# An Alleged Tension between Non-classical Logics and Applied Classical
Mathematics

Sebastian Horvat Faculty of Physics, University of Vienna Iulian D. Toader
Institute Vienna Circle, University of Vienna

###### Abstract

Timothy Williamson has recently argued that the applicability of classical
mathematics in the natural and social sciences raises a problem for the
endorsement, in non-mathematical domains, of a wide range of non-classical
logics. We first reconstruct his argument and present its restriction to the
case of quantum logic (QL). Then we show that there is no problematic tension
between the applicability of classical mathematical models to quantum
phenomena and the endorsement of QL in the reasoning about the latter. Once we
identify the premise in Williamson’s argument that turns out to be false when
restricted to QL, we argue that the same premise fails for a wider variety of
non-classical logics. In the end, we use our discussion to draw some general
lessons concerning the relationship between applied logic and applied
mathematics.

Keywords: applied classical mathematics, quantum logic, logic of vagueness

Word count: 8985

##  1 Introduction

The recent philosophical literature on logic has seen a significant rise in
views that reject some of the epistemological and metaphysical tenets that
have traditionally been attributed to logic, such as its alleged aprioricity,
analyticity and necessity (see e.g. Martin & Hjortland, 2022, and references
therein). It has also been maintained that choosing a logical theory proceeds
along similar abductivist lines to how theories are chosen in the empirical
sciences (Priest, 2016), and more particularly, that developments in the
empirical sciences themselves may (or even should) inform our logical beliefs
(Williamson, 2017).111For a criticism of abductivism about logic, see Hlobil,
2021. A prominent defender of such a view, Timothy Williamson, has recently
put his logical-abductivist convictions in action by advancing an argument -
henceforth referred to as Williamson’s argument or, for short, WA \- to the
effect that the way mathematics is used in the natural and social sciences
provides abductive support for classical logic over some of its non-classical
competitors.

The class of logics specifically targeted by WA includes all those logics
motivated by reasons external to mathematics, as exemplified by quantum and
many-valued logics, which indeed have their origins in empirical
considerations. Logics that reject some classically-valid argument or law due
to issues internal to mathematics – like the intuitionist’s rejection of the
law of excluded middle, motivated by constructivist views about mathematics –
are not within the targeted class. To be sure, it is not a logic’s actual
origin that is of concern here, but rather its intended domain of application,
i.e. the piece of discourse whose logical structure it purports to adequately
capture. Nevertheless, together with Williamson, we will switch
interchangeably between a logic’s origin and its intended domain of
application, as the two typically coincide. It is also worth noting that WA
has nothing to say against so called pure logics, that is, unapplied
mathematical objects that carry ‘logic’ in their name only because they
resemble the mathematical structures of classical logic.

The aim of the present paper is to show that WA is unsound, to explain why we
think this is so, and to expand on what we think there is to learn from this
explanation concerning the general relationship between applied logic and
applied mathematics. Here is, in brief outline, the gist of our analysis. WA
picks up on an apparent tension between applying classical mathematics to a
certain domain (e.g. an empirical domain) and simultaneously endorsing a non-
classical logic in reasoning about that same domain. It then purports to
elevate the latter tension to an actual inconsistency, which is supposed to
corner the deviant logician into a freezing dilemma: either she is to rebuild
mathematics in her own logic, or she is forced to forgo the applicability of
mathematics to empirical domains, thereby losing (at least some of) the
explanatory power characteristic of the natural and social sciences. As this
outline already suggests, the hard work in WA lies in elevating that tension
to an inconsistency. But it is precisely at this point that we part ways with
Williamson, as we argue precisely that the alleged tension, and thus the
alleged inconsistency, does not exist.

The paper is structured as follows. Section 2 starts by carefully
reconstructing WA with the purpose of clarifying what exactly are the
aforementioned undesirable consequences threatening deviant logics, as well as
Williamson’s reasons for believing these threats are significant. Section 3
then focuses on a particular logic targeted by WA, namely on quantum logic,
and shows that at least in this case there is no threat at all, due to the
falsehood of a crucial premise of the argument. After arguing in Section 4
that the same premise fails also in the case of a logic of vagueness, we
generalize our criticism and draw some general lessons concerning the
relationship between applied logic and applied mathematics. We defer a closer
analysis of this relationship to further work.

##  2 Williamson’s argument

A typical deviant logician’s response to the fact that mathematical practice
seems to involve a constant appeal to classical-logical principles is well
expressed in the following passage, quoted by Williamson as well:
“[M]athematical practice is consistent with these reasoning steps [i.e. the
ones present in mathematical reasoning] being instances of mathematical
principles of reasoning, not generalizable to all other discourses. A
fortiori, they may very well be principles of reasoning that are permissible
for mathematics, but not for theorizing about truth.” (Hjortland 2017, 652–3)
This response suggests that just because certain reasoning principles are
validated in mathematical discourse, the deviant logician may contend that the
same principles do not need to extend to other regions of discourse, e.g.
those concerned with truth or, for that matter, mountains or quantum-
mechanical phenomena. While this is tenable as long as the mathematical and
the non-mathematical are kept mutually isolated, Williamson believes that the
applicability of mathematics to non-mathematical domains, as exemplified in
the natural and social sciences, raises a problem. Let us now briefly lay out
this problem, as Williamson understands it.

Let $\mathcal{D}$ be a set of terms denoting the non-mathematical objects that
the deviant logician takes to require a non-classical treatment (e.g. heaps,
electrons, or metalinguistic terms). Also, suppose the deviant logician wants
to keep mathematics classical. Williamson argues that this is problematic for
several reasons. First of all, the deviant logician cannot consistently
substitute elements of $\mathcal{D}$ for variables in the statements of some
classical mathematical theorems. For example, the many-valued logician cannot
apply the theorem $\forall x,y(x=y\vee x\neq y)$ to, say, mountains without
contradicting her thesis that the identity of mountains is possibly
undetermined. Furthermore, the same reason prevents the deviant logician from
appealing to an isomorphism holding between the objects denoted by
$\mathcal{D}$, on the one hand, and some classical purely mathematical
objects, on the other hand: e.g. an isomorphism between a collection of
mountains and a pure set (as defined by standard ZFC set theory). Lastly, not
only does the deviant logician encounter troubles with the application of
classical mathematical theorems, but she also faces problems with the use of
mathematical reasoning in science, for she cannot substitute elements of
$\mathcal{D}$ for variables used in classical derivations from scientific
hypotheses. Since Williamson takes such limitations to imply the impossibility
of applying classical mathematics to the non-mathematical objects denoted by
the terms in $\mathcal{D}$, and since he also takes it to be uncontroversial
that classical mathematics can be applied - and is routinely applied - to the
world outside pure mathematics, he concludes that “in a non-classical world,
pure mathematics is no safe haven for classical logic.” (2018, 21)

As a consequence, the deviant logician would be forced to rebuild mathematics
within her own logic. For example, a quantum logician cannot make sense of the
applicability of classical mathematics to quantum phenomena, unless she
recovers that mathematics in a quantum-logical framework. However, the task of
rebuilding mathematics within a new logic is notoriously difficult, and
Williamson thinks it would rather unavoidably lead to ad-hoc premises, just
like the ones that appear in Hartry Field’s recapture of the least number
principle in a logic without the law of excluded middle (Field
2008).222Actually, Williamson finds Field’s recovery strategy unacceptable for
a couple of reasons: unlike the least number principle, which is routinely
derived from mathematical induction, Field’s recovered version of that
principle is a postulate; also, that version is underivable from a suitable
schema for mathematical induction. In any case, such reasons for blocking
classical recapture are inconsequential to our purposes in this paper. This
would, in turn, make the scientific explanations that involve applications of
classical mathematics – and it is hard to find many that do not – more costly,
since they must include extra assumptions, e.g., the ad-hoc premises required
for classical recapture (2018, 19). Therefore, so Williamson concludes, even
though the path of rebuilding mathematics within a non-classical logic is the
one that a staunch deviant logician ought to take, it nevertheless leads to an
abductively implausible view – implausible, because the usual path trodden by
the classical logician offers a more elegant and less explanatorily costly
alternative.

In order to make sure that the structure of the argument is clear, let us
summarize it as follows. Again, $\mathcal{D}$ is a set of terms denoting non-
mathematical objects that are tentatively taken to require a non-classical
logic $\mathcal{L_{D}}$. Also, suppose mathematics can be (or even has been)
successfully applied to the objects referred to by $\mathcal{D}$. Then WA is
just the following dilemma:

1\. $\mathcal{L_{D}}$ can be taken either to extend to mathematics or not to
extend to mathematics.

2\. If $\mathcal{L_{D}}$ is taken not to extend to mathematics, then it is
inconsistent to hold that mathematics can be applied to the objects referred
to by $\mathcal{D}$.

3\. If $\mathcal{L_{D}}$ is taken to extend to mathematics, then most
explanations in science become more costly.

4\. Therefore, maintaining that reasoning about the objects referred to by
$\mathcal{D}$ can be adequately captured by $\mathcal{L_{D}}$ is either
inconsistent or abductively implausible.

Is WA sound? We think not.333Neil Tennant has recently defended a similar
position, by arguing however that Premise 3 fails in the case of his preferred
substructural logic (Tennant 2022). Let us also note that WA may as well be
countered by maintaining, as Quine suggested, that changing the logic changes
the meaning of the logical constants. On this understanding of deviant logics
and of how they relate to classical logic, it is clear that no tension with
classical mathematics can arise in the first place. However, Williamson is not
concerned with such a view (2018, 12), nor are we in this paper (but see
[reference removed]). As already mentioned, we will problematize the first
horn of the dilemma, i.e., more particularly, we will show that Premise 2
turns out to be false when the argument is restricted to the case of quantum
logic.

##  3 Quantum logic avoids WA

One of the logics that satisfy the presuppositions of WA, and that are thus
allegedly threatened by it, is quantum logic (QL). Williamson suggests this
much: “[S]ince quantum mechanics applies mathematics ubiquitously, those who
propose non-distributive quantum logic as a serious alternative to classical
logic are not excused from the need to reconstruct mathematics on the basis of
their quantum logic.” (2018, 20) Obviously, he is here pushing the second horn
of WA against the quantum logician, while at the same time noting in all
fairness the best attempt so far, by Michael Dunn, to classically recapture
mathematics within non-distributive quantum logic (Dunn 1980). Presumably,
Williamson assumes that the quantum logician could stand no chance against the
first horn of WA. That is, he appears to assume as established already that it
is inconsistent to maintain, on the one hand, that reasoning about quantum
phenomena can be adequately captured by QL, and to believe at the same time,
on the other hand, that classical mathematics can be applied to quantum
phenomena. But as we will now show, this alleged inconsistency is clearly
dissolved once a closer look is taken at both of these hands.

###  3.1 How mathematics is applied to quantum phenomena

It is a matter of scientific practice that, in their modelling of quantum
phenomena, physicists regularly apply classical mathematical tools from linear
algebra, functional analysis, group theory, probability theory, and other
standard mathematical theories. Before sketching what some of these
applications look like, let us say something about the internal structure of
the involved models.444In this section, for convenience, we closely follow
Richard Healey’s terminology and formulation of quantum-mechanical models
(Healey 2017). Needless to say, by doing so we are not committing ourselves to
his, or to any other, particular philosophical interpretation of quantum
mechanics. For a critical discussion of Healey’s quantum inferentialism, see
[reference removed]. A non-relativistic quantum-mechanical model, or simply a
quantum model, can be denoted by an ordered list
$\Theta\equiv\langle\mathcal{H},T,\mathcal{A},\psi,\mu_{\phi}\rangle$, where
the listed symbols stand for the following mathematical objects:

  * •

$\mathcal{H}$ denotes a potentially infinite-dimensional complex Hilbert
space, commonly taken to represent the state space of the modelled system
(analogously to the phase space that is used for the same purpose in classical
mechanics);

  * •

$T\subseteq\mathbb{R}$ is an interval on the real numbers, intended to denote
a time interval;

  * •

$\mathcal{A}$ is a set of self-adjoint operators on $\mathcal{H}$,
corresponding to the dynamical variables associated to the modelled system,
e.g. if the system under consideration is an electron, then $\mathcal{A}$ will
include operators corresponding to its position, momentum, spin, etc.
According to standard quantum mechanics, the values that can be taken by a
dynamical variable are contained in the spectrum Spec($A$) of its
corresponding operator $A$;

  * •

$\psi:T\rightarrow\mathcal{H}$ is a function that assigns a “quantum state”
$\ket{\psi_{t}}\in\mathcal{H}$ to each time instant denoted by $t$. In models
of isolated systems, i.e., systems that are taken not to interact with any
external environment, $\psi$ satisfies Schrödinger’s equation
$i\partial_{t}\ket{\psi_{t}}=H\ket{\psi_{t}}$, for some Hamiltonian operator
$H\in\mathcal{A}$;

  * •

for each unit-norm vector $\ket{\phi}\in\mathcal{H}$, $\mu_{\phi}$ is a
measure on the set of subspaces of $\mathcal{H}$. In particular, for an
arbitrary subspace $S\subseteq\mathcal{H}$:
$\mu_{\phi}(S)=\bra{\phi}P_{S}\ket{\phi}$, where $P_{S}$ is the projector on
$S$.

Thus, a quantum model is an ordered collection of classical mathematical
objects. But how, more exactly, are these objects applied in the modelling of
concrete experiments?

When a physicist is confronted with the task of constructing a quantum model
of a certain experiment of duration $T$, she will start by assigning a Hilbert
space $\mathcal{H}$, a set of self-adjoint operators $\mathcal{A}$ and an
“initial” quantum state $\ket{\psi_{t_{0}}}\in\mathcal{H}$ to the system
involved in the experiment. These assignments are of course not arbitrary, but
highly constrained by the conditions set up in the given experiment. For
example, if at the beginning of the experiment, a device is activated that is
capable of emitting a single electron of a certain energy, this warrants an
assignment of particular objects $\mathcal{H},\mathcal{A}$ and
$\ket{\psi_{t_{0}}}$ to the electron. Furthermore, in order to characterize
the dynamical evolution of the target system, the modeller assigns a
particular Hamiltonian operator $H\in\mathcal{A}$ that generates the function
$\psi$ according to the Schrödinger equation. Up to this point, the physicist
has been merely assigning mathematical objects to her experiment, according to
the rules and constraints of standard quantum theory. The important step that
brings the model closer to empirical reality is provided by the so-called Born
rule, briefly explained as follows.

Suppose that at a certain “final” time $t_{f}$, the physicist performs a
measurement of a collection of dynamical variables associated to the system of
interest (e.g. the position and the spin of an electron). Let
$\mathbf{A}\equiv\left\\{A_{1},...,A_{n}\right\\}\subset\mathcal{A}$, for some
$n\in\mathbb{N}$, be the set of self-adjoint operators that correspond to
these variables. Importantly, according to quantum theory, there is no
possible experiment that could implement a simultaneous measurement of so-
called “incompatible” variables, i.e. variables associated to mutually non-
commuting operators (e.g., a particle’s position and momentum). Consequently,
the operators in $\mathbf{A}$ obey $[A_{i},A_{j}]=0$, for all $i,j=1...,n$.
Next, for each $i$, let
$\mathcal{M}^{A_{i}}_{\Delta^{(i)}}\subseteq\mathcal{H}$ be the eigenspace
corresponding to a subset $\Delta^{(i)}\subseteq\text{Spec}(A_{i})$ of
$A_{i}$’s spectrum. The Born rule then prescribes (or predicts) the
probability $\text{Pr}^{\mathbf{A}}_{t_{f}}(\Delta^{(1)},...,\Delta^{(n)})$
for the outcomes of the measurement of the $n$ dynamical variables performed
at time $t_{f}$ to be contained respectively in subsets
$\Delta^{(1)},...,\Delta^{(n)}$:

$\text{Pr}^{\mathbf{A}}_{t_{f}}(\Delta^{(1)},...,\Delta^{(n)})=\mu_{\psi_{t_{f}}}(\mathcal{M}^{A_{1}}_{\Delta^{(1)}}...\mathcal{M}^{A_{n}}_{\Delta^{(n)}})=\bra{\psi_{t_{f}}}P^{A_{1}}_{\Delta^{(1)}}...P^{A_{n}}_{\Delta^{(n)}}\ket{\psi_{t_{f}}},$
(1)

where $P^{A_{i}}_{\Delta^{(i)}}$ is the projector on subspace
$\mathcal{M}^{A_{i}}_{\Delta^{(i)}}$, for each $i$. The Born rule thus enables
the physicist to use the abstract quantum model – more precisely, its Hilbert
space measures $\mu_{\phi}$ – to generate (classical) probability
distributions $\text{Pr}^{\mathbf{A}}_{t_{f}}$ that can then be compared to
statistical data models extracted from actual experiments.

This brief review of the application of classical mathematics to standard
quantum mechanics will be enough to drive our push back against the first horn
of WA. Before we can do so, let us run through the way QL can be used to
capture reasoning about quantum phenomena.

###  3.2 How QL captures reasoning about quantum phenomena

Despite the empirical success of quantum mechanics, i.e., the fact that
quantum-mechanical modelling of the sort just described yields unprecedented
predictive and explanatory power with respect to the microscopic world,
various problems, such as the measurement problem, Bell’s nonlocality,
contextuality, and others, leave the interpretational conundra of quantum
mechanics still far from resolved. Some realist interpretations urge us, for
instance, that the lesson we ought to take from the empirical success of
quantum mechanics is that there are invisible particles that occasionally
interact non-locally, or that our “world” is just one among the many that
partake in the Everettian “multiverse” (Lewis 2016). Such metaphysical lessons
have to a considerable extent overshadowed a series of attempts – most starkly
exemplified in (Putnam 1968) – at arguing that what quantum mechanics is
primarily teaching us is that the “one true logic”, or at least the logic we
ought to adopt in our reasoning about the microscopic world, is Birkhoff and
von Neumann’s non-distributive quantum logic (or some of its variants).555See
Birkhoff and von Neumann 1936. For more recent presentations, see e.g. Dalla
Chiara et al. 2004\. Some physicists continue to assert that “real quantum
mechanics is not so much about particles and waves as it is about the
nonclassical logical principles that govern their behavior.” (Susskind and
Friedman 2014, 236) With its violation of the distributive law motivated by
the peculiarities of quantum mechanics, QL looks like the perfect target of
WA. On behalf of the quantum logician, let us recall the details of the
quantum-mechanical considerations that led to the adoption of a deviant, non-
distributive logic.

First, we should get clearer on what we mean by ‘QL’ in this paper.
Admittedly, there are many formulations and presentations of QL, and albeit
most of them agree in their core rejection of the distributive law, some of
them yield different relations of logical consequence, which makes it more
accurate to speak of QLs, in the plural (Rédei 1998). Nevertheless, since the
subtleties surrounding these differences do not affect the point of this
paper, we will for simplicity focus on one QL and refer to it in the singular
– thereby meaning that we focus on one particular relation of logical
consequence. We will, however, construct this relation by following two
different routes, one “abstract”, the other “operationalist”, thus called for
reasons that will become clearer in a moment. Even though it would be
sufficient for our purposes to stick to only one of these routes – for,
recall, our aim is to deny the inconsistency of adopting QL in the quantum
domain while still maintaining classical mathematics as applicable therein –
we choose to present both routes, because they will lead to different ways of
refuting Premise 2 in WA, which will prove insightful for the more general
discussion at the end of the paper.

Consider once again an experiment involving a quantum system, say a particle,
and its corresponding quantum model $\Theta$. Since in what follows the
dynamical aspects of the model, i.e. those aspects that have to do with time
dependence, will be irrelevant, let us turn our attention to its “kinematical”
part, $\tilde{\Theta}\equiv\langle\mathcal{H},\mathcal{A},\mu_{\phi}\rangle$.
As stated in the previous section, each element $A$ of $\mathcal{A}$
corresponds to a dynamical variable (e.g., position or momentum) associated to
the system under consideration, and each of these variables can take values in
its operator’s spectrum Spec($A$). Now, roughly speaking, the quantum logician
urges us that there is something peculiar about the logical relations that
obtain between certain sentences that refer to the dynamical variables
associated to our system. The canonical set of sentences that the quantum
logician asks us to focus on is the set of “experimental sentences”
$\mathcal{E}$, i.e., the set that can be generated by conjoining, disjoining
and negating so-called “elementary sentences”, each of which specifies the
range of values taken by a variable associated to our quantum system. More
precisely, an elementary sentence is of the general form
“$\text{S}^{A}_{\Delta}$”, which is shorthand for “The value of the variable
corresponding to operator $A$ lies in interval $\Delta$”, with
$A\in\mathcal{A}$ and $\Delta\subseteq\text{Spec}(A)$. When the informal,
natural language semantics just given to the elements of $\mathcal{E}$ is
regimented in terms of a rigorous, formal semantics, the road bifurcates into
the aforementioned abstract and operationalist routes.666Cf. Rédei 1998, 68-74
and Malament 2002 for the abstract route; Putnam 1968, 192-197 and
Bacciagaluppi 2009, 54-55 for the operationalist one.

Let us take the abstract route first. This involves interpreting the sentences
of $\mathcal{E}$ in terms of an abstract structure borrowed from the
mathematical machinery employed in quantum models. Namely, let $h$ be a
function (an “interpretation”) that maps each sentence in $\mathcal{E}$ to a
closed subspace of the Hilbert space $\mathcal{H}$ pertaining to our
kinematical model $\tilde{\Theta}$. Thus, each elementary sentence of the form
“$\text{S}^{A}_{\Delta}$” is mapped to a closed subspace
$\mathcal{M}^{A}_{\Delta}$, such that the following rules for the QL-
connectives determine the mappings of all composite sentences:

$\begin{split}&h(a\land b)=h(a)\cap h(b)\\\ &h(a\vee b)=h(a)\sqcup h(b)\\\
&h(\lnot a)=h(a)^{\perp},\end{split}$ (2)

where $a$ and $b$ are arbitrary sentences in $\mathcal{E}$. Here the
operations ‘$\cap$’ and ‘$\sqcup$’ denote intersection and linear span,
respectively, whereas the ‘⟂’-operation outputs the subspace orthogonal to its
input. Now we are ready to introduce the quantum-logical consequence relation
as follows. For any $\Gamma\cup\left\\{a\right\\}\subset\mathcal{E}$, the
premises $\Gamma$ are said to quantum-logically imply conclusion $a$, or
$\Gamma\vDash_{QL}a$, if and only if

$\bigcap_{\gamma\in\Gamma}h(\gamma)\subseteq h(a).$ (3)

It can be easily checked that ‘$\vDash_{QL}$’ disobeys the classical-logical
consequence relation, in that there exist triples of sentences
$a,b,c\in\mathcal{E}$ for which distributivity does not hold, i.e., they are
such that $a\land(b\vee c)\centernot\vDash_{QL}(a\land b)\vee(a\land c)$.

The abstract route, just sketched, owes its name to its defining the semantics
of QL in purely mathematical terms, without any further extra-mathematical
interpretation added on top of it. Let us now take the alternative,
operationalist route. This involves an alternative semantics, which is
explicitly defined in terms of idealized operational procedures that may in
principle be carried out in a laboratory. First, let $\mathcal{T}$ be a
possibly uncountable set, whose elements are taken to denote idealized tests,
i.e. idealized operational procedures that may hypothetically be applied to
physical systems. We assume that the application of any test to any physical
system may result in only one of the two mutually exclusive outcomes: either
the system passes the test or it does not. Next, let a function
$\tau:\mathcal{E}\rightarrow\mathcal{T}$ associate one idealized test to each
experimental sentence. In particular, each elementary sentence of the form
“$\text{S}^{A}_{\Delta}$” is mapped to the test that consists in an idealized
measurement of the dynamical variable associated to $A$, such that if the test
is applied, it is passed by a system if and only if the outcome of the
measurement is contained in $\Delta$. Furthermore, tests associated to
compound sentences are determined via the following algorithm, with
$a,b\in\mathcal{E}$ henceforth being arbitrary sentences:

  * •

$\tau(\lnot a)$ is passed by a system with certainty (i.e. unit probability)
if and only if the same system would certainly fail to pass $\tau(a)$;

  * •

$\tau(a\land b)$ is passed by a system with certainty if and only if the same
system would certainly pass both $\tau(a)$ and $\tau(b)$;

  * •

$\tau(a\vee b)$ is passed by a system with certainty if only if the same
system would certainly pass either $\tau(a)$ or $\tau(b)$.

Now, let us define the relation of logical consequence on the sentences in
$\mathcal{E}$ with the use of their associated tests: heuristically, we will
say that the set of premises, $\Gamma$, QL-implies conclusion $b$, if it is
the case that any quantum system that passes each test $\tau(\gamma)$ with
certainty, for all $\gamma\in\Gamma$, also passes $\tau(b)$ with certainty. In
order to formalize this, let us introduce a set of valuations
$\mathcal{V}\equiv\left\\{V_{\phi}|\forall\phi\in\mathcal{H},\text{s.t.}\braket{\phi}{\phi}=1\right\\}$,
where $\mathcal{H}$ is the Hilbert space assigned to the quantum system under
consideration. Each $V_{\phi}:\mathcal{T}\rightarrow\left\\{t,f\right\\}$ is a
bivalent truth-valuation that assigns one of the two truth values to each
test, as follows: $V_{\phi}(g)=t$ if and only if a quantum system prepared in
quantum state $\ket{\phi}$ would pass test $g$ with certainty, where
$g\in\mathcal{T}$ is arbitrary. Whether a system passes a certain test with
certainty is in turn determined by the outcome of hypothetical applications of
quantum models to idealized tests. This means that the value of $V_{\phi}(g)$
is determined by the probability distribution generated from an appropriate
quantum model of the test $g$ performed on a system in state $\ket{\phi}$.
Without getting into more details about the relation between tests, valuations
and quantum models, once the definition of valuations is in place, the logical
consequence relation follows canonically. For any
$\Gamma\cup\left\\{b\right\\}\subset\mathcal{E}$, $\Gamma\vDash_{QL}b$ if and
only if:

$\forall
V\in\mathcal{V}:\quad\text{if}\quad\left(\forall\gamma\in\Gamma,V(\gamma)=t\right)\quad\text{then}\quad
V(b)=t.$ (4)

As it can be easily verified, the hereby constructed logical consequence
relation coincides with the one found above at the end of the abstract route
(i.e., the one given in Eq. (3)), which is the reason we denote both of them
by the same symbol ‘$\vDash_{QL}$’. Now that we have caught a glimpse, cursory
though this may be, into quantum mechanics and into QL, we are ready to get
back to WA, and in particular ready to challenge its second premise.

###  3.3 Why there is no inconsistency

When restricted to the case of QL, Premise 2 of WA can be paraphrased as: “It
is inconsistent to simultaneously hold that (i) QL adequately captures
reasoning about quantum phenomena, and that (ii) classical mathematics can be
applied to quantum phenomena.” As we will now show, this premise is false.

Recall that our quantum logician holds that the logical relations between
experimental sentences require a non-distributive logic. As noted above,
experimental sentences are obtained by logically compounding elementary
sentences, such as $\text{S}^{(X)}_{[1,2]}$, which reads “The value of an
electron’s position lies in the interval $[1,2]$.” But what does this have to
do with the possibility of applying classical mathematics to the quantum
domain? Do the quantum logicians’ claims concerning experimental sentences
have any empirical bearing that could prevent the application of classical
mathematics in quantum modelling? There is no unqualified reply to these
questions, because the informal, natural language semantics does not pinpoint
the precise content of a sentence like $\text{S}^{(X)}_{[1,2]}$. Does this
sentence specify the electron’s location, just like the statement “We are
currently in Vienna” specifies our location? Or is it rather just shorthand
for the conditional statement “If a measurement were performed, its output
would indicate the electron’s position somewhere in the interval $[1,2]$”?
Needless to say, these two possible readings do not have the same empirical
significance: the former, i.e., the statement “The value of the electron’s
position lies in the interval $[1,2]$.” may be understood as a metaphysical
claim about properties instantiated by invisible objects in the world, without
any implications on what we may observe in a performed experiment, whereas the
second reading, as a conditional statement, has a direct bearing on possible
observations in experiments that may be carried out in a laboratory.

Now, notice that the first formal semantics, which led us via an abstract
route to the quantum-logical consequence relation, involves only uninterpreted
classical mathematics, and is thus entirely compatible with any of the two
readings of experimental sentences. In other words, the quantum logicians’
experimental sentences, together with a formal semantics inspired by the
mathematical formalism of quantum mechanics, may as well have no empirical
bearing at all, thus avoiding any conflict with the applicability of classical
mathematics to quantum phenomena. So this abstract route already provides a
counterexample to Premise 2, but it is admittedly rather cheap, in that it
effectively isolates experimental sentences from (the talk about) empirical
reality. For example, it makes no connection between $\text{S}^{(X)}_{[1,2]}$
and possible outcomes of experiments performed on electrons. In contrast, the
second formal semantics, which led us via an operationalist route to the same
quantum-logical consequence relation, makes such a connection, as it equates
the meaning of experimental sentences with operations that can in principle be
performed in a laboratory. This route thus exposes the quantum logician to a
possible inconsistency with the applicability of classical mathematics.
Indeed, now that her experimental sentences are taken to have an empirical
bearing, the non-distributive logical relations between them may appear to be
in tension with the classicality of mathematics employed in quantum modelling.
Nevertheless, as we will see presently, this tension is merely apparent.

The quantum logician that walks along the operationalist route maintains that
certain sentences that concern the outcomes of operational procedures
performable in a laboratory obey a non-distributive logic. So in order to see
whether this conflicts with the applicability of classical mathematics to the
quantum domain, we need to determine what this very applicability implies with
respect to the logical relations between experimental sentences. First,
consider a generic class of quantum models
$\Theta_{\alpha}=\langle\mathcal{H},\mathcal{A},T,\psi,\mu_{\phi}\rangle$,
where the index $\alpha$ indicates that the “initial” quantum state of its
corresponding model is $\ket{\alpha}\in\mathcal{H}$, i.e. that
$\ket{\psi_{t_{0}}}=\ket{\alpha}$. With the help of the Born rule, each of
these models can be used to describe a variety of experiments, as mentioned in
Section 3.1. Secondly, consider a particular experiment
$T^{\mathbf{A}}_{\alpha}$, which consists of the measurement of a set of
dynamical variables associated to operators
$\mathbf{A}\equiv\left\\{A_{1},...,A_{n}\right\\}\subset\mathcal{A}$ on a
quantum system prepared in state $\ket{\alpha}$. Recall that according to
quantum theory, any set of operators corresponding to simultaneously
measurable variables is necessarily commutative, which implies that
$[A_{i},A_{j}]=0$, for all $i,j=1,...,n$. Lastly, let
$\text{Pr}^{\mathbf{A}}_{\alpha}$ be the probability distribution generated
from the quantum model $\Theta_{\alpha}$ in an application to
$T^{\mathbf{A}}_{\alpha}$. Now, we can show that the classicality of
$\text{Pr}^{\mathbf{A}}_{\alpha}$ implies that the logical relations among a
certain subset of experimental sentences obey classical logic.

Let $S^{(i)}_{\Delta}$ be the elementary sentence “The value of the variable
associated to $A_{i}$ lies in $\Delta$”, where $\Delta$ is an arbitrary subset
of $\text{Spec}(A_{i})$. Furthermore, let
$\mathcal{E}_{\mathbf{A}}\subset\mathcal{E}$ be the subset of experimental
sentences generated by logically compounding the elementary sentences
$S^{(i)}_{\Delta}$. Now, notice that the aforementioned experiment
$T^{\mathbf{A}}_{\alpha}$ can be used to furnish idealized tests
$\tau(S^{(i)}_{\Delta})$ for the elementary sentences $S^{(i)}_{\Delta}$.
Indeed, saying that a quantum system prepared in state
$\ket{\alpha}\in\mathcal{H}$ passes test $\tau(S^{(i)}_{\Delta})$ with
certainty is equivalent to asserting that

$\text{Pr}^{\mathbf{A}}_{\alpha}(G^{(i)}_{\Delta})=1,$ (5)

where $G^{(i)}_{\Delta}$ is shorthand for
$(\Omega_{1},...\Omega_{i-1},\Delta,\Omega_{i+1},...,\Omega_{n})$, with
$\Omega_{j}\equiv$ Spec($A_{j}$). Furthermore, the quantum logician’s
operational semantics for disjunction and conjunction implies the following
relations between the tests associated to these connectives and the
probability distribution $\text{Pr}^{\mathbf{A}}_{\alpha}$. Namely, a quantum
system prepared in state $\ket{\alpha}$

  * •

passes test $\tau(S^{(i)}_{\Delta_{i}}\land S^{(j)}_{\Delta_{j}})$ if and only
if $\text{Pr}^{\mathbf{A}}_{\alpha}(G^{(i)}_{\Delta_{i}}\cap
G^{(j)}_{\Delta_{j}})=1$,

  * •

passes test $\tau(S^{(i)}_{\Delta_{i}}\vee S^{(j)}_{\Delta_{j}})$ if and only
if $\text{Pr}^{\mathbf{A}}_{\alpha}(G^{(i)}_{\Delta_{i}}\cup
G^{(j)}_{\Delta_{j}})=1$,

for arbitrary $i,j=1,...,n$, and for $\Delta_{i}\subseteq\text{Spec}(A_{i})$
and $\Delta_{j}\subseteq\text{Spec}(A_{j})$. Note that, being a classical
probability distribution, $\text{Pr}^{\mathbf{A}}_{\alpha}$ obeys the
distributive identity

$\text{Pr}^{\mathbf{A}}_{\alpha}(G^{(i)}_{\Delta_{i}}\cap(G^{(j)}_{\Delta_{j}}\cup
G^{(k)}_{\Delta_{k}}))=\text{Pr}^{\mathbf{A}}_{\alpha}((G^{(i)}_{\Delta_{i}}\cap
G^{(j)}_{\Delta_{j}})\cup(G^{(i)}_{\Delta_{i}}\cap G^{(k)}_{\Delta_{k}})),$
(6)

which in turn, together with the operational definition of the QL-consequence
relation (4), implies the validity of the following argument

$S^{(i)}_{\Delta_{i}}\land(S^{(j)}_{\Delta_{j}}\vee
S^{(k)}_{\Delta_{k}})\vDash_{QL}(S^{(i)}_{\Delta_{i}}\land
S^{(j)}_{\Delta_{j}})\vee(S^{(i)}_{\Delta_{i}}\land S^{(k)}_{\Delta_{k}}).$
(7)

Therefore, by just carefully following her own operationalist route, the
quantum logician seems to be forced to assert the validity of the classical
law of distributivity. This is a consequence of the classicality of the
probability distributions used in modelling quantum-mechanical phenomena.

However, let there be no misunderstanding: the quantum logician is forced to
assert the validity of the classical law of distributivity only for the subset
of sentences $\mathcal{E}_{\mathbf{A}}$ generated by logically compounding the
elementary sentences $S^{(i)}_{\Delta}$, which refer exclusively to dynamical
variables associated to elements in the set of commutative operators
$\mathbf{A}\subset\mathcal{A}$. Does this present a problem for the quantum
logician?

Definitely not! It is perfectly consistent to maintain that the set
$\mathcal{E}$ of all experimental sentences requires a non-classical logic,
while admitting that there are subsets like $\mathcal{E}_{\mathbf{A}}$, which
nonetheless validate classical logic. Importantly, it is no accident that we
started our construction from a set of commutative operators: had we
considered a non-commutative set, such as
$\left\\{X,P\right\\}\subset\mathcal{A}$, where $X$ and $P$ are operators
corresponding to, say, a particle’s position and momentum, the construction
would have been blocked from the get-go. This is because there is no
(idealized) experiment, adequately modelled by a joint classical probability
distribution, that could simultaneously measure variables whose corresponding
operators are mutually non-commuting. Briefly put, since any application of a
quantum model can only be used to generate probability distributions over
values of quantum-mechanically compatible variables, it follows that the
classicality of such distributions can force only some subsets of experimental
sentences $\mathcal{E}_{\mathbf{A}}\subset\mathcal{E}$, for any set of
commutative operators $\mathbf{A}$, to obey classical logic. This is the case
indeed according to QL. Therefore, the alleged tension between QL and the
applicability of classical mathematics in quantum mechanics does not exist.

But the fact that this tension does not exist is not surprising at all.777That
QL does not conflict with the classicality of quantum-mechanical models has
already been suggested e.g. by Dickson (2001, S283), as well as by Putnam
(1968), Bacciagaluppi (2009), and others. Nevertheless, we found it
worthwhile, for the sake of our counterargument, to make the lack of this
conflict, as well as the reasons that underlie such a lack, more explicit. To
insist that it does, as WA has suggested, is actually a curiosity of sorts.
Having explained why neither the abstract, nor the operationalist quantum
logician is threatened by WA, we should recall that the operationalist’s
quantum-logical consequence relation is itself constructed on the basis of
quantum theory and its mathematical formalism. The logical relations between
experimental sentences, as explained in Section 3.2, are related to properties
of tests performable in principle on quantum systems. But these tests and
their properties are not freely posited by a speculative logician: they are
instead explicitly defined in terms of quantum models! This is why we find it
rather odd that an argument such as WA has been levelled against QL, for the
latter has been explicitly motivated by, and formally constructed on the basis
of, the mathematical formalism of quantum theory. That WA is unsound in the
case of QL is, thus, not an accident.

QL was but one target. As noted at the outset, there are others. Since the
motivation and construction of many alternative logics are entirely
independent of mathematical models employed in science, one might feel
inclined to think that these fall prey more readily to WA. Many-valued logics,
for example, have been thought to adequately capture reasoning about vague
objects and phenomena, and they can certainly be constructed independently of
the scientific models of these objects and phenomena. Nonetheless, our
analysis will show that, when WA is restricted to this case, Premise 2 fails
once more.

##  4 Deviant logics of vagueness also avoid WA

Vagueness and its related paradoxes constitute a paramount motivator for non-
classical logics. Indeed, it is often maintained that reasoning about heaps,
mountains and baldness requires the rejection of some classical-logically
valid principle or rule, such as the law of excluded middle or the Cut rule.
Nevertheless, those who endorse deviant logics of vagueness contend that the
latter’s non-classicality does not infect the realm of mathematics:
“Presumably excluded middle holds throughout ordinary mathematics, and indeed
whenever vagueness or indeterminacy is not at issue.” (Field 2008, 101). In
this case, however, WA allows the contention only if the language of
mathematics could be kept isolated from the non-mathematical vocabulary
infected by vagueness. But then it further points out that the mathematical
and the vague are actually not two independent realms, as exemplified in
(natural and social) scientific applications of mathematics to non-
mathematical domains, which are hardly ever immune to vagueness and borderline
cases.888That empirical domains are infected with vagueness can be illustrated
by questions like: What is the maximum distance between a proton and an
electron such that they still constitute a hydrogen atom? If a series of
tumultuous political events count as a revolution, at what precise moment in
time, counted in milliseconds, did it begin? WA thus concludes that the
deviant logician of vagueness ought to reconstruct mathematics in her non-
classical logic, if she wishes not to lose the explanatory power of
scientific-mathematical models.

Williamson acknowledges that the deviant logician may attempt to flee the
dilemma “on the grounds that when applying mathematics in science one can
pretend that the relevant non-mathematical terms are precise, because the gap
between this pretence and reality is irrelevant for scientific purposes.”
(2018, 20) However, this acknowledgement is immediately followed by criticism,
according to which “[i]f there is vagueness in the relevant terms, and the
true mathematics for vague terms is non-classical (because the correct logic
for them is), then one would expect “vagueness errors” in the predictions.”
(loc. cit.) Williamson is, thus, claiming that the “gap between pretence and
reality” is quantifiable and can lead to systematic errors due to the
misalignment between the classical mathematics used in a scientific model and
the non-classical mathematics purportedly inherent to the modelled target
objects or phenomena (i.e. their purported “true mathematics”). In what
follows, we will present a counterexample to Williamson’s claims, by
considering a scientific application of classical mathematics to an object
apparently infected with vagueness, and will explicitly show that reasoning
about the latter in accordance with a deviant logic is perfectly consistent
with the use of classical logic in the mathematical model of that object. More
specifically, we examine a classical mathematical model, based on Newtonian
mechanics, of a planet’s movement.

Consider a team of scientists and engineers, whose aim is to land a space-
probe on a certain planet $X$. Suppose that their strategy for executing the
task is guided by the use of a classical mathematical model constructed on the
basis of Newton’s laws, which is intended to represent the motion of the
planet and its surroundings. Without getting into unnecessary details of what
Newtonian models may generally look like, suppose that our team’s model
consists simply of a “mass-density distribution” function
$\rho:\mathbb{R}^{3}\times\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$, which
satisfies a set of differential equations developed by our team (roughly, by
means of a rather sophisticated application of Newton’s second law). The
interpretation of the model is the following. For each location denoted by a
vector $\vec{x}\in\mathbb{R}^{3}$ and each time instant denoted by
$t\in\mathbb{R}^{+}$, the value $\rho(\vec{x},t)\in\mathbb{R}^{+}$ denotes the
density of mass present at the respective location and time instant. The total
quantity of mass present in a certain volume $V\subset\mathbb{R}^{3}$ at a
time instant $t$ is, thus, given by the integral
$\int_{V}\rho(\vec{x},t)d^{3}x$. Besides representing the matter that
constitutes planet $X$ and its immediate surroundings, the distribution
function $\rho$ may also describe the matter pertaining to other bodies - e.g.
other planets, stars or dust - that the team takes to be relevant for their
task. Suppose that our team also specifies precisely which pieces of matter
pertain to planet $X$ and which do not, thereby modelling the planet as if it
had clearly defined spatial boundaries. Mathematically, this assumption states
that for each time instant $t$, there is a subset
$V^{X}_{t}\subset\mathbb{R}^{3}$, such that the function $\rho(\vec{x},t)$,
restricted to the points $\vec{x}\in V^{X}_{t}$ represents the mass density of
the planet at time instant $t$.

However, as already mentioned above, the term ‘planet’ seems to be
paradigmatically vague, in that it appears that it either fails to refer to a
unique material object, or that its referent lacks sharp boundaries. This
purported (semantic or ontological) vagueness spreads to statements in which
the term ‘planet’ figures, such as $S(X,y)\equiv$ “Material object $y$ is part
of planet $X$”: if $y$ is a rock trapped a few kilometres under $X$’s surface,
then $S(X,y)$ is undeniably true, but if $y$ consists in cosmic dust that has
just landed on $X$’s surface, then $S(X,y)$ seems to be neither true nor
false, thus qualifying as the deviant logician’s potential counterexample to
the classical law of excluded middle or to the principle of bivalence. The
deviant logician, on the one hand, seemingly portrays the mathematical
activities that our scientists and engineers partake in as a game of pretense,
in which the team is representing planet $X$ as if the latter had clear
boundaries, when it in fact does not. Williamson, on the other hand, denies
that our team is playing any games, and holds that $X$ clearly has sharp
boundaries, since the lack of sharpness would otherwise manifest itself
empirically: our team would then certainly notice the vagueness of the term
‘planet’, or they at least should do so, if their execution of the task is to
be as successful as possible. Let us now show why we think Williamson’s take
is incorrect, and more generally, why the deviant logician does not run into
any inconsistencies in cases like the one just described.

As sketched above, our team’s model consists of a mathematical function $\rho$
that is taken to represent the matter that constitutes the planet, its
surroundings, and potentially other celestial bodies that may be of relevance.
Now, this model may be employed, for example, to determine whether some matter
$y$ will be present in some volume $V^{\prime}$ at time $t^{\prime}$, which
may indeed prove to be useful information if our team is to successfully
coordinate the landing of its space-probe (our team might want to minimize
contact with additional matter - e.g. with dust - as the latter could ruin the
space-probe’s equipment). Importantly, notice that it is entirely irrelevant
for our team’s purposes whether $y$ is part of the material body commonly
referred to as ‘planet $X$’, ‘cosmic dust $Y$’, ‘star $Z$’, etc., or whether
there is any such fact of the matter at all. Rather, what is important for our
probe landing is how much matter is roughly present within the space-time
region that is of interest. More generally, the empirical adequacy of model
$\rho$ is entirely independent of whether some of its elements are associated
to a seemingly vague term, such as ‘planet’, ‘dust’, or ‘probe’, and of
whether any such association is to be considered as absolutely (in)correct –
as Williamson would have it – or as a pragmatically justified output of a game
of pretension - as we may put it on behalf of the deviant logician. In other
words, empirical adequacy is independent of whether statements such as
$S(X,y)$ are true, false, both or neither.

Here is another way of making this point. To see that any views concerning the
truth-value (or lack thereof) of statements such as $S(X,y)$ are
inconsequential for the applicability of our model $\rho$ is to recognize the
conceptual possibility of adopting an instrumentalist stance, which conceives
of the mathematical model as a mere device for generating predictions about
data extracted from observations or experiments performed on the modelled
celestial bodies. According to this stance, the classicality of the
mathematical model can, if anything, only force the extracted data to be
classical, but has no say on what logic is adequate for reasoning, in natural
language, about the sources that produced the data. In particular, the
classicality of the mathematical model cannot impose classical-logical
relations onto statements like “This cloud of cosmic dust is part of planet
X”. Even though such a view of scientific models may be explanatorily
unsatisfactory, it is definitely not an inconsistent one, as WA would have it.
Premise 2 thus fails once again: there is no inconsistency between (i) holding
that reasoning about planets, their material constituents, their color and
size, as they appear in our “manifest image”, is adequately captured by a
deviant logic, and (ii) maintaining that classical mathematics can be used in
modelling the dynamics of the matter that constitutes planets and their
surroundings. Let us now conclude by briefly indicating what we take to be a
deeper problem with WA, which we think might explain the two symptomatic
failures of Premise 2 so far discussed.

##  5 Conclusion: A deeper problem with WA?

We have argued that, whether the motivation and construction of a deviant
logic is dependent on mathematical models actually employed in science (as in
the case of non-distributive quantum logic) or not (as in the case of non-
classical logics of vagueness), Premise 2 in WA fails. This would appear to
suggest that there is some deeper problem with Williamson’s attack on deviant
logics, which comes to surface in both cases. But what could that be?

Although classical logic (in the context of one’s mathematical model) and the
deviant logics (in the context of one’s communications outside of the
mathematical model) are both deployed in reasoning about one and the same
system, closer analysis showed that their targets may effectively differ. More
specifically, we have seen that, in quantum mechanics, a classical
mathematical model accounts for quantities that arise in a particular
experimental context, whereas quantum-logical reasoning concerns relations
between experimental propositions, which refer to quantities that arise in
potentially different experimental contexts. Similarly, the classical
mathematical model used by our team in landing their space-probe accounts for
quantifiable aspects of the material constituents of the modelled planet and
of its surroundings, whereas ordinary natural-language discourse about that
same planet disregards such aspects.

This suggests the following general diagnosis. Let T be one’s target system or
phenomenon, and M be a classical-logical mathematical model applicable to T.
The diagnosis is that there are many cases in which, even though M may be used
to account for certain aspects of T, there may be other aspects of T that are
articulable via some discourse D whose structure is adequately captured by a
non-classical logic. In other words, the gulf between M and T is too wide for
the mere fact that M is applied successfully to T to significantly restrict
the logical structure of any meaningful assertoric discourse D about that same
object of investigation T. Such a restriction may incidentally arise only with
further assumptions about what the success of a mathematical model implies
about the modelled object or phenomenon.

For instance, an ontic structural realist, as the one portrayed by Ladyman et
al. (2007), may maintain that at least in some cases the successful
application of a mathematical model to T implies that T itself is “already
structured” in some particular way, which may in turn restrict the logical
structure of other kinds of discourse about T. Conversely, other accounts of
the use of mathematics in science, such as the one articulated by Bas van
Fraassen (2010), may sever such restrictions. Indeed, while on van Fraassen’s
view, a successful mathematical model is taken to be isomorphic (or rather
homomorphic) to the data model generated from a target system (or more
precisely, to the surface model generated from a data model), the relation
between the data model and the target system is in turn determined by
“pragmatic factors” – there need be no kind of morphism between them. This is
another way of saying that “assertions of isomorphism are context-sensitive”
(op. cit., 382), where the different contexts that such assertions are
sensitive to allow for different data models to be constructed from one and
the same target system. If this is correct, then the logic of the data model
need not be the same as the logic used in reasoning about other aspects of the
target system. So it is not really that surprising, if van Fraasen’s view is
adopted, that one can apply M to T, while holding onto the conviction that
reasoning about other aspects of T is adequately captured with a non-classical
logic. Or, with reference to our own examples, it is not that surprising that
one can apply classical-logical quantum-mechanical models to quantum systems,
while holding onto the conviction that drawing inferences from experimental
sentences requires a non-classical logic, just as one can apply classical-
logical Newtonian models to planetary movements, while drawing non-classical
inferences from statements about qualitative aspects of such movements.

The upshot is that, whether one takes the ontic structuralist realist’s, van
Fraassen’s, or some other position, any attempt of characterizing the
relationship between applied logic and applied mathematics, such as the one
embodied in $\mathbf{WA}$, ought to make explicit its background theoretical
account of applications of mathematics in science. Without moving this
background into the foreground, there can be no significant tension arising
between applied logic and applied mathematics, let alone an inconsistency.

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