Topos Theoretic Quantum Realism

Benjamin Eva

(Forthcoming in British Journal for the Philosophy of Science)

Abstract

Topos Quantum Theory (TQT)1 is standardly portrayed as a kind
of ‘neo-realist’ reformulation of quantum mechanics. In this paper, we
study the extent to which TQT can really be characterised as a realist
formulation of the theory, and examine the question of whether the kind
of realism that is provided by TQT satisfies the philosophical motivations
that are usually associated with the search for a realist reformulation of
quantum theory. Specifically, we show that the notion of the quantum
state is problematic for those who view TQT as a realist reformulation of
quantum theory.

1 Introduction
2 Topos Quantum Theory (TQT)

2.1 Phase space
2.2 Hilbert space
2.3 Beyond Hilbert space
2.4 Defining realism
2.5 The spectral presheaf
2.6 The logic of TQT

3 Interpreting States in TQT
4 Interpreting Truth Values and Clopen Subobjects in TQT

4.1 Interpreting the truth values
4.2 Interpreting Subcl(Σ)

5 Neo-Realism
5.1 The covariant approach

6 Conclusion

1 Introduction

The topos theoretic reformulation of quantum theory (TQT) was origi-
nally proposed by Isham ([1997]), and has subsequently been developed

1I use the name ‘topos quantum theory’ to refer to the topos theoretic approach to quantum
theory pioneered by Chris Isham, Jeremy Butterfield and Andreas Döring, which is often
referred to as ‘the contravariant approach’ in the relevant literature. Although various other
contemporaneous projects (some of which are discussed in section 5 of this paper) have applied
topos theory to the study of quantum mechanics, the name ‘topos quantum theory’ appears to
have been claimed by the contravariant approach (see Flori [2013]), and I follow this convention
here.

1



by Isham, Butterfield and Döring amongst others (Isham and Butterfield
[1998], Döring [2012], Döring and Isham [2011]). From its inception, TQT
has been motivated, at least in part, by broadly philosophical considera-
tions. Specifically, TQT is often characterized as an attempt to replace the
traditional ‘instrumentalist’ formulation of quantum mechanics (Hilbert
space) with a new formalism that is more susceptible to a ‘realist’ inter-
pretation. In this paper, we study the extent to which TQT, under such
a characterization, has been successful as a philosophical project.

In section 2, we begin by providing a concise technical overview of some
of the central concepts and results of TQT, together with a summary of
the key philosophical motivations of the project, and a discussion of the
philosophical merits of the definition of realism that is usually employed
in this context.

Each of the following sections focuses on one central aspect of the
quantum realism provided by TQT. Section 3 focuses on the notion of the
quantum state in TQT, and its relationship to the corresponding notion
in the traditional Hilbert space formalism. In particular, we show that
the identification of the spectral presheaf as the quantum state space is
unmotivated and problematic

Section 4 provides an analysis of the logical structure of TQT. Specif-
ically, it examines the physical significance of the truth values used in
the theory, and provides a comparison with other attempts to use many
valued logics in quantum theory, and then raises a significant interpreta-
tional issue concerning the lattice of physical propositions in TQT. Section
5 combines the analyses of the previous sections to provide a general ap-
praisal of the quantum realism of TQT, and gives a brief discussion of the
applicability of the arguments made in this paper to the topos theoretic
approach to quantum theory developed, for example, in (Heunen et al
[2009]). Section 6 concludes.

2 Topos Quantum Theory

We begin this section with a brief summary of the traditional formalisms
of classical and quantum physics, and the asymmetries between them.

2.1 Phase space

The canonical formalism of classical physics can be summarized in the
following way.

Let S be a classical physical system. Then, to S we associate a set S
(the ‘state space of S’) consisting of the set of all possible states of the
system S. To each physical quantity A associated with S (e.g momentum,
position etc), we define a unique corresponding function fA : S → R.
Given s ∈ S, we interpret the value fA(s) as the value that the physical
quantity A would have if the state of S were s.

Next, we define the notion of a physical proposition associated with
S. Specifically, if A is any physical quantity associated with S and ∆ is
any Borel subset of the real numbers, we call the sentence ‘the value of
A lies in the set ∆’(denoted A ∈ ∆) a ‘physical proposition associated

2



with S’. Mathematically, we represent a physical proposition A ∈ ∆ as
the inverse image of ∆ under fA. Generalizing, we can identify physical
propositions about S as the inverse images of Borel subsets of the reals un-
der real-valued functions on S, i.e. physical propositions about S are just
measurable subsets of S. Ordered by inclusion, the measurable subsets of
S form a Boolean algebra, Sub(S).

Now, let s ∈ S. Then s automatically defines a way of assigning clas-
sical truth values to all of the propositions associated with S. Specifically,
given a physical proposition P = A ∈ ∆, we define the truth value of
P relative to S to be 1 if s ∈ P = f−1A (∆) and 0 otherwise. It is easy
to see that this assignment defines a Boolean algebra homomorphism,
hs : Sub(S) → {0, 1} from the set of all physical propositions associated
with S to the two-element Boolean algebra.

To summarize, the canonical formalism of classical physics has two
relevant fundamental properties. Firstly, the algebra of propositions as-
sociated with a classical system forms a Boolean algebra in a natural
way. Secondly, any state of a classical physical system automatically de-
fines a Boolean algebra homomorphism between the algebra of physical
propositions associated with that system and the two element truth value
algebra.

2.2 Hilbert space

In contrast to the canonical formalism for classical physics, the traditional
Hilbert space formalism of quantum theory begins by assigning a complex
Hilbert space H to the physical system S under consideration. The unit
vectors of H are then taken to represent the possible states of S. Any
physical quantity A associated with the system is then represented by a
self-adjoint operator  on H. Then, given an eigenvector |ψ〉 of Â, |ψ〉’s
eigenvalue with respect to  is interpreted as the value that A would have
if the state of S were |ψ〉.

A physical proposition A ∈ ∆ is then represented by the projection
operator corresponding to the closed linear subspace consisting of those
eigenvectors of  with eigenvalues in ∆. Generalizing, we think of all
closed linear subspaces (or, equivalently, all projection operators) of H
as representing physical propositions about S. It is well known that the
lattice of projection operators P(H) onto H will generally fail to form a
Boolean algebra and will only form an orthomodular lattice. This is in
stark contrast to the formalism of classical physics, where the algebra of
propositions about S is always Boolean.

We have seen that in classical physics, a physical state of a system S
automatically defines a consistent assignment of classical truth values to
all of the physical propositions associated with S. This is not the case in
the Hilbert space formalism of quantum theory, where states will only, in
general, provide probabilities for the truth/falsity of any given proposi-
tion. Indeed, it turns out that the existence of a consistent assignment of
classical truth values to all of the physical propositions associated with a
quantum system is generally impossible.

Kochen-Specker Theorem (KST): Let H be a Hilbert space with

3



dimension dim(H) ≥ 3. Then there is no way of assigning classical truth
values to all of the projections operators on H in a way that preserves the
functional relationships2 between those projection operators.

Given the formalism outlined above, where physical propositions are
identified with projection operators and physical quantities are identi-
fied with self-adjoint operators, KST can be interpreted as saying that
it is impossible to simultaneously assign classical truth values to all of
the propositions associated with a quantum system in a way that pre-
serves the functional relationships between them, or equivalently, that it
is impossible to assign definite values to all of the physical quantities as-
sociated with a quantum system in a way that preserves the functional
relationships between them.

Again, this is radically disanalogous with the formalism of classical
physics, where it is always possible, in principle, to assign classical truth
values to all of the propositions associated with a physical system in a way
that respects all of the algebraic relationships between those propositions.

2.3 Beyond Hilbert space

The fundamental motivating precept behind TQT is nicely summarized
in the following passage:

‘When dealing with a closed system, what is needed is a realist interpretation
of the theory, not one that is instrumentalist. The exact meaning of “realist” is
infinitely debatable, but, when used by physicists, it typically means the follow-
ing:

(1). The idea of “a property of the system” (i.e. “the value of a physical
quantity”) is meaningful, and representable in the theory.

(2). Propositions about the system are handled using Boolean logic. This
requirement is compelling in so far as we humans think in a Boolean way.

(3). There is a space of “microstates” such that specifying a microstate leads
to unequivocal truth values for all propositions about the system. The existence
of such a state space is a natural way of ensuring that the first two requirements
are satisfied.

The standard interpretation of classical physics satisfies these requirements,

and provides the paradigmatic example of a realist philosophy in science. On

the other hand, the existence of such an interpretation in quantum theory is

foiled by the famous Kochen-Specker theorem’. (Döring and Isham [2008])

The primary motivating problem for TQT3 is that, unlike clas-
sical physics, quantum physics is not amenable to a realist interpre-

2By ‘functional relationships’, we will mean the famous non-contextuality condition FUNC,
which states that for any real valued function f : R → R and any assignment V of classical
truth values to projection operators, we should have V (f(P)) = f(V (P)), where P is any
projection operator.

3At this point it should be noted that TQT has tended to focus primarily on the interpre-
tational problems that occur even in synchronic contexts, and has been largely unconcerned

4



tation, in the sense defined above. TQT can be construed as the
attempt to provide a new formalism for quantum mechanics that,
like the formalism of classical physics, has an unproblematically re-
alist interpretation. The main obstacle to the availability of such a
realist interpretation is KST. For, as we have seen, KST rules out
the existence of a quantum state space satisfying condition (3).

By citing classical physics as the ‘paradigmatic example’ of a re-
alist physical theory, Isham and Doering implicitly articulate one of
the key methodological principles of TQT: to provide a formalism
for quantum theory that, as far as possible, mimics and parallels the
formalism of classical physics, and in doing so, inherits its archety-
pally realist properties.

Now, we have already had occasion to note two fundamental dis-
analogies between the formalisms of classical physics and quantum
mechanics. Firstly, the algebra of propositions associated with a
classical system, unlike the algebra of propositions associated with
a quantum system, is Boolean. Secondly, a classical state, unlike a
quantum state, automatically assigns classical truth values to all of
the propositions associated with the system to which it corresponds.
Indeed, classical states can actually be thought of as nothing more
than Boolean algebra homomorphisms between the algebra of mea-
surable subsets of the state space and the two element Boolean alge-
bra. KST essentially tells us that such homomorphisms never exist
for the algebra of physical propositions associated with a quantum
system. As Bub puts it,

‘while a classical property state is selected by a two-valued homomorphism

on the Boolean lattice of classical properties, a quantum property state ... is not

selected by a two-valued homomorphism on the non-Boolean lattice of subspaces

representing the properties of a quantum mechanical system. There are no such

two-valued homomorphisms’. (Bub [1999])

2.4 Defining realism

Isham and Döring’s definition of realist physical theories employs
three closely related criteria. We will consider each in turn.

The first criterion, that ‘the idea of “a property of the system”
(i.e. “the value of a physical quantity”) is meaningful, and repre-
sentable in the theory’ will henceforth be referred to as ‘PROP’.
Of course, PROP is a very natural condition to employ in a def-
inition of what it means for a physical theory to be realist. For,
if a purportedly realist theory allows us to talk about some physi-
cal quantity Q associated with a physical system S, then of course

with the dynamics of quantum theory. Specifically, it has not yet been argued that TQT
addresses the measurement problem.

5



that theory should allow us to talk about S having a value for Q,
since how else could we conceive of Q as really representing a feature
of physical reality? One of the central interpretational problems of
quantum theory, namely the problem of what one should say about
those physical quantities that are represented by self-adjoint oper-
ators for which the state of the system is not an eigenstate, can
be characterised as the problem of reconciling quantum theory with
PROP.

The second criterion imposes the condition that in any realist
physical theory, ‘Propositions about the system are handled using
Boolean logic’. We will refer to this condition as ‘BOOLE’. BOOLE
is really just an application to physics of Michael Dummett’s prin-
ciple that whenever one is realist about the subjects of a particular
domain of discourse, they are thereby committed to using classical
logic when reasoning in that domain (see e.g chapter 10 of Dum-
mett [1978] ). This kind of position is often motivated by the idea
that the law of excluded middle encodes a form of realism, in so
far as it requires that the disjunction of any proposition with its
own negation must be true. The idea is that the law of excluded
middle prevents the world from being indifferent to the truth of any
given proposition, since it requires that either the proposition or its
negation must be true. In contrast, if a physical theory allows the
disjunction of a proposition with its negation to have a non-maximal
truth-value, then it looks like the theory allows for the possibility
that physical reality is indifferent to the truth of the proposition in
question.

However, this argument is flawed on two counts. Firstly, the law
of excluded middle alone is not sufficient to motivate classical logic.
There are many non-classical substructural logics that satisfy this
law. Pertinently in this case, orthodox quantum logic satisfies the
law of excluded middle, so this argument could equally well be used
to motivate the adoption of non-distributive quantum logic. Sec-
ondly, as we will see in a later section, the absence of excluded mid-
dle need not entail any kind of indeterminism. In TQT, all physical
propositions are assigned definite truth values in a way that violates
excluded middle. These truth-values contain concrete physical in-
formation and do not require that the propositions to which they
are assigned be viewed as ‘indeterminate’ or non-realist.

Isham and Doering claim that BOOLE ‘is compelling in so far as
we humans think in a Boolean way’. Of course, this claim is easily
countered by the observation that there is no obvious reason that
physical reality should be amenable to human thought. However,
this perspective does gain some traction when one compares classi-
cal logic with orthodox quantum logic. For, non-distributive quan-
tum logic suffers from numerous pathologies that render its status
as a logic dubious. For example, quantum logic generally lacks a de-

6



duction theorem (see Malinowski [1990]) and does not allow for the
definition of a natural implication connective. This renders reason-
ing with quantum logic highly problematic, and arguably impossible,
and not just for humans. Now, it seems reasonable to require that
the logic associated with a realist physical theory should at least sat-
isfy certain minimal structural conditions. For example, it should
contain some kind of meaningful implication connective satisfying
a list of necessary formal properties and be subject to a deduction
theorem. Although this is not enough to motivate BOOLE, it is at
least a motivation for imposing some kind of formal restriction on
the logical structure of realist physical theories.

The third criterion, requiring the existence of ‘a space of “mi-
crostates” such that specifying a microstate leads to unequivocal
truth values for all propositions about the system’ will be referred
to as ‘STATE’. Like PROP, STATE is a highly intuitive condition
to impose on realist physical theories. For, the notion of a physical
state can be seen as a formalisation of the concept of ‘how things
are’ with respect to the system in question, and it is natural to char-
acterise realist physical theories as telling us ‘how things are’ with
respect to the systems they describe. Indeed, interpreters of quan-
tum theory often ask for an account of ‘what the world would be
like if quantum mechanics were true’. The fact that this question is
problematic is indicative of the failure of quantum theory to satisfy
STATE.

2.5 The spectral presheaf
4

It is well known that, from a mathematical perspective, KST
is a consequence of the existence of non-commuting observables on
Hilbert spaces (or, to be more precise, that non-commuting opera-
tors are necessary for the application of KST). So, for example, if we
let B(H) represent the set of all bounded self-adjoint operators on a
Hilbert space H (with dim(H) ≥ 3), then, because B(H) will gener-
ally contain some non-commuting observables, KST will apply, and
it will be impossible to simultaneously assign definite values to all of
the self-adjoint operators in B(H) in a satisfactory manner. How-
ever, if we take any Abelian Von-Neumann subalgebra V of B(H),
KST generally will not apply to V since all of the operators in V are
pairwise commuting (and the algebra P(V ) of projection operators

4Although we try to keep the use of category theoretic notions to a minimum, some famil-
iarity with the basic notions of category theory and topos theory are assumed throughout the
rest of the paper. For a thorough exposition of the relevant notions and a detailed survey of
the technical structure of TQT, see (Flori [2013]). Also, for a detailed conceptual introduction
to TQT, see (Isham [2011])

7



in V will be Boolean). So it will, in general, be possible to assign
definite values to all of the operators in V without destroying any
of the algebraic/functional relationships between them. Intuitively,
V corresponds to a kind of ‘classical perspective’ on the quantum
system, in the sense that if we talk only about physical quantities
that are represented by operators in V , the situation will be analo-
gous to the situation in classical physics, i.e. the algebra of physical
propositions will form a Boolean algebra and KST will not apply.
This means that it will be possible to define a kind of ‘state of the
system’ from the perspective of V , i.e. a two-valued Boolean algebra
homomorphism on P(V ).

We can give these intuitive notions a rigorous generalisation in
the following way: Let V (H) represent the set of all Abelian Von-
Neumann sub-algebras of B(H). Then, to each V ∈ V (H), we can
associate the set ΣV of all homomorphisms from V into C, known as
the ‘Gelfand spectrum’ of V . An element of the Gelfand spectrum of
V is just a way of assigning a definite value to each of the operators in
V in a way that preserves all of the algebraic and functional relations
between them. In the terminology established above, the Gelfand
spectrum consists of all of the possible states of the system being
described, from the perspective of V . In TQT, ΣV is customarily
referred to as the ‘local state space’ of the system at V .

Note that V (H) can be turned into a partially ordered set by the
‘is a subalgebra of’ relation, denoted by ⊆. Given V ′ ⊆ V , we think
of V ′ as a ‘smaller’ classical context than V , in the sense that any
physical property that can be talked about from the perspective of
V ′ can also be talked about from the perspective of V . Viewed as
a poset, V (H) is often referred to as the ‘context category’. We are
now ready to define the quantum state space in the formalism of
TQT.

Def 2.5.1: The spectral presheaf5 Σ on V (H) is defined by

Objects: Given V ∈ V (H), the component ΣV of Σ at V is the
Gelfand spectrum of V .

Arrows: Given i : V ′ ⊆ V , ΣiV ′,V : ΣV → ΣV ′

λ 7→ ΣiV ′,V (λ) = λ|V ′
6

Intuitively, Σ is an assignment that takes each classical perspec-
tive V to its associated local state space. Furthermore, if V ′ is a
smaller classical perspective than V , then Σ will take the inclusion

5By a ‘presheaf on V (H)’, we will mean a contravariant set-valued functor on V (H).
6Where λ|V ′ represents the restriction of λ to V ′.

8



arrow from V ′ into V and return the function that takes a state λ
from the local state space of V , and returns the state λ|V ′ , which can
be thought of as the result of throwing away all the extra physical
information that is contained in V , but not V ′.

The following theorem is perhaps the most fundamental result of
TQT.

Theorem 2.5.2: The Kochen-Specker theorem is equivalent to
the fact that the spectral presheaf on V (H) (for dim(H) ≥ 3) has
no global elements7

Theorem 2.5.2 shows that, if we interpret the spectral presheaf
on V (H) as the state space of a quantum system S described by
the Hilbert space H, and correspondingly interpret the individual
quantum states as the global elements of the spectral presheaf, then
KST is actually equivalent to the fact that there are no quantum
states for S (the spectral presheaf has no global elements). Later,
we will provide a detailed discussion of the notion of the quantum
state space in TQT, but for now, it suffices to note that one of
the key interpretational moves in TQT is the interpretation of the
spectral presheaf as the quantum state space.

We have seen that in TQT, the quantum state space is formalised
as a kind of amalgamation of the local, classical state spaces of each
of the ‘classical perspectives ’ on V (also referred to as ‘contexts’).
Thus, we read

‘The topos approach emphasises the role of classical perspectives onto a

quantum system...One of the main ideas is that all classical perspectives should

be taken into account simultaneously’ (Döring [2011])

This emphasis on classical perspectives is deeply reminiscent of
Bohr’s famous ‘principle of complementarity’(PC), which is neatly
summarised by the claim that

‘Talk of the position of an electron has sense only in the context of an ex-

perimental arrangement for making a position measurement.’ (Bohr, quoted
in Gibbins [1987])

The philosophical upshot of PC is that physical propositions
about a quantum system can only be made with reference to some
fixed classical perspective on that system. This notion is taken seri-

7Intuitively, a global element of the presheaf Σ is the generalisation of the usual set theoretic
notion of element to Σ. Formally, a global element of Σ is an assignment g : V (H) → SET
satisfying (i) g(V ) ∈ ΣV , ∀V ∈ V (H), and (ii) Given V,V

′ ∈ V (H), and any arrow i : V ′ → V ,
g satisfies ΣiV ′,V

(g(V )) = g(V ′).

9



ously in TQT, and is evident in the way that physical propositions
are eventually formalised.

2.6 The logic of TQT:

Now that we have access to a suitable notion of a quantum state
space, we need to define a corresponding formalisation of physical
propositions. Continuing the analogy with the formalism of classical
physics, we need to represent physical propositions as a special class
of ‘subsets’ of the quantum state space. But of course, the quantum
state space is now formalised as a presheaf, not a set, so we need
to generalise this requirement. Specifically, we want to represent
physical propositions as a special class of sub-presheaves of Σ.

Now, in classical physics, the physical proposition A ∈ ∆ is rep-
resented by the subset of the state space that makes the proposition
true. This suggests a natural generalisation. Specifically, we rep-
resent a physical proposition A ∈ ∆ in the following way: for each
V ∈ V (H), we take the subset of the local state space at V that
makes the physical proposition A ∈ ∆ true, i.e. we take the set
{λ ∈ ΣV |λ(P) = 1}, where P is the projection operator that corre-
sponds to A ∈ ∆, in the usual sense. But this is problematic, since it
will not generally hold that P ∈ V , and so the local states of V might
not be defined on P . This problem is the mathematical realisation
of PC. A physical proposition will not generally be meaningful for
all classical perspectives at once.

So, since we cannot generally be certain that a physical proposi-
tion will be meaningful at every classical perspective, we will attempt
to ‘simulate’ the physical proposition in question by approximating
to it whenever it is not included in the classical perspective in which
we are working. Specifically, we define

Def 2.6.1: The outer daseinisation of a projection operator P
for a classical perspective V ∈ V (H) is defined to be the projection
operator given by

δo(P)V =
∧
{Q ∈ P(V )|Q ≥ P}

Where P(V ) is the set of all projection operators in V , and ≥ is
the usual ordering on the lattice of projections.

Intuitively, the outer daseinisation of P at V is V ’s best approx-
imation to P ‘from above’. Of course, if P ∈ P(V ), then this ap-
proximation will just be P itself. If P /∈ V , then this approximation
will be the strongest proposition that can be asserted from the clas-
sical perspective represented by V that is implied by P . The idea is
that in this case, V does not contain enough information about the

10



system to allow us to assert P , but the outer daseinisation of P at
V is the closest thing to P that we are allowed to say, given only the
information contained in V . It may be that P is just a proposition
A ∈ ∆, where A is the momentum of the system. Now, if V contains
the projection Q corresponding to B ∈ Θ, where B represents the
position of the system, and Θ is a sufficiently small interval, then
P and Q won’t commute (by the uncertainty principle) and so we
will have P /∈ V . In this case, δo(P)V might be the projection cor-
responding to the proposition A ∈ Γ, where ∆ ⊆ Γ and Γ is a large
enough interval to allow us to assert A ∈ Γ at V without violating
the uncertainty principle.

Returning to the analogy with classical physics, we think of each
such approximation to a physical proposition A ∈ ∆ at a classical
perspective V as the ‘local proposition’ to which A ∈ ∆ corresponds
at V , and we represent each such local proposition with the subset
of the corresponding local state space that makes it true.

Def 2.6.2: Let V ∈ V (H), P ∈ P(H) (where P(H) is the lattice
of all projection operators on H). Then define

Sδo(P)V = {λ ∈ ΣV |λ(δ
o(P)V ) = 1}

So Sδo(P)V is the set of the possible states of V that make V ’s
approximation to P true.

We are now ready to represent any quantum proposition as a
sub-presheaf of the spectral presheaf.

Def 2.6.3: Let P ∈ P(H). Then we define P ’s outer daseinisa-
tion presheaf δo(P) on V (H) by

Objects: Given V ∈ V (H), the component δo(P)
V

of δo(P) at

V is given by δo(P)
V

= Sδo(P)V

Arrows: Given i : V ′ ⊆ V , δo(P)
iV ′,V

: Sδo(P)V → Sδo(P)V ′

λ 7→ λ|V ′

Note that this is well defined since if λ ∈ Sδo(P)V , then λ(δ
o(P)V ) =

1. Now, since V ′ ⊆ V , we have

δo(P)V =
∧
{Q ∈ P(V )|Q ≥ P} ≤

∧
{Q ∈ P(V ′)|Q ≥ P} =

δo(P)V ′ . So δ
o(P)V ≤ δo(P)V ′ .

11



So, since λ is a homomorphism and λ(δo(P)V ) = 1, we have that
λ(δo(P)V ′ ) = 1, proving that the presheaf is well defined.

Note that we refer to the general fact that δo(P)V ≤ δo(P)V ′ for
V ′ ⊆ V as ‘coarse graining’.

Thus, in analogy with classical physics, we represent a propo-
sition as a collection of subsets of local state spaces, one for each
V ∈ V (H). Specifically, for V ∈ V (H), we choose the subset of V ’s
local state space that makes V ’s approximation to the proposition
true. The following definition will be useful,

Def 2.6.4: A sub-object (sub-presheaf) S of the spectral presheaf
Σ is a presheaf on V (H) such that (i) ∀V ∈ V (H)(SV ⊆ ΣV ) (ii)
Given i : V ′ ⊆ V , ΣiV ′,V (SV ) ⊆ SV ′

Now, it should be noted that for any P ∈ P(H) and for any
V ∈ V (H), the set Sδo(P)V ⊆ ΣV is always a clopen subset of V ’s
Gelfand spectrum ΣV (where ΣV is given the weak * topology). To-
gether with 2.6.4, this fact motivates the following definition,

Def 2.6.5: A clopen sub-object S of the spectral presheaf is a
subobject S of Σ such that ∀V ∈ V (H) SV is a clopen subset of ΣV

So, in TQT, we represent propositions as clopen sub-objects of
the spectral presheaf. Note that for each V ∈ V (H), the set cl(ΣV )
of clopen subsets of V ’s Gelfand spectrum is in bijective correspon-
dence with the set P(V ) of all projection operators in V . Thus,
any clopen subobject S of the spectral presheaf defines a ‘local
proposition’ PSV ∈ P(V ) for each classical perspective V . It is
easily seen that coarse graining will apply to these local proposi-
tions. For, if V ′ ⊆ V then ΣiV ′,V (SV ) ⊆ SV ′ . So, given λ ∈ ΣV
such that λ(PSV ) = 1, we have that λ ∈ SV and hence λ|V ′ ∈ SV ′ ,
which implies λ|V ′ (PSV ′ ) = 1, i.e. λ(PSV ′ ) = 1. This proves that
PSV ≤ PSV ′ .

So we can think of S as a kind of ‘global proposition’ whose
local components get more general as we lose information by mov-
ing to smaller classical perspectives. Note that it will not generally
hold that S = δo(P) for some P ∈ P(H). The outer daseinisation
presheaves are only a special subclass of the clopen sub-objects of
the spectral presheaf. By interpreting the set Subcl(Σ) of all clopen
sub-objects of the spectral presheaf as the set of all physical propo-
sitions that can be made about the quantum system, we are making
a significant interpretational leap in order to obtain a logical struc-
ture that is strictly richer than that of traditional quantum logic.
We consider the philosophical merits of this interpretational move

12



in detail in section 4.
We have now defined the algebra of propositions for TQT, and

are in a position to examine whether it satisfies Isham and Döring’s
stringent condition for realist physical theories, BOOLE. The fol-
lowing result is important in this respect:

Theorem 2.6.6: Define the ordering relation ≤ on Subcl(Σ) as
follows: Given S,T ∈ Subcl(Σ): S ≤ T ↔ (∀V ∈ V (H) : SV ⊆ TV ).
Then ≤ turns Subcl(Σ) into a complete, bounded distributive lat-
tice. Specifically, Subcl(Σ) is a complete Heyting algebra.

We will consider the philosophical significance of 2.6.6 in detail
later in the paper. For now, there is one more salient aspect of the
formalism of TQT that we need to be acquainted with.

Recall that in classical physics, it was always possible to assign
classical truth values to all of the physical propositions associated
with a classical system in an unproblematic way. Specifically, this
was achieved by assigning a state to the system in question. Simi-
larly, KST tells us that it is not possible, in the Hilbert space formal-
ism, to assign truth values unproblematically to all of the physical
propositions associated with a quantum system, it is generally possi-
ble, given a state vector |ψ〉, to assign truth values unproblematically
to some physical propositions. Specifically, any physical proposition
A ∈ ∆ that is represented by a projection operator P that has |ψ〉 as
one of its eigenvectors will be assigned a classical truth value by |ψ〉.
Now, in TQT, we do not yet have access to any notion of a quantum
state that can assign truth values to physical propositions (i.e. to
elements of Subcl(Σ)). But this is a problematic state of affairs. For,
surely, a basic condition that any meaningful physical theory should
fulfill is that it should have some mechanism for assigning truth val-
ues to at least some of the physical propositions with which it is
concerned. Indeed, this requirement is implied by the first criterion
of realism that was posited by Isham and Döring, PROP. So, in order
to satisfy its own philosophical motivations, TQT needs to supply a
procedure for assigning truth-values to physical propositions.

At this stage, the topos-theoretic structure of the formalism of
TQT becomes indispensable. In TQT, we represent both the quan-
tum state space and physical propositions as presheaves over the con-
text category. Generalising, we can define the category SETV (H)

op

of all presheaves over the context category.8 Now, it is well known
that SETV (H)

op

, as a presheaf category, is actually a topos. The
technical details of this fact are mainly unimportant for the pur-
poses of this article. Intuitively, what this means is that SETV (H)

op

8Morphisms in this category are just presheaf morphisms, in the standard sense. The
explicit definition will be unimportant in what follows

13



can be thought of as a kind of mathematical universe that models
all intuitionistically valid mathematics. The elements of this uni-
verse can be thought of as sets that ‘vary’ over the context category,
i.e. for each element V of the context category, they give you some
(classical, non-varying) set. In this sense , we can interpret the stan-
dard model of classical set theory as a presheaf topos. Specifically,
the classical set theoretic hierarchy can be reformalised as the topos
of presheaves over a trivial one-element poset P. So any set x is
uniquely associated with the presheaf that takes the only element
p ∈ P to the set x. Thus, SETV (H)

op

can be seen as a natural
generalisation of the set theoretic hierarchy, i.e. one in which the
sets are allowed to ‘vary’ over the context category. The main thing
that is lost in this generalisation is that we are no longer working in
a model of the whole of classical mathematics. Rather, we are work-
ing in a model of all of intuitionistic mathematics, and the logic that
holds in SETV (H)

op

is intuitionistic rather than classical.
The relevance of all this to our current problem, i.e. defining a

procedure for assigning truth values to physical propositions in TQT,
is that SETV (H)

op

has a very rich internal structure that allows us
to simulate much of the structure of the set-theoretic hierarchy. In
particular, like the set-theoretic hierarchy, SETV (H)

op

has its own
algebra of truth values. Recall that we can reformulate the set-
theoretic hierarchy as the topos of presheaves SETP

op

over a trivial
one-element poset P . Now, in classical set theory, the truth value
algebra is just the two element Boolean algebra {0, 1}. By means
of the following definition, we can find the natural generalisation of
{0, 1} to SETV (H)

op

.

Def 2.6.7: Let P be a poset, and let p ∈ P . A sieve on p is
a subset S ⊆ P such that (i) ∀q ∈ S(q ≤ p), (ii) If q ∈ S and
r ∈ P,r ≤ q, then r ∈ S, i.e. S is a collection of elements of P ‘be-
low’ p that is closed downwards under the ordering on P . We call the
maximal sieve on p, {q|q ≤ p}, the ‘principal sieve on p’, denoted ↓ p.

Now, if P is the trivial one-element poset with a sole element
p ∈ P , then there are exactly two sieves on p, i.e. the empty sieve
and the maximal sieve S = {p}. We know that the set-theoretic
universe is just the presheaf topos SETP

op

, and that the truth value
algebra for this universe is just {0, 1}, where 0, 1 are some arbitrary
sets. So, since there are only two sieves on p ∈ P , we take 0 and 1
to be the empty and maximal sieves on p, respectively. Thus, the
classical truth values of set theory can be identified with the sieves
on the sole element of the trivial one-element poset.

For our current purposes, the most salient consequence of the
fact that SETV (H)

op

is a topos is that there exists a distinguished
object (presheaf) in SETV (H)

op

, called ‘the subobject classifier’ and

14



denoted Ω, whose global elements are interpreted as the truth-values
of the mathematical universe SETV (H)

op

. Intuitively, Ω is the trans-
lation of the two-element Boolean algebra in SETV (H)

op

. The fol-
lowing standard result gives an explicit description of Ω.

Proposition 2.6.8: The subobject classifier in SETV (H)
op

is
the presheaf, Ω, defined by

Objects: Given a Context V , the set ΩV is defined to be the set
of all sieves on V .

Arrows: Given a morphism iV,V ′ : V → V ′ in V (H), the asso-
ciated function ΩV,V ′ : ΩV ′ → ΩV is defined to take a sieve S on V ′
to ΩV,V ′ (S) = {V ′′ ∈ S|V ′′ ⊆ V}.

Proposition 2.6.8 tells us that the subobject classifier in SETV (H)
op

is a direct generalisation of the two-element Boolean algebra. For,
just as the two-element Boolean algebra can be interpreted as an
assignment that takes each element of the one-element poset to the
set of all sieves on that element, the subobject classifier is just the
presheaf that takes each classical context V ∈ V (H) and returns
the set of all sieves on that context. Furthermore, just as a classical
truth value can be interpreted as a sieve on the sole element of the
one-element poset, a truth value in SETV (H)

op

is really just a way
of picking, for each V ∈ V (H), one sieve on V .

Now, one of the fundamental ideas behind TQT is that SETV (H)
op

is somehow the natural mathematical habitat for a formalisation of
quantum theory. Working on the basis of this assumption, it is
natural to conclude that when assigning truth values to physical
propositions in TQT, we should use the truth-values of SETV (H)

op

,
not the classical truth-values of set theory. Thus, our problem has
taken a new form. We are no longer attempting to assign classi-
cal truth-values to physical propositions. Rather, we are trying to
assign truth values ‘inside’ of SETV (H)

op

.
In the Hilbert space formalisation of quantum theory, any state-

vector |ψ〉 automatically assigned classical truth values to all of the
physical propositions represented by projection operators of which
|ψ〉 is an eigenvector. Similarly, in order to assign truth-values to
physical propositions in TQT, we begin by defining use state-vectors
from the relevant Hilbert space.

Def 2.6.9: The truth value of a physical proposition, repre-
sented in TQT by an element S of Subcl(Σ), relative to a pure
state |ψ〉, is defined context-wise: given a context, V ∈ V (H), we
define the truth value of S relative to |ψ〉 to be ‖S ∈ Tψ‖V =
{V ′ ⊆ V |δo(|ψ〉〈ψ|)

V ′
⊆ SV ′} = {V ′ ⊆ V |δo(|ψ〉〈ψ|)V ′ ≤ PSV ′} ⊆

15



{V ′ ⊆ V | 〈ψ|PSV ′ |ψ〉 = 1}

The truth value assigned to S at a context V by the pure-state
|ψ〉 is the collection of all sub-contexts of V whose corresponding
‘local proposition’ PSV ′ is implied by that context’s approximation
to the proposition that is normally interpreted as ‘the state of the
system is |ψ〉’. The crucial point to note at this stage is that these
truth value assignments are in bijective correspondence with the one
dimensional subspaces of the original Hilbert space H. Intuitively,
the idea is that the truth value of a clopen subobject S with respect
to a state vector |ψ〉 tells you at which classical perspectives the
proposition ‘the state of the system is |ψ〉’ implies the proposition
represented by S. At first blush, the physical significance of these
truth values seems obscure. In section 4, we will give a detailed
philosophical analysis of their meaning, and attempt to provide a
natural physical interpretation.

3 Interpreting States in TQT

We have seen that TQT is motivated largely by KST and the way it
excludes the possibility of the Hilbert space formalisation of quan-
tum theory ever satisfying Isham and Döring’s third criterion for re-
alist physical theories (the existence of a space of microstates whose
elements assign truth values to all of the relevant physical propo-
sitions in a consistent manner), STATE. We are now in a position
to assess whether or not TQT fares any better than the orthodox
Hilbert space formalism with respect to satisfying STATE.

In the literature, the spectral presheaf of a quantum system is
often described as ‘the analogue of the state space of a classical
system’ (Döring [2010]). This idea can really be thought of as the
fundamental interpretational conceit of TQT, as much of the overall
interpretation of the formalism is justified by the conception of the
spectral presheaf as the quantum state space. This all leads to the
following natural question: if the spectral presheaf plays the role of
the state space in TQT, then what plays the role of the individual
quantum states?

The first, and most intuitive, possible response to this question is
to claim that the global elements of the spectral presheaf represent
the individual quantum states. For, what is a state space other than
an object whose elements are states? However, we have already seen
that the non-existence of global elements of the spectral prehseaf
(for dimension greater than 2) is equivalent to KST. So, if we decide
to take this line and interpret the individual states of the system as
global elements of the spectral presheaf, then we will be forced to
conclude that, in general, there are no individual quantum states,

16



and that the quantum state space is, in this specific sense, empty.
At this point, one might be inclined to discount this interpre-

tational move as implausible. For, surely, whatever formal objects
we use to represent quantum states, we should always at least be
sure of their existence. However, there is a possible response to this
kind of objection. Specifically, recall that in classical physics, states
can be identified with homomorphisms between the lattice of phys-
ical propositions and the lattice of truth values. If, taking analogy
with classical physics as our guide, we characterise quantum states
in the same way, then KST can be interpreted as saying that there
are no quantum states. Then, given the interpretation of global ele-
ments of the spectral presheaf as individual quantum states, the fact
that KST is equivalent to the non-existence of these global elements
appears very natural.

Let us suppose for the moment that this response is adequate.
We are still faced with the question of whether or not this inter-
pretational move satisfies STATE. In particular, we want to know
whether a global element of the spectral presheaf will generally as-
sign truth-values to physical propositions in an unproblematic way.
But since, in general, there are no such global elements, this condi-
tion is trivially satisfied.

Philosophically, this looks uninformative. Although, technically,
this representation of individual quantum states does satisfy STATE,
the victory is a Pyhrric one. For, the notion of Realism with which
we have been working has been defined primarily in terms of the
possibility of assigning truth values to physical propositions. So, we
should surely require that our theory should contain some mecha-
nism for assigning truth values to physical propositions (indeed, this
requirement is really just Isham and Döring’s first criterion, PROP).
But any such mechanism will make no mention of global elements
of the spectral presheaf, since there are none. Following our guiding
principle of analogy with classical physics, surely we should interpret
these truth-value assignments as the individual states of the system,
contradicting the interpretation of the global elements of the spec-
tral presheaf as the individual quantum states. The very principle
that justified this interpretational manoeuvre (analogy with classical
physics) also rules it out.

In fact, in the previous section, we saw the mechanism that is gen-
erally used to assign truth values to physical propositions in TQT.
This provides us with another candidate for the formal representa-
tion of individual quantum states, i.e. the truth value assignments.
Again, this kind of move is justified by analogy with classical physics,
where states and truth value assignments can be unproblematically
identified.

The mechanism in question uses pure state vectors in the origi-
nal Hilbert space H to assign truth values to physical propositions.

17



Indeed, truth-value assignments in TQT are in bijective correspon-
dence with pure-state vectors in H. So, under this proposal, the
individual quantum states of TQT correspond to one-dimensional
subspaces of H.

This is essentially the same as the interpretation that is preva-
lent in the literature (see e.g Döring [2010]). To be precise, the
usual proposal is that we represent individual quantum states by
the daseinisation presheaves corresponding to projections onto one-
dimensional subspaces of the original Hilbert space H. So, given
some |ψ〉 ∈ H, we interpret the object δo(|ψ〉〈ψ|) as a possible state
of the system, and call it a ‘pseudo-state’. Since the daseinisation
operation is injective, this is equivalent to interpreting the one di-
mensional subspaces of the original Hilbert space as the possible
states of the system.

Although this is the dominant approach to representing individ-
ual states in TQT, it is immediately open to several possible crit-
icisms. Firstly, if the individual states of the quantum system are
really just the one-dimensional subspaces of H, then what reason
do we have to interpret the spectral presheaf as the quantum state
space? For, the one-dimensional subspaces of H do not correspond
to elements of the spectral presheaf (since there are none). Indeed,
there is no obvious relationship between the one-dimensional sub-
spaces of H and the spectral presheaf that independently justifies
the interpretation of the former as the individual states of the quan-
tum system and the latter as the state space in which they live.

This leads to a further, and much graver criticism. If, in TQT,
the individual states are represented by one-dimensional subspaces of
H, then surely H is the most natural candidate for representing the
quantum state space. But then, we’re right back where we started,
in orthodox Hilbert space quantum theory, and the whole enterprise
collapses.

The advocate of TQT might respond that they are only attempt-
ing to provide a reformulation of quantum theory, and so it should be
expected that the quantum states of TQT can be bijectively mapped
back to the one-dimensional subspaces of the relevant Hilbert space.
But this seems unsatisfactory. For, if we want our reformulation
to exactly preserve the quantum states of the original Hilbert space
formalism, then surely we should also want it to preserve the re-
lationship between those states and the state space to which they
correspond. As we have seen, this relationship is not preserved. In
the Hilbert space formalism, quantum states are elements of the state
space, in TQT they are not. This raises the question ‘why should
we interpret the spectral presheaf as the quantum state space?’.

Furthermore, there is a definite sense in which TQT is more than
just a straightforward reformulation of Hilbert space quantum the-
ory. Consider, for example, the physical propositions of TQT. As we

18



will discuss in section 5, there are strictly more physical propositions
in TQT than there are in the Hilbert space formalism. In particu-
lar, TQT posits the existence of physical propositions that cannot
be represented by projection operators. This is a clear example of
a way in which TQT does more than simply provide a new face
for the Hilbert space formalism. It introduces fundamentally new
structure that cannot be reduced to the familiar structure of the
old formalism. What’s more, this new structure is used to license
bold interpretational claims by TQT’s proponents. Returning to the
example, TQT’s extra new physical propositions are necessary for
obtaining the intuitionistic logical structure of the formalism, which
is central to the project’s philosophical vision.

In order to avoid these criticisms, one could decide to weaken
the condition STATE and require only that states correspond to
probability measures on the clopen subobjects, rather than requiring
that they correspond to truth-value assignments. Philosophically,
this looks like a strange move for the proponent of TQT to make,
since states in the original Hilbert space formalism also satisfy this
weakened form of STATE (i.e., they induce probability measures
on the lattice of projections), which means that a large part of the
motivation is lost. However, let us suppose that the advocate of
TQT has an answer to this kind of worry, and is willing to argue for
a weaker form of realism. Then, they might be inclined to propose
that individual states in TQT should be represented by probability
measures on the lattice of clopen subobjects.

Since we are in a topos-theoretic setting, and the advocate of
TQT wants everything to be defined in the topos of presheaves over
V (H), we want our probability measures to take values in an object
that lives in this topos (just as we wanted to use truth-values that
live in this topos). Towards this end, it is possible to define an ana-
logue of the closed unit interval in SETV (H)

op

as follows,

Def 3.1: The unit interval object in SETV (H)
op

is the presheaf,
[0, 1]≤, defined by

Objects: Given a Context V , the set [0, 1]≤
V

is defined to be

the set of all order reversing maps from ↓ V to [0, 1].
Arrows: Given an inclusion morphism iV,V ′ : V

′ → V in V (H),
the associated function [0, 1]≤

V,V ′
: [0, 1]≤

V
→ [0, 1]≤

V ′
is the re-

striction map that given f ∈ [0, 1]≤
V

, returns the restriction f|↓V ′
of f to ↓ V ′.

For current purposes, the technical details of this definition are
unimportant (see Döring and Isham [2012] for a full technical expla-
nation). What matters is that this object plays the role of the closed

19



unit interval in SETV (H)
op

. Accordingly, the notion of a probability
measure on the lattice of clopen subobjects of the spectral presheaf
is defined as taking values in [0, 1]≤, not [0, 1]. Generally, a proba-
bility measure is defined to be a presheaf morphism from Subcl(Σ)
to [0, 1]≤ satisfying the category-theoretic translations of the usual
probability axioms (for details see Isham and Döring [2012]). The
suggestion then is that these presheaf morphisms play the role of
individual quantum states in TQT.

However, it turns out that this approach leads directly to the
same problem that occurred when we tried to interpret truth-value
assignments on Subcl(Σ) as quantum states. For, in (Isham and
Döring [2012]), it was shown, using Gleason’s theorem, that prob-
ability measures on Subcl(Σ) are in bijective correspondence with
density matrices on the original Hilbert space H. In particular, any
such probability measure corresponds to the trace measure of some
density matrix ρ. So, the proposal that the quantum states of TQT
should be represented by probability measures is really equivalent
to the proposal that quantum states should be represented by den-
sity matrices. But then, we arrive back again at the Hilbert space
formalism, where mixed states are represented by density matrices.
Again, given this interpretation, it seems natural to claim that the
original Hilbert space H is really playing the role of the quantum
state space, not the spectral presheaf. Certainly, it seems that given
this interpretation, H is at least as good a candidate for the title of
‘quantum state space’ as the spectral presheaf. For, as we have seen,
density matrices, probability measures on the clopen subobjects of
the spectral presheaf, and probability measures on the lattice of pro-
jection operators are all really the same thing.

At any rate, one might be inclined to argue, even if the propo-
nent of this interpretation does provide an answer to this objection,
they are only providing a notion of quantum state that satisfies the
weakened version of STATE, where states correspond to probability
measures rather than truth value assignments. They have still not
shown that TQT satisfies the realist criteria that have been set out.
However, it turns out that this kind of objection is easily countered.
For, in TQT, mixed states can also be used to define truth value
assignments that generalize the truth value assignments that corre-
spond to pure states in a natural way. Indeed, in (Isham and Döring
[2012]), it was shown that any probability measure on Subcl(Σ) can
be replaced, without loss of information, by a truth value assignment
defined by the mixed state corresponding to that measure. Although
we do not have the room to explore this impressive result in detail
here, it is worth noting that it hs been used to justify a form of real-
ism about quantum probabilities in TQT. Since, the argument goes,
probability measures on Subcl(Σ) correspond to mixed states which
correspond (uniquely) to truth value assignments, we can think of

20



probability measures on Subcl(Σ) as truth value assignments. And,
furthermore, since we are realists about truth values (i.e. we think
that they represent genuine, real properties of the system that ex-
ist at all times), we can be realists about probability measures, and
treat them as genuine properties of the system. As Isham and Döring
put it,

‘this approach to probability theory allows for a new type of non-instrumentalist

interpretation that might be particularly appropriate in propensity schemes’

(Isham and Döring [2012])

Unfortunately, we cannot provide a detailed analysis of this kind
of claim here. The key fact to bear in mind is that although, at
first blush, it looks like probability measures on Subcl(Σ) will be too
weak to provide a realist notion of quantum states, it turns out that
they do actually correspond to truth value assignments, and so do
satisfy STATE. One interesting consequence of this is that TQT does
not appear to draw any kind of fundamental ontological distinction
between pure state vectors in the original Hilbert space and density
matrices acting on that Hilbert space. Both allow us to define truth
value assignments on Subcl(Σ), and so provide notions of quantum
state that satisfy the key criterion STATE.

In summary, we have considered three possible candidates for the
representation of individual quantum states in TQT. Firstly, we saw
that interpreting global elements of the spectral presheaf as quantum
states is problematic because of KST, which tells us that they do not
generally exist, and so can play no role in the assignment of probabil-
ities or truth-values to physical propositions. Secondly, we saw that
the common convention of interpreting the daseinisations of projec-
tions onto one-dimensional subspaces of the original Hilbert space is
also problematic because it reduces to interpreting pure state vec-
tors in the original Hilbert space as quantum states, which suggests
that the original Hilbert space, not the spectral presheaf, should
be interpreted as the quantum state space. Finally, we saw that
interpreting probability measures on Subcl(Σ) as quantum states is
equally problematic because it reduces to interpreting density matri-
ces as quantum states, which again suggest that the original Hilbert
space is really playing the role of the quantum state space. Gener-
ally, the problem is that TQT proposes a new formal representation
of the quantum state space, without providing a new formal rep-
resentation of individual quantum states. Conceptually, this seems
problematic because it seems to imply that the new representation
of the quantum state space is not actually capable of carrying out
the work done by its predecessor.

21



4 Interpreting Truth Values and Clopen
Subobjects in TQT

4.1 Interpreting the truth values

In section 2, we saw that physical propositions in TQT are assigned
truth values ‘inside’ of SETV (H)

op

. These truth values are a natural
generalisation of the classical truth values familiar from set theory,
obtained by extending the poset over which we define our presheaves.
However, we have not yet discussed the physical meaning of these
truth values, i.e. what does it mean to assign a physical proposition
a truth value within SETV (H)

op

? In this section, we will examine
the physical significance of these truth values and attempt to provide
them with a natural physical interpretation.

The first fact to note is that for any context V , the set ΩV of all
sieves on V is a Heyting algebra under some appropriately chosen set
theoretic operations, with maximal and minimal elements ↓ V and ∅,
respectively. Recall that this is a direct generalization of classical set
theory, where V will be the only element of a trivial poset and ΩV
will be the two-element Boolean algebra. So we can think of local
truth values as belonging to an algebraic structure that generalizes
the familiar two element Boolean algebra of classical set theory.

It turns out that we can also obtain a Heyting algebraic structure
when we consider ‘global truth values’ (global elements of Ω). For, it
is a basic fact of topos theory that the collection, Γ(Ω), of all global
elements of the subobject classifier in a topos will always form a
Heyting algebra. So, we obtain Heyting algebras of truth values at
both the local and the global levels. So on a mathematical level,
these truth values appear unproblematic. They inhabit algebraic
structures that are familiar from intuitionistic logic, and can be seen
as direct generalisations of classical truth value algebras.

The fact that these truth value algebras generalize the classi-
cal two-element Boolean algebra can also be seen to have physical
significance. In classical physics, if we consider the collection of
all physical quantities associated with a system (corresponding to
B(H)), we will not find any incompatible observables, and so we
will not have to move to the category of commutative sub-algebras
of observables. We only need one context, i.e., the collection of all
relevant physical quantities. So our poset will be trivial, and we
will only obtain classical truth values. So there is a definite sense in
which the generalisation of classical truth values in TQT is neces-
sitated by the existence of incompatible observables. If there were
no incompatible observables, we would only need one context, and
every proposition would obtain a classical truth value. In this sense,
TQT represents a mathematical generalisation of classical set theory

22



and a physical generalisation of classical physics.
Recall the explicit form of the truth value assignments in TQT,

i.e. given a clopen subobject S and a pure state vector |ψ〉, the truth
value of S relative to |ψ〉 at a context V is {V ′ ⊆ V |δo(|ψ〉〈ψ|)V ′ ≤
PSV ′}. For now, let us focus on the case in which S is δ

o(P) for some
projection P . In this case, PSV = δ

o(P)V . So, what the truth value
is telling you here is at which classical perspectives the approxima-
tion to the proposition ‘the state of the system is |ψ〉’ implies the
corresponding approximation to the proposition P . Whenever this
holds, the expectation value of the approximation of P with respect
to |ψ〉 will always be 1.

In orthodox Hilbert space quantum theory, a proposition having
expectation value 1 at a state is interpreted as meaning that the
proposition is completely true at that state. So the truth value of
a proposition δo(P) relative to a state-vector |ψ〉 at a context V
tells you about those sub-contexts of V whose approximation to P
is completely true at |ψ〉, in the sense of orthodox Hilbert space
quantum theory. Intuitively, what this kind of truth value tells you
is how far you have to generalize V ’s approximation to P before it
becomes true with respect to |ψ〉. Clearly, then, these truth values
encode a great deal of physically significant information.

Compare this with, for example, Reichenbach’s proposal of a
third ‘indeterminate’ truth value for problematic quantum propo-
sitions:

‘The starting point of Reichenbach’s considerations was Heisenberg’s princi-

ple of indeterminacy, which states that a microphysical quantity has no definite

value if a complementary quantity has already been measured. Heisenberg and

other physicists considered statements about these unmeasurable quantities as

“meaningless”. For Reichenbach, however, this term was unacceptable , since as

a logical empiricist he subscribed to the ideal of a scientific language not con-

taining any meaningless statements at all...To solve this difficulty, Reichenbach

now proposed to reserve the label “meaningless” to propositions about quanti-

ties unmeasurable in any physical situation, and to call those propositions which

might have been true or false under different circumstances “indeterminate” ’

(Kamlah [1981]).

The philosophical advantages of the truth values used in TQT
over those offered by an approach like Reichenbach’s are manifold:
Firstly, Reichenbach’s truth values are very uninformative compared
to the sieve valuations outlined above. All propositions about quan-
tities that are incompatible with observables about which we already
have information are assigned the same truth value: ‘indeterminate’.
In contrast, in TQT, these kinds of propositions will generally be
assigned varying truth values, all of which encode a great deal of

23



information about the ways in which the proposition can be approx-
imated at different contexts, and how these approximations behave.
Reichenbach’s third truth value is completely blind to the differences
between propositions of this sort, and seems to function as nothing
more than a generic label for problematic quantum propositions.
Simply labeling these propositions as ‘indeterminate’ does not teach
us anything interesting, and it only serves to mystify these proposi-
tions further. In contrast, the sieve truth values of TQT have a clear
meaning. We know exactly what it is that they are telling us, i.e.
whether it is possible to generalise (by moving to a smaller context)
an approximation to a quantum proposition in a way that makes the
generalisation true, and if so, how.

Secondly, the truth values of TQT are not arbitrarily selected.
They are a fundamental part of the mathematical framework of this
formalisation of quantum mechanics. The truth values in a topos
are always uniquely determined. As Butterfield puts it,

‘One common reason for being suspicious of notions of partial truth, and

of many valued logics, is the idea that the proposed notions are arbitrary: that

other definitions, e.g. about how to define truth functional connectives, would be

just as well motivated by the logico-semantic phenomena or intuitions appealed

to, as are the proposed definitions. But this reason does not apply here. For ...

in any topos, the collection of generalized truth-values (the sub-object classifier

Ω) is completely fixed by the structure of the topos; (and in general, there is

no other topos with a similar but different Ω that one can argue to be as well

motivated as the given one). In short, the many valued logics that arise in a

topos are not arbitrary’ (Butterfield [1999]).

Finally, we have seen that the non-classical partial truth values of
TQT arise as a natural generalization of the truth values in classical
physics and set theory. Indeed, the truth values of TQT are deter-
mined by the mathematical structure of the context category, and
hence of the Hilbert space associated with the quantum system. Dif-
ferent Hilbert spaces will give rise to different context categories and
different truth values. So the truth values of TQT bear a close rela-
tionship to the mathematical structure of the quantum formalism.
Again, Reichenbach’s ‘indeterminate’ truth value appears arbitrary
and uninformative in contrast. It bears no relationship to the for-
malisation of the properties of a quantum system, and does not arise
as a natural generalization of the truth values of classical physics.

Thus, we have seen that the truth values employed by TQT are
well motivated and intuitive from both a mathematical and a phys-
ical point of view.

24



4.2 Interpreting Subcl(Σ)

In section 2, we saw that TQT represents physical propositions by
clopen subobjects of the spectral presheaf, so that Subcl(Σ) repre-
sents the lattice of physical propositions, analogous to the lattice of
measurable subsets of the phase space in classical physics, or the
lattice of projection operators in orthodox Hilbert space quantum
theory. One of the fundamental results of TQT is that this lattice
is actually a Heyting algebra under some very natural operations,
and so the logic that corresponds naturally to the theory is intu-
itionistic. Recall that one of Isham and Döring’s realist criteria,
BOOLE, was that the logic of our theory should be Boolean. So
TQT fails to satisfy BOOLE. However, the fact that the lattice of
physical propositions is now a distributive Heyting algebra rather
than a non-distributive orthomodular lattice is commonly portrayed
as a major advantage of TQT. The idea is that intuitionistic logic
is, in some intuitive sense, ‘closer’ to classical logic than traditional
quantum logic is.

This kind of claim is difficult to evaluate, since there is no uni-
versally accepted metric for measuring the proximity of one logic
to another. Although intuitionistic negation allows us to recover
distributivity and all the corresponding structural properties that
are absent from orthomodular quantum logic, it forces us to sur-
render double negation elimination and the law of excluded middle,
which are both still present in orthomodular quantum logic, where
the negation operation behaves classically.

Certainly, though, quantum logic suffers from many well publi-
cised formal pathologies (like the absence of a deduction theorem
(see Malinowski [1990]) or a canonical implication operation), and
it is probably fair to say that it is technically not as well understood
as intuitionistic logic. It could be argued that these technical de-
ficiencies render quantum logic incapable of supporting meaningful
physical reasoning, and that this provides a justification for viewing
the intuitionistic logic of TQT as superior. Unfortunately, this kind
of claim cannot be adequately addressed here, but it looks like the
most plausible defense of the claim that the intuitionistic logic of
TQT represents genuine progress against the orthomodular quan-
tum logic of the Hilbert space formalism.

So, let us suppose for now that this kind of claim is indeed jus-
tified, and that the intuitionistic logic of TQT represents a genuine
philosophical advance. There are still major interpretational issues
that need to be answered regarding Subcl(Σ). For, recall the ar-
gument that lead to the interpretation of Subcl(Σ) as the lattice of
physical propositions. Specifically, we saw that in TQT, we can rep-
resent physical propositions by elements of Subcl(Σ). We did this
by taking a physical proposition, associating it with the appropriate

25



projection operator P , and then taking the daseinisation presheaf
δo(P) corresponding P , which is an element of Subcl(Σ). Essen-
tially, we began by identifying physical propositions with projection
operators (as we do in the Hilbert space formalism), and then de-
fined an injection of the lattice of projection operators into Subcl(Σ).
However, this injection is not surjective. In fact, the image of the
assignment that takes each projection operator P to δo(P) is only
a small subalgebra of Subcl(Σ). The vast majority of clopen sub-
objects of the spectral presheaf do not correspond to any projection
operator. Now, given that we are supposed to interpret Subcl(Σ) as
the lattice of physical propositions, it seems pertinent to ask what
kind of physical propositions these clopen subobjects are supposed
to represent. We already know that they cannot possibly represent
any proposition of the form A ∈ ∆, asserting that the value of an
observable A lies in a range ∆, since, by the spectral theorem, any
such physical proposition will correspond to a particular projection
operator P , and so will be represented by δo(P). So, the task of in-
terpreting the vast majority of the elements of Subcl(Σ) looks like a
difficult one. The elements of this algebra are supposed to represent
physical propositions, but most of them do not seem to represent
any kind of proposition that is actually used in physics.

For the purposes of illustration, let’s consider an example. First,
note that the daseinisation operation does not preserve meets . To
see this, let P ∈ P(H) and V ∈ V (H) be such that P /∈ V .
Then we know that δo(P)

V

 P and δo(1 −P)

V

 1 − P . So

δo(P)
V
∧ δo(1 −P)

V
6= 0. But of course, P ∧ (1 − P) = 0 and

δo(P ∧ (1 −P))
V

= 0. So δo(P ∧ (1 −P)) 6= δo(P) ∧ δo(1 −P).
So, let P,Q ∈ P(H) be such that δo(P) ∧ δo(Q) 6= δo(P ∧Q). In
this case, TQT can easily account for the physical significance of
δo(P ∧Q). Specifically, we interpret this object as TQT’s repre-
sentation of the physical proposition that is the conjunction of the
physical propositions that correspond to P and Q in the Hilbert
space formalism. This is unproblematic. However, the same cannot
be said for δo(P)∧δo(Q). For, this object cannot be taken to repre-
sent the conjunction of the physical propositions that correspond to
P and Q. This role is already filled by δo(P ∧Q), which is a com-
pletely distinct entity. So we need a new physical interpretation for
δo(P)∧δo(Q). But it is very difficult to see what kind of satisfactory
interpretation we can possibly give in this case. In particular, it can
be shown 9, that there cannot exist any other projection R such that

9We can argue in the following way: First, define the map ε : Subcl(Σ) → P(H) by ε(S) =∨
{P ∈ P(H)|δo(P) ≤ S} (this definition and the proof of the properties of ε can be found in

unpublished work by Döring and Cannon). ε has the important properties that (i) It preserves
meets, (ii) ε(δo(P)) = P , for any P ∈ P(H). Now, if R is such that δo(R) = δo(P) ∧ δo(Q),
then, using (i) and (ii) we obtain R = ε(δo(R)) = ε(δo(P) ∧ δo(Q)) = ε(δo(P)) ∧ε(δo(Q)) =

26



δo(P) ∧δo(Q) = δo(R). So if we are to interpret δo(P) ∧δo(Q) as a
physical proposition, it will have to be an entirely new kind of phys-
ical proposition that cannot be represented by any single projection
operator.

This looks like a major problem. For, even if intuitionistic logic
does turn out to be philosophically preferable to orthomodular quan-
tum logic, it looks like this advantage would have been bought at
the cost of introducing a whole new class of phantom propositions
that have no natural physical interpretation. Until such an inter-
pretation has been provided, the purported philosophical benefits
of the intuitionistic logic of TQT look to have been bought at an
unreasonably high cost, i.e. the introduction of a class of physical
propositions with no actual physical significance.

5 Neo-Realism

In this section, we will draw together the analyses of the preceeding
sections in order to assess the relationship of TQT to the realist
criteria imposed by Isham and Döring.

Recall that the first realist criterion, PROP, is not generally sat-
isfied by orthodox quantum mechanics, where the Kochen-Specker
theorem problematizes the notion of a ‘property’ of a quantum sys-
tem. However, in TQT, this criterion becomes less problematic.
For, we have seen that, in TQT, it is always possible to assign truth
values (in SETV (H)

op

) simultaneously to all physical propositions
associated with a quantum system without violating any algebraic
relationships holding between those propositions. In this sense, the
notion of a ‘property of the system’ is always meaningful in TQT.
For, if you give me a physical property and a state-vector (or a
density matrix), I can always give you a meaningful answer about
whether or not the system in question has that physical property.
Specifically, I can tell you the truth value of the physical proposi-
tion asserting that the system in question has that property. In this
sense, TQT satisfies PROP.

Clearly, the notion of a ‘property of the system’ being employed
here differs from the usual notion in a fundamental sense. In TQT,
we cannot simply state whether or not the system in question has
a given physical property. This is reflected in the complexity of the
truth values assigned to physical propositions. Of course, this means
that the notion of a ‘physical property’ loses much of its usual meta-
physical weight, and there will be many who are dissatisfied with the
apparent lack of any metaphysically substantial notion of a physical
property in TQT. One possible response to this concern is to argue

P ∧Q, and so δo(P ∧Q) = δo(R) = δo(P)∧δo(Q), which contradicts our original assumption
about P and Q. So there can be no such R.

27



that the notion of a physical property is actually just a by-product
of the notion of a physical proposition. Specifically, one could argue
that to attribute a physical property to a system is equivalent to as-
serting one of the physical propositions associated with that system,
and that by assigning any extra philosophical weight to the notion of
a physical property, one is really just introducing unnecessary meta-
physical baggage into their physical theories. This argument then
reduces the problem of interpreting the notion of a physical property
in TQT to the philosophically simpler problem of interpreting the
truth values of TQT, which was addressed in the previous section.
At any rate, it seems that the advocate of TQT can plausibly claim
that the theory satisfies PROP, given a metaphysically thin notion
of ‘physical property’.

Isham and Döring’s second criterion, BOOLE, was that the phys-
ical propositions of the theory should obey classical logic, i.e. should
form a Boolean algebra. This criterion is not generally satisfied by
TQT. Theorem 2.6.6 only tells us that the physical propositions in
TQT form a Heyting algebra, i.e. they obey intuitionistic logic.
Thus, TQT does not qualify as a realist theory, under the defini-
tion offered by Isham and Doering. Furthermore, we saw that, in
obtaining the Heyting algebraic structure of the lattice of physical
propositions, TQT introduced a class of purportedly physical propo-
sitions that do not have any natural physical interpretation. So, not
only does TQT fail to satisfy BOOLE, it also introduces a serious
new interpretational difficulty into the logical structure of quantum
theory.

So overall, TQT fails to satisfy Isham and Döring’s criteria for
realist physical theories. Specifically, it fails to satisfy BOOLE and,
as we saw in section three, the third criterion cannot be unproblem-
atically applied to it. However, we have seen that TQT does possess
the following properties:

(PROP) The idea of ‘a property of the system’ (i.e. ‘the value of
a physical quantity’) is meaningful, and representable in the theory.

(HEYT) Propositions about the system are handled using intu-
itionistic logic.

(TRUTH) It is always possible to assign determinate truth-values
to all of the physical propositions associated with a system in a con-
sistent way

Now, it is clear that HEYT and TRUTH represent approxima-
tions to BOOLE and STATE. The difference between TRUTH and
STATE is that TRUTH does not invoke any notion of a physical
state, which, as we have seen, is problematic in TQT. In light of the
fact that TQT satisfies these weaker forms of the criteria for realist
physical theories, it is sometimes described as a ‘neo-realist theory’.

28



We are now in a position to assess the philosophical significance
of the ‘neo-realism’ offered by TQT. One intuitive way of describing
what it means for a physical theory to be realist is to say that realist
theories tell us ‘how things are’ with respect to the systems they
describe. So one might be in inclined to ask ‘does TQT tell us how
things are with quantum systems?’. But this is not an easy question
to answer. For, there are multiple possible answers that all, at first
blush, appear to have an element of truth about them. Firstly, one
could reply that, since TQT satisfies TRUTH, it certainly does tell
us how things are with respect to the systems it describes. For, given
any physical proposition that one could want to assert of a system
described by TQT, the theory will return a definite truth value. On
the other hand, one could reply that, since TQT appears to have
significant interpretational problems concerning quantum states and
state spaces, it does not tell us ‘how things are’ with respect to the
systems it describes, since what this really amounts to is specifying
the states of the systems in question. Both of these replies are well
motivated, but by different intuitions concerning what it means for
a theory to be realist.

The advocate of TQT will likely be inclined to argue that what
really matters when we are considering whether a theory ‘tells us
how things are’ with respect to the systems it describes is whether
or not the theory satisfies TRUTH. In this respect, TQT’s realism
has something of a pragmatist flavour. It is concerned primarily
with being able to evaluate what we can say about the values of the
physical quantities associated with the systems it describes. TQT
achieves this, but at a price. For, as we have seen, TQT introduces
new interpretational issues of its own. Specifically, it introduces a
completely new kind of quantum state space without providing a cor-
responding new kind of individual quantum state. Thus, TQT does
satisfy STATE (“There is a space of microstates such that specifying
a microstate leads to unequivocal truth values for all propositions
about the system”) if we take the ‘space of microstates’ to be the
original Hilbert space of the relevant system, and not the spectral
presheaf. But this contradicts one of the basic interpretational as-
sumptions of TQT, that the spectral presheaf plays the role of the
quantum state space.

So, although TQT satisfies PROP and TRUTH, it’s unclear whether
or not it really lives up to the philosophical ideals associated with
realist scientific theories. Indeed, it could be argued that TQT even
undermines traditional realist metaphysical notions like ‘property’
and ‘state’, by focusing primarily on the more pragmatic (and tech-
nically complicated) matter of assigning topos theoretic truth values
to physical propositions. Furthermore, while it could reasonably be
argued that TQT succeeds in constructing a quantum logic that is
more amenable to realism, we have seen that this new logic is bought

29



at the price of introducing a whole new class of uninterpreted (and
apparently uninterpretable) physical propositions.

This all leads naturally to the conclusion that the distinction
between realist and non-realist physical theories is actually rather
crude and, ultimately, for the case of TQT at least, unhelpful. This
is somewhat ironic in so far a TQT is commonly advertised as a kind
of realist reformulation of quantum mechanics. But the preceding
discussion seems to show that realism is not an absolute notion. It
admits of various degrees and dimensions.

5.1 The covariant approach

What has been referred to so far as ‘topos quantum theory’ (TQT)
is actually only one of several attempts to use topos theory to gain
new insight into the mathematical and conceptual structure of quan-
tum theory (see e.g Heunen et al [2009], Wolters [2013], Adelman
and Corbett [1995]). In particular, there is one other approach,
commonly referred to in the literature as the ‘covariant approach’
(because of its use of co-variant, as opposed to contravariant func-
tors), that is very closely related both technically and conceptually
to TQT. Indeed, the covariant approach drew much of its inspiration
from TQT, and was developed over the past few years by Chris He-
unen, Bas Spitters and N.P Landsman, amongst others. Although
we cannot give any kind of detailed account of the subtleties of the
covariant approach here (for a full formal development and concep-
tual discussion, see Heunen et al [2009], and for a detailed technical
comparison of the covariant approach and TQT, see Wolters [2013]),
it is worthwhile to give some preliminary comments regarding the
applicability of the arguments offered in this paper to the covariant
approach.

In section 2, we noticed that TQT could be seen as a kind of for-
mal embodiment of the Bohrian principle of complementarity (PC).
In the covariant approach, the centrality of PC is made very explicit,
and Bohr is often cited as the primary philosophical influence of the
project. Thus, we read

‘Niels Bohr’s “doctrine of classical concepts” states that we can only look

at the quantum world through classical glasses, measurement merely providing

a ‘classical snapshot of reality’. The combination of all such snapshots should

then provide a complete picture... This doctrine has a transparent formulation

in algebraic quantum theory, to the effect that the empirical content of a quan-

tum theory described by a certain noncommutative C∗-algebra A is contained

in suitable commutative C∗-algebras associated to A.’ (Heunen et al [2009])

Conceptually, the idea is fundamentally the same as in TQT.
We formalise PC by somehow ‘breaking up’ a non-commutative al-

30



gebra of observables, corresponding to a quantum system, into its
constituent commutative parts, which represent the various classical
perspectives that one can take on the system. The fundamental tech-
nical difference is that, rather than considering the Von-Neumann
algebraic structure of the operator algebras, the covariant approach
focuses on the C∗ algebraic structure. As in TQT, topos theory is
the tool that is used to form the ‘combination of all such snapshots’
that ‘provide a complete picture’. To be precise, in the covariant ap-
proach, we form the poset C(A) of commutative C∗ subaglebras of
the non-commutative algebra A, and then define a covariant functor
from C(A) into SETS that takes each element of C(A) to its Gelfand
spectrum, and this functor is interpreted as the quantum state space.
Clearly, this functor is directly analogous to the spectral presheaf in
TQT, and so is denoted by Σ. Just as TQT works in the topos of
contravariant set-valued functors over V (H), the covariant approach
works in the topos of covariant set-valued functors over C(A). So,
in a similar way, it uses the internal logic (which is intuitionistic)
and the internal truth-values of the topos.

Now, it turns out that the relevant features of TQT that have
been discussed in this paper are mainly all reproduced in the covari-
ant approach, i.e.

(1) KST tells us that Σ has no global elements, which makes
the interpretation of the global elements of Σ as the states of the
covariant approach problematic.

(2) In the covariant approach, the lattice of physical propositions
is represented by the lattice of ‘open subobjects’ of Σ, and there
is a non-surjective injection of the lattice of projections in A into
this lattice of open subobjects, analogous to the daseinisation map.
Also, the lattice of open subobjects forms a Heyting algebra.

(3) In the covariant approach, individual states are represented
by probability measures on Σ. It was shown (see theorem 14 of He-
unen et al [2009]) that such probability measures are in bijective
correspondence with the quasi states on A, which are standardly
used to represent quantum states in the C∗ algebraic approach to
quantum theory.

It seems that properties (1)-(3) mean that the covariant approach
is subject to the same arguments that have been made about TQT
in this paper. Specifically, since the covariant approach represents
quantum states in the same way as an existing formalisation (the
C∗ algebraic formalism), which makes no reference to Σ, it seems
pertinent to ask in what sense Σ is really playing the role of a quan-
tum state space. Also, since the logic of the covariant approach is
intuitionistic, not classical, the philosophical advantages of the new
logical structure will be doubted by many. Finally, the covariant
approach is subject to the same criticism that was made of TQT in

31



section 4, i.e. it seems to introduce a new class of physical propo-
sitions that do not correspond to projection operators without sup-
plying them with a natural physical interpretation. So, in terms of
providing a realist reformulation of quantum theory, it seems that
the covariant approach is in a very similar situation to TQT.

6 Conclusion

In conclusion, we have seen that although TQT fails to satisfy the
criteria for realist theories posited by Döring and Isham, the theory
does approximate to realism in various key respects. Crucially, the
theory does possess a logical structure that allows truth-values to
be assigned to all physical propositions in an unproblematic way. In
section 4, we argued that the truth-values used in TQT are physi-
cally significant, and philosophically preferable to the kinds of truth
values that are standardly used in many-valued logical approaches
to quantum theory. However, it was also argued that the problem
of interpreting those clopen subobjects of the spectral presheaf that
are not obtained by the daseinisation of projection operators is a
pressing philosophical obstacle for the proponent of TQT, and that
before a satisfactory solution to this problem is given, the purported
philosophical benefits of TQT’s new logical structure seem shallow.

In section 3, it was argued that the identification of the spec-
tral presheaf as the quantum state space of TQT is unjustified, and
showed that there is currently no single formal representation of the
quantum state-space in TQT that has a satisfactory relationship to
the rest of the formalism (in particular, to the representations of
individual quantum states in TQT). In section 5, we saw that anal-
ogous conclusions are likely to hold with respect to the covariant
approach to quantum theory, due to the structural similarities of
that approach with TQT.

Overall, we have seen that to describe TQT as a ‘realist’ refor-
mulation of quantum theory is actually quite misleading insofar as
the theory lacks some of the strong metaphysical properties that are
usually associated with realist physical theories. However, we have
also seen that the theory overcomes some of the most problematic
‘non-realist’ properties of orthodox quantum theory in an intuitive
and elegant way, and thereby challenges the notion of what kind of
realism we should expect from our scientific theories.

Acknowledgements

Many thanks to the U.K Arts and Humanities Research Council,
whose funding supported the author during the writing of this pa-
per. Special thanks also to Richard Pettigrew and the annonymous

32



referees for their helpful and patient comments on earlier versions of
this paper. Finally, thanks to the British Society for the Philosophy
of Science, who gave the author the opportunity to present a draft
of this paper at their 2014 annual conference, and to the audience
at that conference for their insightful questions.

University of Bristol, Department of Philosophy, Cotham House,
Bristol, BS6 6JL, UK

be0367@bristol.ac.uk

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