What Was the Syntax-Semantics Debate
in the Philosophy of Science About?

Sebastian Lutz∗

Preprint: 2015-02-26

Abstract

The debate between critics of syntactic and semantic approaches to the
formalization of scientific theories has been going on for over 40 years. I
review the debate in light of a recent exchange between Hans Halvorson,
Clark Glymour, and Bas van Fraassen and argue that the only remaining
disagreement concerns the alleged difference in the language dependence of
syntactic and semantic approaches. This difference turns out to be illusory.

Keywords: semantic approach, Suppes predicate, constructive empiricism,
syntactic approach, received view, logical empiricism, model theory, defini-
tional equivalence

1 Introduction

The basic controversy about syntactic and semantic formalizations of theories
seems simple: In syntactic approaches theories are described by sets of sentences,
in semantic approaches they are described by classes of model-theoretic struc-
tures, and the question is which is preferable. But the debate has been going
on since at least the 1960s and has spanned such diverse topics as the problems
of formalization in first order logic, the need for formalization in general, the
role of models in the sciences, the dependence of scientific theories on their for-
mulation, and the right way of identifying phenomena. It has recently come to
a head, with Suppe (2000), Suppes (2002) and Muller (2010) criticizing syntac-
tic approaches, Halvorson (2013) and Lutz (2014d) defending them, Halvorson
(2012) criticizing semantic approaches, and Glymour (2013) and van Fraassen
(2014) defending them. By the standards of the debate, this most recent exchange
has a comparably narrow focus: The question is whether either of the two kinds

∗Munich Center for Mathematical Philosophy, Ludwig-Maximilians-Universität München. se-
bastian.lutz@gmx.net. I thank Holger Andreas, Richard Dawid, Adam Caulton, Samuel Fletcher,
Hans Halvorson, Thomas Pashby, and Karim Thébault for helpful discussions. Research for this
article has been supported by the Alexander von Humboldt Foundation.



Sebastian Lutz The Syntax-Semantics Debate in the Philosophy of Science—Preprint

of approach is in principle unable to describe specific scientific theories. How-
ever, discussing only this most recent exchange is well-nigh impossible without
taking the history of the debate into account because the recent exchange still
contains so many rudiments of previous ones. Even when focused on the cen-
tral participants and further streamlined, the debate is somewhat unstructured.
It goes a bit like this:1

Carnap (1939, 1956): I think that we need to get some precision into the dis-
cussion about scientific theories. I suggest formalizing theories—and their
connections to experiments and observations—using sentences in higher
order logic, with one set of terms distinguished as observational and inter-
preted directly.

P. Suppes (1967, 2002): Well, one should certainly not formalize theories in
first order logic. Doing so would often be incredibly difficult and cum-
bersome. It would be better to define the models of the theory directly
using a set theoretic predicate.

Van Fraassen (1970): Or one could describe theories as restrictions on their
state-spaces. These state-spaces could then be used to interpret simple ele-
mentary sentences like ‘quantity m has value r at time t ’ that can connect
to observations in some more or less difficult way.

F. Suppe (1974b): I think Carnap’s suggestion cannot work because it leads to
distinctions without a difference: The same theory can be described by two
different sets of sentences. By contrast, Suppes predicates and state-space
restrictions stay the same. So we should use either of those for formalizing
theories.

Van Fraassen (1980, 1989): You know what? We don’t need elementary sen-
tences. We can be wholly independent of the language of predicate logic by
just sticking with state-space descriptions. We can describe observational
data by structures, theories by classes of structures, and the relation be-
tween data and theory by the embedding relation. Of course we still need
some language to describe all these structures, but at least it does not have
to be predicate logic.

F. Suppe (2000): That’s good, because most theories cannot even be described
syntactically: The Löwenheim-Skolem theorem entails that sentences can-
not describe the models of a theory as precisely as is often necessary.

Halvorson (2012): Wait a minute. It seems to me that structures are no good at
all for describing theories. Ignoring the relation of a theory to the world for
now, I can come up with examples in which the same theory is described

1. I thank Karim Thébault for the suggestion of reproducing the debate as a conversation.

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by different classes of structures, and examples in which two different the-
ories have the same class of structures.

Glymour (2013): I don’t think you need to ignore the relation of a theory to
the world. Since embedding, for instance, is a purely formal relation, your
examples still work. But they fail to work once you take into account that
in model theory, structures already come with a language of predicate logic.
So one has to take language into account.

Halvorson (2013): But if one uses a language of predicate logic, then there is
nothing to the much touted language independence of structures. In fact,
then things look downright syntactic to me.

Van Fraassen (2014): I think you are missing the point. I always said that we
need some language to describe structures, and my state-space account in-
cluded elementary sentences anyway. As to the counterexamples: Since
they ignore the relation of a theory to the world, they cannot work as a
criticism of accounts that include the relation to the world.

In the following, I aim at showing that the above dialogue is an accurate summary
of the historical debate and that it contains a number of weak arguments. I will
further argue that the exact point of disagreement as it emerges from the debate
is so minor that the debate seems to have been about barely anything at all.

2 The Received View

One recurring problem of the debate is that it mixes criticisms of syntactic ap-
proaches in general with criticisms of the logical empiricists’ position, and the re-
cent exchange is no exception. Hence in this section, I will give a short overview
of the logical empiricists’ position on the formalization of theories. As it turns
out, this is already enough to counter some criticisms.

From the 1920s to the 1960s, the logical empiricists, especially Carnap and
Hempel, developed their formal approach to the analysis of scientific theories
that Putnam (1962, 240) would later call the “Received View”. It is the most-
developed and best known syntactic approach. In the Received View, a scientific
theory is formalized as a set of sentences ( called theoretical sentences) that con-
tain only logical symbols, mathematical symbols, and the symbols of the theory
( theoretical terms). The theoretical terms are connected to terms that refer to
observable properties ( observation terms) through sets of sentences that contain
both theoretical and observation terms ( correspondence rules). Both kinds of sen-
tences are assumed to be in predicate logic of any order: Carnap and Hempel
used higher order logic for the formalization of scientific theories ( the former
more often than the latter), and a fortiori neither demanded a restriction to first
order logic ( Lutz 2012, §2).

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It is thus a historical puzzle that the Received View and ( which is even more
puzzling) syntactic approaches in general have been charged with demanding
formalizations in first order logic.2 For instance, Suppes (2002, 3) criticizes the
“standard sketch”, by which he means the Received View, for the “strong ten-
dency on the part of many philosophers to speak of the [theoretical sentences]
as a logical calculus purely in syntactical terms” because theories like quantum
mechanics “need to use [ . . . ] many results concerning the real numbers. For-
malization of such theories in first-order logic is utterly impractical” (4). This
argument rests essentially on the alleged restriction to first order logic because
of “the success of second order logic to capture directly most all mathematical
practice” ( Leivant 1994, §5.2) and might indeed be more natural a formalization
of mathematical practice than set theory ( Andrews 2002, xi–xiv).

Van Fraassen (2014, 281) criticizes the Received View’s reliance on predicate
logic in general, starting from Toulmin’s exposition of the logical empiricists’
project ( Toulmin 1974, 602):

At its most grandiose, the ambition was to establish epistemologi-
cal foundations [ . . . ] for positive scientific knowledge in its entirety.
What [ . . . ] Peano and Russell had done by founding pure mathemat-
ics in formal logic [ . . . ] the advocates of unified science hoped to do
for the whole of natural science. By adding further primitive terms,
postulates, and correspondence rules, they hoped to incorporate all
genuine branches of science into a single axiomatic edifice[.]

And van Fraassen (2014, 281) continues himself: “This revival of Leibniz’ math-
esis universalis is what Fred Suppe took on in his dissertation[.] It was equally
what I saw as being replaceable[.]” It is clear from the above, however, that the
Received View ( as opposed to the logical empiricists’ aim for the whole of sci-
ence) does not require one language for all theories, but only one language for a
theory and its correspondence rules. It was just the logical empiricists’ ambition
( and only at its most grandiose) to establish a unified language. Furthermore, a
theory and its correspondence rules could be described in different languages—as
long as there is at least one language of predicate logic, the assumptions of the
Received View are fulfilled.3

Besides relying on predicate logic, the Received View assumes that the ob-
servation terms are given a direct semantic interpretation, which, through the
correspondence rules and theoretical sentences, restricts the possible semantic
interpretations of the theoretical terms ( Carnap 1939, §24; Feigl 1970, 5–6). The
resulting semantics can either be given informally by stating, for example, that
the function “‘lg(x , t)’ designates the length in centimeters of the body x at the
time t ” ( Carnap 1939, 58) or formally using, for example, Tarski semantics. The

2. In a recent example, Thompson (2007, 488–489) incorrectly claims a restriction to first order
logic also for the foundations of the Received View ( Whitehead and Russell, 1927/1929) and one
of its applications ( Woodger 1939).

3. In thank Richard Dawid for this point.

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latter would be natural because Carnap was an early defender of Tarski seman-
tics ( Carnap 1936). Such formal semantics for the Received View were indepen-
dently developed by Przełęcki (1969), Suppe (1971), and Andreas (2010). That
they all turned out to be equivalent or almost equivalent ( Lutz 2014c) shows
how strongly implied they are by Carnap’s elucidations of the Received View.

It is clear that a set of sentences whose words are given no interpretation
whatsoever cannot be about the world in any interesting sense. Formally, this
point was made by Przełęcki (1969, 27–28) who, using an argument akin to the
ones by Newman (1928) and Putnam (1989, 353), points out that a set of sen-
tences whose terms have not been given an interpretation can at best determine
the cardinality of its domain ( cf. Lutz 2014d, §4). Thus the Received View and
indeed any syntactic approach must, on pain of triviality, contain a semantic
interpretation of some sort. This is why the Received View included semantic
interpretations, as Suppe (1974a, II.E) also notes.

3 Criticisms of Syntactic Approaches

The criticisms discussed in the preceding section do not even apply to the Re-
ceived View, let alone syntactic approaches in general. The question is now
which criticisms do apply to syntactic approaches and thus what makes an ap-
proach syntactic. It is clear that the Received View has to be one specific syn-
tactic approach. If it were not, the main question of the debate would become
whom critics of syntactic approaches have been criticizing for 40 years.

A good starting point for the discussion of the criticisms of syntactic ap-
proaches is an overview of the fall of the Received View and syntactic approaches
in general by Suppe (2000, s103). He considers the following problems of the Re-
ceived View severe:

1. Correspondence rules were a heterogeneous confusion of meaning relation-
ships, experimental design, measurement, and causal relationships some of
which are not properly parts of theories.

2. Symbolic logic is an inappropriate formalism.

3. Theories are not linguistic entities and thus theories are individuated incor-
rectly.

The first problem is clearly directed at the Received View in particular, not at
syntactic approaches in general, since it only applies to syntactic approaches that
rely on correspondence rules. Their use has many detractors ( e. g., Suppes 2002,
7) and few defenders ( Lutz 2014a, §4). But it is clear that syntactic approaches in
general may describe the relation between theories and observations without the
use of correspondence rules. Indeed, Suppe (1974a, 108, n. 225) himself summa-
rizes Suppes’s competing semantic approach to the theory-observation relation

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“in terms of theories of which [Suppes’s] models are realizations”, that is, syntac-
tically, and Turney (1990) and Lutz (2014a, §3) prove equivalences between their
respective syntactic concepts and van Fraassen’s semantic concept of empirical
adequacy, which is also meant to describe the theory-observation relation. The
Received View’s related distinction between theoretical and observational terms
has also been a focus of criticism ( e. g., Putnam 1962; van Fraassen 1980, §3.6),
although it has some defenders ( Suppe 1971, 1972; Chang 2005). As in the case
of correspondence rules, however, syntactic approaches clearly do not have to
rely on this distinction; the accounts by Turney (1990) and Lutz (2014a, §3), for
example, do not.

The second problem seems to apply to syntactic approaches in general. But
Suppes argument for the severity of the second problem relies on the incorrect as-
sumption that the Received View is restricted to first order logic: He refers to the
Löwenheim-Skolem theorem ( which holds only in first order logic) for arguing
that syntactic approaches in general must lead to non-standard models, which
makes symbolic logic an inappropriate formalism ( Suppe 2000, s104). Clearly,
the argument is therefore also not applicable to syntactic approaches in general
( cf. Lutz 2014d, 1478–79). Van Fraassen’s criticism of the use of predicate logic
mentioned above relies on the logical empiricists’ search for a mathesis universalis.
As noted, the Received View does not require a single language for all scientific
theories, and it is also clear that not all syntactic approaches require a single
language for all scientific theories.

The third problem applies to all syntactic approaches, and will be the focus
of my discussion. Suppe argues for this point as follows: Understanding ‘propo-
sitions’ “in the medieval sense of the term”, that is, as “interpreted sentences of
some particular language”, Suppe (1974a, 204–5) claims that theories cannot be
the propositions in which they are formulated because

quantum theory can be formulated equivalently as wave mechanics
or as matrix mechanics; whichever way it is formulated, it is the same
theory, though its formulations as wave mechanics will constitute a
collection of propositions which is different from the collection of
propositions resulting from its formulations as matrix mechanics.

Therefore van Fraassen (1989, 222) can claim that “in discussions of the structure
of theories [language] can largely be ignored”. But syntactic approaches always
carry with them a language, which is why they must fail. As van Fraassen (1989,
221) writes: “In any tragedy, we suspect that some crucial mistake was made
at the very beginning. The mistake, I think, was to confuse a theory with the
formulation of a theory in a particular language”. Halvorson (2012, 201–2) cites
more claims by van Fraassen and others according to which the reliance on a
particular language is a crucial mistake.

The question is how independent syntactic approaches can be from language.
In the Received View ( and in general), two theories are identified when they
are described by logically equivalent sets of sentences. One could also go further

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and identify all analytically equivalent sets of sentence, that is, all sets whose
equivalence follows logically from a set of meaning postulates ( Lutz 2014d, §3).
Halvorson (2012, 191) suggests the following equivalence relation between sets of
sentences: Two sets T and T ′ of sentences are definitionally equivalent if and only
if they can both be extended by sets ∆ and, respectively, ∆′ of explicit definitions
such that T ∪∆ and T ′∪∆′ are logically equivalent. Mutual definability is an-
other, more inclusive equivalence relation ( cf. Andréka, Madarász, and Németi
2005). Two sets of sentences are mutually definable if and only if each set can
be extended by explicit definitions such that it entails the other set. Both defini-
tional equivalence and mutual definability lead to equivalence classes of sentences
and thus to syntactic approaches. There is no obvious reason not to take other
equivalence relations, like mutual interpretability ( Hodges 1993, ch. 5), into ac-
count. And it is not obvious that wave and matrix mechanics are so different in
their formulation that they are not definitionally equivalent, mutually definable,
or mutually interpretable.

Summing up, the Received View did not rely on first order logic, nor did
it distinguish between theories with logically equivalent axiomatizations. Syn-
tactic approaches in general can rely on more general equivalence relations be-
tween a theory’s axiomatizations, and they can do away with correspondence
rules and the distinction between observational and theoretical vocabulary. This
leaves two aspects of the Received View that make an approach syntactic: the use
of predicate logic ( possibly of higher order) and semantic interpretations ( possi-
bly formal). The restriction to predicate logic is necessary to not trivialize the
notion of a syntactic approach.4 As van Fraassen (1989, 222) notes, “to present a
theory, we must present it in and by language. That is a trivial point”. The use of
semantic interpretations is necessary to not trivialize all syntactic approaches, as
the consideration by Przełęcki (1969, 27–28) shows. Thus an approach is syntac-
tic if every theory is described by an interpreted set of sentences in higher order
logic, where the interpretation does not have to be formal. Taking into account
that different sets of sentences can be equivalent in one way or another, the fol-
lowing seems like the most plausible circumscription of syntactic approaches for
the sequel:

Definition 1. An approach is syntactic if and only if in the approach every the-
ory is described by an equivalence class of sets of sentences of higher order logic,
where at least some of the sentences are interpreted.

This definition leaves open which equivalence relation generates the equiv-
alence class, as different syntactic approaches can rely on different equivalence
relations.

4. Carnap (1956, 64) in fact allows modal logic as well, but this will not be necessary for the
following discussion.

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4 Semantic Approaches

The semantic approaches, even the formal ones, were never as unified as the
logical empiricists’ positions. In this section, I will survey those approaches that
are important for the exchange between Halvorson, Glymour, and others, and
suggest a general conception of semantic approaches that arguably underlies the
debate.

4.1 State Spaces

In an outline of his version of the Semantic View ( endorsed, for example, by
Suppe 1974a, 222–23), van Fraassen (1970, 328–29) focuses on

the formal structure of nonrelativistic theories in physics [ . . . ]. A
physical system is conceived of as capable of a certain set of states,
and these states are represented by elements of a certain mathemat-
ical space, the state-space. [ . . . ] To give the simplest example, a clas-
sical particle[’s] state-space can be taken to be Euclidean 6-space,
whose points are the 6-tuples of real numbers (qx , qy , qz , px , py , pz).

Van Fraassen (1970, 330) then distinguishes between laws of coexistence, laws
of succession, and laws of interaction. In the non-statistical case ( §5.1), laws of
coexistence select the physically possible subset of the state-space (330), laws of
succession select, in the instantaneous state picture, the physically possible trajec-
tories in the state-space (331), and laws of interaction at least in principle reduce
to the above (332). The interpretation of physical theories relies on “elementary
statements” ðU(m, r, t)ñ formulating propositions “to the effect that a certain
such physical magnitude m has a certain value r at a certain time t ” (328). Such
a proposition is true if and only if the system is in state-space region h(U), where
h is some satisfaction function. Finally, the “exact relation between U(m, r, t) and
the outcome of an actual experiment is the subject of an auxiliary theory of mea-
surement, of which the notion of ‘correspondence rule’ gives only the shallowest
characterization” (328). In van Fraassen’s account, the state-space description is
given in the metalanguage and semantically related to the elementary statements
in the object language, which are then related to measurement outcomes via an
auxiliary theory of measurement. Van Fraassen does not state whether the aux-
iliary theory is given in the object language and, relatedly, he is silent about
the semantic interpretation of the elementary sentences. Van Fraassen’s account
of scientific theories therefore leaves out what the logical empiricists attempted
to analyze: The connection of the formalism of the theory with the world.5

5. This is not an oversight: Van Fraassen (1987, 108) states that “the correct view about the
meaning and understanding of newly introduced terms [ . . . ] makes no practical difference to
philosophy of science, as far as one can tell. It is a good problem to pose to philosophers of
language, and to leave them to it.”

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This connection could thus be given ( formally or informally) as in syntactic ap-
proaches.

Van Fraassen (1970, 326) notes that there “are natural interrelations between
the two approaches” ( his and the syntactic one) given by the possibility of de-
scribing a set of sentences by its models and a structure by the sentences of
which it is a model. “These interrelations [ . . . ] would make implausible any
claim of philosophical superiority for either approach.” This claim of peaceful
coexistence is gone in van Fraassen’s exposition of constructive empiricism, his
later approach to scientific theories and their relations to observations. Here van
Fraassen (1980, 56) claims that the “syntactically defined relationships are simply
the wrong ones” for the description of the relation between theory and observa-
tion. Worrall (1984, 71) points out the tension of this passage with van Fraassen’s
earlier remarks, and van Fraassen (1989, 212–213) replies:

[W]e must advance the conception of natural language as not being
constituted by any one realization of any such logical skeleton [of
predicate logic]. [ . . . ] In my opinion, it would be a poor choice to
try and describe a whole language in which a given theory can be for-
mulated. The reason is that descriptions of structure in terms of satis-
faction of sentences is much less informative than direct mathemati-
cal description. [Instead] we may choose a very small fragment, such
as what I have called the fragment of elementary statements. [Their
syntactic form is trivial] and therefore the semantic study alone has
some significance.

And in an endnote, (365, n. 34; cf. van Fraassen 1987, 121) adds: “It should be
added however that I soon found it much more advantageous to concentrate
on the propositions expressible by elementary statements, rather than on the
statements themselves. [ . . . ] At later points there is not even a bow in the direc-
tion of syntactic description.” Having started out with an account of scientific
theories that includes an object language ( for elementary sentences) and a met-
alanguage ( for the description of the state-space), van Fraassen has thus arrived
at an account that contains only what used to be the metalanguage. Since ‘ob-
ject language’ and ‘metalanguage’ have only a relative sense ( cf. Tarski 1944, §9),
van Fraassen’s account at this point is simply phrased in a single language. Since
natural language is interpreted, the only reason why van Fraassen’s account of
scientific theories is not syntactic is thus that he does not assume that theories
are formalized in predicate logic, but rather informally in natural language and
the semi-regimented natural language of mathematics.

As noted, higher order logic can capture mathematical practice quite directly
( Leivant 1994, 260). Thus as far as the mathematical component of scientific the-
ories is concerned ( that is, the component described by van Fraassen’s account),
it is not clear why there should be a principled problem with the use of higher
order logic as opposed to informal mathematics. It may seem that syntactic ap-
proaches still have the disadvantage of dealing with equivalence classes of sets of

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sentences, but the state-space approach is in the same position: The same scien-
tific theory can be described in different state-spaces, or in the same state-space
but with differently formulated restrictions. Therefore the state-space approach
also has to deal with equivalence classes of sentences, albeit not necessarily in
predicate logic.

4.2 Structures

As van Fraassen notes, his later approach to scientific theories and their rela-
tions to observations, constructive empiricism, does not rely on elementary state-
ments. The formal aspect of constructive empiricism is phrased completely in
terms of model theory: A theory is given by a family of structures ( called ‘mod-
els of the theory’), each appearance is given by a single structure, and a theory
is empirically adequate if and only if all appearances are isomorphic to specially
designated substructures ( “empirical substructures”) of a member of the family
of structures ( van Fraassen 1980, §3.9; cf. Lutz 2014d, §3.1). It is not obvious
how this approach relates to van Fraassen’s state-space approach: Elementary
sentences cannot be embedded in state-spaces in the normal sense of the word,
and the embeddability of the propositions expressed by elementary sentences in
state-spaces is even more questionable as long as ‘proposition’ has not received a
mathematical definition. But with his new formalism, van Fraassen (1989, 366)
can claim independence from the syntax of predicate logic, because he considers
structures to be so independent:

The impact of Suppes’s innovation [i. e., switching to models] is lost
if models are defined, as in many standard logic texts, to be partially
linguistic entities, each yoked to a particular syntax. In my terminol-
ogy here the models are mathematical structures, called models of a
given theory only by virtue of belonging to the class defined to be
the models of that theory.

A standard text in which models are “yoked to a particular syntax” is the text-
book by Chang and Keisler (1990, §1.3), where structures are defined as pairs
〈A,I〉 of a non-empty set A of objects ( a domain) and an interpretation I , which
is a mapping from each n-ary relation symbol to an n-ary relation R ⊆ An on A,
each m-place function symbol F to an m-place function G : Am −→ A on A, and
each constant symbol c to a constant x ∈ A. From the notation of the relation
and constants on A it is obvious that these are extensional concepts such that, for
instance, R = R′ when for all x , x ∈ R if and only if x ∈ R′. Relations with the
same extension are identical, while there can be two different relation symbols
with the same extensions of their interpretations. Hodges (1993, 1–4) explicitly
stresses this reliance on a vocabulary.

It is thus somewhat unfortunate that in his discussion of the role of language,
van Fraassen (2014, 277) refers to Chang and Keisler (1990) for the detailed def-
inition of ‘structure’. For their reliance on a vocabulary is essential and cannot

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be avoided by identifying “the structure [ . . . ] as the range of the interpretation”
( van Fraassen 2014, n. 1): If, for instance, a structure 〈A,I〉 contains an interpre-
tation solely for multiple relation symbols Pi , i ∈ I , and maps all of them to the
same R, taking its image would lead to 〈A,{R}〉, so that only one relation would
be left. However, van Fraassen also refers to Robinson (1963, 9–10), who uses an
informal version of a conception of ‘structure’ that is used by Bell and Slomson
(1974, §3.2). They define a relational structure A as a pair 〈A,{Ri}i∈I〉 of a do-
main A and a family ( an indexed set) {Ri}i∈I of extensional relations.6 Similarly,
Robinson (1963, 9) does “not identify the relations as such with sets of ordered n-
ples of individual objects of [structure] M so that two relations may hold at the
same n-ples.” That is, Robinson uses non-extensional relations that correspond
to pairs of ( extensional) relations and their index in the formalism of Bell and
Slomson. Robinson then interprets the vocabulary by a bijection between the
vocabulary and the set of non-extensional relations or, in the formalism of Bell
and Slomson (1974, §3.2), by a bijection between the vocabulary and the index
set I of the family of relations.7

In the following, I will call structures formalized via a vocabulary ‘labelled
structures’ and structures formalized via an index set ‘indexed structures’. A set
containing the symbols of a structure and their associated types is a signature.
Note that labelled structures uniquely determine a signature, while indexed struc-
tures only determine the types of the signature, not the associated symbols. The
difference between indexed and labelled structures is of central importance for
van Fraassen ( as detailed above) and others ( e. g., French and Ladyman 1999, 114–
15). It marks the difference between a notion of structure that is independent
from any vocabulary of predicate logic and a notion of structure that uniquely
determines such a vocabulary.

Van Fraassen (2014, 280) also cites a text by Suppes (1957, §12.2) to show
that structures can be defined independently of a vocabulary. Here, structures
are written as tuples containing a domain and relations, functions, and con-
stants. On the face of it, such a tuple has neither a vocabulary nor an index
set, but when interpreted as a labelled structure, the vocabulary is implicitly un-
derstood ( Hodges 1993, 4), and interpreted as an indexed structure, the position
in the tuple takes the role of the index set, so that 〈A, R1, . . . , Rn〉 can be read as
〈A,{Ri}i∈{1,...,n}〉 ( cf. Lutz 2014d, 1480–84). Thus one cannot say that “reducing
a [labelled] model to a bare set-theoretical n-tuple would be akin to reducing a
function f : A−→ B to its image f (A). Obviously, the function f contains more
data than its image f (A), and in exactly the same way, a model contains more
data than just the resulting set-theoretical structure” ( Halvorson 2013, 477). For,

6. It is clear from their use of the definition that Bell and Slomson (1974) intend ‘{Rn | n ∈ω}’
to be a family ( rather than the set of the family’s elements).

7. To be precise, Robinson assumes an injective mapping from the relations to the vocabulary,
which has the advantage that a structure can interpret vocabularies with symbols that do not occur
in the structure. Bell and Slomson (1974, §3.2) use a family of symbols and a family of relations
with the same index set, which leads to a natural bijection.

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as discussed in connection with van Fraassen’s remark above, an indexed struc-
ture contains more information than the image of an interpretation in the case
of a labelled structure. In general, if I : Pi 7→ Ri , then taking the image of
I leads to 〈A,{Ri | i ∈ I}〉 6= 〈A,{Ri}i∈I〉. Specifically, it can be the case that
〈A,{Ri}i∈I〉 6= 〈B ,{Si}i∈I〉 while 〈A,{Ri | i ∈ I}〉= 〈B ,{Si | i ∈ I}〉 ( the opposite
case is impossible).

( Van Fraassen (2014, 277) calls the combination of an indexed structure with
an appropriate interpretation a ‘model’, and can thus state that “a model is [ . . . ]
a structure with an interpretation”. But it is unclear why van Fraassen would
want to use this definition of ‘model’ for his semantic approach, since, save for
some renaming of indexes, the definition makes a model into nothing else but
a labelled structure. Furthermore, his claim that according to this definition “a
model is [ . . . ] an interpreted structure” is still highly misleading, for it is not the
structure that is interpreted, but the vocabulary; a model in this sense contains
an interpreting structure.)

4.3 A Fairly General Account of Semantic Approaches

To avoid the yoke of a vocabulary, semantic approaches should rely on indexed
rather than labelled structures. This is the first aspect of the fairly general account
of semantic approaches that I want to develop now. Its second aspect springs
from van Fraassen’s observation that the structures describing a theory are them-
selves “describable only up to structural isomorphism” ( van Fraassen 2008, 238;
cf. van Fraassen 2002, 22). Furthermore, often not all of a theory’s structures
are isomorphic to each other, because many theories are not categorical. Thus
the second aspect of semantic approaches is that a theory is described by a class
of indexed structures, not a single structure. A theory can also typically be de-
scribed by different classes of structures. In the case of van Fraassen’s state-spaces,
different state-spaces for the same theory are not isomorphic, and when Hendry
and Psillos (2007, 137, my emphasis) state in connection with the equivalence of
matrix and wave mechanics that “a ( semantic) model of one could be turned into
a model of the other”, they effectively point out that the structures are not the
same ( cf. Halvorson 2012, 188). Finally, it is clear that a theory cannot consist
solely of purely formal structures, since they have to connect to empirical results
in some way ( see also Lutz 2014d, §4). In publications relying on semantic ap-
proaches, this connection is typically not called ‘interpretation’ as in syntactic
approaches, but ‘representation’ ( e. g., Suppes 2002). Thus for the sequel, it is
appropriate to describe semantic approaches as follows:

Definition 2. An approach is semantic if and only if in the approach every
theory is described by an equivalence class of classes of indexed structures, where
at least some of the structures represent.

Often, a class of indexed structures is described using a set theoretical or
Suppes predicate ( see Suppes 2002, §2.3), which is a set theoretic description of

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the domains, relations, functions, and constants of the structures as well as the
formulas the relations, functions, and constants have to fulfill. It is clear that,
as in the case of state-spaces, the same class of structures can be described in
different ways, so that semantic approaches fulfilling definition 2 also have to
deal with equivalence classes of sentences.

At this point, it might seem as if semantic approaches are nothing but specific
syntactic approaches that, unlike syntactic approaches in general, are wedded to
the use of the language of set theory. The reasoning goes as follows: The set theo-
retical sentences describing the Suppes predicate of a theory can be given either in
a set theory axiomatized in higher order logic, in a set theory axiomatized in first
order logic, or in informal, naive set theory. First order set theory is essentially
higher order logic ( see, e. g., Väänänen 2001, §2) and thus if the set theoretical
sentences are axiomatized at all, the theory described by the Suppes predicate is
described in higher order logic and thus according to a syntactic approach. For
semantic approaches to be something other than specific syntactic approaches,
the set theoretical sentences describing the Suppes predicate must therefore be
given in naive set theory. As already noted in connection with van Fraassen’s dis-
avowal of any syntactic leanings ( §4.1), the difference is that between predicate
logic and the semi-regulated language of mathematics. But as Suppes (2002, 30)
writes:

The only important distinction between axiomatic and naive set the-
ory is that in axiomatic set theory one continually must consider
questions about the existence of sets[.] Although questions about
the existence of sets shall not be considered explicitly here, it is a
simple and routine matter to embed everything that is said [ . . . ]
within some well-defined axiomatic set theory[.]

It seems, then, that only a simple and routine matter keeps semantic approaches
from being specific syntactic approaches.

If the preceding argument were sound, my thesis that there is no essential dif-
ference between syntactic and semantic approaches would have been established,
and I could end here. But the argument leaves out the interpretation of syntacti-
cally described theories and the representation relation of semantically described
theories. In analogy to the interpretation of syntactic approaches, one can leave
the representation relation informal; in that case, there is indeed no principled
difference between syntactic and semantic approaches. But the predominant for-
mal approaches assume that a structure represents via an isomorphism or some
generalization thereof, for example an embedding ( cf. Halvorson 2012, 185). Like
the use of Tarski semantics, this demand requires that objects in the world can be
members of sets, and that it is possible to map from abstract objects like numbers
to objects in the world and sets thereof ( Lutz 2014d, §4). But the isomorphism
between two structures is different from the interpretation of a set of sentences
by a structure, simply on formal grounds. An isomorphism relates objects in
one domain to objects in another, and, for instance, elements of one relation to

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elements of another. An interpretation, on the other hand, relates primitive rela-
tion symbols to (Cartesian products of) classes of objects of some domain. Thus
the difference between interpretations and representations distinguishes between
syntactic and semantic approaches. Specifically, a theory described by a set of
sentences as in a syntactic approach can be formally interpreted by a structure
as given in semantic approaches, but a structure is not isomorphic to a set of
sentences. Note, however, that this does not mean that either approach is easier
to connect to objects in the world. In fact, an isomorphism relation between the
world and an indexed structure can be used to construct an interpretation of a
set of sentences, and an interpretation of a set of sentences by the world can be
used to construct an isomorphism relation to a structure ( Lutz 2014d, §4).

There are many semantic approaches in the wider sense that are completely
unrelated to the above discussion, namely those that interpret ‘model’ not in the
sense of model theory but as ‘scientific model’. As Frigg and Hartmann (2008,
§4.1) put it: “Although different [semantic approaches] assume a different notion
of model [. . . ,] they all agree that models are the central unit of scientific theoriz-
ing.” In other words, semantic approaches in the wider sense share with semantic
approaches in the narrow sense only the commitment to calling the central units
of scientific theorizing ‘models’. Suppes (1960) provided an influential argument
that the meaning and use of ‘model’ in the sciences is the same as in model theory
( cf. Suppes 2002, §2.1). But his argument at best establishes that the formal aspect
of scientific models can be reformulated in set theory, and it relies on the tacit
( and problematic) identification of set theoretic descriptions with descriptions in
model theory ( Lutz 2012, 97–99). Hence semantic approaches in the wider sense
may very well be as close to syntactic approaches as to semantic approaches, and
the relevance of scientific models for the debate between the two approaches
may very well be nil. Thus the claim by van Fraassen (2014, 278) that “models
in the sciences are not exactly what are called models in meta-mathematics” is
something of an understatement.

5 Criticism of Semantic Approaches

Criticisms of semantic approaches usually have been limited to very specific ap-
proaches ( e. g., Suárez and Cartwright 2008, §5) or have been aimed at all at-
tempts at formalization ( see Halvorson 2012, §7). The critique by Halvorson
(2012) therefore stands out in that it criticizes specifically semantic approaches.
At the core of Halvorson’s critique lies the question of theory individuation. In
a reversal of Suppe’s criticism, Halvorson argues that semantic approaches in-
correctly distinguish between different formulations of the same theory, that is,
make distinctions without a difference. But Halvorson goes further and argues
that semantic approaches also fail to distinguish theories that are different.

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5.1 An Inability to Distinguish Between Different Theories

At the core of Halvorson’s argument that semantic approaches identify theories
that are different lies his notion of isomorphism for semantic approaches. It is
explicated by Glymour (2013, 291) and accepted by Halvorson (2013, 476) in
roughly the following form:

Definition 3. Two structures 〈A, R1, . . . , Rn〉 and 〈B , S1, . . . , Sn〉 are H-isomorphic
just in case there is a bijection j : A −→ B and a bijection k : {Ri}i∈{1,...,n} −→
{Si}i∈{1,...,n} such that 〈a1, . . . , an〉∈ Ri if and only if 〈 j(a1), . . . , j(an)〉∈ k(Ri).8

This definition is not obviously uncharitable; in fact, it is the definition devel-
oped and endorsed by Turney (1990, def. 2) in his explication of van Fraassen’s
notion.

Halvorson now uses the notion of H-isomorphism to argue that semantic
approaches fail to distinguish different theories. To do this, he assumes that in se-
mantic approaches a theory is described by an equivalence class of classes of struc-
tures, as is also assumed in definition 2. And he assumes that a class of structures
is equivalent to another if and only if the two are pointwise H-isomorphic, that is,
there is a bijection F from one to the other such that F(A) is H-isomorphic to
A ( Halvorson 2012, 190). The argument Halvorson (2012, 192) uses for showing
that pointwise H-isomorphisms identify different theories can be slightly short-
ened as follows:

Claim 1. Let T and T ′ have the same signature containing one-place first-order
predicates P1 and P2. Let T = {∃x∀y(x = y)} and T

′ = T ∪{∀x(P1 x → P2 x)}.
Then T and T ′ are not logically equivalent, but they have H-isomorphic models.

Proof. Every model of T ′ is ( and thus is H-isomorphic to) a model of T . Without
loss of generality, assume that for T ′, the syntactic predicate P1 is interpreted by
semantic relation S1 and P2 by S2. Then the models of T

′ are given by the class
�

〈{a}, S1, S2〉 | S1 ⊆ S2, a arbitrary
	

. Now let A = 〈{a}, R1, R2〉 be any model of
T . If a ∈ R1, R2 or a 6∈ R1, R2, then A is a model ( and thus H-isomorphic to a
model) of T ′. If Ri ={a}, and R j =∅, {i , j}={1, 2}, choose k of definition 3 so
that k(R j)= S1 and k(Ri)= S2.

Thus two theories that can be distinguished in syntactic approaches cannot
be distinguished in semantic approaches, making semantic approaches less ex-
pressive. Since furthermore T and T ′ are also intuitively different, semantic ap-
proaches are intuitively inadequate.

Glymour’s response to Halvorson’s example is that pointwise H-isomor-
phism is the wrong equivalence relation between classes of structures, and
specifically that H-isomorphism is the wrong relation between structures. In-
stead, Glymour (2013, 291) argues and Halvorson (2013, 476) agrees, semantic

8. Halvorson omits k and instead writes ‘〈 j(a1), . . . , j(an)〉∈ S j ’, where j is meant to stand for
π(i), with π a permutation on {1, . . . , n} ( email from 27 May 2014).

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approaches must rely on labelled structures and the standard model theoretic
notion of isomorphism for labelled structures. Specifically, the existence of an
isomorphism between two structures must entail their elementary equivalence,
as it does in standard model theory. The use of labelled structures, of course,
forces the semantic view under the yoke of language, since it determines a
specific signature of predicate logic ( cf. Glymour 2013, 288–289; Halvorson 2013,
476).

But labelled structures and the associated notion of isomorphism are not nec-
essary to block Halvorson’s example. The example is also blocked by indexed
structures and their associated notion of isomorphism. For relational first order
structures, it is given by Bell and Slomson (1974, 73) and can be written as fol-
lows:

Definition 4. An isomorphism from an indexed structure A = 〈A,{Ri}i∈I〉 to
B= 〈B ,{Si}i∈I〉 is a bijection h from A to B such that for any n-ary relation Ri ,
i ∈ I and all a1, . . . , an ∈ A, 〈a1, . . . , an〉∈ Ri if and only if 〈h(a1), . . . , h(an)〉∈ Si .

The difference to H-isomorphisms ( definition 3) is clear: An H-isomorphism
is an isomorphism if and only if k(Ri)= Si for all 1 ≤ i ≤ n. And this difference
suffices for blocking Halvorson’s argument against pointwise H-isomorphisms:

Claim 2. Let T and T ′ have the same signature containing one-place first-order
predicates P1 and P2. Let T = {∃x∀y(x = y)} and T

′ = T ∪{∀x(P1 x → P2 x)}.
Then the classes of models of T and T ′ are not pointwise isomorphic.

Proof. In T , let P1 be assigned to Ri and P2 assigned to R j , with {i , j}= {1, 2}.
Then the class of models of T contains every structure 〈{a}, R1, R2〉 with a single-
ton domain. In T ′, let P1 be assigned to Sk and P2 assigned to Sl . Then the class
of models of T ′ contains all and only structures with a singleton domain and
Sk ⊆ Sl , and thus for domain {b} no structures with Sk ={b} and Sl =∅. There
are then two possibilities. If i = k and j = l , there is a T -model with Ri = {a}
and R j =∅ but no isomorphic T

′ model. If j = k and i = l , there is a T -model
with R j ={a} and Ri =∅ but no isomorphic T

′ model.

Note also that two isomorphic indexed structures are only elementary equiv-
alent if they have the same vocabulary, with each symbol mapped to the same
index of the index set of the structure’s family.

With the definition of isomorphism for indexed structures, it is easy to define
the notion of embedding that is so central to constructive empiricism:9

Definition 5. An embedding of an indexed structure A = 〈A,{Ri}i∈I〉 in B =
〈B ,{Si}i∈I〉 is an isomorphism from A to a substructure 〈C ,{Si ∩C

ni}i∈I〉 of B,
where C ⊆ B and ni is the arity of Si .

9. Halvorson’s argument that semantic approaches cannot rely on the model theoretic notion
of embedding is discussed in appendix A.

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It remains to be seen why Glymour and Halvorson both assume that there
cannot be an appropriate notion of isomorphism for structures without a unique
signature. An argument by Glymour (2013, 288) for the need for labelled struc-
tures provides a hint:

Suppose a theory has multiple models, one of which is 〈D1;ϕ,χ ,ϕ〉,
and there is a data model that maps three measured quantities to
the three relations. To predict what would happen were the con-
ditions of measurement changed, another model, say 〈D2;τ,δ,ω〉,
must be chosen and the measured quantities mapped to relations of
that model, but the radical semantic view has insufficient resources
to specify which relations among τ,δ,ω in this second model corre-
spond to ϕ,χ ,ϕ, respectively.

However, 〈D1;ϕ,χ ,ϕ〉 and 〈D2;τ,δ,ω〉 are tuples, and hence there is a perfectly
fine way to specify which relations among τ,δ,ω correspond to which relations
among ϕ,χ ,ϕ: The first relation in the one, ϕ, corresponds to the first relation
in the other, τ, and analogously for the second and third relations.

I surmise that while Glymour is explicit about his use of a non-standard
notion of isomorphism in his discussion, he implicitly assumes a non-standard
notion of structure as well. His notion does not contain an ordering of the rela-
tions, but only their number:

Definition 6. The H-structure corresponding to an indexed structure 〈A,{Ri}i∈I〉
is 〈A,[Ri | i ∈ I]〉.

[Ri | i ∈ I] is the multiset resulting from the family {Ri}i∈I .10 A multiset
is like a set, except that elements can occur any finite number of times ( Blizard
1989). The notion of an H-structure fits very well with Glymour’s argument
for the need for labelled structures: As long as structures are expressed by tu-
ples ( or families), each relation in the structure 〈D1,ϕ,χ ,ϕ〉 is associated with
exactly one relation of the structure 〈D2,τ,δ,ω〉. It is only when each struc-
ture is expressed by a domain and a multiset, 〈D1,[ϕ,χ ,ϕ]〉 and 〈D2,[τ,δ,ω]〉,
that the correspondence is lost. And this loss of correspondence is analogous to
the loss of correspondence incurred through the change from isomorphisms to
H-isomorphisms:

Claim 3. Structure A is H-isomorphic to B if and only if there are a structure A′
with the same corresponding H-structure as A and a structure B′ with the same
corresponding H-structure as B such that A′ is isomorphic to B′.

Proof. Two structures 〈A,{Ri}i∈I〉 and 〈A
′,{R′i}i∈I〉 have the same correspond-

ing H-structure if and only if A = A′ and there is a bijection k : {Ri}i∈I −→
{R′i}i∈I such that k(Ri)= R

′
j for some j ∈ I . Thus it is straightforward to show

10. I follow Hein (2003, 29) in designating multisets by square brackets.

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that f is an isomorphism from A to B if and only if f is an H-isomorphism
from some A′ to some B′ that have the same corresponding H-structures as A
and B, respectively.

Thus for two H-structures A and B, the claim that some structures corre-
sponding to A and B are isomorphic is equivalent to the claim that some struc-
tures corresponding to A and B are H-isomorphic. Phrased more casually, the
notion of isomorphism is indistinguishable from the notion of H-isomorphism
after the move from structures to H-structures. This seems to be the most plau-
sible justification of Glymour’s and Halvorson’s claim that without a language,
only H-isomorphisms are possible: The claim follows from the assumption that
without language, only H-structures are possible.

5.2 Distinctions Without a Difference

Halvorson not only argues that semantic approaches identify different theories,
but also that semantic approaches make distinctions without a difference. Un-
derlying his argument is the assumption that the equivalence relation for syntac-
tic approaches is definitional equivalence ( Halvorson 2012, 191). As Halvorson
(2013, 193–94) notes, the theory of groups can be given a syntactic description
with a signature containing a first order two-place relation symbol ‘◦’ and a first
order constant symbol ‘e ’, but also a definitionally equivalent syntactic descrip-
tion with a signature containing only ‘◦’. The classes of indexed models of these
descriptions are not pointwise H-isomorphic.

Glymour’s response is straightforward: The equivalence classes of semantic
approaches can also be taken to be determined by definitional equivalence, which
in first order model theory can be phrased equivalently in terms of structures
using coalescence ( Glymour 2013, 292–93). Glymour again assumes labelled struc-
tures for his argument, but the concept of coalescence can also be defined using
indexed structures.11 There is also another, very cheap way of defining defini-
tional equivalence without reference to a vocabulary: An explicit definition of
n-place relation symbol P has the form ð∀x1 . . . xn[P x1 . . . xn ↔ϕ(x1, . . . , xn)]ñ,
where x1, . . . , xn are pairwise different variables and ϕ is some formula with n free
variables. If the relation R is assigned to P , then the model theoretic definition
of truth entails a corresponding set theoretic condition ∀a1, . . . , an[〈a1, . . . , an〉∈
R ⇔ ϕ∗(a1, . . . , an)], where ϕ

∗ is the set theoretic formula that corresponds to
ϕ ( cf. Przełęcki 1969, 14–15; Schurz 2014, 95).12 This procedure is also possible
in higher order logic. The sets ∆ and ∆′ used to establish definitional equiva-
lence ( T ∪∆��T ′∪∆′) of two theories T and T ′ can thus be translated into set

11. These definitions rely on the notions of expansion and reduct: An indexed structure A =
〈A,{Ri}i∈I〉 is a reduct of B= 〈B ,{S j} j∈J〉 if and only if A= B , I ⊆ J , and Ri = Si for all i ∈ I . A
is an expansion of B if and only if B is a reduct of A.
12. Definitions of function and constant symbols require additional considerations that are tech-

nically important but do not change the conclusions of this discussion.

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theoretic conditions on all expansions of the elements of two classes of indexed
structures T and T′. If these expansions lead to the same class of structures, T
and T′ can be treated as in the same equivalence class. The classes of models of
two syntactic theories T and T ′ are so equivalent if and only if the two syntactic
theories are definitionally equivalent. Being somewhat more subtle, one could
define set theoretic restrictions on the expansions of structures in T and T′ so
that each structure can be expanded in exactly one way. If there is then a class
of structures that fulfills these restrictions and has T and T′ as reducts, call that
class the common unique expansion class of T and T′.

Claim 4. If T and T ′ are definitionally equivalent, the classes of their models have
a common unique expansion class. If two classes T and T′ of indexed structures are
models of first order theories (finite higher order theories) T and T ′, respectively, and
have a common unique expansion class that is the class of models of some first order
theory (finite higher order theory) T ∗, then T and T ′ are definitionally equivalent.

Proof. An explicit definition of a symbol uniquely determines the symbol’s ex-
tension in any expansion ( Hodges 1993, 61). Since the restrictions on the expan-
sions can be expressed in set theory, the first part of the claim follows. For the
second part, note that by assumption every model of T and T ′, respectively, can
be expanded to a model of T ∗, and this expansion is unique. For first order logic,
it follows from Beth’s theorem that T ∗ entails explicit definitions for all symbols
not in T or in T ′. For finite theories of higher order logic, this was shown by
Tarski (1935, theorem 3; cf. Leivant 1994, §5.1).

Halvorson (2013, 475) agrees with Glymour’s claim that with labelled struc-
tures, the corresponding notion of isomorphism, and definitional equivalence,
Halvorson’s examples of incorrect identifications of theories are blocked. But
with definitional equivalence as the identity criterion for theories, so Halvorson
(2013, 478), one central application of semantic approaches, structural realism, is
in trouble:

Recall that some structural realists want to subscribe to the view
that:

(p1) To believe a theory T means to believe that the world is iso-
morphic to a model of T .

But structural realists should also be committed to the claim that:

(p2) There is no good reason to believe T over an equivalent theory
T ′.

[ . . . ] But p1 and p2 [ . . . ] entail that believing T means believing
that the world is isomorphic to a model of T but also that the world
is isomorphic to a model of T ′. But nothing is isomorphic both to a
model of T and to a model of T ′.

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In this argument, Halvorson assumes that the models of T are the structures
in which T is true, that is, he assumes that ‘model of T ’ is understood as in
model theory. With definition 2 and definitional equivalence as the equivalence
relation, however, the models of T are those in which T or any T ′ definitionally
equivalent to T is true. ( Hence in semantic approaches it would be advantageous
to speak of the models that describe T to distinguish this notion from the notion
of ‘model of T ’ in model theory.) Thus Halvorson’s argument does not establish
that structural realism is in trouble, but rather the following: If structural realists
believe a theory T , the structures that they can ascribe to the world are not
restricted to the models of T , but to the models that describe T , that is, models
of theories T ′ that are definitionally equivalent to T .

5.3 H-structures, Indexed Structures, and Labelled Structures

Glymour and Halvorson agree that H-isomorphisms ( and thus H-structures) are
insufficient for an analysis of scientific theories. They also agree that therefore, se-
mantic approaches must rely on labelled structures, which include a vocabulary
and therefore a full language. But this poses a problem, as Halvorson (2013, 475)
observes, for “if we add language to the semantic view, then we have a syntac-
tic view”.13 I have argued above that the use of labelled structures in a semantic
approach indeed makes it into a syntactic approach, but proponents of semantic
approaches can very well disagree with Glymour’s and Halvorson’s claim that
because H-structures do not suffice for the description of theories, semantic ap-
proaches have to rely on labelled structures. As argued above, the semantic view
relies on indexed structures, and proponents of semantic views have put a lot
of weight on the distinction between indexed and labelled structures in their
arguments for the superiority of semantic approaches ( see §4.2).

How important, then, is the distinction? I claim that formally, there is no
relevant distinction at all and thus that Glymour and Halvorson are correct in
their conclusion ( cf. Lutz 2014d, 1480). The case is especially obvious for rela-
tional structures: Here the labelled structure is given by a pair of a domain and
an interpretation of all relation symbols:

〈A,J〉 with J : V −→{Ri | i ∈ I} . (1)

An indexed relational structure is given by a pair 〈A,{Ri}i∈I〉, containing a do-
main and a family of relations. The standard definition of a family {Ri}i∈I is
that of a mapping f : I −→{Ri | i ∈ I} from the family’s index set to the set of
relations ( Bourbaki 1968, 81–82). Substituting the definiens for the definiendum,
the indexed structure is therefore given by

〈A, f 〉 with f : I −→{Ri | i ∈ I} . (2)

13. Halvorson adds, incorrectly as argued above, that such a syntactic view would be “indistin-
guishable from the oft-reviled ‘received view of theories’”.

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Thus the family of an indexed structure is an interpretation with I =V , since
then 〈A,{Ri}i∈I〉= 〈A,J〉. Therefore an indexed relational structure is formally
indistinguishable from a labelled relational structure.

More generally, a labelled structure 〈A,I〉 is a pair of a domain A and a map-
ping I from symbols to functions, relations, and constants. An indexed struc-
ture, 〈A,{Ri}i∈I ,{F j} j∈J ,{ck}k∈K〉 is a tuple of a domain A, a family of relations,
a family of functions, and a family of constants. Assuming that the index sets I , J ,
and K are disjoint, this means that the indexed structure is definitionally equiva-
lent to the pair 〈A,{Wl}l∈I∪J∪K〉, where {Wl}l∈I∪J∪K is a mapping from I ∪J ∪K
to the set {Wl | l ∈ I ∪ J ∪K} := {Ri | i ∈ I}∪{F j | j ∈ J}∪{ck | k ∈ K}.14 Even
if I , J , and K are not disjoint, the indexed structure is definitionally equivalent
to 〈A,{Wl}l∈I∗∪J∗∪K∗〉, where I

∗ := {〈i ,∅,∅〉 | i ∈ I}, J∗ := {〈∅, j ,∅〉 | j ∈ J},
and K∗ :={〈∅,∅, k〉 | k ∈ K}. In short, the families induce a mapping from their
index sets to the relations, functions, and constants. And this mapping is indistin-
guishable from an interpretation, which is also a mapping from a set of symbols
to the relations, functions, and constants.

In effect, the difference between an indexed structure and a labelled structure
is that an indexed structure’s index sets are not meant to be symbols of predicate
logic. Of course, that does not hinder anyone to use them as such, and it also
leaves nothing of the claim that when theories are given by indexed structures,
there is some greater language independence. For two indexed structures are iso-
morphic given their index sets, so that changing the index sets of two isomorphic
structures can render them non-isomorphic, and changing the index sets of two
non-isomorphic structures and can sometimes render them isomorphic. A de-
pendence on a vocabulary is thus exchanged for a dependence on an index set
( which could be taken to be a vocabulary). Therefore there is no substantial mid-
dle ground between H-structures ( which are, as Halvorson shows, not expressive
enough) and labelled structures.15

6 Van Fraassen’s Defense of Semantic Approaches

Van Fraassen (2014) makes three “gentle remarks” about the discussion between
Glymour and Halvorson:

1. The alleged dependence of any plausible semantic view on language is the
trivial one already identified by himself (279),

2. Halvorson misrepresents ( a) syntactic approaches and ( b) semantic ap-
proaches (280–82), and

14. To be precise, and understanding families as functions, {Wl}l∈I∪J∪K|I = {Ri}i∈I ,
{Wl}l∈I∪J∪K|J ={F j} j∈J , and {Wl}l∈I∪J∪K|K ={ck}k∈K .
15. Another suggested middle ground that does not fall prey to Halvorson’s argument equally

turns out to be equivalent to labelled structures ( Lutz 2014d, 483–84).

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Sebastian Lutz The Syntax-Semantics Debate in the Philosophy of Science—Preprint

3. Halvorson’s arguments do not affect semantic approaches because they ig-
nore the role of representation ( van Fraassen 2014, §2).

Van Fraassen’s first remark starts from Glymour’s claim that “Newton, Einstein,
Schrodinger, and so on, had no way of specifying the class of relational structures
they intended, except indirectly, as those structures satisfying their theoretical
claims. Nor do we today” ( Glymour 2013, 288). And these theoretical claims
are given in a language. “Equations are sentences or formulas in a language”, for
instance (288–89). With a reference to his “trivial point” that “to present a theory,
we must present it in and by language”, van Fraassen (2014, §3) responds that

there must be a serious point to Glymour’s iteration of this triv-
ial point. Either he wants to strongly endorse Halvorson’s take on
what the semantic view really is, or else he spends those several im-
passioned paragraphs to warn Halvorson away from tilting at wind-
mills. I think of Professor Glymour as a person of serious intent, so
the latter seems more likely.

Thus Glymour either makes van Fraassen’s point clear to Halvorson, or he is
not being serious. A third option is that Glymour and Halvorson agree on the
role of language in semantic approaches and make a non-trivial point different
from van Fraassen’s. This more charitable interpretation of their exchange is not
quite as obvious because Glymour (2013, 287) begins his discussion noting that
a Suppes predicate is a predicate and thus is linguistic, which does suggest van
Fraassen’s trivial point. But Glymour’s discussion of the relation between the
two structures 〈D1;ϕ,χ ,ϕ〉 and 〈D2;τ,δ,ω〉 leads him to the stronger conclu-
sion that the structures involved in semantic approaches must be labelled ones.
And this means that even if the Suppes predicate can be described by a multitude
of languages different from predicate logic, the structure that it describes must
determine a unique signature for predicate logic. Thus if van Fraassen (1989, 366)
indeed thinks that the “impact of Suppes’s innovation is lost if models are de-
fined [ . . . ] to be partially linguistic entities, each yoked to a particular syntax”,
then Glymour argues that the impact of Suppes’s innovation is lost. Recall that
van Fraassen (1989, 212–13) claimed that “we must advance the conception of nat-
ural language as not being constituted by any one realization of any such logical
skeleton [of predicate logic]. [ . . . ] In my opinion, it would be a poor choice to
try and describe a whole language in which a given theory can be formulated.”
But if every structure is labelled, then each description of a theory contains the
complete signature for a predicate logical realization in which the theory can be
formulated. Hence there is much more than “a bow in the direction of syntactic
description” in van Fraassen’s semantic approach

Based on Toulmin’s description of the Received View’s ambition at its most
grandiose, remark 2( a) relies on the assumption that syntactic approaches assume
a mathesis universalis. Van Fraassen does not spell out Halvorson’s misrepresen-
tation, but it is presumably his assumption that syntactic approaches can be ap-

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plied to single theories. As noted above, this is not a misrepresentation, neither
for syntactic approaches nor for the Received View.

For remark 2( b), van Fraassen (2014, §3) refers to his earlier account of ele-
mentary sentences and satisfaction functions to show that he did not intend to
avoid language altogether ( “a trivially absurd idea”) and that Halvorson therefore
misrepresents van Fraassen’s semantic account. But since van Fraassen (1989, 365,
n. 34) “soon found it much more advantageous to concentrate on the proposi-
tions expressible by elementary statements, rather than on the statements them-
selves”, this historical remark does not touch Halvorson’s criticism, which ap-
plies to van Fraassen’s current position with indexed structures and without el-
ementary sentences. Of course, to make van Fraassen’s trivial point, a reference
to elementary sentences is not necessary ( since even Suppes predicates must be
phrased in some language). But neither the trivial point nor the use of elemen-
tary sentences addresses Halvorson’s ( and Glymour’s) conclusion, which is that
semantic approaches require the language of predicate logic and hence cannot
rely only on natural language or elementary sentences alone.

Van Fraassen’s remark 3 starts from an analogy: Treating a theory as a class
of structures is like treating the Mona Lisa as a piece of wood. A more direct
formulation is as follows: Semantic approaches assume that every theory can be
described by an equivalence class of classes of structures; they do not assume
that every such equivalence class describes a theory, for only those equivalence
classes that represent do. This point is well taken, but it does not obviously
invalidate Halvorson’s criticisms if the representational aspect of the theories ( the
“theoretical hypothesis”) is not further specified. The question is then whether
van Fraassen’s account of the theoretical hypothesis, empirical adequacy, escapes
Halvorson’s criticism.

Oddly, van Fraassen does not respond to Glymour’s argument to the con-
trary. Glymour (2013, 289–90) points out that empirical adequacy is defined via
embeddability, which is preserved under isomorphism. So a purely formal dis-
cussion like Halvorson’s should be enough. More explicitly, one can directly
address van Fraassen’s criticism that “in the cases Halvorson discusses [ . . . ] it
is not clear what the theoretical hypothesis is supposed to be. Halvorson does
not address the problem of defining empirical adequacy” ( van Fraassen 2014,
n. 2). The problem can be addressed easily by supplying theoretical hypothe-
ses. In van Fraassen’s case, this means identifying specific substructures as em-
pirical substructures. Their isomorphism with the appearances then establishes
empirical adequacy. For the group example, the models of the theory formal-
ized with ‘◦’ and ‘e ’ include structure 〈R,+, 0〉. Thus assume that the empiri-
cal substructure for this specific model is given by 〈N,+, 0〉. When formalized
with ‘◦’ alone, the models of the theory do not contain 〈R,+, 0〉 with empiri-
cal substructure 〈N,+, 0〉, but rather 〈R,+〉 with empirical substructure 〈N,+〉.
Therefore the two classes of models of definitionally equivalent axiomatizations
of the same theory are not empirically adequate for the same appearances, and
are therefore incorrectly treated as describing different theories by van Fraassen’s

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Sebastian Lutz The Syntax-Semantics Debate in the Philosophy of Science—Preprint

semantic approach. The models of Halvorson’s theories T ={∃x∀y(x = y)} and
T ′= T ∪{∀x(P1 x → P2 x)} each have only one substructure, namely themselves.
Therefore the models are their own empirical substructures and T and T ′ are
theories only about observable phenomena. Since the models of T and T ′ are
pointwise H-isomorphic, T has a model that is H-isomorphic to an appearance
if and only if T ′ has such a model, and thus the classes of the models of the two
theories are empirically adequate for the same appearances. Thus two different
theories are incorrectly treated as the same. Therefore taking the representational
aspect of theories into account does not avoid Halvorson’s criticism. This also
means that van Fraassen cannot rely on H-structures, but must rely on indexed
structures, which, I have argued, amount to labelled structures.

7 Not With a Bang, But a Whimper

Halvorson’s arguments have shown that plausible semantic approaches cannot
rely on a notion of structure that consists of a domain and a multiset, but must
rely on the notion of an indexed structure. The analysis of the notion of an in-
dexed structure then shows that the alleged language independence of semantic
approaches boils down to a decision not to call a specific mapping an interpreta-
tion, but a family. This makes no formal difference, and informally it is at best
something of an indicator where one’s interests lie: In the set theoretic aspects
of the formalism or in the predicate logical aspects. Accordingly, there have been
philosophers of science working on both aspects ( e. g., Kemeny 1956; Przełęcki
1969; Demopoulos 2008; Andreas 2010; van Benthem 2012) without therefore
being incoherent.

In spite of these results, a lot of effort has gone into disparaging semantic
and especially syntactic approaches. Even van Fraassen (2014, 282), who wants
to end his remarks “on a positive note” with the suggestion to let “a hundred
flowers blossom” as far as the formalization of theories is concerned, cannot re-
frain from a Parthian shot at “the poverty-stricken syntactic approach of Carnap
and his cohort”. But van Fraassen’s own account of empirical adequacy can be
described in the Received View ( Lutz 2014a), which is thus at least not more
poverty-stricken than van Fraassen’s account. Similarly, the partial structure ap-
proach, which has been presented as a generalization of the semantic approach
based on model theory ( da Costa and French 1990) and as decisively superior to
the Received View ( French and Ladyman 1999) can be described in the Received
View as well ( Lutz, forthcoming). So in spite of over 40 years of infighting in the
philosophy of science, the syntax-semantics debate was about a distinction that
marks no difference.

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Sebastian Lutz The Syntax-Semantics Debate in the Philosophy of Science—Preprint

A Theory Independence of Embeddings

Halvorson’s argument discussed in §5.1 is meant to establish that semantic ap-
proaches have no adequate notion of isomorphism because the only possible
notion of isomorphism that fits semantic approaches fails ( cf. Halvorson 2013,
476). Since an isomorphism can be defined as a surjective embedding, one could
thus conclude that the semantic approach cannot explicate the relation of embed-
ding between theories either. Halvorson (2012, 199) also argues directly for this
latter claim. He states that, while the term ‘embeddability’ “has a technical use
in model theory”, “the semantic approach cannot explicate the relation of em-
bedding between theories”. This claim has some prima facie plausibility, given
that van Fraassen himself seems to be working with a definition of isomorphism
other than that of model theory: He switched from elementary statements and
satisfaction functions to embeddings while retaining his commitment to formal-
izations in state-spaces ( van Fraassen 1980, §3.9; van Fraassen 1989, §9.3,§9.4),
without however spelling out how embeddings work in state-spaces. If there is
indeed no semantic account of embedding and since an embedding can be defined
as an isomorphism to a substructure, Halvorson can conclude that the semantic
approach cannot explicate the relation of isomorphism between theories either
( Halvorson 2012, 201).16

Assuming that Halvorson’s direct argument against the semantic approach’s
ability to explicate embedding succeeds, it provides a reason for van Fraassen to
rely on the model theoretic notion of embedding. And in spite of initial appear-
ances, van Fraassen must mean the model theoretic conceptions of isomorphism
and embedding when using these terms in his definition of empirical adequacy
( Lutz 2014b, 3206–8), the most obvious being his reference to model theoretic
structures as the meaning of ‘model’ discussed above ( van Fraassen 2014, 277).

Glymour and I have argued that the notions of isomorphism and embedding
defined in model theory ( whether for indexed or labelled structures) are appro-
priate for semantic approaches. Halvorson (2012, 199) to the contrary claims that
model theoretic embeddability describes “a relation between models of a single
theory”, which would render both embeddings and isomorphisms incapable of
relating different theories. But this claim is false: The models of one theory ( say,
‘∃x∀y(x = y)’) may be embeddable in the models of an incompatible theory ( say,
‘∃x y[x 6= y ∧∀z(z = x ∨ z = y)]’), and two non-equivalent theories ( say, ‘∃x P x ’
and ‘∀x P x ’) can share a model and thus have isomorphic models. More gen-
erally, neither the definition of embedding nor that of isomorphism references
theories. At the most, embeddings and isomorphisms describe relations between
structures of a single signature or index set ( cf. Hodges 1993, 1, 5).

16. This argument assumes that even outside of model theory, embeddings should be defined as
isomorphisms to substructures, whatever these terms mean.

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	Introduction
	The Received View
	Criticisms of Syntactic Approaches
	Semantic Approaches
	State Spaces
	Structures
	A Fairly General Account of Semantic Approaches

	Criticism of Semantic Approaches
	An Inability to Distinguish Between Different Theories
	Distinctions Without a Difference
	H-structures, Indexed Structures, and Labelled Structures

	Van Fraassen's Defense of Semantic Approaches
	Not With a Bang, But a Whimper
	Theory Independence of Embeddings