohm theory requires a preferred relation of distant simultaneity for its for- mulation. There is a series of theorems that show that any theory of this sort, on which the quantum state is supplemented by extra vari- ables that are required to have a probability distribution given by the Born rule, must employ a distinguished relation of distant simultane- ity, as it is not possible to satisfy the postulate about probabilities on arbitrary spacelike hypersurfaces. See Berndl et al. (1996); Dickson and Clifton (1998); Arnztenius (1998); Myrvold (2002, 2009). This has the consequence that such theories require a dynamically distin- guished relation of distant simultaneity; see Myrvold (2021, §5.5.1) for the argument. Dynamical collapse theories, on the other hand, do not require a preferred relation of distant simultaneity for their formulation. There is an extension of the GRW theory to a relativistic context (Dove, 1996; Dove and Squires, 1996; Tumulka, 2006), which involves a fixed, finite number of noninteracting particles. There are also extensions of the CSL theory to the context of relativistic quantum field theories (Bedingham, 2011a,b; Pearle, 2015). These theories involve probabilistic correlations between spacelike separated events that are not attributable to events in their common past. That is, they involve a rejection of the Reichenbach Common Cause Principle. The question of whether a theory such as this is in violation of any restriction on causal relations that is motivated by considerations of special relativity has been a hotly debated one. Several authors have argued over the years, in different ways, in favour of the compatibility of theories like that with the requirements of special relativity; these include Shimony (1978, 1984, 1986), Jarrett
(1984), Skyrms (1984), Redhead (1987), Ghirardi and Grassi (1996), and Ghirardi (2012). See Myrvold (2016) for a recent argument for compatibility of special relativity with violations of Bell inequalities.
6 Ontological questions concerning quan-
tum states
6.1 The question of quantum state realism
We have introduced quantum states via their operational significance: they encode probabilities of outcomes of experiments. Should we think of them as representing some feature of the system to which they are ascribed? Positions that deny that quantum states represent features of phys- ical reality have a history as old as quantum theory itself. This is one thing that Bohr and Einstein agreed upon. For Bohr, all description of physical reality must be couched in classical terms, and the limits of classical physics are the limits of physical description; quantum wave functions have only “symbolic” status (see Bohr 1934, 17). Einstein argued, in several places (see, e.g., Einstein 1936), that quantum states should be regarded as akin to the probability distributions of classical statistical mechanics, that is, as representing incomplete knowledge of some deeper underlying physical state. The chief locus of difference between the two had to do with the propriety of seeking a deeper level of description. The idea that quantum states are like that is an attractive one. It faces considerable obstacles, and it should be non-controversial that quantum states are not just like classical probability distributions. A useful way of sharpening the question of realism about quan- tum states is afforded by the framework constructed by Harrigan and Spekkens (2010). This framework makes explicit some principles that are deeply embedded in our reasoning about the world. Suppose that Alice has a choice of two or more preparation pro- cedures that she can subject a system to. Having made the choice, she passes the system on to Bob, who can do an experiment on the system, and, from the outcome, reliably identify the procedure Alice has chosen. We would take this as an indication that distinct choices of preparation on Alice’s part result in physical differences in the sys- tem being prepared, and that the outcome of Bob’s experiment is
ψ-epistemic model (Barrett et al., 2014). On such a model, the indis- tinguishability of quantum states is fully explained by overlap of the corresponding probability distributions on ontic state space. There are a number of theorems concerning the viability of the programme of constructing a theory that is maximally ψ-epistemic, or, failing that, a theory that is not ψ-ontic. In particular, Barrett et al. (2014) show that no theory that reproduces quantum probabilities for outcomes of experiments and fits into the framework just sketched can be maximally ψ-epistemic, or even come close to being so. Pusey, Barrett, and Rudolph (PBR) show that, provided that the theory satisfies a postulate called the Preparation Independence Postulate, it must be ψ-ontic in order to reproduce quantum probabilities for outcomes of experiments (Pusey, Barrett, and Rudolph 2012). The Preparation Independence Postulate is a postulate to the ef- fect that it is possible to subject a pair of distinct systems A and B to preparation procedures that render their ontic states probabilistically independent of each other. This postulate involves an assumption, called the Cartesian Product Assumption, to the effect that, for a preparation of that sort, the state of the composite system AB can be fully represented by specifying a state of A and a state of B. This is a non-trivial restriction on the state spaces employed in the the- ory. A weaker assumption, called the Preparation Uninformativeness Condition, which makes no assumptions about the structure of the state spaces, was suggested by Myrvold (2018c, 2020). On the basis of this weaker assumption, a weaker conclusion can be derived. The conclusion that is derived from this condition is that, on any theory that satisfies it, pure quantum states |ψ⟩ and |ϕ⟩ that are not too close to each other are ontically distinct. Here, the condition of not being too close is that the absolute value of their inner product be less than 1 /
6.2 The ontological status of quantum states
Suppose that we are realists about quantum states. This means that distinct pure quantum states represent physically distinct states of affairs. This still leaves open the question of what sorts of physical reality these states represent. In this section we briefly discuss some options.
6.2.1 Quantum state monism
Could there be nothing more to the world than what is represented by a quantum state? Recall that a quantum theory is not an uninterpreted formalism. A quantum theory involves an identification of physical quantities to be represented, and an association of operators with those quantities. The eigenstate-eigenvalue link yields property attributions in the spe- cial case of eigenstates. If we had a dynamical collapse theory that produced eigenstates of the right sorts of dynamical quantities—if, for example, it yielded definite mass or energy content for regions of space that are small on the macroscopic scale—then such a theory could, in a straightforward way, be a quantum state monist theory. Sometimes skepticism is expressed about this, but this skepticism seems aimed at a different project, a project that would involve starting with a math- ematical formalism devoid of physical interpretation and attempting to interpret it physically. Things are not so simple, because dynamical collapse theories do not produce eigenstates of appropriate physical quantities, and there are principled reasons for not expecting a dynamical collapse theory to do that. For this reason, Ghirardi and collaborators proposed a weakening of the eigenstate-eigenvalue link, according to which a sys- tem is to be ascribed a property if its quantum state is sufficiently close to being an eigenstate of the corresponding operator (Ghirardi, Grassi, and Pearle 1990, 1298; see also Ghirardi, Grassi, and Benatti 1995, 13). This modification has been dubbed, by Clifton and Monton (1999), the fuzzy link.^3 For a defense of quantum state monism along the lines proposed by the originators of the GRW and CSL theories, see Myrvold (2018a, 2019). Everettian theories seem to be best interpreted along these lines. Since such theories eschew collapse, on such a theory a quantum state will not typically be anywhere near an eigenstate of familiar macro- scopic variables. However, the quantum state of any bounded region (^3) Peter Lewis (2016, 86–90) distinguishes between a fuzzy link, according to which there
is some precise threshold p such that a system possesses the property A = a if and only if the probability of finding some other value is less than p, and a vague link according to which possession of a definite property is a matter of degree. It is hard to imagine what arguments there could be (or even what it might mean) for there to be a precise threshold. Albert and Loewer (1996) argue, correctly in my opinion, that there could be no such precise threshold, and that the modified link must be somewhat vague. This is what I mean by a fuzzy link.
