representations like the position representation (with position eigenstates,
but no operator for momentum) or the momentum representation (with momentum
eigenstates, but no operator for position). Such non-regular representations
are unitarily inequivalent to the Schrödinger representation, and also to one
another. Thus, the uniqueness (up to isomorphism) of any local model, assigned
to a system relative to a physical situation, fails in QM. This is a
categoricity problem for a representationalist metasemantics, and I argued
that it is conceivable that this was the problem at the heart of Einstein’s
incompleteness argument.

But non-regularity poses a categoricity problem for an inferentialist
metasemantics as well. This is because, if there is no inferentialist way of
eliminating non-regular states, then the modal claims that make explicit what
is implicit in material inferences of QM would turn out to be all false! For,
if there is no inferentialist way of eliminating non-regular states, no set of
backing conditions would be sufficient to justify assigning to a system,
relative to a physical situation, a regular state in the Schrödinger
representation, rather than, say, a position state (or a momentum state) in a
non-regular representation.

One might think that this could be easily fixed by reformulating the modal
claims as follows: “Necessarily, if the backing conditions are such and such,
then the assignment of either a regular or a non-regular quantum state is
justified, and so the Born probabilities are such and such.” But in fact this
only emphasizes the problem more clearly, for the latter modal claim actually
expresses different material inferences, starting from the same set of backing
conditions, relative to the same system and physical situation:

> Backing conditions are such and such. Thus, the assignment of a regular
> quantum state is justified and the Born probabilities are such and such.
>
> Backing conditions are such and such. Thus, the assignment of a non-regular
> quantum state is justified and the Born probabilities are not such and such.

Despite starting from the same set of backing conditions, relative to the same
system and physical situation, the Born probabilities will be different
depending on whether the assigned state is regular or non-regular, as we will
see exemplified below. Which of these material inferences should an agent
choose to endorse? Which one determines the meaning of the quantum and non-
quantum expressions used in the inferences? It seems reasonable to believe
that, according to quantum inferentialism, different material inferences
determine different meanings for the same expressions. So this is a
categoricity problem for quantum inferentialism: the same QM rules of material
inference allow different meanings for the same QM expressions. The problem
shows that semantic attributes cannot be determined in exactly the way the
inferentialist claims they are determined: inferentially.

If the problem can admit of no inferentialist solution, then QM expressions
would turn out to be semantically indeterminate. But furthermore, this problem
would also have undesirable epistemological consequences for Healey’s view,
regarding the objectivity of facts described by probabilistic statements and,
thus, for his prescriptivist interpretation of Born probabilities. Before I
consider what a quantum inferentialist might do in order to address the
categoricity problem, let me spell out these metasemantic and epistemological
consequences by returning to Alice in Wigner’s friend scenario.101101101Just
in case what I have said so far may not be worrisome enough, recall the fact
mentioned in chapter 2, that the Stone-von Neumman theorem fails in QFT, which
allows an infinity of unitarily inequivalent Hilbert space representations.
Also consider related worries expressed by Abhay Ashtekar: “How can there be
an unique diffeomorphism state? Surely, quantum gravity admits an infinite
number of diffeomorphism invariant states!” (Ashtekar 2009, 1929) If it is
reasonable to believe that the non-uniqueness of quantum states allows a
plurality of material inferences starting from the same set of backing
conditions and relative to the same physical situation, then inferentialism
has a categoricity problem in these quantum theories as well.

On an inferentialist approach, Alice would follow the rules for the
introduction of quantum states, and given the experimental setup in her lab,
she would assign a quantum state to her target system. Then, following a
legitimate application of elimination rules, this state would yield
probabilities for her experimental statements $p_{a}$ and $p^{\prime}_{a}$ via
the following material inference:

> Backing conditions are such and such. Thus, the assignment of a unique
> quantum state is justified and the Born probabilities are “$Pr(p_{a})$ =
> 0.5” and “$Pr(p^{\prime}_{a}$) = 0.5”.

These two probabilistic statements are, on Healey’s view, prescriptions for
Alice’s degrees of belief. She is therefore justified to believe that $a$ = +1
and $a$ = –1 are equally likely to obtain. But suppose that Alice realizes
that, given the assumptions of the Stone-von Neumann theorem, there is a
variety of quantum states that can be assigned to her system, including non-
regular ones. Accordingly, she is considering extending the class of her
models by introducing nonregular $\ket{\psi}^{nr}$ besides the regular ones
$\ket{\psi}^{r}$. As she well knows, there is no unitary transformation
between such states. This allows her to formulate different material
inferences, say,

> Backing conditions are such and such. Thus, the assignment of
> $\ket{\psi}^{r}$ is justified and the Born probabilities are “$Pr(p_{a})$ =
> 0.5” and “$Pr(p^{\prime}_{a}$) = 0.5”.
>
> Backing conditions are such and such. Thus, the assignment of
> $\ket{\psi}^{nr}$ is justified and the Born probabilities are “$Pr(p_{a})$ =
> 0.49” and “$Pr(p^{\prime}_{a}$) = 0.51”.

If it is reasonable to believe that these different material inferences
determine different meanings for the expressions used in them, then this is a
problem for an inferentialist metasemantics of QM, since it shows that the
meaning of QM expressions cannot be determined by the material rules of QM,
just like the meaning of logical connectives cannot be determined by the
formal rules of CL.

There are two easy ways out, which we need to acknowledge and reject before we
consider more serious ones. First, the quantum inferentialist might be
inclined to ignore the categoricity problem and insist on the uniqueness of a
quantum state assignment, by assuming regularity of states from the get-go.
That is, she would take the Stone von Neumann as a fundamental result, but
ignore its problematic assumptions, since it seems safe to believe that in
practice they are unimportant in any case. Secondly, the quantum
inferentialist might want to bite the bullet of semantic indeterminacy, on the
ground that it is unlikely to have any consequences for the practice of QM.
After all, there are even logical inferentialists who are ready to do so in
the face of Carnap’s categoricity problem:

> After all, inferential rules are the only thing that matter to an
> inferentialist. If criticized for ignoring something vital –- namely that
> inferentialism is not able to render the truly logical constants –- the
> inferentialist can reply that such criticism is on a par with criticizing an
> atheist for ignoring the secrets of the Holy Trinity. … True, inferentialism
> and classical logic do not form an ideal couple, but this does not undermine
> inferentialism as such. (Peregrin 2014, 179)

One could similarly say, paraphrasing this quotation, that only the QM rules
of material inference matter to a quantum inferentialist. If criticized for
being unable to single out the genuinely quantum states – $\ket{\psi}^{r}$ or
$\ket{\psi}^{nr}$ – the quantum inferentialist can reply that such criticism
is on a par with criticizing an atheist for ignoring the secrets of the Holy
Trinity, and then admit that inferentialism and QM do not form an ideal
couple. Needless to say, however, the first easy way out is begging the very
question at stake here, while the second turns out to recommend a very big
bite: indeed, I think that Peregrin’s suggestion is tantamount to giving up
inferentialism as metasemantics, i.e., as an explanation of semantic facts.

Note that, although detrimental to quantum inferentialism, this suggestion
might not be damaging to other neo-Bohrian approaches to QM, like QBism. The
latter endorses the former’s rejection of the representational capacities of
quantum models, as well as the view that the semantic attributes of the
quantum formalism are embodied, as Bohr had emphasized, in its rules. However,
QBism regards these rules, and especially the Born rule, along with
probability theory as a whole, in a general normative sense, rather than
specifically as rules of material inference. More exactly, what is normative
here is the relation expressed by the Born rule, not the individual relata:
“The normative thrust of quantum theory is that it is a kind of glue for
probability assignments over and above the requirements of raw probability
theory. From the QBist perspective, it is this glue which indicates the
physical content of quantum theory.” (Fuchs 2023) This seems to suggest a
different metasemantics than inferentialism, one according to which the
semantic attributes of the quantum formalism are determined in virtue of the
normative properties of the relation between probability assignments –
properties that are constrained primarily by the condition of Dutch-book
coherence. In any case, a closer consideration of the metasemantic commitments
of QBism must be deferred to future work.

Let us now turn to the epistemological consequences of the categoricity
problem for Healey’s view. Since on his account Born probabilities give
prescriptions for Alice’s degrees of belief, if she can materially infer
different probabilistic statements, and they are all justified on the basis of
her application of QM, then she has an epistemological problem: what is Alice
now really justified to believe? What set of credences is she supposed to
adopt? Furthermore, recall that on Healey’s account, non-quantum magnitude
claims or experimental statements, like Alice’s $p_{a}$ and $p_{a}^{\prime}$,
have a primary representational role. This is, recall, Healey’s solution to
the no-exit problem, discussed above. What it means is that, even if they do
not do any metasemantic work, we can specify their truth conditions:

“$a$ = +1” is true iff $a$ = +1,

“$a$ = –1” is true iff $a$ = –1,

in the same way in which “The atom has decayed, the flask has shattered, and
Findus is alive” is true if and only if the atom has decayed, the flask has
shattered, and Findus is alive. Similarly, and even if their primary role is
to provide prescriptions, probabilistic claims about non-quantum magnitude
claims are also representational, though only “weakly representational”
(Healey 2017, 135), so we can specify their truth conditions as well:

“$Pr(p_{a})=0.5$” is true iff $Pr(p_{a})=0.5$,

“$Pr(p_{a})=0.49$” is true iff $Pr(p_{a})=0.49$.

Despite Healey’s emphasis on the objectivity of such probabilistic claims,
which of their truth conditions obtain appears to be a fact that depends on
Alice’s choice between $\ket{\psi}^{r}$ and $\ket{\psi}^{nr}$. More generally,
if an observer can extend her class of models as suggested, then her choice of
models yields different probabilistic claims about the same experimental
statement. This implies, against Healey’s view, that a significant category of
facts is observer-dependent after all.

What else might a quantum inferentialist say in response to the categoricity
problem? As far as I can see, there is a variety of options, some better than
others. So let me take them in turn, considering each with respect to whether
they are compatible with inferentialism.

First, one can insist on physical grounds, that for any given experimental
setup, relative to a physical situation, there is always a uniquely correct
state – a regular state – which can be assigned to a target system. Indeed,
one might wonder, are non-regular states even admissible in QM? As mentioned
above in chapter 2, they are arguably physically significant and may be taken
to give a precise form to Bohr’s complementarity. However, as we have noted
there, their elimination might be justified on the ground that the unitary
time evolution described by the Schrödinger equation does not sit well on non-
regular representations, which are dynamically mutually inaccessible
(Feintzeig et al. 2019, 2022). Nevertheless, Healey cannot justify regularity
in this way, by endorsing the elimination of non-regular states for dynamical
reasons. This is not only because his dissolution of quantum ontology indeed
blocks the inferentialist’s access to this justification. But even if one
allowed QM its own beables, this justification for the elimination of non-
regular states would not be acceptable to the inferentialist: the elimination
of non-regular states should be justified inferentially, that is decided only
on the basis of QM’s material rules of inference, and not by appealing to the
semantic features of quantum models. Furthermore, if non-regular states turn
out to be useful in QM, then any reason for eliminating them as spurious
concepts would conflict with Brandom’s inferential non-conservativity
argument, endorsed by Healey.

Secondly, one might want to restrict, on pragmatist grounds, the probability
valuations of non-quantum magnitude claims such that an agent cannot assign
different probability values to the same claim. Here, as should be obvious,
pragmatism would again conflict with inferentialism. As we have seen above, in
the context of a relativistic extension of Wigner’s friend scenario, the
inferentialist illicitly restricts the truth valuations of experimental
sentences to solve the objectivity problem raised by the elimination of
superfacts, and thereby avoid truth-relativism. But such restrictions do not
help the inferentialist against truth-relativism, since they are semantic
constraints, rather than constraints that follow from the inferential rules.
For the same reason, pragmatist restrictions on probability valuations should
not be appealed to in the present context, either.

My own suggestion of a potential solution to the categoricity problem, on
behalf of the quantum inferentialist, takes seriously the idea that the
elimination of non-regular states should be justified inferentially, that is
decided only on the basis of QM’s material rules of inference, and not by
appealing to semantic features of quantum models and dynamics or to semantic
restrictions on probability valuations of magnitude claims. The suggestion is
that the inferentialist should attempt to reformulate the Schrödinger equation
as a rule of material inference, that is more precisely as a rule for the
introduction of quantum states, instead of as a semantic constraint on the
time evolution of quantum states. Moreover, the inferentialist should take the
Schrödinger equation as a permanent or open-ended rule, i.e., as universally
valid for all mathematically possible extensions of the language of standard
QM. This, by itself, should be enough to block any assignment of non-regular
states to physical systems. It seems to me that this is the most promising way
of dissolving the problem of categoricity for quantum inferentialism and,
thus, averting the semantic indeterminacy of the standard formalism of QM due
to this problem.102102102This inferentialist strategy, which takes rules to be
open-ended in the sense explained in the text, has been applied already in the
metasemantics of classical logic and mathematics, and it is arguably a
successful strategy. More exactly, inferentialists about CL have taken
introduction and elimination rules for the logical connectives to be open-
ended, i.e., valid for all possible extensions of the language of CL, and
inferentialists about PA have taken first-order mathematical induction to be
an open-ended rule for the introduction of arithmetical concepts, i.e., a rule
valid for all possible extensions of the language of PA. For more on open-
endedness in philosophy of logic and mathematics, see e.g. Warren 2020, Murzi
and Topey 2021. Of course, an alternative strategy, for the case of QM, is to
change the standard formalism by replacing the Weyl algebra with a different
mathematical structure, as proposed in Feintzeig and Weatherall 2019,
Feintzeig 2022. Such alternatives will be explored elsewhere, as will the
complex relationship between categoricity and the permanence of rules,
suggested by a reconstruction of the latter as a precursor of open-endedness.

But this is not enough. The quantum inferentialist should adopt a similar
strategy not only with respect to introduction rules, but with respect to
elimination rules as well. The only QM rule that Healey suggests must be
understood inferentially, as an elimination rule for quantum states, is the
Born rule. But I think that the inferentialist should attempt to reformulate
decoherence as an elimination rule as well. This is because it is not
compatible with inferentialism to apply decoherence, as Healey does, as a
semantic constraint on the application of the Born rule, i.e., as a constraint
justified by the need to eliminate the magnitude claims that, in a certain
physical situation, are meaningless. Healey argues that the Born rule should
only be applied to stably decohered states, for instance, to position states
that decohere at detection, i.e, after a suitable interaction with another
system. This allows the selection of magnitude claims, e.g., about position,
that have a determinate truth value, and are thus meaningful as a guide for an
agent’s expectations. However, as already mentioned, this is unacceptable to
the inferentialist: just like the Schrödinger equation, decoherence should be
formulated as a rule of material inference, rather than taken as a semantic
constraint.

Healey, himself, suggests what might be considered as a normative approach to
what I posed as a categoricity problem:

> There are at least three respects in which … probabilistic statements … are
> objective: There is widely shared agreement on them within the scientific
> community. A norm is operative within that community requiring resolution of
> any residual disagreements. This norm is not arbitrary but derives directly
> from the scientific aims of prediction, control, and explanation of natural
> phenomena. … Flouting this norm would leave one unable to account for the
> patterns of statistical correlation among events described by true magnitude
> claims to which quantum theory is ultimately responsible. (Healey 2017, 135)

The demand that I think the quantum inferentialist should satisfy is that this
norm, whatever it is, be compatible with an inferentialist metasemantics,
i.e., with the view that the semantic attributes of all linguistic expressions
in QM are determined by QM rules of material inference. This is the ultimate
criterion for the admissibility of any solution to the categoricity problem
for quantum inferentialism.

The inferentialist might, of course, also want to insist that the very
question about the categoricity of QM is well-posed only for a rational
reconstruction of QM, i.e., a reformulation of a theory in a formal (ideally,
first-order) language, rather than for its standard Hilbert space formalism.
In other words, the inferentialist could demand that the metasemantic analysis
should start by first verifying that QM can be rationally reconstructed as a
formal system in which QM rules are expressed formally, and only then one
should verify whether the semantics of this system is unique up to the
relevant isomorphism. After all, Carnap’s metasemantic analysis of CL also
started with formal axiomatization, before its outcome would be articulated as
a categoricity problem for logical inferentialism. Since Healey rejects, as we
will see in section 6.2, the Carnapian task of a rational reconstruction of
any scientific theory, it seems plausible to believe that this should be
enough to reject the categoricity problem as a problem for standard QM. But I
will argue below, in the concluding chapter, when I will finally turn to the
problem of the viability of a rational reconstruction of QM, that Healey’s
challenge to the Carnapian task can be met on inferentialist grounds.

##  6 The logical perfection of physical theories

Whether we adopt a representationalist or an inferentialist metasemantics for
standard quantum mechanics (QM), categoricity turns out to be a serious
problem. One might be inclined to think that the ensuing semantic
indeterminacy of the standard formalism of QM is inevitable. This concluding
chapter makes two suggestions: first, categoricity, as a property apt to
characterize physical theories, should nevertheless be pursued as a
metasemantic ideal; and secondly, a viable rational reconstruction of QM, as a
Carnapian language, should not be thought impossible as a basis for a
metasemantic analysis of QM.

###  6.1 Categoricity as a metasemantic ideal

In the first chapter, I mentioned Hilbert’s distinction between two
development strategies based on the axiomatic method: a progressive strategy,
leading from the axioms of a theory to what can be derived from them by
following its own rules, and a regressive strategy, leading to deeper axioms,
uncovered by a critical examination of that theory. In the philosophy of
physics, this distinction is usually attributed to Einstein, who in his
article for The London Times (Einstein 1919), similarly distinguished between
constructive and principle theories, or rather, between progressive and
regressive strategies for developing theories.103103103For a recent account of
the history of Einstein’s distinction in his own work, see Giovanelli 2020. Of
course, the British audience of the Times would have been already familiar
with that very distinction, since Russell had advocated and extensively
discussed it in writing, as we will presently see.104104104For Russell’s
influence on Hilbert and his school, see Mancosu 2010, chapter 4. As for
Einstein, he declared: “I owe innumerable happy hours to the reading of
Russell’s works, something which I cannot say of any other contemporary
scientific writer, with the exception of Thorstein Veblen.” (Einstein 1944,
18sq) But in truth, British scientists would have been familiar with the two
strategies at least since Peacock’s 1833 Report (see above, section 3.1.1),
where he noted

> a marked distinction between those sciences which, like algebra and
> geometry, are founded upon assumed principles and definitions, and the
> physical sciences: in one case we consider those principles and definitions
> as ultimate facts, from which our investigations proceed in one direction
> only … whilst in the physical sciences there are no such ultimate facts
> which can be considered as the natural or the assumable limits of our
> investigations. It is true, indeed, that … we assume certain facts or
> principles … making them the foundation of a system of propositions … such
> assumed first principles, however vast may be the superstructure which is
> raised upon them, form only one or more links in the great chain of
> propositions, the termination and foundation of which must be for ever
> veiled in the mystery of the first cause. … The first principles, therefore,
> which form the foundation of our mathematical reasonings in the physical
> sciences … can never cease to be more or less the subject of examination and
> inquiry at any point of our researches. (Peacock 1833, 186)

On Peacock’s view, the continuous examination of the principles of a physical
theory is demanded by their very nature, since they cannot be considered as
ultimate facts. But the result of examination may be expected to take the
examiner as close as possible to principles that can be assumed as such facts.
Reconstructions of theories attempted in this vein are, thus, metaphysically
and epistemologically motivated. What I want to suggest in this chapter is
that starting with Russell’s examination of the principles of mathematics,
reconstructions came to be motivated metasemantically as well (or even
primarily so).

Let me start nevertheless by recalling Einstein’s famous distinction, before
we turn to Russell’s own, since it is Einstein’s notion of the logical
perfection of a reconstructed theory that is at the center of this kind of
motivation. Here is the most widely quoted passage from the Times article
mentioned above:

> We can distinguish various kinds of theories in physics. Most of them are
> constructive. They attempt to build up a picture of the more complex
> phenomena out of the materials of a relatively simple formal scheme from
> which they start out. … When we say that we have succeeded in understanding
> a group of natural processes, we invariably mean that a constructive theory
> has been found which covers the processes in question. Along with this most
> important class of theories there exists a second, which I will call
> ‘principle-theories.’ These employ the analytic, not the synthetic, method.
> … The advantages of … the principle theory are logical perfection and
> security of the foundations. (Einstein 1919, 228)

On Einstein’s view, both these strategies (or methods) start from (or apply
to) the same formal scheme – the same set of axioms – but they are not
exclusive or mutually inconsistent, only driven by different ideals, which
lead them accordingly in different directions. Einstein always emphasized
logical perfection as an ideal of physics: “our notions of physical reality
can never be final. We must always be ready to change these notions – that is
to say, the axiomatic basis of physics – in order to do justice to perceived
facts in the most perfect way logically.” (Einstein 1931, 266) Such passages
have always raised a nagging question: what is logical perfection? Einstein
never seems to have paused to define this notion more explicitly (Dongen 2010,
59).

My suggestion, an admittedly speculative one, is that we can get an idea of
what Einstein might have had in mind if we consider Russell’s own
reconstruction of arithmetic from logical principles. This reconstruction, we
will presently see, was primarily driven by the need to provide a unique
semantics for arithmetic, i.e., to give arithmetical expressions a precisely
determined meaning, rather than allowing an infinity of models of the
arithmetical axioms. This view transpires most explicitly in Russell’s
Introduction to Mathematical Philosophy, where he wrote:

> Mathematics is a study which, when we start from its most familiar portions,
> may be pursued in either of two opposite directions. The more familiar
> direction is constructive, towards gradually increasing complexity: from
> integers to fractions, real numbers, complex numbers; from addition and
> multiplication to differentiation and integration, and on to higher
> mathematics. The other direction, which is less familiar, proceeds, by
> analysing, to greater and greater abstractness and logical simplicity;
> instead of asking what can be defined and deduced from what is assumed to
> begin with, we ask instead what more general ideas and principles can be
> found, in terms of which what was our starting-point can be defined or
> deduced. It is the fact of pursuing this opposite direction that
> characterises mathematical philosophy as opposed to ordinary
> mathematics.105105105Cf. Russell 1919, 1. Of course, Russell had been saying
> similar things for at least a decade: “In every science, we start with a
> body of propositions of which we feel fairly sure. These are our empirical
> premises, commonly called the facts, which are generally got by observation.
> We may then ask either: What follows from these facts? or, what do these
> facts follow from?” (Russell 1907, 573) Note also that, like Russell in
> 1919, Einstein sometimes referred to logical simplicity as, presumably, a
> form of logical perfection (e.g., in Einstein 1949).

Russell compared these two pursuits with the use of different tools, “one to
take us forward to the higher mathematics, the other to take us backward to
the logical foundations” (2). After discussing Peano’s achievements with
respect to the latter pursuit, Russell famously noted “some of the reasons why
Peano’s treatment is less final than it appears to be”. (7) First and
foremost, “Peano’s three primitive ideas – namely, ‘0,’ ‘number,’ and
‘successor’ – are capable of an infinite number of different interpretations,
all of which will satisfy the five primitive propositions.” (7) In more
detail:

> Every progression verifies Peano’s five axioms. It can be proved,
> conversely, that every series which verifies Peano’s five axioms is a
> progression. Hence these five axioms may be used to define the class of
> progressions: ‘progressions’ are ‘those series which verify these five
> axioms’. … The progression need not be composed of numbers: it may be
> composed of points in space, or moments of time, or any other terms of which
> there is an infinite supply. Each different progression will give rise to a
> different interpretation of all the propositions of traditional pure
> mathematics; all these possible interpretations will be equally true. In
> Peano’s system there is nothing to enable us to distinguish between these
> different interpretations of his primitive ideas.” (8-9)106106106In his
> Principles of Mathematics, Russell had already criticized Dedekind on
> similar grounds (Russell 1903, §242), which as is well known, attracted
> Cassirer’s criticism (Cassirer 1953, 39sq).

Following Frege, Russell believed that one could do better than this, that one
should be able to derive the Peano axioms from logical principles, which then
Russell hoped would make it possible to single out the intended interpretation
of arithmetic: “our numbers should have a definite meaning, not merely …
certain formal properties.” (10) Mathematical philosophy, as he conceived of
it, is a project that aims at defining the “definite”, i.e., intended, meaning
of arithmetical terms, as opposed to the “indefinite” meaning they would have
if interpretations could be determined up to isomorphism only. Summarizing
Frege’s and his own achievements, Russell concluded:

> We have thus reduced Peano’s three primitive ideas to ideas of logic: we
> have given definitions of them which make them definite, no longer capable
> of an infinity of different meanings, as they were when they were only
> determinate to the extent of obeying Peano’s five axioms.
> (24)107107107Interestingly, Russell also suggested a more general project in
> metasemantics: “What are the possible meanings of a law expressed in terms
> of which we do not know the substantive meaning, but only the grammar and
> syntax?” (55)

Russell’s hope to single out the intended interpretation of the arithmetical
axioms, by pursuing their reconstruction from logical principles, turned out
to be illusory: one can only single out the intended interpretation of the
arithmetical axioms up to isomorphism at best (for more on this, see section
2.2.1 above). But the significant point here is that Russell’s primary
motivation for pursuing the reconstruction of arithmetic was clearly
metasemantic.

Taking this seriously, one may indeed consider categoricity as a
metatheoretical ideal capable of reinforcing a metasemantic motivation for the
reconstructionist strategy that Einstein envisaged. As mentioned already in
the introduction, I think Hilbert had shared a similar idea: “The goal is to
formulate the physical requirements so completely that the [meaning of the]
analytical apparatus is uniquely determined [up to isomorphism].” (Hilbert et
al. 1928) Furthermore, I want to suggest that categoricity can and should be
pursued, in physics, not only in logic and mathematics, as a form of logical
perfection. More recently, Boris Zilber and his collaborators have attempted
to rekindle philosophical interest in logical perfection:

> Although the expression [“logical perfection”] is quite often used
> (informally) in mathematical practice and even sometimes in more formal
> discussion around mathematics, we construe it here for the first time as an
> independent philosophical notion. Informal use of the expression often
> happens in the form of an (implicit) aesthetic criterion; it is arguably one
> of the strongest drivers of mathematical activity, as one of the main tests
> for its relevance. Since the advent of mathematical logic as an independent
> discipline, it has become possible to investigate the formal notion of
> categoricity by mathematical means. We use this notion as the main base of
> our notion of logical perfection. (Morales et al. 2021)

The notion of logical perfection that Zilber investigates is formally defined
in terms of the modern model-theoretical concept of k-categoricity, which
applies to a theory just in case, for any k, all its models of cardinality k
belong to the same isomorphism class. This is different than the standard
notion of categoricity (which is independent of cardinality constraints), but
it may turn out to be more suitable for theories like QM, given the necessity
to consider local, rather than global, models of quantum systems. Importantly,
the notion of k-categoricity is also a notion apt to characterize first-order
theories. Zilber’s general view is that one should not give up the attempt to
reconstruct physical theories as first-order theories, despite the latter’s
Löwenheim-Skolem properties: “a formal theorem like Löwenheim-Skolem could not
undermine the quest for a univocal [i.e., categorical] theory of physics.”
(lecture in Bogotá, May 13, 2020; quoted in Kennedy 2022, 82). Nevertheless,
one may still harbor doubts as to whether the standard Hilbert space formalism
can be given a first-order formalization. Such doubts may, however, be
diminished if one considers logics that are minimally distant from first-order
logic.

Take, for instance, continuous first-order logic, which is an extension of
first-order logic whose vocabulary consists of constant symbols, n-ary
function symbols and n-ary relation symbols, and which replaces the set of
possible truth values from {T, F} to the bounded interval [0, 1], and Boolean
functions as connectives by continuous functions from [0, 1]n to [0, 1] (Ben
Yaacov et al. 2008). Negation, disjunction, conjunction, and implication are
defined as follows:

$\neg a:=1-a$; $a\vee b:=min(a,b)$; $a\wedge b:=max(a,b)$; $a\rightarrow
b:=min(1-a,b)$.

First-order quantifiers $\forall x$ and $\exists x$ are replaced by the
operations $sup_{x}$ and $inf_{x}$, respectively, which are defined on the
complete linear ordering of elements in the unit interval. Continuous first-
order logic allows, for example, a formalization of C∗-algebras (Farah et al.
2021). Similarly, one can search for a formalization of Hilbert space
representations (or Rieffel’s G-modules) as continuous first-order models.
Then an appropriately formulated version of the Stone-von Neumann theorem
might be rigorously interpreted as an appropriately conceived categoricity
result for QM.

###  6.2 Imperfect knowledge of logical perfection

But have we not learned already that the 20th century project of formalizing
scientific theories has been an utter failure? Are the philosophers of science
not in agreement that the Carnapian project of rationally reconstructing
physical theories like QM is generally uninformative and perhaps incoherent as
well? Indeed, but this last section of the book will suggest otherwise. We
will see that at least two of the most widely endorsed criticisms of this
project can actually be met if the Carnapian is willing to give up
representationalist metasemantics.

Let me start by considering some of Carnap’s remarks on QM, as given in his
“Indeterminism in Quantum Physics” – the last chapter of Philosophical
Foundations of Physics: An Introduction to Philosophy of Science, where he
considered whether “a change in the form of logic used in physics” (Carnap
1966, 288) can be motivated by QM. Unlike Putnam, however, Carnap was not
ready to take lessons in logic from QM. The reasons he gave for this is as
follows: “Physicists seldom present their theories in a form that logicians
would like to see. … the postulates of the entire field of physics stated in a
systematic form that would include formal logic.” (op. cit., 291) But how did
he think this might ever be achieved? A systematic presentation of the entire
field of physics should be achieved through “the application of modern logic
and set theory, and the adoption of the axiomatic method in its modern form,
which presupposes a formalized language system.” (loc. cit.) The suggestion
here is, of course, that questions about the logic of QM cannot be properly
addressed until a rational reconstruction of the whole of physics becomes
available, which seems easy to reject as preposterous. But a watered-down
suggestion, which only requires a rational reconstruction of QM, might be more
difficult to reject. In any case, what did Carnap actually mean by rational
reconstruction?

The notion of rational reconstruction first appears in his Aufbau: “A theory
is axiomatized when all statements of the theory are arranged in the form of a
deductive system whose basis is formed by the axioms, and when all concepts of
the theory are arranged in the form of a constructional system whose basis is
formed by the fundamental concepts.” (Carnap 1928, §2)108108108For the
development of the notion of rational reconstruction, see Beaney 2013. For a
discussion focused on Carnap on rational reconstruction, see Demopoulos 2007.
For my own reading of Carnap’s reconstructivist project in the Aufbau, see
Toader 2015. Thus, a theory is rationally reconstructed if a Hilbert-style
axiomatization – a framework of concepts – is turned into a constructional
system. This requires the application of what Susan Stebbing would call
“directional analysis” to uncover the fundamental concepts of the
system.109109109Cf. Stebbing 1935, where she characterized the system of
Russell and Whitehead’s Principia Mathematica as a directional postulate
system, rather than an axiomatic system. In his “Axiomatic Thinking”, Hilbert
famously wrote: “When we assemble the facts of a definite, more-or-less
comprehensive field of knowledge, we soon notice that these facts are capable
of being ordered. This ordering always comes about with the help of a certain
framework of concepts… [which] is nothing other than the theory of the field
of knowledge.” (Hilbert 1918, §2) After 1934, a rational reconstruction is
supposed to use the tools developed in the Logical Syntax: one can then say
that a theory is rationally reconstructed if it is constructed as a Carnapian
language: a consequence relation is first defined, logical and descriptive
terms must then be distinguished, etc. As is well known, Carnap’s view on the
rational reconstruction of scientific theories evolved from the early partial
interpretation approach, to an approach based on formulating Ramsey and Carnap
sentences, and then to his mature view that deploys Hilbert’s epsilon
operator.110110110This evolution has been the subject of intense analysis and
there is no need to recall it here. For a most recent discussion, see
Demopoulos 2022.

As we have noted in the section 5.3.1, Carnap argued in 1943 that submitting a
rational reconstruction of CL – Hilbert and Bernays formalization of classical
logical reasoning – to metasemantic evaluation shows that this is not
categorical, since it lacks a unique semantics up to a isomorphism. This
entails, as we noted, that the meaning of logical connectives is not fixed by
its formal rules, since some of them turn out to be not truth-functional under
valuations that Carnap constructed. This categoricity problem led him to
propose a revision of the rational reconstruction that formalizes valid
arguments by allowing multiple conclusions. Taking the hint, one may surmise
that a rational reconstruction of QM should also be required for a
metasemantic evaluation: investigating whether this rational reconstruction is
categorical should tell us something about the metasemantics of QM. More
exactly, if it turns out that no rational reconstruction of QM is categorical,
then the formal rules do not fix the semantic attributes of quantum concepts.
Quantum inferentialism is, therefore, false. But the inferentialist can reply,
as suggested at the end of section 5.3.2, that there cannot be any viable
rational reconstruction of QM, and so this objection against quantum
inferentialism misfires.

Can there be a rational reconstruction of QM? Might standard QM already count
as such? Standard QM, the axiomatization given by von Neumann in 1932,
certainly looks like a Hilbertian framework of concepts. But is it a Carnapian
constructional system? It’s not clear that its basic concepts are fundamental:
they do not seem to be the result of any directional analysis. There is no
systematic construction of concepts, either. In contrast, recent re-
axiomatizations of QM are results of directional analysis, since they derive
standard QM from information-theoretical principles, considered to be more
fundamental than the von Neumann axioms (e.g., Hardy 2001). But they are not
formulated as Carnapian languages, and do not consider the definition of a
consequence relation as a central task. But perhaps this is not surprising
after all: maybe there is no rational reconstruction of QM because there
cannot be any. Consider Healey’s reasons for rejecting the Carnapian project:

> For Carnap, the task of the philosopher is to seek clarification of what a
> scientific theory says by means of a logical reconstruction of the theory
> within a precisely defined language. … there is now a consensus among
> philosophers of science that the labor involved in re-expressing a
> significant theory in a formal language and then giving its semantics by
> means of correspondence rules would make this neither a practicable nor a
> useful technique for revealing its structure and function. (Healey 2017,
> 123)

Rational reconstruction is said to be neither practicable, nor useful, but
note that Healey does not take it to be impossible. More importantly, note
that a rational reconstruction is understood to require representationalism:
the semantics of its language is supposed to be given by correspondence rules.
This immediately suggests that a rational reconstruction were possible, and
might be even practicable and useful, if it should not require
representationalism, but, instead, inferentialism. Healey, for one, could
definitely embrace this suggestion. However, others reject Carnap’s project
precisely because they understand it to require non-representationalism, at
least for theoretical terms (which is what Carnap already suggested in 1927):

> The non-representationalist strategy … is not new, nor is it specific to
> quantum theory. It is, rather, the central idea in the logical-positivist
> and logical-empiricist pictures of science. … It is almost universally
> accepted today that these approaches are not viable. But the predominant
> reason, historically, that they fell from grace was … the increasingly clear
> realization – notably … in the recognized failure of Carnap’s project in the
> Aufbau (1928) – that observation is theory-laden.” (Wallace 2020, 92)

Rational reconstruction is considered not viable because it is taken to assume
a sharp division between observational terms (with a representationalist
semantics) and theoretical terms (with a non-representationalist semantics).
But no such division is defensible. However, this may further suggest that a
viable rational reconstruction should not require non-representationalism only
for theoretical terms: if inferentialism were globally required, for all
terms, then no problematic division between observational and theoretical
terms would be required. Nevertheless, putting together these two suggestions,
we seem to have a circularity problem: Testing inferentialism requires
rational reconstruction, but a viable rational reconstruction requires
adopting inferentialism. Thus, testing inferentialism requires adopting
inferentialism. But, of course, there really is no circularity, for in fact we
have the following situation: testing inferentialism for the language of QM
requires a viable rational reconstruction of QM, but a viable rational
reconstruction of QM requires adopting inferentialism for the language of the
RR. In other words, testing inferentialism for the language of QM requires
adopting inferentialism for the language of the RR. No circularity.

One final obstacle that I want to consider for the very possibility of a
rational reconstruction of QM is inspired by a more general and seemingly
devastating criticism raised by Michael Potter (2000, 277):

1\. It is an experimental fact that QM makes adequate empirical claims, i.e.,
there is inductive evidence for this fact.

2\. Thus, we can know that QM makes adequate empirical claims.

3\. Suppose a rational reconstruction of QM can be given.

4\. We cannot know that QM makes adequate empirical claims unless one proves
the consistency of the rational reconstruction of QM.

5\. Due to Gödel’s second incompleteness theorem, the consistency of the
rational reconstruction of QM cannot be proved.

6\. Thus, we cannot know that QM makes adequate empirical claims.

7\. Therefore, QM cannot be rationally reconstructed.

However, I think that one can easily overcome this obstacle by making a
distinction between two types of knowledge: fallible knowledge, based on
inductive evidence (premise 2), and infallible knowledge, based on
metamathematical proof (premises 4 and 6). If we do so, then it is open to us
to say that we can learn something about QM from a rational reconstruction of
QM. Submitting it to metasemantic evaluation by investigating its categoricity
would perhaps allow us, for example, to determine the validity of quantum
inferentialism. But for the very same reason why we cannot have infallible
knowledge that QM makes adequate empirical claims, the outcome of metasemantic
evaluation cannot provide infallible knowledge that the rules of QM precisely
determine the semantic attributes of quantum concepts. Necessarily, we can
only have fallible, imperfect knowledge about the logical perfection of QM.

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