realism.tex

\chapter{Metalogic in Analytic Philosophy} \label{chap-realism}

%% TO DO: people like Psillos complain that it is no longer clear what
%% the scientific realism debate is even about.  Yes, I agree with
%% this.  But we can now see that it has a lot to do with how finely
%% one draws the net of "equivalence".  

Much of 20th century analytic philosophy was concerned --- when not
explicitly, then implicitly --- with theories, and with the
relationships between them.  For example, is every spacetime theory
equivalent to one with Euclidean background geometry?  Or, is folk
psychology reducible to neuroscience?  Or, can there be a good reason
to choose a theory over an empirically equivalent rival theory?

But what is a theory?  And what does it mean to say that two theories
are equivalent, or that one theory is reducible to another?  Carnap
had the audacious idea that philosophy can follow mathematics' method
of explication: to take an intuitive notion, and to find a nearby
neighbor in the realm of precisely defined mathematical concepts.  In
this book, we've tried to follow Carnap's lead; and indeed, we've
hoped that we could do a bit better than Carnap, because mathematics
has come a long way in the past 75 years.  We now have mathematical
concepts --- such as adjoint functors, or categorical equivalence ---
the likes of which Carnap could not even have dreamed about.

In this book, we've attempted to explicate the concept of a theory, as
well as some of the relations between theories that scientists and
philosophers find it useful to discuss.  With these explications in
the background, we can now return to some of the big questions of
philosophy of science, such as, ``what is the proper attitude to take
toward a successful scientific theory?''



\section{Ramsey sentences} \label{ramsey}

%% van fraassen = Beth shows that there is no mysterious notion of
%% implicit definition

%% Lewis on theoretical terms

%% Functionalism (Bealer)

%% Dewar on Ramsey equivalence

No analytic philosopher's education is complete until she learns the
magic of the Ramsey sentence.  The idea was proposed by Frank
\cite{ramsey}, and was reinvented by Carnap in the 1950s --- or, more
accurately, Carnap forgot that he learned about it from Herbert
Bohnert \cite[see][]{psillos2000}.  Most contemporary philosophers
know of the idea because David \cite{lewis-terms} argued that it
solves the \emph{problem of theoretical terms}.  In the years since
Lewis' seminal paper, Ramsey sentences have become a sort of {\it deus
  ex machina} of analytic philosophy.

Let's start with a simple example.  Suppose that $P$ is a theoretical
predicate, and that $O$ is an observational predicate.  (Or in Lewis'
preferred terminology, $O$ is antecedently understood vocabulary, and
$P$ is new vocabulary.)  Now suppose that our theory $T$ consists of a
single sentence $P(c)\to O(c)$, which might be paraphrased as saying
that $O(c)$ is an empirical sign that $P(c)$.  (Here $c$ is a constant
symbol.  We omit first-order quantifiers to keep things simple.)  To
form the Ramsey sentence of $T$, we simply perform an instance of
second-order existential generalization:
\[ \begin{array}{c c} P(c)\to O(c) \\ \hline \exists X(X(c)\to
    O(c)) \end{array} \] The sentence below the line is called the
Ramsey sentence $T^R$ of the theory $T$.  Thus, while the original
theoretical statement $T$ mentions some particular property $P$, the
Ramsey sentence $T^R$ simply says that there is some or other property
that plays the appropriate role.  It may feel --- and has felt to many
philosophers --- that the truth of $T^R$ somehow magically endows the
term $P$ with meaning.  In particular, philosophers are wont to say
things like, ``$P$ is whatever it is that plays the role described by
$T^R$.''

Since Ramsey sentences draw upon the resources of second-order logic,
the neophyte is left to wonder: does the philosophical magic here
depend on something special that happens in second-order logic,
something that only the most technically sophisticated philosophers
can understand?  We think that the answer to this question is No.  In
fact, Ramsifying a theory simply weakens that theory in the same way
that existentially quantifying a first-order sentence weakens that
sentence.  Consider the following pedestrian example.

\begin{example} Let $\Sigma =\{ m\}$, where the name $m$ is a
  theoretical term.  Let $T$ be the theory $\exists x(x=m)$ in
  $\Sigma$.  Then the Ramsey sentence $T^R$ of $T$ is the sentence
  $\exists x(x=x)$, which is just a tautology.  That is, $T^R$ is the
  empty theory in the empty signature.  It is easy to see that the
  inclusion $I:T^R\to T$ is conservative, but is not essentially
  surjective.  In particular, there is no formula $\phi$ of $\Sigma$
  such that $(I\phi )(x)\equiv (x=m)$.  The fact that $I$ is not
  essentially surjective corresponds to the fact that
  $I^*:\mathrm{Mod}(T)\to \mathrm{Mod}(T^R)$ is not full.  Here $I^*$
  is the functor that takes a model of $T$, and forgets the extension
  of $m$.  In general, then, $I^*M$ has more symmetries than $M$.

  We can be yet more precise about the differences between
  $\mathrm{Mod}(T)$ and $\mathrm{Mod}(T^R)$.  In short, a model of
  $T^R$ is simply a nonempty set $X$ (and two such models are
  isomorphic if they have the same cardinality).  For each $p\in X$,
  there is a corresponding model $X_p$ of $T$ where $X_p(m)=p$.  For a
  fixed $X$, and $p,q\in X$, there is an isomorphism $h:X_p\to X_q$
  that maps $p$ to $q$.  However, the automorphism group of $X_p$ is
  smaller than the automorphism group of $X$.  Indeed,
  $\mathrm{Aut} (X_p)$ consists of all permutations of $X$ that fix
  $p$, hence is isomorphic to $\mathrm{Aut}(X\backslash \{
  p\})$.

  We can see then that $T$ and $T^R$ are not intertranslatable (or
  definitionally equivalent).  Nonetheless, there is a sense in which
  mathematicians would have no qualms about passing from $T^R$ to the
  more structured theory $T$.  Indeed, once we've established that the
  domain $X$ is non-empty (which, of course, is a presupposition of
  first-order logic), we could say, ``let $m$ be one of the elements
  of $X$.'' This latter statement does not involve any further
  theoretical commitment over what $T^R$ asserts.  \end{example}

Our advice then to the neophyte is not to allow herself to be
intimidated by second-order quantification.  In fact, we will argue
that passage from a theory $T$ to its Ramsified version $T^R$ either
forgets too much of what the original theory said, or says {\it more}
than what the original theory said --- depending on which notion of
second-order logical equivalence one adopts.  Before we do this, let's
pause to recall just how much philosophical work Ramsey sentences have
been asked to do.  We will look at three applications.  First, Carnap
claims that Ramsey sentences solve the problem of dividing the
analytic and synthetic parts of a scientific theory.  Second, Lewis
claims that Ramsey sentences solve the problem of theoretical terms,
and in particular the problem of giving meaning to ``mentalese'' in a
physical world.  Third, contemporary structural realists claim that
Ramsey sentences give a way of isolating the structural claims of a
scientific theory.

\subsubsection*{Carnap's irenic realism}

One theme running throughout Carnap's work is a rejection of what he
sees as false dilemmas.  In one sense, Carnap is one of the most
pragmatic philosophers ever in the western tradition, as he places
extreme emphasis on questions such as: {\it what questions are worth
  asking, and what problems are worth working on?}  Now, one can
imagine a philosophy graduate student asking herself: what question
should I try to answer in my dissertation?  If she's a particularly
ambitious (or perhaps overconfident) student, she might decide to
determine whether materialism or dualism is true.  Or she might decide
to determine whether scientific realism or instrumentalism is true.
Carnap's advice to her would be: to work on such questions is not a
good use of your time.

In the early 20th century, the debate between scientific realism and
instrumentalism centered around the question: do theoretical entities
--- i.e.\ the things named by scientific theories, but which are not
evident in our everyday experience --- exist?  Or, shifting to a more
explicitly normative manner of speech: are we entitled to believe in
the existence of these entities, and perhaps even obliged to do so?
Carnap thinks that these questions are ill-posed.

Toward the end of his career, Carnap hoped that Ramsey sentences could
help show why there is no real argument between realism and
instrumentalism.  In particular, if $T$ is a scientific theory
containing some theoretical terms $r_1,\dots ,r_n$, then Carnap parses
$T$ into two parts: the Ramsey sentence $T^R$, and the sentence
$T^R\to T$ that has since been dubbed the ``Carnap sentence''.  Carnap
claims that the Ramsey sentence $T^R$ gives the empirical (synthetic)
content of $T$, whereas $T^R\to T$ gives the definitional (analytic)
part of $T$.  The latter claim can be made plausible by realizing that
$T^R\to T$ is trivially satisfiable, simply by stipulating appropriate
extensions for $r_1,\dots ,r_n$.

\cite{psillos2000} argues that Carnap's equation of synthetic content
with the Ramsey sentence makes him a structural realist --- in which
case he is subject to Newman's objection, which impales him on the
horns of the realism-instrumentalism dilemma.  \cite{friedman2011}
disagrees, arguing that Carnap's invocation of the Ramsey sentence
successfully implements his neutralist stance.  Debate on this issue
continues in the literature --- see e.g.\ \cite{uebel2011,beni}.


\subsubsection*{Ramsey sentence functionalism}

In the philosophy of mind, Ramsey sentences came to play a central
role through the work of \cite{lewis-id,lewis-psy,lewis1994}, and more
generally in a point of view known as \emph{functionalism}.  To be
sure, Lewis claims not to know whether or not he is a functionalist;
and most functionalists don't talk explicitly about Ramsey sentences.
However, by the 1980s, the connection between functionalism and Ramsey
had been firmly established \cite[see][]{shoemaker}.

Around 1970, materialist reductionism had gone out of style.
Philosophers concluded that folk psychology cannot, and should not, be
reduced --- neither to descriptions of behavior, nor to physiological
descriptions.  However, philosophers weren't ready to give up the
physicalist project, and in particular, they didn't want to entertain
the possibility that there is an autonomous realm of mental objects or
properties.  The goal then is to explain how mental properties are
anchored in physical properties, even if the former cannot be
explicitly defined in terms of the latter.

Functionalism, and functional definitions, are supposed to provide a
solution to this problem.  According to functionalism, mental
properties are defined by the role that they play in our total theory
$T$, which involves both mental concepts (such as ``belief'' and
``desire'') and physical concepts (such as ``smiling'' or ``synapse
firing'').  How then are we supposed to cash out this notion of being
``defined by role''?  It's here that Ramsey sentences are invoked as
providing the best formal explication of functional definitions.

Contemporary analytic philosophers routinely mention Ramsey sentences
in this connection.  Nonetheless, long ago, \cite{bealer} argued that
this attempt to define mental properties --- call it ``Ramsey sentence
functionalism'' --- is inconsistent.  According to Bealer,
functionalism has both a negative and a positive theses.  On the
negative side, functionalism is committed to the non-reductionist
thesis: mental properties (m-properties) cannot be explicitly defined
in terms of physical properties (p-properties).  On the positive side,
m-properties {\it are} defined in terms of the role they play
vis-a-vis each other and the p-properties.

Let $T$ be a theory in signature $\Sigma \cup \{r_1,\dots ,r_n \}$,
where we think of $\Sigma$ as p-vocabulary, and of $r_1,\dots ,r_n$ as
m-vocabulary.  We then adopt the following proposal (which defenders
of functionalism are welcome to reject or modify):
\begin{quote} $T$ provides functional definitions of $r_1,\dots ,r_n$
  in terms of $\Sigma$ just in case in each model $M$ of the Ramsey
  sentence $T^R$, there are unique realizing properties
  $M(r_1),\dots ,M(r_n)$. \end{quote} It's easy to see then that $T$
provides functional definitions of $r_1,\dots ,r_n$ in terms of
$\Sigma$ only if $T$ implicitly defines $r_1,\dots ,r_n$ in terms of
$\Sigma$.  Indeed, if $M$ and $N$ are models of $T$, then
$M|_{\Sigma}$ and $N|_{\Sigma}$ are models of $T^R$, and it follows
from the uniqueness clause that $M(r_i)=N(r_i)$.  It then follows from
Beth's theorem that $T$ explicitly defines $r_1,\dots ,r_n$ in terms
of $\Sigma$.

Bealer's argument, if successful, shows that functionalism is
inconsistent: the positive thesis of functionalism entails the
negation of the negative thesis.  Surprisingly, however, functionalism
lives on, apparently oblivious of this little problem of
inconsistency.  In fact, functionalism hasn't just survived, it is
flourishing and spreading its tendrils --- indeed, it has become an
overarching philosophical ideology: \emph{the Canberra plan}.  The
goal of the Canberra plan is to find a place in the causal nexus of
physical properties for all the stuff that makes up our daily lives
--- things like moral and aesthetic values, laws, society, love, etc..
(For further discussion, see \cite{menzies}.)  One has to wonder: how
attractive would the Canberra program be if it were clearly recognized
that what it seeks is nothing more or less than the kind of full-blown
physicalistic reduction that Carnap tried to achieve in his {\it
  Aufbau} program?



\subsubsection*{Structural realism}

In more recent times, Ramsey sentences have been invoked in support of
a trendy view in philosophy of science: \emph{structural realism}.  In
the early 1990s, structural realism was the new kid on the block in
discussions of scientific realism and antirealism.  As forcefully
recounted by \cite{worrall}, there are good arguments against both
scientific realism, and against scientific antirealism.  Against
scientific realism, there is the {\it pessimistic metainduction},
which points to the long history of failed scientific theories as
evidence that our current favorite scientific theories will probably
also fail.  Against scientific antirealism, there is the {\it no
  miracles argument}, which points to the success of scientific
theories as something crying out for an explanation.  In good Hegelian
fashion, Worrall seeks a synthesis of the extremes of realism and
antirealism --- a position which offers the best of both worlds.  His
proposal is \emph{structural realism}, according to which the part of
a theory to take seriously is it's pronouncements on issues of {\it
  structure}.

Worrall illustrates the idea of ``preserved structure'' with a
specific example.  In particular, before Einstein's special theory of
relativity, it was thought that there was a substance, the ``aether,''
in which electric and magentic waves propagated.  After the
Michelson-Morley experiment, and the success of special relativity,
there was no longer any use for the aether.  Thus, the transition to
special relativity might be taken to be a particularly clear example
of failed reference --- showing, in particular, that pre-Einsteinian
physicists ought not to have taken their theory so seriously.

Nonetheless, says Worrall, it would have been a mistake for
pre-Einsteinian physicists to treat their theory instrumentally,
i.e.\ merely as a tool for making predictions.  For the form of the
equations of motion was preserved through the transition to special
relativity --- hence, they would have done well to trust their
equations.  The general lesson, says Worrall, is to trust your
theory's structure, but not the underlying stuff it purports to be
talking about.

Worrall's example is highly suggestive, and we might like to apply it
in a forward-looking direction.  In particular, take one of our
current-day successful scientific theories $T$, such as quantum
mechanics.  The pessimistic metainduction suggests that $T$ will be
wrong about something.  But can we already make an educated guess
about which parts of $T$ will be preserved, and which part will go on
the scrap heap with other rejected theories?

\cite{zahar2001,cruse,zahar2004} provide a specific proposal for
picking out the structural commitments of a theory $T$: they are given
by its Ramsey sentence $T^R$.  This idea certainly has some intuitive
appeal --- trading on an analogy to ``coordinate free'' descriptions
of space.  For a naive or straightforward description of physical
space, we might use triples of real numbers, i.e.\ the mathematical
space $\7R ^3$.  But now our description of space has superfluous
structure.  In particular, we assigned the origin $0\in\7R ^3$ to some
particular point in space --- but we didn't mean to indicate that the
denoted point is any different than any other point in space.  Thus,
our description breaks the natural symmetry of space, and it would be
natural to look for another description that respects these
symmetries.  Indeed, that's precisely the idea behind the move from
using vector spaces to using affine spaces to describe space.

Now, just as a vector-space description of space breaks its symmetry,
so our theoretical descriptions in general might fail to respect the
symmetry between properties.  For example, we didn't need to use the
word ``electron'' to describe those things that are found in the
energy shells around an atom's nucleus --- we could simply say that
something or other plays the relevant role.  And that's exactly what
the Ramsified theory says.  Thus, it might seem that $T^R$ provides a
more intrinsic description than the original theory $T$.

Nonetheless, the intuitive appeal of Ramsey sentences fades quickly in
the light of critical scrutiny.  Most famously, already in 1928,
Newman argued that Bertrand Russell's structuralism trivializes, for
these structural claims are true whenever their observational
consequences are true \cite[see][]{newman1928}.  The so-called Newman
objection to structural realism has been the center piece of recent
debates about Ramsey-sentence structuralism.  But even before we get
to that level of scrutiny, there is something quite strange in the
idea of passing to the Ramsified theory $T^R$ to get rid of
redundancy.  Let's recall that a formal theory $T$ doesn't actually
refer to thinks like electrons or protons --- it's formulated in an
uninterpreted calculus.  Hence, $T$ doesn't actually have any
referring terms.

It seems that the impulse to Ramsify is no other than the original
impulse to use uninterpreted mathematical symbols to represent
physical reality.  You'll recall that one of the key maneuvers in the
development of non-euclidean geometries was de-interpreting words like
``line'', thereby liberating mathematicians to focus attention solely
on the relation that ``line'' plays relative to other (uninterpreted)
terms in their formal calculus.

In any case, what's really at stake here is the question of what
attitude we should take toward the best scientific theories of our day
and age.  At one extreme, radical scientific realists assert that we
should give nothing less than {\it full} assent to these theories,
interpreted literally.  To draw an analogy (that scientific realists
will surely eschew), the extreme scientific realist is akin to the
radical religious fundamentalist, and in particular to those
fundamentalists who say that one must interpret scriptures literally.
The point of that injunction, we all know, is to enable religious
leaders to foist their opinions on others.  At the opposite extreme,
an extreme scientific antirealist sees science as having no epistemic
authority whatsoever --- i.e.\ a successful scientific theory doesn't
call for any more epistemic attention on my part than, say,
Zoroastrianism.

In the light of this somewhat hyperbolic characterization of the
anti/realism debate, we can see various alternative positions as
granting a selective epistemic authority to successful scientific
theories.  Consider an analogy: suppose that you know a highly skilled
car mechanic Jacob.  You completely trust Jacob when it comes to his
opinions on automobile-related issues.  For example, if he says that
you need a new alternator, then you won't doubt him, even if it costs
you a lot of money.  Nonetheless, if Jacob tells you that you need a
new kidney, or that you should vote for a certain candidate, you might
well ignore his opinion --- since he's speaking on a topic that lies
outside his proper expertise.

Now, selective scientific realists consider successful scientific
theories to be epistemically authoritative, but only when they speak
on topics within their expertise.  The different brands of selective
realism are distinguished by how they understand the expertise of
science.  For example, a constructive empiricist (such as van
Fraassen) trusts a successful scientific theory $T$ when it makes
predictions about empirical phenomena (presupposing, as he does, that
it makes sense to speak of predictions and empirical phenomena ---
precisely the point to which Boyd and Putnam object).  Similarly, a
structural realist (such as Worrall) trusts a successful scientific
theory $T$ on its structural pronouncements.  But if $T$ says
something about things in themselves (or whatever is {\it not}
structure), then the structural realist treats it as no more of an
authority than your auto mechanic is on politics.

The previous considerations suggest that varieties of selective
scientific realism can be classified by means of different notions of
theoretical equivalence.  For example, the strict empiricist thinks
that the important part of a theory is its empirical content; and
hence, if two theories $T_1$ and $T_2$ agree on empirical content,
then there is no epistemically relevant difference between them.
Similarly, a structural realist thinks that the important part of a
theory is its pronouncements about structure; and hence, if two
theories $T_1$ and $T_2$ agree on structure, then there is no
epistemically relevant difference between them.  In the particular
case of Ramsey-sentence structuralism, the structural pronouncement's
of a theory $T_i$ are captured by its Ramsey sentence $T_i^R$.  Hence,
if $T_1^R\equiv T_2^R$, then there is no epistemically relevant
difference between $T_1$ and $T_2$.

Unfortunately, the statement ``$T_1^R\equiv T_2^R\,$'' doesn't have an
obvious meaning, since there is no single, obviously correct notion of
second-order logical consequence.  What this means is that we get
different notions of ``same structure'' depending on which notion of
second-order consequence we adopt.  Let's review, then, some salient
notions of second-order logical consequence.

Second-order logic is a complicated subject in its own right, and has
been the source of much dispute among analytic philosophers.  We refer
the reader to studies such as \citep{shapiro,bueno} for more details.
For present purposes, it will suffice to make some minor modifications
of first-order logical grammar: first, we add a list of second-order
variables $X,Y,\dots $.  Each second-order variable has a specific
arity $n\in\7N$, which means that it can stand in the place of an
$n$-ary relation symbol.  We then permit formulas such as
$X(x_1,\dots ,x_n)$, with a second-order variable of arity $n$ applied
to $n$ first order variables.  We also add an existential quantifier
$\exists X$ that can be applied to quantify over second-order
variables.

Now there are two important facts to keep in mind about second-order
logic.  The first fact to keep in mind is that second-order logic has
is intrinsically incomplete --- hence there is no tractable syntactic
relation ``$\vdash$'' of second-order provability.  The second fact to
keep in mind is that there are several candidates for the semantic
relation ``$\vDash$'' of entailment.  Depending on which choice we
make for this relation, we will get a different notion of logical
equivalence.

\begin{defn} A second-order $\Sigma$-frame $\2F=(M,(\2E )_{n\in\7N })$
  consists of a first order $\Sigma$-structure $M$, and for each
  $n\in \7N$, a subset $\2E _n$ of $\2P(M^n)$.  We let
  $\2E ^{\2F}=\bigcup _{n\in\7N}\2E _n$.  Here the sets in
  $\2E ^{\2F}$ will give the domain of the second-order quantifiers in
  frame $\2F$. \end{defn}

In order to define the relation $\vDash$, we will also make use of the
notion of a variable assignment.  Given a $\Sigma$-frame $\2F$, a
first-order variable assignment $g$ assigns each variable $x$ to an
element $g(x)\in M$.  A second-order variable assignment $G$ assigns
each variable $X$ of arity $n$ an element $G(X)\in \2E _n$.  We then
define:
\begin{quote} $M[G,g] \vDash \exists X\phi$ iff for some $E\in \2E_n$,
  $M[G^E_X,g]\vDash \phi$, where $G^E_X$ is the second-order variable
  assignment that agrees with $G$ on everything besides $X$, which it
  assigns to $E$.  \end{quote} Now to define the relation $\vDash$
between sentences, we have to decide which second-order
$\Sigma$-frames to quantify over.  We get three different notions,
depending on the family we choose:
\begin{enumerate}
\item For \emph{full semantics}, we permit only those $\Sigma$-frames
  in which $\2E _n=\2P(M^n)$.
\item For \emph{Henkin semantics}, we permit all $\Sigma$-frames in
  which $\2E _n$ is closed under first-order definability.
\item For \emph{frame semantics}, we permit all $\Sigma$-frames.
  \end{enumerate}
  Recall that the more structures, the more counterexamples, and hence
  the fewer implications.  Accordingly, full semantics has more
  entailments than Henkin semantics, and Henkin semantics has more
  entailments than frame semantics; hence, full semantics yields a
  more liberal notion of equivalence than Henkin semantics, which
  yields a more liberal notion of equivalence than frame semantics.

  In the following discussion, we will take $T_i$, for $i=1,2$, as a
  theory in signature $\Sigma\cup \Sigma _i$, where $\Sigma _i$ is
  disjoint from $\Sigma$.  We let $T_i^*$ be the result of replacing
  terms in $\Sigma _i$ with (possibly second-order) variables, and we
  let $T_i^R$ be the corresponding Ramsey sentence of $T_i$.  We now
  give a general schema for Ramsey equivalence of theories:
  \begin{defn} Two theories $T_1$ and $T_2$ are \emph{Ramsey
      equivalent} if $T_1^R$ is logically equivalent to
    $T_2^R$. \end{defn} \noindent The three choices of frames above
  give rise to three notions of Ramsey equivalence.
\begin{itemize}
\item RE$_1$ = loose Ramsey equivalence = Ramsey sentences are
  equivalent relative to full semantics.
\item RE$_2$ = moderate Ramsey equivalence = Ramsey sentences are
equivalent relative to Henkin semantics.
\item RE$_3$ = strict Ramsey equivalence = Ramsey sentences are
  equivalent relative to frame semantics.
\end{itemize}
Obviously then we have RE$_3$ $\Rightarrow$ RE$_2$ $\Rightarrow$
RE$_1$.

We can now give a sharpened formulation of the Newman problem --- in
the spirit of \cite{ketland} and \cite{dew-ram}.  Recall that on the
old-fashioned syntactic view of theories, two theories $T_1$ and $T_2$
are considered to be empirically equivalent if they have the same
consequences in the observation language.  If we now think of $\Sigma$
as the observation vocabulary, then we could formulate this criterion
as saying that $\cn{T_1}|_\Sigma=\cn{T_2}|_\Sigma$, where
$\cn{T_i}|_\Sigma$ indicates the restriction of the set of
consequences to those that contain only observation terms.

One might also wish to formulate a more semantically oriented notion
of empirical equivalence.  For example, we might say that two theories
$T_1$ and $T_2$ are empirically equivalent if their models agree on
$\Sigma$-structure.

\begin{defn} We say that $T_1$ and $T_2$ are $\Sigma$-equivalent just
  in case for each model $M$ of $T_1$, there is a model $N$ of $T_2$
  and an isomorphism $h:M|_\Sigma\to N|_\Sigma$, and vice
  versa. \end{defn}

The following result shows that this semantic notion of empirical
equivalence implies the syntactic notion.

\begin{prop} If $T_1$ and $T_2$ are $\Sigma$-equivalent then
  $\cn{T_1}|_\Sigma=\cn{T_2}|_\Sigma$. \end{prop}

\begin{proof} Suppose that $T_1$ and $T_2$ are $\Sigma$-equivalent.
  Let $\phi$ be a $\Sigma$-sentence such that $\phi\not\in\cn{T_2}$.
  By completeness, there is a model $M$ of $T_2$ such that
  $M\not\vDash\phi$.  Since $T_1$ and $T_2$ are $\Sigma$-equivalent,
  there is a model $N$ of $T_1$ and an isomorphism
  $h:M|_\Sigma\to N|_\Sigma$.  But then $N\not\vDash\phi$, hence
  $\phi\not\in \cn{T_1}$.  It follows that
  $\cn{T_1}|_\Sigma\subseteq \cn{T_2}|_{\Sigma}$.  The result follows
  by symmetry.
\end{proof}

However, this implication cannot be reversed, i.e.\ the syntactic
notion of empirical equivalence doesn't imply the semantic notion.

\begin{example} Let $\Sigma$ be the empty signature (with equality).
  Let $\Sigma _1 = \{ c_r\mid r\in \7R \}$, and let $T _1$ be the
  theory in $\Sigma\cup\Sigma _1$ with axioms $c_r\neq c_s$, for all
  $r\neq s$.  Let $T_2$ be the theory in $\Sigma$ that says there are
  infinitely many things.  Then $\cn{T_1}|_\Sigma=\cn{T_2}|_\Sigma$.
  However, $T_2$ has a countable model $M$, and $T_1$ has no countable
  model.  Therefore, $T_1$ and $T_2$ are not $\Sigma$-equivalent.
\end{example}

The Newman problem for structural realism is usually phrased as saying
that it's too easy for a theory's Ramsey sentence to be true --- that
the Ramsey sentence is ``trivially realizable''.  We can make precise
what is meant here by ``too easy'' in terms of the notion of
theoretical equivalence.  In short, Ramsey equivalence --- i.e.\
having logically equivalent Ramsey sentences --- is too liberal a
notion of equivalence.  In particular, empirically equivalent theories
are Ramsey equivalent.

\begin{prop}[Dewar] If $T_1$ and $T_2$ are $\Sigma$-equivalent, then
  $T_1^R$ and $T_2^R$ are logically equivalent relative to full
  semantics. \end{prop}

\begin{proof} Suppose that $T_1$ and $T_2$ are $\Sigma$-equivalent.
  Now let $\2F$ be a full $\Sigma$-frame such that $\2F\vDash T_1^R$.
  Thus, there is a second-order variable assignment $G$ such that
  $\2F [G]\vDash T_1^*$.  Let $M$ be the $\Sigma\cup\Sigma _1$
  structure obtained by assigning $M(R)=G(X)$, where $X$ is the
  variable in $T_1^*$ that replaces $R$ in $T_1$.  Clearly $M$ is a
  model of $T_1$.  Since $T_1$ and $T_2$ are $\Sigma$-equivalent, $M$
  is $\Sigma$-isomorphic to a model $N$ of $T_2$.  This model $N$ of
  $T_2$ defines a second order variable assignment $G'$ such that
  $\2F [G']\vDash T_2^*$, and hence $\2F\vDash T_2^R$.  \end{proof}

The notion of empirical equivalence imposes no constraints whatsoever
on what the theories $T_1$ and $T_2$ say in their theoretical
vocabulary --- and for this reason, nobody but the most extreme
empiricist should adopt weak Ramsey equivalence as their standard.

Moving back toward the right wing side of the spectrum of theoretical
equivalence, one might hope that moderate Ramsey equivalence would
provide a more reasonable standard.  But the following result shows
that any two mutually interpretable theories satisfy RE$_2$.

\begin{prop}[Dewar] If $T_1$ and $T_2$ are $\Sigma$-equivalent and
  mutually interpretable, then $T_1^R$ and $T_2^R$ are logically
  equivalent relative to Henkin semantics.  \end{prop}

\begin{proof} Suppose that $T_i$ is a theory in
  $\Sigma\cup \Sigma _i$.  We will show that if $F:T_1\to T_2$ is a
  translation (which is the identity on $\Sigma$), then
  $T_2^R\vDash T_1^R$, where the $\vDash$ symbol is entailment
  relative to Henkin semantics, and $T_i^R$ is the result of
  Ramsefying out $\Sigma _i$.  Suppose then that $F:T_1\to T_2$ is a
  translation, and that $\2H$ is a Henkin structure (of signature
  $\Sigma$) such that $\2H\vDash T_2^R$.  Thus, $\2H [G]\vDash T_2^*$
  relative to some second-order variable assignment $G$.  Consider
  then the first-order structure $M$ for signature
  $\Sigma\cup \Sigma _1$ that agrees with $\2H$ on $\Sigma$, and such
  that $M(P)=G(X_P)$, for each $P\in \Sigma _2$, where $X_P$ is the
  second-order variable that replaces $P$ in $T_1^*$.  It is clear
  then that $M\vDash T_2$.  Now we will use the fact that the
  translation $F:T_1\to T_2$ gives rise to a functor
  $F^*:\mathrm{Mod}(T_2)\to \mathrm{Mod}(T_1)$ (\ref{mfunc}).  In
  particular, $(F^*M)(Q)=M(F(Q))$ for each relation symbol
  $Q\in\Sigma\cup\Sigma _1$.  Now define a second-order variable
  assignment $G'$ by setting
    \[ G'(X_Q) \: = \: (F^*M)(Q) \: = \: M(F(Q)) ,\] for each variable
    $X_Q$ that occurs in $T_1^*$.  (Again we use $X_Q$ to denote the
    variable that replaces a relation symbol $Q$ that occurs in
    $T_1$.)  To see that $G'$ is a Henkin-admissible assignment, note
    that $F(Q)$ is a $\Sigma _2$-formula, and so $M(F(Q))$ is a
    first-order definable subset of $M$.  By construction, each
    first-order definable subset of $M$ is an element of $\2E ^{\2H}$.
    Now, it's clear that $\2H [G']\vDash T_1^*$, and hence that
    $\2H\vDash T_1^R$.  Since $\2H$ was an arbitrary Henkin frame, it
    follows that $T_2^R\vDash T_1^R$.  By symmetry, if there is a
    translation $G:T_2\to T_1$, then $T_1^R\vDash T_2^R$.  Therefore,
    if $T_1$ and $T_2$ are mutually interpretable, then $T_1^R$ and
    $T_2^R$ are Henkin equivalent. \end{proof}

  There is one last hope for the Ramsifier: that strict Ramsey
  equivalence (RE$_3$) will provide the right notion of structural
  equivalence.  Unfortunately, RE$_3$ proves to be the worst candidate
  for structuralism, since intertranslatable theories need not satisfy
  RE$_3$.
  
  \begin{example} Let $\Sigma _1 = \{ r\}$, and let
    $\Sigma _2=\{ r'\}$, where both $r$ and $r'$ are unary predicates.
    Let $T_1=\{ \exists xr(x) \}$, and let
    $T_2=\{ \exists x \neg r'(x)\}$.  The reconstrual
    $F(r)=\neg r'(x)$ induces a homotopy equivalence between $T_1$ and
    $T_2$, i.e.\ $T_1$ and $T_2$ are intertranslatable.  However, the
    Ramsey sentences of $T_1$ and $T_2$ are not frame equivalent.  In
    particular, consider any frame $\2F$ with first-order domain $M$,
    and $\2E ^{\2F}_1=\{ M\}$, i.e.\ $M$ is the only admissible subset
    of $M$.  Then $\2F \vDash T_1^R$ but $\2F \not\vDash
    T_2^R$.  \end{example}
  
  Since strict Ramsey equivalence (RE$_3$) is more conservative
  (``right wing'') than definitional equivalence, we don't expect
  structural realists to find it congenial.  But what about those
  hard-core realists --- like David Lewis or Ted Sider --- who pin
  their theoretical hopes on natural properties and reference
  magnetism?  Might they actually want a criterion of equivalence that
  is even more conservative than definitional equivalence?  In fact,
  it seems that frame semantics might be a good way to capture the
  idea that to describe a possible world, you need to say not only
  what things exist, but also what the natural properties are.  We
  should note, however, that adopting a first-order signature $\Sigma$
  already goes some way to picking out natural properties.  When we
  specify a $\Sigma$-structure $M$, we get a natural property
  $M(\phi )$ for each formula $\phi$ of $\Sigma$.  It's not clear then
  why a theorist who has adopted a first-order signature $\Sigma$
  would need to additionally specify a notion of natural properties.

The previous results can be summarized in the following diagram:
\[ \begin{tikzcd}
    \text{EE}   \arrow[Rightarrow]{d} &  \text{MI} \arrow[Rightarrow]{l}{}  \arrow[Rightarrow]{d} & \text{IT} \arrow[Rightarrow]{l}{}  \arrow[strike through,Rightarrow]{dr}{} \\
    \mathrm{RE_1} & \mathrm{RE_2} \arrow[Rightarrow]{l}{} & &
    \mathrm{RE_3} \arrow[Rightarrow]{ll}{} \end{tikzcd} \] Here ``EE''
is empirical equivalence (explicated semantically), ``MI'' is mutual
interpretability over $\Sigma$, and ``IT'' is intertranslatability
over $\Sigma$, which is equivalent to definitional equivalence.  It
appears then that none of the notions of Ramsey equivalence gets us
near the promising area in the neighborhood of intertranslatability.
Most philosophers, we think, would agree that intertranslatability is
a reasonable --- if somewhat strict --- explication of the idea that
two theories have the same logical structure.


\section{Counting possibilities}


%% counting the number of possibilities

%% are they the same or different?  A question that doesn't really
%% have a good answer

%% need to account for relations between worlds

If you page through an analytic philosophy journal, it won't be long
before you see the phrase ``possible world''.  Many philosophical
discussions focus on this concept, and it is frequently used as a
basis from which to explicate other concepts --- Humean supervenience,
counterfactuals, laws of nature, determinism, physicalism, content,
knowledge, etc..  When the logically cautious philosopher encounters
this concept, she will want to know what rules govern its use.  Where
things get really tricky is when philosophers start invoking facts
about the structure of the space of possible worlds --- e.g.\ how
many worlds there are, which worlds are similar, and which worlds are
identical.  These sorts of assumptions play a significant role in
discussions of fundamental ontology.  To take a paradigm example,
\cite{baker2010} argues that if two models are isomorphic, then they
represent the same possible world.

Analytic philosophers might be the primary users of the phrase
``possible world'', but they aren't the only ones using the concept.
Scientists talk about possible worlds all the time.  However, at least
in the exact sciences, there are explicit rules governing the use of
possible worlds talk.  Indeed, these rules are built into the
structure of their theories, and more particularly, in the structure
of those theories' spaces of models.  Following \cite{elvis}, we think
that philosophers ought to try to understand the way that scientists'
theories guide their use of modal concepts.

Nonetheless, it's not hard to find philosophers scratching their
heads, and asking themselves questions like the following:
\begin{quote} ($\star$) Consider two general relativistic spacetimes,
  $M$ and $N$, and suppose that $h:M\to N$ is an isomorphism (e.g.\ a
  metric preserving diffeomorphism).  Do $M$ and $N$ represent the
  same possible world? \end{quote}
\begin{quote} ($\ast$) Consider two Newtonian spacetimes, $M$ and $N$,
  and suppose that $h:M\to N$ is an isomorphism (e.g.\ a shift).  Do
  $M$ and $N$ represent the same possible world?  \end{quote}
\cite{elvis} helpfully classifies philosophers into two groups
according to how they answer these questions: the {\it shiftless}
claim that isomorphisms do not generate new possibilities, and the
{\it shifty} claim that isomorphisms do generate new possibilities.
In particular, the shiftless philosopher says that if $h:M\to N$ is an
isomorphism, then $M$ and $N$ represent the same possibility.  In
contrast, the shifty philosopher allows that $M$ and $N$ might
represent different possibilities, even though they are isomorphic.
While the majority of philosophers of physics and metaphysicians have
become shiftless, Belot champions the heterodox, shifty point of view.
As we will now argue, all parties to the dispute have adopted a
questionable presupposition, viz.\ that it makes sense to count
possibilia.

But first, what hangs on this dispute between the shifty and the
shiftless?  In the first place, shiftless philosophers believe that
they are on the right side of history, ontologically speaking.  In
particular, they believe that it would be wrong to countenance the
existence of two possibilities, represented by $M$ and $N$, when a
single one will do the job.  This way of thinking trades on vague
associations with Leibniz's principle of the identity of
indiscernibles: since $M$ and $N$ are indiscernible, there is no
reason to regard them as different.  Belot points out, however, that
shiftless philosophers have trouble making sense of how theories can
guide the use of modal concepts.  In particular, he argues that the
shiftless view necessarily collapses the distinction between
deterministic and indeterministic theories.

The mind boggles at the existence of shiftless philosophers ---
because the view so patently conflicts with the standard reading of
physical theories, not to speak of plain common sense.  Take, for
example, a Galilean spacetime $M$, and let $\gamma :\7R\to M$ be an
inertial worldline in $M$.  Now, a boost $x\mapsto x+vt$ for some
fixed $v>0$ is represented by an isomorphism $h:M\to M$.  Does this
boost generate a new possibility?  The question might seem confusing
because the model on the right side of $h:M\to M$ is the same as the
model on the left side.  It might seem to be trivially true, then,
that $h:M\to M$ does {\it not} generate a new possibility.  But let's
see what happens if we adopt the shiftless view.  If $h$ does {\it
  not} generate a new possibility, then we ought to say, of a particle
in inertial motion that it could {\it not} be in some other state of
inertial motion (because there is no other such state of inertial
motion).  But that claim is completely contrary to the way that
physicists use this theory to guide their modal reasoning.  When a
physicist adopts Galilean relativity, she commits to the claim that
there are many distinct possible states of inertial motion, and that a
thing which is in one state of inertial motion {\it could be} in some
other state of inertial motion.  In other words, it matters to
physicists that the isomorphism $h:M\to M$ is not the identity
isomorphism, and in particular, that the worldline $h\circ \gamma$ is
not the same as the worldline $\gamma$.  Nonetheless, shiftless
philosophers can't make sense of these modal claims, because they
insist that isomorphisms don't generate new possibilities.

Despite the implausibility of the shiftless view, there are some very
serious and smart philosophers who defend it.  What is it, then, that
really drives their insistence on saying that isomorphism (at the
level of representations) implies identity (at the level of the
represented)?  We suspect that the shiftless are fumbling their way
toward an insight --- but an insight that is difficult to articulate
when one is operating with mistaken views about mathematical objects,
and in particular about the relation between abstract and concrete
objects.  Of course, we blame a lot of this confusion on Quine and his
followers, who decided that we have no need of the abstract-concrete
distinction --- in particular, that belief in the existence of
abstracta is no different in principle from belief in the existence of
concreta.  But without that distinction, we are bound to say some
silly things about the representation relation.

At risk of oversimplifying, we will first give a simple formulation of
the basic insight toward which we think the shiftless philosophers are
fumbling:
\begin{quote} ($\dagger$) A theory $T$ is indifferent to the question
  of the identity of its models.  In other words, if $M$ and $N$ are
  models of $T$, then $T$ neither says that $M=N$, nor that $M\neq N$.
  The only question $T$ understands is: are these models isomorphic,
  or not?
\end{quote}
Now please don't get us wrong: ($\dagger$) does not say that
isomorphic models are identical, nor does it say that the theory $T$
treats isomorphic models as if they were identical.  No, from the
point of view of $T$, the question, ``are they identical?'' simply
does not make sense.  And claims of identity, or non-identity of
models, play {\it no} explanatory role in the theory.

We realize that this claim is controversial, and that it might take
some time for philosophers to become comfortable with it.  The problem
is that we learned a little bit of set theory in our young years, and
we seem to assume that everything lives in a world of sets --- where
questions of the form ``is $M$ equal to $N$'' always have a definite
answer.  Indeed, the rigid grip of set theory makes philosophers
profoundly uncomfortable with contemporary mathematics, which likes to
play a fast and loose game with identity conditions.  Consider a
simple example (due to John Burgess): suppose that we ask two
different mathematicians two different questions:
\begin{quote} (Q1) How many groups are there with two
  elements? \end{quote}
\begin{quote} (Q2) Inside the group $\7Z _2\oplus \7Z _2$, how many
  subgroups are there with two elements? \end{quote} What we are
likely to find is that mathematicians will give apparently conflicting
answers.  On the one hand, they will tell us that there is only one
group with two elements.  On the other hand, they will tell us that
$\7Z _2\oplus \7Z _2$ has two distinct subgroups with two elements.
Obviously, if taken literally, these two answers contradict each
other.  But of course, there is no genuine conflict, and
mathematicians are not in crisis about the number of groups with two
elements.  No, the fact is, mathematicians use words and symbols in a
different way than we use them in everyday life --- e.g.\ when we
count the number of apples in a basket.

To reinforce this point, recall that categorical equivalence doesn't
respect the number of objects in a category.  Consider for example the
following two categories: let $\cat{C}$ be the category with one
object, and one identity morphism.  Let $\cat{D}$ be the category with
two objects $a,b$, one identity morphism from each object to itself,
and a pair of morphisms $f:a\to b$ and $g:b\to a$ that are inverse to
each other.  Then $\cat{C}$ and $\cat{D}$ are equivalent categories
--- which entails that ``this category'' doesn't really have a
definite number of objects.  It is not correct to say that it has one
object, and it's not correct to say that it has two.  Or perhaps
better: it is just as correct to say that it has one object as it is
to say that it has two.

Here then is our positive proposal:
\begin{quote} For the purposes of interpreting a theory $T$, the
  collection $\mathrm{Mod}(T)$ of its set-theoretic models should be
  treated as nothing more nor less than a {\it category}.  In
  particular, the philosopher of science shouldn't say things about
  $\mathrm{Mod}(T)$ that are not invariant under categorical
  equivalence, nor should they argue over questions --- such as ``how
  many models does $T$ have?'' --- whose answer is not invariant under
  categorical equivalence.  \end{quote} If this proposal is adopted
then there is no debate to be had between the shifty and the
shiftless.  The question they are asking --- do isomorphisms generate
new possibilities? --- depends on a notion (the number of isomorphic
possibilities) that is not invariant under categorical equivalence.

The rationale for this proposal is our belief that models of a theory
$T$ in $\cat{Sets}$ are {\it representations} of that theory; the
set-theoretic description of these models is not {\it not} itself a
further theory that attempts to describe the world at an even finer
grained level of detail than was done by $T$.  We can further clarify
these points by means of a simple example.

\begin{example} Suppose that Berit is a scientist with a very simple
  theory.  Her language $\Sigma$ has a single predicate symbol $P$,
  and her theory $T$ says that there are exactly two things, one of
  which is a $P$:
  \[ \exists x\exists y(P(x)\wedge \neg P(y)\wedge \forall z((z=x)\vee
    (z=y))) .\] Now we metatheorists know that a set-theoretic model
  $M$ of $T$ consists of a two-element set, say $X=\{ a,b\}$, with a
  singleton set $M(P)$.  Let $M$ be the model such that $M(P)=\{ a\}$,
  and let $N$ be the model such that $N(P)=\{ b\}$.  Then the
  permutation $h(a)=b,h(b)=a$ gives a $\Sigma$-isomorphism $h:M\to N$.
  (But the permutation $h$ is not an automorphism of $M$.)

  Let's consider the shifty-shiftless dilemma with regard to the
  models $M$ and $N$, with the isomorphism $h:M\to N$.  The shifty
  philosopher (e.g.\ Belot) says that $M$ and $N$ represent distinct
  possibilities.  The shiftless philosopher (e.g.\ Baker) says that
  $M$ and $N$ represent the same possibility.  Who is on the side of
  truth?

  In our opinion, both the shifty and the shiftless say misleading
  things about this example.  On the one hand, the shifty claim is
  misleading, because the user of $T$ doesn't have the language to say
  what would be different between $M$ and $N$.  She cannot say, ``in
  $M$, $a$ is $P$, and in $N$, $a$ is not $P$,'' because she herself
  doesn't have the name ``$a$''.  The shiftless wants us to start
  counting how many models there are, but the theory $T$ doesn't
  answer that question.

  On the other hand, the shiftless would insist that there is only one
  possibility, represented redundantly by $M$ and $N$.  But that claim
  is misleading for the following reason.  Berit's theory $T$ is an
  extension of the theory $T_0$, in empty signature, that says there
  are exactly two things.  Let $I:T_0\to T$ be the translation of
  $T_0$ into $T$, and let $I^*:\mathrm{Mod}(T)\to\mathrm{Mod}(T_0)$ be
  the functor that forgets the assignment of $P$.  Here $I^*M$ and
  $I^*M$ are both the bare two-point set $X$, and the isomorphism
  $I^*h=h:X\to X$ is the non-trivial permutation.  Recall, though,
  that functors map identity morphisms to identity morphisms.  Hence,
  if the isomorphism $h:M\to N$ is considered to be an identity (as
  the shiftless seem to do), then it would follow that $I^*h$ is the
  identity morphism.  Thus, contra the shiftless, we cannot identify
  $M$ and $N$, and forget that there was a non-identity isomorphism
  $h:M\to N$.  If we do that, then we won't be able to see how the
  theory $T$ is related to the theory $T_0$.

  The confusion here is somewhat similar to Sk{\o}lem's paradox (about
  the existence of uncountable sets in models of ZF set theory), where
  we run into trouble if we don't distinguish between claims made in
  the object language and claims made in the metalanguage.  In the
  present case, one might be tempted to think of the theory $T$ as
  saying things such as:
  \begin{quote} In model $M$, $a$ is a $P$. \end{quote} Of course, $T$
  says no such thing, since it doesn't have names for models or for
  elements in models.

  The other problem here is in the way that we've set up the problem
  --- by speaking as if the representation relation holds between $M$
  (or $N$) and the world.  To the contrary, the representation
  relation holds between Berit's language and the world, and we (the
  metatheorists) are representing Berit's theorizing using our own
  little toy theory (which presumably includes some fragment of set
  theory, because that's a convenient way to talk about collections of
  formulas etc.).  Berit herself doesn't claim that $M$ (or $N$)
  represents the world --- rather, the metatheorist claims that $M$
  and $N$ represent ways that Berit's language could represent the
  world.  Accordingly, Berit doesn't claim that $M=N$, or that
  $M\neq N$; those are metatheoretical assertions --- and do not add
  to the stock of knowledge about the world.  \end{example}

Before proceeding, we should deal with an obvious objection to the
view we've put forward.  Some philosophers will point out that it is
simply false to say that physicists don't count the number of
possibilities.  Indeed, it's precisely by counting the number of
possibilities that physicists derive notions such as entropy.

We do not disagree with this point, but it doesn't conflict at all
with our positive proposal (to talk about models of a theory as a
category).  Just remember that category theory is a framework that is
almost infinitely flexible: what we can talk about in a
categorically-invariant way depends on how we --- or physicists ---
define the relevant category.  For the case at hand, if $X$ is a
classical phase space, then it is assumed that $X$ is a discrete
category --- i.e.\ that there are no non-trivial isomorphisms between
elements of $X$.  Thus, in this case, there is no question about
whether to count two isomorphic possibilities as the same; because we
(or better, the physicists) have chosen not to admit isomorphic
possibilities.

To be clear, we explicitly reject the idea that there is a single
relation ``being isomorphic'' that either holds or does not hold
between concrete objects.  On the contrary, the notion of isomorphism
applies to abstractions, and different notions of isomorphism are
valid for different levels of abstraction.  It's up to us to decide
which level of abstraction serves our purposes in reasoning about
concrete, physical reality.  (In particular, models of a theory are
not concrete realities, and that's why they cannot either be identical
or non-identical.)

For all of its other virtues, one of the defects of the semantic view
of theories is that it obscures the object/meta-language distinction,
a distinction that is absolutely necessary to make sense of the notion
of symmetry of representations.  To be more accurate, the target of
this criticism are advocates of the ``language-free'' or
``semantic-L'' view \cite[see][]{halvorson2013}. The picture we get
from the language-free semantic view is that mathematical structures
are out there in the world, and that they are either isomorphic to
each other, or they are not.  Of course, that picture completely
ignores the fact that isomorphisms are defined in terms of language;
or, to put it more accurately, that isomorphisms relate mappings
$M:\Sigma\to\cat{Sets}$ and $N:\Sigma\to\cat{Sets}$, which have a
common domain $\Sigma$.  Thus, in particular, arbitrary mathematical
structures are neither isomorphic nor non-isomorphic.

The object language $\Sigma$ serves as the reference point in defining
a notion of symmetry.  The object language tells us what must be held
fixed, and the metalanguage tells us what can be varied.  In
particular, a model $M$ of a theory $T$ can have a non-trivial
automorphism group because of two features of the formal setup:
\begin{enumerate}
\item The metalanguage describes the world in finer-grained language
  than the object language.
\item Distinctions that are not made by the object language are not
  significant for the kinds of explanations that the theory $T$ gives.
\end{enumerate}
If we drop either one of those components, then we will most likely
make a hash of the notion of symmetry.  Without the metalanguage,
there is no way to see any difference between $a$ and $b$, and so no
way to express the change the occurs in the permutation $a\mapsto b$.
But if we think of the metalanguage as a better object language, then
we shouldn't count $a\mapsto b$ as a symmetry, since these two things
are distinguishable in the metalanguage.  Thus, it's precisely the
mismatch between object language and metalanguage that provides us
with a rich notion of symmetry; and conversely, the importance of the
notion of symmetry gives us reason to maintain a distinction between
object language and metalanguage.

The distinction between object- and meta-language is one of the most
interesting ideas in 20th century logic and philosophy --- and it
remains one of the least well understood.  Obviously, Carnap made a
lot of this distinction, and in fact, he seems to use it as his
primary analogy in formulating the distinction between internal and
external questions, and more generally in understanding the
relationship between theories in the exact sciences and our other,
non-scientific beliefs and attitudes.  In contrast, Quine seems to
reject the idea that there is an important difference of status
between object and metalanguage.  He seems to propose, instead, that
the ascent to metalanguage should be seen as an extension of one's
object language --- and so assertions in the metalanguage have exactly
the same force as assertions in the object language.\footnote{I fear
  that I may be missing the nuance in Quine's position.  I would
  encourage the interested reader to consult more careful historical
  studies such as \cite[][pp 182ff]{becker} and \cite{hintikka}.}

\section{Putnam's paradox} \label{putnam}

Perhaps the most notorious argument from logical metatheory to
philosophy is Putnam's model-theoretic argument against realism
\citep{putnam1977,putnam-models}.  Here is how the argument goes.
\begin{quote}
  Suppose that theory $T$ is consistent, i.e.\ $T$ does not imply
  $\bot$, or equivalently, $T$ has a model.  Now let $W$ represent the
  collection of all actually existing objects, i.e.\ $W$ represents
  ``the world''.  Besides consistency, we will make two other minimal
  mathematical assumptions about $T$: First, the cardinality of the
  language is not so large as to force belief in the existence of too
  many objects.  In short, we require that $|\Sigma |\leq |W|$.
  Second, the theory $T$ doesn't entail that there are at most $n$
  things, for $n\in\7N$.

  We then proceed as follows: by the L{\"o}wenheim-Sk{\o}lem theorem,
  there is a model $M$ of $T$ such that $|M|=|W|$.  This means, of
  course, that there is a bijection $f:M\to W$.  Now we define another
  model of $T$, still called $W$, by setting $W(p)=f(M(p))$ for each
  relation symbol $p$ in the theory $T$.  But then the the world is a
  model of $T$.  That is, $T$ is true. \end{quote}
The above argument is intended to show that if $T$ is consistent, then
$T$ is true --- actually true, in the real world.  There is one
obvious way to try to block this argument, and that's to say that the
model $W$ may not be the ``intended'' assignment of relation,
function, and constant symbols to things in the real world.  However,
Putnam tries to block that response essentially by calling upon your
charity.  Imagine that $T$ is the theory held by some other person,
and that you're going to try your best to believe that what that
person says is true.  In other words, you are going to give her the
benefit of the doubt whenever possible.  Then what Putnam has shown is
essentially that there is a way of giving her the benefit of the
doubt.

This simple-looking argument is so subtle, and there are many ways we
might respond to it.  But let me be completely clear about my view of
this argument: it is absurd.  This version of Putnam's argument is not
merely an argument for antirealism, or internal realism, or something
like that.  This version of the argument would prove that all
consistent theories should be treated as equivalent: there is no
reason to choose one over the other.  Thus, Putnam's paradox is
essentially an argument for one of the most radically liberal views of
theoretical equivalence imaginable.  The only more radical view is the
Zenonian view, according to which all theories are equivalent.

To keep things concrete, let's suppose that $T$ is Mette's theory.
The goal of Putnam's argument is to show that Mette's theory is true.
In my view, the problematic assumption in the argument is the
following:
\begin{quote}
  (S) The world can be described as an object $W$ in the universe of
  $\cat{Sets}$.  \end{quote} The question to be raised here is: {\it
  who} is using the theory of sets to describe the world?  Putnam's
presentation makes it seem that either: (1) it's unproblematic and
theory-neutral to describe the world as a set, or (2) a realist must
describe the world as a set.  I don't agree with either claim.

Let's remember that nobody here --- including Putnam --- is free from
language and theory.  When Putnam describes the world as a set, it
might seem that he is making minimal assumptions about it.  But the
opposite is true.  When you have a set, you have all of its subsets;
and when you have two sets, you have all of the functions between
them.  To even say these things, we need the rich and expressive
language of set theory.

Thus, Putnam has set things up in a misleading way by (1) describing
the world as a set, but (2) failing to note who is responsible for
this description of the world as a set.  Suddenly it becomes clear why
Putnam's argument goes through, and why it's completely trivial.
Putnam assumes that Mette's theory $T$ is set-theoretically
consistent, which simply means that Mette's theory can be translated
into the background theory $T_0$ that was used to describe the world.
That is, there is a translation $F:T\to T_0$.  Putnam rightly
concludes that the $T_0$-theorist could take Mette's theory $T$ to be
true.  What Putnam does not show is that {\it anybody}, regardless of
their background theory, could take Mette's theory to be true.

%% putnam1

Putnam's argument should actually not make any assumption about $W$,
i.e.\ it should be like a black box.  However, Putnam begins by
assuming that there is already a fixed interpretation of ZF into the
world, i.e.\ we know what objects are, and collections of objects, and
functions between objects, etc.  He then asks whether $T$ has one (or
perhaps even many) interpretations into this already understood
domain.  And of course, the answer is Yes.

Thus, Putnam assumes that he is permitted a trans-theoretical language
to speak of the domains $M$ and $W$.  By ``trans-theoretical'' here, I
mean simply that the language of $T_0$ (in this case, ZF) is not the
same as the language of the theory $T$.  In particular, for Putnam's
argument to go through, he needs to be able to make distinctions in
$W$ that simply cannot be made by users of the theory $T$.

To make these ideas more concrete, let's consider an example: Let
$\Sigma = \{ c,d\}$, where $c$ and $d$ are constant symbols.  Let $T$
be the theory in $\Sigma$ that says $c\neq d$, and
$\forall x((x=c)\vee (c=d))$.  (This example violates the strictures
of Putnam's L{\"o}wenheim-Sk{\o}lem based argument, but the point will
not depend on those details.)  Of course, there is only one model of
$T$ up to isomorphism.  And yet, a skeptical worry arises!  Imagine
two people, Mette and Niels, both of whom accept $T$, and both of whom
think that the world is the set $\{ a,b\}$.  And yet, Mette says that
$c$ denotes $a$, whereas Niels says that $c$ denotes $b$.  Do Mette
and Niels disagree?  The answer is Yes and No.

Note, first of all, that we have already misdescribed the situation.
Mette cannot say that ``$c$ denotes $a$'', because $a$ is not a name
in her language.  Similarly, Niels cannot say that ``$c$ denotes
$b$''.  Thus, it is actually Putnam sitting in his ivory tower and
saying: ``Mette uses $c$ to denote $a$'' and ``Niels uses $d$ to
denote $b$.''  But how does Putnam's language here get a grip on the
world?  How can he tell what Mette and Niels are denoting, and that
they are different things?  Now, Putnam might claim that it is not he,
but the realist, who thinks that the world is made of things, and that
when our language-use is successful, our names denote these things.  So
far I agree.  The realist does think that.  But the realist can freely
admit that even he, the great philosopher, has just another theory,
and that his theory cannot be used to detect differences in how other
people's theories connect up with the world.  All of us, Mette, Niels,
Hilary, you, and I, are on the same level when it comes to language
use.  None of us has the metalinguistic point of view that would
permit us to see a mismatch between language and world.

Now, I suspect that some people might think that I've simply affirmed
Putnam's conclusion --- i.e.\ that I have embraced internal realism.
Given that I don't know what ``internal realism'' means, I can't
either affirm or deny that claim.  But I do insist that Putnam's
argument doesn't work; for if it works, then we have no reason to
discriminate between (ideal) consistent theories, and we are forced to
adopt an absurdly liberal account of theoretical equivalence.

Consider another scenario, where now I, rather than Putnam, get to
choose the rules of the game.  In other words, I have my own theory
$T_0$ of which I believe the world $W$ is a model.  Then along comes
Putnam and says that any consistent theory can be interpreted into the
world $W$.  But if my background theory $T_0$ is not ZF, then I don't
see $W$ as a set, and Putnam's argument cannot even get started.  In
particular, I don't necessarily grant that there is an isomorphism
$f:M\to W$ between a model $M$ of $T$ and this model $W$ of my theory
$T_0$.  For one, what would I even mean by the word ``isomorphism''?
I, the user of the theory $T_0$, know about isomorphisms between
models of my theory.  However, $M$ is a model of a different theory
$T$, written in a different signature $\Sigma$, and so there may be no
standard of comparison between models of $T$ and models of $T_0$.

There is still another more severe problem for Putnam's argument.  For
a scientific theory to be ``ideal'' it's really not enough for it to
correctly report every actual fact.  In must do more!  There are a few
ways to get a handle on what more a good scientific theory must do.
David Lewis recognized that the ``best theory'' is not simply one that
gets every fact correct.  Instead, the ``best'' achieves an ideal
balance of strength and simplicity.  Here ``strength'' means reporting
the facts, and simplicity means \dots well, we all know it when we see
it, right?  Whether or not we philosophers have a good account of
simplicity, the fact is that Lewis was right that there is (at least)
a second component to theory evaluation, and it has something to do
with systematicity, or choosing the right language, or cutting nature
at the joints.

Thus, when I'm looking at a scientific theory, I'm not just interested
in whether it's true.  You could write down every truth in a massive
encyclopedia, and I wouldn't consider it to be the best scientific
theory.  There are better and worse ways to say the truth.  And what
this means for our considerations here is that not all true theories
are created equal; thus, certainly not all ideal consistent theories
are equal.

We might want to go to the trouble of explaining when I, user of
theory $T_0$, would grant that $T$ can be interpreted into a model $M$
of my theory.  In the simplest sort of case, I would require that for
each relation symbol $p$ of $T$, there is a formula $Fp$ of the
appropriate arity of my language $\Sigma _0$ such that $p$ can be
interpreted as $M(Fp)$.  As a user of theory $T_0$, I only recognize
those subsets of $M$ that can be described via the predicates of my
theory.  In particular, I don't necessarily have the resources to name
the elements of the domain $M$, and I don't necessarily have the
resources to collect arbitrary elements of $M$ and form subsets out of
them.  I can only talk about ``things that satisfy $\phi$'', where
$\phi$ is one of the predicates of my language.

So, suppose then that $T$ is consistent relative to my theory $T_0$:
for each model $M$ of my theory, there is a model $M^*$ of $T$ with
the same domain as $M$, and such that for each relation symbol $p$ of
$\Sigma$, $M^*p = M(Fp)$ for some formula $Fp$ of my language
$\Sigma _0$.  However, even in this scenario, I wouldn't necessarily
consider the theory $T$ to be adequate, for it may fail to pick up the
relationships between various models of my theory.  I'd want to know
that the user of $T$ recognizes the same connections between models
that I do.  In particular, where I see an elementary embedding
$h:M\to N$, I would require the user of $T$ to see a corresponding
elementary embedding $h^*:M^*\to N^*$ between models of his theory
$T$.  And that just means that $h\mapsto h^*$ completes the definition
of a functor from $\mathrm{Mod}(T_0)$ to $\mathrm{Mod}(T)$, where the
object part is given by $M\mapsto M^*$.  We then have the following
result.

\begin{prop} Let $F$ be a map of $\Sigma$-formulas to
  $\Sigma _0$-formulas such that the map $M(\phi )\mapsto M(F\phi )$
  defines a functor from $\mathrm{Mod}(T_0)$ to $\mathrm{Mod}(T)$.
  Then $F:T\to T_0$ is a translation. \end{prop}

\begin{proof} Define a reconstrual $G:\Sigma\to\Sigma _0$ by setting
  $Gp=Fp$.  We claim that $T_0\vdash G\phi\lra F\phi$ for all formulas
  $\phi$ of $\Sigma$.  For this, it suffices to run through the
  clauses in the definition of $F$.  For example, we need to check
  that $F(\phi _1\wedge \phi _2)\equiv F(\phi _1)\wedge F(\phi _2)$,
  where $\equiv$ means provable equivalence modulo $T_0$.  But this is
  easy to check: let $M$ be an arbitrary model of $T_0$.  Then
  \[ \begin{array}{l l l} M(F(\phi _1\wedge \phi _2)) & \equiv & M^*(\phi _1\wedge \phi _2) \\
    &\equiv & M^*(\phi _1)\cap  M^*(\phi _2) \\
     & \equiv & M(F(\phi _1))\cap M(F(\phi _2) \\
     &\equiv & M(F(\phi _1)\wedge F(\phi _2)) .\end{array} \]
(Here I've ignored for simplicity the fact that $\phi _1$ and $\phi _2$
might have different free variables.)  The clauses for the other
connectives and for the quantifiers are similar.  \end{proof}

The upshot of this result for Putnam's argument is as follows: a user
of a theory $T_0$ should only grant that $T$ can be true if there is a
translation of $T$ into $T_0$.  This result is not surprising at all.
In real life, this is the sort of criterion we do actually employ.  If
I hear someone else speaking, I judge that what they are saying
``could be true'' if I can reconstrue what they are saying in my
language.  If there is no way that I can interpret their utterances
into {\it my} language, then I am forced to regard those utterances as
false or meaningless --- unless I decide to adopt a new language.

\newcommand{\3}{\cal}

As Otto Neurath pointed out, and as Quine liked to repeat, we cannot
start the search for knowledge from scratch.  Each of us already has a
theory, or theories.  And we have a notion of permissible translations
between theories which regulates (or describes) our attitude about
which other theories could potentially be correct.  If a theory $T$
can be conservatively translated into my theory $T_0$, then I will
think that $T$ might possibly say something true (perhaps if its terms
are charitably interpreted).  But even then, I would not necessarily
judge $T$ to be true.  Indeed, if my standard of theoretical
equivalence is weak intertranslatability, then I will judge $T$ to be
potentially true (even under the most charitable interpretation) only
if $T$ and $T_0$ are weakly intertranslatable.  (And do recall that
weak intertranslatability is a fairly conservative criterion of
equivalence.)

What Putnam has shown, at best, is that {\it relative} to a background
theory $T_0$ of bare sets, a theory $T$ that has a model in
$\cat{Sets}$ could be charitably interpreted as true by a user of
$T_0$.  The result is really not very interesting --- except insofar
as it reminds us of the dangers of uncritically accepting set theory
as our background metatheory.  Indeed, set theory makes non-trivial
existence claims, e.g.\ the claim that any two points in a model are
related by a permutation.

The things I've just said might sound quite similar to Lewis'
\citeyearpar{lewis-putnam} response to Putnam's argument.  You'll
recall that Lewis attempts to block the argument precisely by denying
the permissibility of the relevant permutation --- or, what's the
same, of denying that each subset of \textsc{World} picks out a
genuine property.  But Lewis' response is not, by itself, sufficient
to block Putnam's argument.  Suppose indeed that we've identified a
privileged subclass $\cal{N}$ of natural properties among the subsets
of \textsc{World}.  We can also require, as Lewis does, that a
predicate symbol $p$ of the signature $\Sigma$ must be assigned to a
set $M(p)\in \cal{N}$.  In other words, $M$ cannot assign $p$ to any
old subset of \textsc{World}.

What Lewis has done here, in effect, is to propose an extension of
Putnam's background theory $T_0$, by means of adding predicate symbols
to the signature $\Sigma _0$ in order to designate the subsets in
$\cal{N}$.  Let $T_1$ be Lewis' strengthened background theory --- the
theory that describes the world as a set \textsc{World}, with a
privileged family $\3N$ of subsets of \textsc{World} to represent the
natural properties.  Then Lewis' requirement that the predicates of
$T$ be interpreted as elements of $\3N$ is simply Lewis saying that
for him to count the theory $T$ as true, there would have to be an
interpretation of $T$ into his background theory $T_1$.  Since Lewis'
background theory $T_1$ is expressively weaker than Putnam's
background theory $T_0$, it is more demanding to ask for an
interpretation of $T$ into $T_1$ than it is to ask for an
interpretation of $T$ into $T_0$.

It's easy to see that this maneuver does indeed block complete
trivialization: for some choices of $\3N$, there are theories $T$ that
are set-theoretically consistent, but that cannot be translated into
Lewis' background theory $T_1$.  To take one trivial example, suppose
that $T_1$ has three natural properties: the emptyset, the entire
world, and some proper subset $A$ of the world.  Suppose that $T$ says
that
\[ \exists xPx\wedge \exists yQy\wedge \neg\exists z(Pz\wedge Qz) .\]
Then $T$ is set-theoretically consistent, but $T$ cannot be translated
into $T_1$.

Nonetheless, Lewis' demands here are not strong enough.  In general,
for any sufficiently rich family of natural properties $\3N$, too many
theories $T$ will be interpretable into Lewis' background theory
$T_1$.  And hence, if Lewis grants Putnam's call for charitable
interpretation, then Lewis will have to say that those theories are
true.  That concedes too much.  It is easy to think of examples that
would make a realist choke.  For example, suppose that Gargamel has a
theory that says there are many gods, and there are no electrons.  If
Lewis countenances just a single natural property with instances, then
Gargamel's theory can be translated into Lewis' background theory ---
and, by the principle of charity, should be counted as true.

We will not engage now in further formal investigation of these
matters, e.g.\ to ask how many natural properties there need to be in
order for a given theory $T$ to be interpretable into Lewis'
background theory.  We don't think that question is very interesting
--- because we've already gone off on a bad track.  There are two
interrelated problems here.  The first problem is that Lewis'
background theory $T_1$ has little to recommend it, even if we are
inclined to accept that there are ``natural properties.''  (And anyone
who uses first-order logic implicitly does accept the existence of
natural properties --- they are precisely the properties that are
definable in her language.)  The second, and deeper, problem is that
Lewis, like Putnam, seems to be supposing that all parties --- or at
least all metaphysical realists --- can agree on some particular fixed
background theory $T_\ast$.  We reject that assumption, and as a
result, Putnam's paradox simply dissolves.








%% I do want to say something now about translations, and about the
%% idea of interpreting $T$ into a model of $T'$.  The point is that
%% for a user of $T$, we judge that $T'$ is true if there is an
%% appropriate interpretation of $T'$ into $T$






%% bring into question Putnam's idea of "ideal theory".  And this is
%% important because it brings into question that idea tout court

%% TO DO: Lewis and natural properties


\section{Realism and equivalence}

According to the standard stereotype, the logical positivists were
{\it antirealists} or {\it instrumentalists} about scientific
theories.  Moreover, this antirealist stance was facilitated by means
of the syntactic analysis of scientific theories, according to which a
theory $T$'s language has some purely observational terms $O$, and its
empirical content can be identified with $T|_O$.  With this formal
analysis, the positivists could then articulate their particular
versions of epistemic and semantic empiricism:
\begin{itemize}
\item Epistemic empiricism: the reasons we have to believe $T$ derive
  from reasons we have for believing $T|_O$.
\item Semantic empiricism: the meaning of terms in
  $\Sigma\backslash O$ derives exclusively from the meaning of terms
  in $O$.
\end{itemize}
The extreme instrumentalist would say that the terms in
$\Sigma\backslash O$ have no meaning: there are merely instruments to
facilitate making predictions.  The attenuated instrumentalist tries
to find a way for terms in $\Sigma$ to inherit meaning from terms in
$O$.

In the 1960s and 1970s, the syntactic view of theories was
discredited, and the tide seemed to have turned decisively against
antirealism --- or at least against this stereotyped antirealism.
Without a clear delineation of the empirical part of a theory, it was
no longer possible to think that warrant or meaning could flow upward
from the observationally relevant parts of a theory.

Van Fraassen characterized the state of play in 1976: ``After the
demise of logical positivism, scientific realism has once more
returned as a major philosophical position'' \citep[623]{bas1976}.  He
goes on to characterize scientific realism as commitment to the
following thesis:
\begin{quote}
  The aim of science is to give us {\it a literally true story of what
    the world is like}; and the proper form of acceptance of a theory
  is to believe that it is true.  \citep[623]{bas1976} \end{quote} As
is well-known, van Fraassen then gave several strong arguments against
scientific realism, before going on to develop his positive
alternative: {\it constructive empiricism}.

In the years that followed, there was much back and forth debate: van
Fraassen on the side of constructive empiricism --- and dozens of
other philosophers on the side of scientific realism.  Interestingly,
however, the terms of this debate had been set by van Fraassen, and
these terms were rarely (if ever) questioned.  In particular, the
scientific realists seem to have been happy enough with van Fraassen's
characterization of their position; their job was merely to bring out
its merits.

However, if we look more closely, it becomes apparent that the debate
wasn't so clear cut.  During the 1960s and 1970s, scientific realists
were fond of saying that the philosophical position of scientific
realism is itself a scientific hypothesis, and that the reasons for
believing it are of the same nature as the reasons for believing any
other scientific theory.  In particular, they claimed that the
hypothesis of scientific realism is the {\it best explanation} for the
success of the scientific enterprise.

Now, van Fraassen certainly questioned the latter claim.  But more
interestingly, he chose not to play by the same game as the scientific
realists.  For van Fraassen, the reasons for being a constructive
empiricist are different in kind from the reasons for accepting a
scientific theory.  For those who were following the debate closely,
it became clear that the choice between realism and antirealism about
science was not a simple disagreement about which hypothesis better
explains a common domain of phenomena.  There was a deeper and more
elemental disagreement about the goals of philosophical reflection.

For many philosophers of the next generation, the question of
scientific realism versus scientific antirealism had receded too far
into the upper reaches of metaphilosophy.  The simple ``pro and con''
arguments of the 1970s and 1980s were not going to get us anywhere,
seeing that the opposed parties were using different standards to
evaluate these arguments.  Thus, the next generation of philosophers
of science moved downward --- back to the analysis of specific
scientific theories.  Although they may not openly use these words, I
suspect that many philosophers of science now feel that ``realism or
antirealism?'' is a pseudoquestion, or at least not a particularly
interesting question.

Speaking of pseudoquestions, what makes a question pseudo?  Here is
one criterion: a question is pseudo if getting an answer to it
wouldn't change anything you do.  By that standard, it's easy to see
why the realism-antirealism debate might seem like a pseudodebate.
Would a scientist do anything differently tomorrow if he converted to
constructive empiricism?  Wouldn't he go on looking for the elegant
and powerful theories, and using them to make predictions and give
explanations?

This last thought suggests a better way to understand what's really at
stake in the realism-antirealism debate.  I suggest that the debate
can be fruitfully reconceived as a battle over standards of
theoretical equivalence.  In particular, a realist is somebody who
adopts --- or recommends that people adopt --- stricter standards of
theoretical equivalence.  Conversely, an antirealist is somebody who
adopts --- or recommends that people adopt --- looser standards of
theoretical equivalence.  In short, realists are conservatives about
theoretical equivalence, and antirealists are liberals about
theoretical equivalence.

This construal of the realism-antirealism debate matches well with
various well known cases.  Consider, for example, the case of the
logical positivists.  We tend to think that they were antirealists
because they said: ``the content of the theory $T$ resides in its
observable part $T|_O$''.  But there are a lot of unclear words here,
such as ``content'' and ``residing'' and ``observable'', and so this
doesn't make for a very sharp statement of a philosophical thesis.
However, one concrete implication of these positivist words is that if
$T|_O=T'|_O$, then we should treat $T$ and $T'$ as equivalent.  For
example, suppose that two scientists, say Werner and Erwin, have
apparently conflicting theories $T$ and $T'$ with the same empirical
content, i.e.\ $T|_O=T'|_O$.  Then the positivist would recommend that
Werner and Erwin reconcile, for there can be no reason to prefer $T$
over $T'$ or vice versa.  The difference between their theories is no
more important than the difference between theories written in German
and French.  In contrast, if $T$ says {\it anything} that conflicts
with $T'$, then the scientific realist thinks that one of the two must
be better than the other, and that we should actively pursue inquiries
to determine which it is.

This picture of the realism debate also makes sense of what structural
realists were trying to achieve.  In short, structural realists urge
that not every single detail of a successful scientific theory should
be taken with equal seriousness.  In particular, they argue that if
two theories $T$ and $T'$ differ only with respect to content, and not
with respect to structure, then one can have no reason to prefer $T$
over $T'$, or vice versa.  The normative core of structural realism,
then, is to propose a notion of theoretical equivalence that lies
somewhere to the left of the dangerously right-leaning realist view,
and the dangerously left-leaning views of the logical positivists and
the later Putnam.

Scientists and philosophers, and in fact everyone, has implicit
standards of equivalence that they employ to judge between truth
claims, especially when those claims seem {\it prima facie} to
conflict.  If you believe ``God doesn't exist'', and your French
colleague believes ``{\it Dieu n'existe pas}'', then you know that
there is no dispute to be settled.  Not only are those two sentences
compatible with each other, they are equivalent.  Even within a single
language, we can say the same thing in different ways.  Imagine that
your friends Anne and Bent disagree about the number of roses in the
vase on the counter.  Anne says, ``there are six roses'', and Bent
says, ``there are a half dozen roses''.  In such a case, you would
surely advise Anne and Bent to kiss and make up, since their dispute
is merely verbal.

Those cases are easy.  But there are more difficult cases in life ---
especially as we move into the more abstruse regions of the sciences.
(And that's not even to speak of cases such as differences in matters
of politics or religion.)  For example, there is a debate among
evolutionary biologists about the units of selection: is it the
individual, or the species?  Many a friendship between scientists has
been broken because of disagreement on issues like this one.  But what
if there really was no dispute between them?  What if they were saying
the same thing in different terms?

You might think such a scenario is unimaginable.  But if the history
of science can be trusted, then there have been numerous cases where
{\it prima facie} disagreement has later been judged to be spurious.
For example, in the mid 1920's, Werner Heisenberg developed a theory
that made use of non-commutative algebra in order to predict the
outcomes of measurements.  This theory, called {\it matrix mechanics}
was hailed by many as a breakthrough, for it unified the {\it ad hoc}
recipes that plagued the old quantum theory of Bohr and Sommerfeld.
However, others abjured matrix mechanics, on the grounds that it was
incomprehensible, unvisualizable, and entailed bizarre claims, most
notably the existence of ``quantum jumps''.  Thus, a competing theory
was developed by Erwin Schr{\"o}dinger, a theory based on completely
different ideas and mathematical techniques.  According to
Schr{\"o}dinger's theory, there are waves moving through physical
space (or a higher dimensional configuration space), and particles
such as electrons are simply harmonic resonances in these waves.

Thus, Heisenberg presented one theory, $T_1$, to account for the
quantum phenomena, and Schr{\"o}dinger presented another theory,
$T_2$, to account for the same phenomena.  While both these were
empirically adequate, the battle between Heisenberg and
Schr{\"o}dinger was fierce, including name calling, a battle for
prominence at professional meetings, and competition for funding and
university positions.  The behavior of Heisenberg and Schr{\"o}dinger
clearly indicated that they saw this debate as {\it genuine}, and in
need of resolution.

The conclusion of this story is typically told as follows.  Based on
some suggestions that Schr{\"o}dinger himself made, a young
mathematician John von Neumann formulated a conjecture: Heisenberg's
matrix mechanics $T_1$ is equivalent to Schr{\"o}dinger's wave
mechanics $T_2$.  Von Neumann then went on to prove this theorem, to
the great satisfaction of most participants involved, especially those
like Niels Bohr, who didn't want to choose between Heisenberg and
Schr{\"o}dinger.  As a result, {\it the debate came to an end}.  Since
$T_1$ and $T_2$ are equivalent theories, there is no question about
which one is better, at least not in any epistemically or
ontologically relevant sense.  There is no decision to be made about
whether to accept $T_1$ or $T_2$.

Such is the nature of judgments of theoretical equivalence.  When one
judges that theories $T_1$ and $T_2$ are equivalent, one judges that
accepting $T_1$ is tantamount to accepting $T_2$.  Conversely, if one
feels that $T_1$ might be favored over $T_2$, or vice versa, then one
judges that these theories are {\it not} equivalent.

Are there equivalent theories?  Setting aside the
Heisenberg-Schr{\"o}dinger theory as controversial, still every sane
person will admit that at least some theories are equivalent.  For
example, say that theory $T_1$ is the version of general relativity
written down in the textbook {\it General Relativity} by Robert Wald
that is sitting on the shelf in my office.  Let $T_2$ be the version
of general relativity written down in the textbook {\it General
  Relativity} by Robert Wald that is sitting on the shelf in Carlo
Rovelli's office.  Of course, we all judge that $T_1$ and $T_2$ are
equivalent.  In fact, our initial reaction is to say that these are
the {\it same} theory, and that's why we use a definite description
for it: ``{\it the} general theory of relativity''.  But if we boil
everything down to fundamental physics, then it's hard to see what
these two theory-tokens have in common.  One consists of ink marks on
paper in Princeton, NJ, and the other consists of ink marks on paper
in Marseilles.  The ink marks are different, and hence whatever they
constitute is different was well.

In my experience, philosophers' immediate reaction to this sort of
example is to fly to the realm of abstract entities.  They say
something like this: your book and Rovelli's book contain sentences
that pick out the same {\it propositions}; hence the sentences in
these two books both represent the same theory.  Now, I don't disagree
with their claim, I only doubt its utility.  If you give me two
languages I don't understand, and theories in the respective
languages, then I have no idea initially whether those theories pick
out the same propositions.  And that's precisely the sort of case we
face with something like matrix and wave mechanics.  Employing new
formalism that is not yet very well understood, it is unclear whether
these theories say the same thing.  Thus, we need some criterion for
equivalence that is {\it checkable}, at least in principle.  In other
words, we need to know when two sentences pick out the same
proposition.

There are essentially two ways to proceed from here.  On the one hand,
we can ask: what features must two theories have in common in order to
be equivalent?  In philosophical jargon: what are the necessary
conditions for theoretical equivalence?  This question can also be
given a mathematical gloss: what are the {\it invariants} of
theoretical equivalence?  For example, some people would say that for
two (single sorted) first-order theories $T_1$ and $T_2$ to be
equivalent, they must agree on the number of existing objects.  There
are other conditions we might try to impose, but which are a bit more
difficult to cash out in terms of formal logic.  For example, many
contemporary philosophers would say that two equivalent theories must
have the same {\it primitive notions}, i.e.\ those objects,
properties, etc.\ that ground the other things that the theory
mentions.  However, it is not obvious, to me at least, what this
particular idea of primitive notions could be explicated by means of
the structure of first-order theories.

The second question we could ask has a more top-down flavor: could we
simply define an equivalence relation on the collection $\mathbf{Th}$
of all theories.  Disregarding the fact that $\mathbf{Th}$ is a proper
class and not a set, there will be {\it many} such equivalence
relations, all of which yield some notion of theoretical equivalence.
Among these untold number of equivalence relations, there are some
that have relatively simple mathematical definitions.  For example,
the notion of {\it intertranslatability} gives rise to one such
equivalence relation.

An ideal method, I think, is to take both procedures into account.  On
the one hand, we need not accept a definition of equivalence if it
violates necessary conditions to which we are committed.  On the other
hand, some of us might feel compelled to abandon an intuitive
necessary condition of equivalence --- i.e.\ some intuitive invariant
of theoretical equivalence --- if it conflicts with what otherwise
seems the most reasonable formulation of an equivalence relation on
$\mathbf{Th}$.

We can see these sorts of choices and trade-offs being made all the
time in philosophy.  On the more conservative side, philosophers such
as David Lewis and Ted Sider lay heavy stress on choosing the right
primitives.  At times it seems as if they would go so far as to say
that there is a {\it privileged language} for metaphysics, so that no
theory in this language could be equivalent to a theory that is not in
this language.  (Of course, one wonders then how they individuate
languages.)

One could imagine an even more conservative stance on theoretical
equivalence.  For example, suppose that $\Sigma$ is a fixed signature
(say, the preferred signature for metaphysics), and $T_1$ and $T_2$
are theories in $\Sigma$ that have the same consequences
(equivalently, have the same models).  Should we then consider $T_1$
and $T_2$ to be equivalent.  I suspect that Sider would say Yes.  But
I also suspect that some philosophers would have said No, for they
might have thought that there are even preferred ways of axiomatizing
a theory.  Indeed, if you really believe that some facts are more
basic than all the others, then shouldn't those facts be the ones
enunciated in the axioms, so that all other facts are seen as flowing
from them?  Thus, we get an even finer-grained equivalence relation on
$\mathbf{Th}$ if we demand that equivalent theories are in the same
signature, {\it and} have the same axioms.

Even that requirement --- having the same axioms --- is not the most
conservative imaginable.  We might even require that the theories
literally have the same notation.  For example, in formulating group
theory, we could use the symbol $\circ$ for the binary relation, or we
could use the symbol $\bullet$.  Who knows, perhaps one of these two
symbols more perspicuously represents the structure of the binary
function in the world that we are trying to represent.  At the
farthest end of this spectrum, one could adopt a pure ``Heraclitean''
account of theoretical equivalence, according to which no two theories
are the same.  In other words, the criterion of theory identity could
be made out to be {\it literal identity} --- of symbols, axioms, etc..

Conservative views of theoretical equivalence tend to align with
``realist'' views about science or metaphysics.  Roughly speaking, if
you think that the world has real structure, then you'll think that a
good theory has to represent the structure that is out there.  If two
theories disagree about that structure, then they cannot be
equivalent.  Going in the opposite direction, liberal views of
theoretical equivalent tend to align with ``antirealist'' views about
science and metaphysics.  We see this tendency with Nelson Goodman in
the 1960s, and with Hilary Putnam in the 1970s.  Putnam's move toward
antirealism was augured by his giving many examples of theories which
he says are equivalent, but which realists regard as being
inequivalent.  For example, Putnam claims that Euclidean geometry
based on points is equivalent to Euclidean geometry based on lines ---
even though the models of these two theories can have different
cardinalities.

Long before Putnam turned in this direction, the connection between
antirealism and liberal views of theoretical equivalence had already
been established.  I'm thinking here of the logical positivists and
their infamous notion of {\it empirical equivalence}.  The idea here
is that two theories $T_1$ and $T_2$ are empirically equivalent just
in case they share the same observable consequences --- and regardless
of what else these might say.  So, to take an extreme example, if
$T_2$ is $T_1$ plus the sentence, ``there is a new unobservable
particle,'' then $T_1$ and $T_2$ are empirically equivalent.

Now, for the logical positivists --- or at least, for some of them ---
empirical equivalence is equivalence enough.  For they identified the
content of a proposition with that proposition's empirical
consequences; and it follows from this that if two propositions $\phi$
and $\psi$ have the same empirical consequences, then they have the
same content, i.e.\ they are the same proposition.  Stepping back up
to theories, as collections of propositions, the positivist view of
content entails that two theories are equivalent {\it tout court} if
they are empirically equivalent.

The positivist view of theoretical equivalence is quite liberal, and
certainly unacceptable to scientific and metaphysical realists.  Most
of us have the intuition that theories can say different things about
unobservable things, even if those theories agree in all their
observational consequences.  But in this case, we have to reject
empirical equivalence as a sufficient condition for theoretical
equivalence.

A case can be made that Putnam's view of theoretical equivalence
eventually became --- at least tacitly and in practice --- even more
liberal than empirical equivalence.  In putting forward the
model-theoretic argument, Putnam essentially makes an argument for the
following claim:
\begin{quote} If $T$ is consistent (and has other virtues such as
  completeness), then $T$ ought to be taken as true. \end{quote} Now,
in application to {\it two} consistent theories $T$ and $T'$, we have
the following result:
\begin{quote} If $T$ and $T'$ are consistent (and have other virtues
  such as completeness), then $T$ and $T'$ ought both to be taken as
  true.  \end{quote} In other words, consistent, ideal theories are
true in all conditions, hence in all the same conditions, and so they
are equivalent.  That is a radically liberal view, almost Zeno-like in
its implications.  For in this case, there is only one equivalence
class of consistent theories.

What I've left out from this story so far are all the intermediate
(and more plausible) views of theoretical equivalence --- views that
we have been discussing throughout this book, such as definitional
equivalence or Morita equivalence.  To put everything together,
consider the diagram below, which places the different views of
theoretical equivalence on a one-dimensional spectrum from maximally
liberal (Zenonian) to maximally conservative (Heraclitean).
\begin{quote} Zeno $\leftarrow$ categorical $\leftarrow$
  w-intertranslatable $\leftrightarrow$ Morita $\leftarrow$
  s-intertranslatable $\leftrightarrow$ CDE $\leftarrow$ logical
  $\leftarrow$ Heraclitus \end{quote} So, given this wide range of
different notions of equivalence, how are we to choose among them?
And do we need to choose among them?  I would say that we don't have
to explicitly choose among them --- but that our attitudes towards
them mirror our attitudes towards real life cases, or at least to
cases that come up in other philosophical discussions.  Consider, for
example, North's \citeyearpar{north2009} argument for the
inequivalence of Hamiltonian and Lagrangian mechanics.  She says,
``Hamiltonian and Lagrangian mechanics are not equivalent in terms of
statespace structure.  This means that they are not equivalent,
period.''  In other words, she's putting a model of Hamiltonian
mechanics next to a model of Lagrangian mechanics, and comparing
structure.  Seeing that these structures are not ``equivalent'', she
declares that the theories are not equivalent.  We see then that, at
the very least, North adopts a criterion that is more conservative
than categorical equivalence, which has no interest in the internal
structure of individual models.  (In fact, \cite{barrett-hamiltonian}
shows that Hamiltonian and Lagrangian mechanics are categorically
equivalent.)  Most likely, North's criterion is further to the right
than even Hudetz's constructible categorical equivalence, for she
doesn't consider questions as to whether Lagrangian structure can be
constructed from Hamiltonian structure, and vice versa.

We can see a similar thought process going on with critics of
quantifier variance.  Indeed, we can think of debates about quantifier
variance as debates about which notion of theoretical equivalence to
adopt.  The opponents of quantifier variance insist that
equinumerosity of models is a necessary condition for theoretical
equivalence.  Thus, they draw the line short of Morita equivalence,
which allows that equivalent theories can have models of different
cardinalities.  In contrast, defenders of quantifier variance
realize that there is certainly a sense in which theories can be
intertranslatable even if they violate that cardinality constraint.
The question boils down to which criterion of theoretical equivalence
is the better one to adopt?

I believe that this is one of the most interesting questions that
philosophers can ask, precisely because it's a non-factual question.
Or, to put it more accurately, the answer that one gives to such a
question determines what one thinks is a factual question --- and so
it's not the kind of question that two parties can easily resolve by
appeal to a shared stock of facts.  Nonetheless, we've made a lot of
progress on the technical side, so that we now have a much more clear
sense of what's at stake, and of the price we must pay for adopting
some particular formal notion of equivalence as an intuitive guide to
our practice of judging between theories.

Consider, for example, the locus of debate between definitional
equivalence and Morita equivalence, or what is essentially the same,
between s-intertranslatability and w-intertranslatability.  This is
precisely the flashpoint for recent debates in metametaphysics.  The
first thing to notice is that both of these views of theoretical
equivalence are quite moderate and centrist!  Supposing that
quantifier variantists are advocating something like Morita as their
preferred notion of theoretical equivalence, they are nowhere near as
liberal as the Putnam of the model-theoretic argument.

We can also see that the ontology of w-intertranslatable theories can
never be radically different from each other.  If $F:T\to T'$ is a
homotopy equivalence (between single-sorted theories), then for each
model $M$ of $T'$, there is a model $F^*M$ of $T$, whose domain is
explicitly constructed by the recipe:
\[ (F^*M)(\sigma ) \: = \: M(\sigma ')\times\cdots\times M(\sigma
  ')/\sim ,\] where $\sim$ is an equivalence relation defined by the
theory $T'$.  There are a couple of important points here.  First, the
ontology of $F^*M$ results from simple logical constructions of the
ontology of $M$.  Borrowing terminology from Bertrand Russell, we
could say that the elements of $F^*M$ are {\it logical constructs} of
elements of $M$.  Second, the recipe for constructing $F^*M$ from $M$
is {\it uniform}, i.e.\ it doesn't depend on $M$.  In other words,
it's not just that each model of $T$ consists of logical constructs of
elements of a model of $T'$, it's that the type of construction is
uniform.  It's in this extended, and yet nonetheless quite strong
sense, that $T$ has the same ontology as $T'$.

Moreover, since $F$ is assumed to be a homotopy equivalence, we can
say the same thing in reverse order: each model of $T'$ consists of
logical construct of elements of a model of $T$, and this construction
is uniform on models.  One bonus insight here is seeing how the
relation ``being a logical construct of'' differs from the
mereological parthood relation.  Consider the specific example of the
point and line formulations of affine geometry (see Section
\ref{go-geometry}).  Here the points are logical constructs of lines,
and the lines are logical constructs of points.  It's tempting to
think then that points are logical constructs of points --- but that
would be incorrect.  The reason that inference doesn't go through is
that ``being a logical construct of'' is not like the mereological
notion of parthood.  To get a line, we don't simply take two points,
we take an equivalence class of two points.  Thus, there is no sense
here in which a line results from taking a composite of points.  The
opposite direction is even more clear.  We can construct points from
lines, but certainly a point is not made out of lines.

The upshot of these considerations is that moving from definitional
equivalence to Morita equivalence is not as radical a generalization
(or liberalization) as it might seem at first.  Even for the
ontological purists, a case could be made that Morita equivalence
involves only the slightest relaxation of the constraint that
equivalent theories should have equinumerous domains.

In contrast, categorical equivalence is extremely liberal from an
ontological point of view.  It's possible, indeed, to have
categorically equivalent theories where there is no reasonable sense
in which the ontology of the first's models can be constructed from
the ontology of the second's models.

There are, however, some intermediate cases that are worth
considering.  Some of these are discussed by \cite{hudetz-new}.  Here
we just look at one example that will be familiar from Chapter
\ref{cat-prop}.  Consider the categories $\cat{Bool}$ of Boolean
algebras, and $\cat{Stone}$ of Stone spaces.  As we proved,
$\cat{Bool}$ is equivalent to the opposite of $\cat{Stone}$, where the
arrows have been flipped.  Moreover, the functors relating these two
categories do have a strongly constructive flavor.  The functor
$F:\cat{Bool}\to\cat{Stone}^{op}$ is the representable functor
$\mathrm{hom}(-,2)$, where $2$ is the two-element Boolean algebra.
The functor $G:\cat{Stone}^{op}\to\cat{Bool}$ takes the clopen
subsets.  In both cases, the functor involves construction an object
of one category out of an object of the second category, and possibly
some reference object, such as $2$.

Could these latter sorts of functors be taken as representing genuine
theoretical equivalences?  There are two clarifications we need to
raise for that question.  First, the question doesn't even make sense
until we say something more about how a category, which may not be of
the form $\mathrm{Mod}(T)$ for a first-order theory, can represent a
theory.  Second, for many physical theories --- and {\it pace} Quine
--- the elements of a mathematical domain $X$ are not necessary meant
to represent objects in the physical world.  Consider the following
example which, besides being extremely interesting in its own right,
illustrates several of these points.

General relativity (GTR), qua mathematical object, can roughly be
taken to be the category $\cat{Lor}$ of Lorentzian manifolds, equipped
with an appropriate collection of smooth mappings between them.  There
has been a longstanding debate --- stimulated, no doubt, by Quine's
criterion of ontological commitment --- about whether accepting GTR
demands that one accept the existence of spacetime points.  Perhaps
partially in response to that claim, John Earman noted that GTR could
also be formulated in terms of mathematical objects called ``Einstein
algebras''.  The relationship between Lorentzian manifolds and
Einstein algebras is suggestively parallel to the relationship between
Stone spaces and Boolean algebras.  This parallel was confirmed by
\cite{rosenstock}, who showed that $\cat{Lor}$ is equivalent to the
category $\cat{EAlg}$ of Einstein algebras.

If one takes categorical equivalence as the criterion for theoretical
equivalence, then the Einstein algebra formulation of GTR is no better
nor worse than the Lorentzian manifold formulation.  However, one
might also wish to draw a stronger conclusion: one might wish to say
that Rosenstock et al.'s proof shows that accepting GTR does {\it not}
involve ontological commitment to spacetime points.

However, that conclusion would be hasty.  The implicit argument
pattern here would run as follows:
\begin{quote}
  Let $T$ be a theory with a sort $\sigma$.  If $T$ is equivalent to
  $T'$, and $T'$ doesn't quantify over $\sigma$, then to accept $T$
  cannot involve ontological commitment to things of type
  $\sigma$. \end{quote} To see that this inference pattern proves too
much, we can consider some simple examples.  First, consider the
example of the theory $T$ in sort $\Sigma =\{\sigma \}$ that says
there are exactly two things; and consider the theory $T'$ in sort
$\Sigma '=\{\sigma '\}$ that says there are exactly two things.  By
the above inference rule we would have to conclude that accepting $T$
does not demand ontological commitment to things of type $\sigma$,
merely because there is another sort symbol $\sigma '$.  This is
silly.  The difference between $\sigma$ and $\sigma '$ could be merely
notational.

Perhaps then the argument pattern above is meant to be a bit more
nuanced.
\begin{quote} Let $T$ be a theory with sort $\sigma$.  If $T$ is
  equivalent to a theory $T'$, and $T'$ has no sort $\sigma '$ that is
  ``isomorphic'' to $\sigma$, then accepting $T$ does not involve
  ontological commitment to things of type $\sigma$. \end{quote} The
word ``isomorphic'' was put into quotes because we would still need to
explicate what we mean by it.  But that could be done.  e.g.\ we might
say that an equivalence $F:T\to T'$ shows that $\sigma$ and $\sigma '$
isomorphic if $F(\sigma )=\sigma '$ and $E_{x,y}\equiv (x=_\sigma y)$
for variables $x,y$ of sort $\sigma$.  But in this case, the proposed
criterion simply begs the question against the idea that Morita
equivalent theories can have the ``same ontology''.  To take Morita
equivalence seriously as a criterion of theoretical equivalence means
simply that there is no cross-theoretical reference point for counting
objects or quantifying over them.


\section{Flat versus structured views of theories} \label{sec:flat}

For the past fifty years, philosophers' discussions of the nature of
scientific theories has been dominated by the dilemma: are theories
sets of sentences, or are theories collections of models?  But the
point of this debate has become less and less clear.  Most of us these
days are non-essentialists about mathematical explications.  For
example, most of us don't think that scientific theories really are
sets of axioms, or that they really are collections of models.
Instead, we think that different explications are good for different
purposes.  There is, nonetheless, a big question lurking in the
background --- viz.\ the question of whether we should conceive of
theories as ``flat'', or whether we should conceive of them as
``structured''.  And this question comes up whether one thinks that
theories are made of sentences, or whether one thinks that they are
made of models.

The syntactic view of theories is usually formulated as follows:
\begin{quote} A theory is a {\it set} of sentences. \end{quote} This
formulation provides a {\it flat} view: a theory consists of a
collection of things, and not in any relations between those things,
or structure on those things.  In contrast, a {\it structured} view of
theories says that scientific theories are best represented by
structured mathematical objects.  For example, a structured syntactic
view of theories might say that a theory consists of both sentences
and inferential relations between those sentences.

A flat version of the semantic view might be formulated as:
\begin{quote} A theory is a {\it set} (or {\it class}) of
  models. \end{quote} In contrast, a structured version of the
semantic view will say that a theory consists of a structured
collection of models.  For example, a theory might consist of models
with certain mappings between these models (such as elementary
embeddings), or a theory might consist of models and certain
``nearness'' relations between those models.

Both the syntactic and the semantic views of theories are typically
presented as flat views.  In the latter case, I suspect that the flat
point of view is accidental.  That is, most proponents of the semantic
view are not ideologically committed to the claim that a theory is a
bare set (or class) of models.  They may not have realized the
implications of that claim, or that there is an alternative to it.

In contrast, in the case of syntactically oriented views, some 20th
century philosophers were ideologically committed to a flat view ---
perhaps due to their worries about intensional and/or normative
concepts.  The main culprit here is Quine, of course ...  whose
criticism of the analytic-synthetic distinction is directed precisely
against a structured view of theories.  On a structured syntactic view
of theories, the essential structure of a theory includes not just
some number of sentences, but also the logical relations between those
sentences.  In this case, commitment to a theory would involve claims
about inferential relations, in particular, claims about which
sentences are logical consequences of the empty set.  In other words,
a structured syntactic view of theories presupposes an
analytic-synthetic distinction.

Quine's powerful criticisms of the analytic-synthetic distinction
raise worries for a structured syntactic picture of theories.  But is
all well with the unstructured, or flat, syntactic view?  I maintain
that the unstructured view has {\it severe} problems that have never
been addressed.  First of all, if theories are sets of sentences, then
what is the criterion of equivalence between theories?  A
mathematically minded person will be tempted to say that between two
{\it sets}, there is only one relevant condition of equivalence,
namely equinumerosity.  But certainly we don't want to say that two
theories are equivalent if they have the same number of sentences!
Rather, if two theories are equivalent, then they should have some
further structure in common.  What structure should they have in
common?  I would suggest that, at the very least, equivalent theories
ought to share the same inferential relations.  But if that's the
case, then the content of a theory includes its inferential relations.



\section{Believing a scientific theory}

It is often said that the difference between scientific realists and
antirealists is that the former {\it believe} scientific theories, and
the latter do not --- or at least they don't believe everything that
these theories say.  (e.g.\ the constructive empiricist doesn't
necessarily believe what theories say about unobservable things.)
This classification is based on a presupposition, viz.\ that the
notion of ``believing a scientific theory'' is clear --- or is at
least as clear as the notion of believing, say, that my desk is brown.
But this presupposition doesn't quite fit the facts.  On none of the
reasonable analyses of ``scientific theory'' is a theory nothing more
than a set of claims about the world.  Therefore, {\it believing a
  scientific theory} cannot be analyzed without remainder into
believing a set of claims about the world.

To see what's at issue here, it will be helpful to revisit one of the
earliest objections to the semantic view of theories.  According to
the semantic view of theories, a scientific theory is a class of
models.  Now, the objector to the semantic view points out that there
is a grammatical problem: in the phrase ``$S$ believes that $X$'', the
second argument $X$ needs to be filled by something toward which a
person can bear a propositional attitude.  The argument $X$ cannot be
replaced by a name such as ``Thor'', or predicate such as ``purple'',
much less by a name for a class of things, such as ``the set of \dots
''.  In particular, it makes no grammatical sense to say that ``$S$
believes that $\2M$'', where $\2M$ is a class of models.

The semanticists have a ready reply to this objection:
\begin{quote}
  Semantic Analysis of Belief (SAB): When a theory $T$ is given by
  means of a class $\2M$ of models, then belief in $T$ means belief
  that the world is isomorphic to one of the models in
  $\2M$. \end{quote} There are many, many problems with SAB, most
notably the opacity of the notion of a model being isomorphic with the
concrete world.  (That issue is discussed extensively by
\cite{bas2008}.)  However, there is another problem with SAB that we
find even more serious, because it bears directly on questions of a
normative nature, e.g.\ to what one commits oneself when one accepts a
scientific theory.  In particular, we think that ``believing $T$''
involves more commitments beyond those that are expressed by ($\ast$).

Consider a specific example.  Let $T$ be Einstein's general theory of
relativity (GTR).  According to SAB, a person believes GTR iff she
believes that the world is isomorphic to one of the models of GTR.
But that analysis is inaccurate in both directions: it captures both
more, and less, than physicists actually believe when they accept GTR.
First, it captures more, because it seems to commit physicists to the
belief that there is some privileged model of GTR that gives the best
overall picture of the physical world.  Of course, if you know how GTR
works, you'll find that picture to be laughable.  Just imagine two
relativists --- a cosmologist, and a black hole theorist ---
sitting down to argue over whose model gives a more perspicuous
representation of reality.  They won't do that, because they are well
aware that these models are accurate representations for certain
purposes, and not for others.  And what's more, it's we --- the users
of physics --- who choose the intended application of the theory.
Thus, SAB says more than physicists will actually want to say about
their theories.

Second, the semantic analysis of belief (SAB) also omits some of the
content that physicists pack into their theories.  Indeed, SAB locates
the content of a theory in one or other particular model, ignoring the
fact that physicists routinely invoke the existence of other models,
not to speak of a rich system of relations between models.  Indeed, if
a model $M$ is removed from its context in $\mathrm{Mod}(T)$, then it
can no longer do the representational and explanatory work that it's
expected to do.  Consider again the case of GTR.  As we noted above,
GTR is a powerful theory not because it is overly specific, but
because it is widely applicable --- offering different, but related,
models for a wide variety of situations.  GTR finds the unity between
these situations, including counterfactual situations.  (Fans of David
Lewis might be reminded of his insight that a good theory balances
informativeness and simplicity.  We only note that there are two
different ways to be informative: by saying what is unique about your
situation, or by saying what is common among many different
situations.)

Furthermore, some of the most powerful explanations in GTR draw on
facts about how a model sits inside the space of all models, some of
which we known not to represent the actual world.  For example, what
explains the fact that our universe began in a singularity?  According
to GTR, singular spacetimes are generic, i.e.\ they densely pack the
space of cosmological solutions to Einstein's field equations; hence,
the reason our universe begins in a singularity is because most
nomologically possible universes begin this way.

The fact that GTR uses all of its models, and the relations between
them, is only reinforced by looking at simple examples from
first-order logic.  If we take a first-order theory $T$, then
typically a single model $M$ of $T$ does not contain enough
information to reconstruct $T$.  In other words, if you give me a
model $M$ of $T$, I couldn't reliably reconstruct the theory $T$ of
which it was a model.  What that means is that $M$ contains less
information than the theory $T$ itself.  The content of the syntactic
object $T$ is not contained in a single model $M$, but in the
structured collection $\mathrm{Mod}(T)$ of all its models.  What this
means in turn is that accepting $T$ cannot be reduced to a claim about
one of the models in $\mathrm{Mod}(T)$; instead, accepting $T$ must
involve some sort of attitude toward the entire collection
$\mathrm{Mod}(T)$.

The point we are making here ties all the way back to the preface of
the book, where we tried to justify our omission of modal logic.
There we claimed that accepting a first-order theory --- with no
explicit modal operators --- involves modal commitments.  We're making
the same point here.  To accept a theory $T$ isn't just to take a
stand on how the world {\it is}, it is also to take a stand on how the
world {\it could be}.  More is true.  To accept a theory $T$ involves
choosing a language $\Sigma$, and this language determines how we
parse the space of possibilities --- e.g.\ which possibilities we
consider to be isomorphic, and which we consider not to be isomorphic.
(If you've read the previous chapters carefully, you're also aware
that the language $\Sigma$ determines the topological structure of
$\mathrm{Mod}(T)$.)  In short, the syntactic approach to theories had
the advantage (largely unnoticed by its proponents), that the
syntactic object $T$ packs in a lot of information --- about what is
possible, and about how to classify possibilities.  One of the dangers
of the semantic view is forgetting how much scientific theories say.

The fault here doesn't lie completely with the semantic view of
theories.  In fact, there's an analogous problem for those, such as
Quine, who accept a flat syntactic view of theories (see Section
\ref{sec:flat}).  According to the flat syntactic view, a theory $T$
is a {\it set} of sentences.  Indeed, Quine --- among other flat
syntacticists --- sometimes equates belief in $T$ with belief in a set
of sentences.  But that cannot be quite right, as we can see again
from actual scientific theories, as well as from simple examples from
first order logic.

As for examples from first-order logic, let $\Sigma _1$ be the empty
signature, and let $T_1$ be the theory in $\Sigma _1$ that says there
are exactly two things.  Let $\Sigma _2=\{ c\}$, where $c$ is a
constant symbol, and let $T_2$ be the theory in $\Sigma _2$ that says
there are exactly two things.  Here $T_1$ and $T_2$ share the same
axiom, but they aren't equivalent theories by any reasonable standard
--- not even by categorical equivalence.  The first theory's models
have automorphism group $\7Z _2$, whereas the second theory's models
have trivial automorphism group (since the denotation of $c$ is
fixed).  Nonetheless, $T_1$ and $T_2$ agree on the statements that
they make about any particular model: they both say that there are two
things.  The user of $T_2$ has an extra name $c$, but her using this
name does not amount to any claim about how things are.  Thus, we have
a puzzle: on a world-by-world basis, $T_1$ and $T_2$ say the same
thing; and yet, it's not reasonable to think that $T_1$ and $T_2$ say
the same thing.

The solution to this little puzzle is to recognize that believing a
theory cannot be reduced to believing that a certain collection of
sentences is true.  At the very least, believing a theory also
requires that we adopt a language --- or an ``ideology'', as Quine
liked to call it.  However, Quine wasn't completely clear on what the
reasons might be for accepting an ideology.  The issues became
slightly more clear when Lewis suggests that our choice of ideology
corresponds to our beliefs about which properties are ``natural'', and
when \cite{sider2013} suggests that choice of ideology is tantamount
to assertion that the world has a certain structure.  While we don't
necessarily agree with this way of describing the situation, we agree
that ideology plays a theoretical role.

If we claim that a theory is a collection of sentences, then we ought
also to accept the claim that theories are equivalent only if they
contain the same sets of sentences.  Or, to be more accurate, two
theories are equivalent just in case each sentence in the first is
equivalent to a sentence in the second, and vice versa.  But now, what
standard of equivalence should we use for the sentences?  The only
reasonable standard --- two sentences are equivalent if they express
the same proposition --- is of no use in comparing actual scientific
theories.  Thus, the only reasonable account of the identity of
scientific theories treats theories as a {\it structured} objects, in
which case equivalence means having the same structure.  And then we
have a challenge question: what does it mean to believe or accept a
structured object?

It might be illuminating to compare a scientific theory with the kinds
of beliefs for which people live and die --- e.g.\ religious beliefs.
As you know, many western religions have creeds that are supposed to
capture the key tenets of the system of belief.  Now, suppose that you
were to try to write down the central tenets of a scientific theory as
a creed.  For example, you might take a copy of Robert Wald's {\it
  General Relativity}, and start searching through it for the basic
``truth claims'' of the theory.  However, you'll quickly grow
frustrated, as it doesn't seem to make any specific claims.  GTR
doesn't say what happened on December 7, 1941, nor does it say how
many planets are in our solar system, nor does it say (before one
selects a particular model for application) how old the universe is.
Instead, GTR consists of some mathematics, and some recommendations
about how to apply this mathematics to various situations.  And yet,
there is never any hint that GTR is a bad theory because it's not
specific enough.  Quite to the contrary, GTR is a good theory
precisely because it is so general.

One might be tempted to think that the creed of GTR is summed up in
its basic equations, Einstein's Field Equations (EFE).  In this case,
to accept GTR would be to say:
\begin{quote} ($\dagger$) I believe that
  $R_{ab} - {1 \over 2}g_{ab}\,R = T_{ab}$. \end{quote} This is an
interesting possibility to consider, and there are two attitudes we
could take to it.  I will call these two attitudes the physicist's
attitude, and the metaphysician's attitude (almost certainly
caricaturing both).  In my experience, physicists don't say things
like ($\dagger$).  Certainly, they write down EFE, and they use it as
a rule for generating descriptions of situations that they take to be
accurate.  But I've never heard a physicist say, ``I believe that
Einstein's Field Equations are true.''  These physicists seem to have
a positive attitude toward EFE --- perhaps we should call it
``acceptance,'' but I don't think we could call it ``belief''.

In contrast, metaphysicians encourage us to say things such as, ``the
success of GTR gives us reason to think that EFE are true.''  In order
to make sense of EFE being true, the metaphysician might search for
referents for the terms that occur in it.  For example, in the spirit
of David Armstrong, he might say something like, ``The symbols
$R_{ab}$ and $T_{ab}$ refers to natural properties, and and EFE is the
statement that a second-order relation holds between these
properties.''  This kind of metaphysician seems to think that there
aren't enough concreta to account for the meaning of the abstract
statements of science.  Thus, he transmogrifies abstracta into
concreta, and then uses these new hybrid abstract-concreta as the
referents for the symbols in scientific statements.  After all this
hocus pocus, we're supposed to understand what it means to say that
EFE is {\it true}, and hence, what it would mean to {\it believe} EFE.
However, physicists haven't yet bought into this way of speaking or
thinking.  Physicists who haven't been corrupted by analytic
philosophy still think of the theoretical statements of science as
different in kind from more mundane claims such as, ``there is an
apple on my desk.''

It might sound like I am saying that physicists are instrumentalists,
i.e.\ that they treat the theoretical statements of science as mere
instruments from which to derive predictions.  I believe that this is
yet another false dilemma --- in this case, a false dilemma suggested
by formal semantics.  In formal semantics we have a simple, black and
white distinction between interpreted and uninterpreted terms.  So,
the metaphysician might ask: are the terms in EFE interpreted or
uninterpreted?  The dilemma is false.  The terms in EFE are sometimes
interpreted by users of the theory, e.g.\ physicists.  But
interpretation presupposes an interpreter, and so it doesn't make
sense to ask of a theory, {\it sans} theorist, whether its terms are
interpreted.

The existence of more than one model --- or, to speak more accurately,
of more than one application --- is not a bug of scientific theories,
it is a feature.  What is lost in informativeness is gained in
applicability.  But the more flexible a theory is in its applications,
the less sense it makes to think of our attitude towards that theory
as simple ``belief''.  Perhaps this is one reason why we need another
word, such as ``acceptance''.  As van Fraassen pointed out long ago,
to accept a theory cannot be reduced to an attitude that the theory
somehow mirrors the world.  Acceptance of a theory involves a sort of
appropriation, where the theory serves as a guide to future action.

I've been considering the question, ``what does it mean to accept a
scientific theory?'' and have found ample reason to reject the idea
that it's nothing more than a special case of belief.  Accepting a
scientific theory may involve believing that some things are true, but
it also involves a more complex set of attitudes --- such as adopting
certain standards for explanation, certain rules for reasoning about
counterfactual scenarios, etc..  Perhaps these additional aspects of
theory acceptance are what Carnap was trying to capture when he talked
about a theory's ``$P$-rules''.

What I'm saying here might be anathema to doctrinaire Quineans who
have no place in their picture for any of these rules.  Indeed, we
think that accepting a scientific theory is similar to accepting the
validity of an argument.  To say that $\phi\vdash\psi$ is valid is not
(on my picture) a claim about how things are; rather, it's a claim
about about how people ought to reason.  In the same way, to accept a
scientific theory isn't simply believing that something is true; it's
taking a stance on how one ought to reason about nature.

% The semantic view of theories is typically wedded with what I will
% call the \emph{atomic theory of scientific representation}.  According
% to TARS, a theory represents a domain by means of its individual
% models.  The idea here is that the theory hooks up to the world at a
% single node (one of its models).  But as we have seen, the
% mathematical content (or representational capacity) of a theory
% includes structure on the class of models.  If this structure on the
% class of models is a part of this theory's representational capacity,
% then why doesn't the theory use it to represent?

% First, a scientific theory represents by means of its entire class of
% models, and the structure thereupon.  Of course, a good theory will
% also have close connections between individual models and the
% phenomena --- such is the nature of applications of the theory.  For
% example, GTR is a structured class of models, but one of its
% individual models (the Schwarzschild solution) is a particularly good
% representation for a collapsing star.

% Now, our positive proposal (that $\mathrm{Mod}(T)$ as a whole
% represents) raises one concern: if $\mathrm{Mod}(T)$ does the
% representing, then what does it represent?  Indeed, we tend to think
% of representation as involving some kind of correspondence between the
% representor and the represented.  To put the point in another way, how
% could $\mathrm{Mod}(T)$ be a ``correct representation'' unless there
% are correlates ``out there'' for all the structure we find in
% $\mathrm{Mod}(T)$?  In particular, there's a danger that our account
% entails that to accept a theory $T$ demands that one accept a very
% robust form of Lewisian modal realism.

% However, there's yet another false --- or at least questionable ---
% presupposition in this line of thought, viz.\ the presupposition that
% successful representation demands isomorphism, in particular, that it
% demands a one-to-one correlation between elements of the
% representation and the things that are represented.  The point we are
% making here is an old one, for it goes back at least as far as William
% of Ockham.  To his great credit, Ockham realized that our language is
% filled with words that play a role in our representations without
% themselves purporting to represent any particular thing.  The view we
% want to endorse here is a sort of Ockhamism vis-a-vis the mathematical
% structure of theories.



\section{Notes}

\begin{itemize}
  \item For more technical details on second-order logic, see
  \cite{shapiro,manzano-book}.  Philosophers have argued quite a bit
  about the advantages and disadvantages of second-order logic.  For
  example, Quine argued that second-order logic is, ``set theory in
  sheep's clothing.''  See e.g.\ \cite{bueno}.
\item Carnap gives his mature view of Ramsey sentences in
  \citep{carnap-pfp}.  For more on the role of Ramsey sentences in
  Carnap's philosophy of science, see
  \cite{psillos2000,psillos2006,friedman2011,demo2013}.
\item For more on
  the Ramsey sentence functionalism, see \cite{shoemaker}.
\item For a detailed, but older, discussion of the technical issues
  surrounding Ramsey sentences, see \cite[][Chap.\ 3]{tuomela}.  For
  recent discussion of the prospects of Ramsey sentence structuralism,
  see \cite{ketland,melia,ainsworth,dew-ram}.
\item For general surveys of structural realism, see
  \cite{frigg2011,ladyman}.  The idea behind structural realism goes
  much further back than the 1980s.  Something similar had been
  proposed by Poincar\'e and Russell in the early 1900s, and then
  again by Grover Maxwell in the 1960s.  What's new about the 1990s
  reincarnation of structural realism is (1) the explicit claim that
  it can solve the pessimistic metainduction, and (2) the explication
  of structure in terms of Ramsey sentences.  Needless to say, the
  idea behind structural realism could survive, even if --- as we've
  argued --- Ramsey sentences don't provide a useful explication of
  the structure of a theory.
\item Putnam's model theoretic argument first appeared in
  \cite{putnam1977,putnam-models}, with antecedents in Quine's
  permutation arguments for ontological relativity.  The most
  influential response to Putnam is Lewis' \citeyearpar{lewis-putnam},
  which is the {\it locus classicus} of his version of metaphysical
  realism which emphasizes the notion of {\it natural properties}.
  That torch has been taken up by \cite{sider2013}.  The response we
  gave to Putnam's argument follows the spirit of \cite{bas-putnam}.
  For an excellent overview of Putnam's arguments, see \cite{button}.
 \end{itemize}


%% continuum diagram

%% invariants of theoretical equivalence



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