## 1. Introduction



In philosophy, three families of perspectives on scientific theory are
operative: the *Syntactic View*, the *Semantic View*,
and the *Pragmatic View.* Savage distills these philosophical
perspectives thus:



> 
> The *syntactic* view that a theory is an axiomatized collection
> of sentences has been challenged by the *semantic* view that a
> theory is a collection of nonlinguistic models, and both are
> challenged by the view that a theory is an amorphous entity consisting
> perhaps of sentences and models, but just as importantly of exemplars,
> problems, standards, skills, practices and tendencies. (Savage 1990,
> vii-viii)
> 



Mormann (2007) characterizes the Syntactic and Semantic Views in
similar terms, and is among the first to use the term "Pragmatic
View" to capture the third view (137). The three views are
baptized via a trichotomy from linguistics deriving from the work of
Charles Morris, following Charles S. Peirce. In a classic exposition,
the logical positivist Carnap writes:



> 
> If in an investigation explicit reference is made to the speaker, or,
> to put it in more general terms, to the user of a language, then we
> assign it to the field of *pragmatics*. (Whether in this case
> reference to designata is made or not makes no difference for this
> classification.) If we abstract from the user of the language and
> analyze only the expressions and their designata, we are in the field
> of *semantics*. And if, finally, we abstract from the designata
> also and analyze only the relations between the expressions, we are in
> (logical) *syntax*. The whole science of language, consisting
> of the three parts mentioned, is called *semiotic*. (1942, 9;
> see also Carnap 1939, 3-5, 16)
> 



To summarize, syntax concerns grammar and abstract structures;
semantics investigates meaning and representation; and pragmatics
explores use. Importantly, while no view is oblivious to the syntax,
semantics, or pragmatics of theory, the baptism of each is a product
of how one of the three aspects of language is perceived to be
dominant: theory as syntactic logical reconstruction (Syntactic View);
theory as semantically meaningful mathematical modeling (Semantic
View); or theory structure as complex and as closely tied to theory
pragmatics, i.e., function and context (Pragmatic View). Each of these
philosophical perspectives on scientific theory will be reviewed in
this entry. Their relations will be briefly considered in the
Conclusion.


### 1.1 Syntactic, Semantic, and Pragmatic Views: The Basics



It will be helpful to pare each perspective down to its essence. Each
endorses a substantive thesis about the structure of scientific
theories.



For the Syntactic View, the structure of a scientific theory is its
reconstruction in terms of sentences cast in a metamathematical
language. Metamathematics is the axiomatic machinery for building
clear foundations of mathematics, and includes predicate logic, set
theory, and model theory (e.g., Zach 2009; Hacking 2014). A central
question of the Syntactic View is: in which logical language should we
recast scientific theory?



Some defenders of the Semantic View keep important aspects of this
reconstructive agenda, moving the metamathematical apparatus from
predicate logic to set theory. Other advocates of the Semantic View
insist that the structure of scientific theory is solely mathematical.
They argue that we should remain at the mathematical level, rather
than move up (or down) a level, into foundations of mathematics. A
central question for the Semantic View is: which mathematical models
are actually used in science?



Finally, for the Pragmatic View, scientific theory is internally and
externally complex. Mathematical components, while often present, are
neither necessary nor sufficient for characterizing the core structure
of scientific theories. Theory also consists of a rich variety of
nonformal components (e.g., analogies and natural kinds). Thus, the
Pragmatic View argues, a proper analysis of the grammar (syntax) and
meaning (semantics) of theory must pay heed to scientific theory
complexity, as well as to the multifarious assumptions, purposes,
values, and practices informing theory. A central question the
Pragmatic View poses is: which theory components and which modes of
theorizing are present in scientific theories found across a variety
of disciplines?



In adopting a *descriptive* perspective on the structure of
scientific theories, each view also deploys, at least implicitly, a
*prescriptive* characterization of our central topic. In other
words, postulating that scientific theory is \(X\) (e.g., \(X\) = a
set-theoretic structure, as per Suppes 1960, 1962, 1967, 1968, 2002)
also implies that what is not \(X\) (or could not be recast as \(X\))
is not (or could not possibly be) a scientific theory, and would not
help us in providing scientific understanding, explanation,
prediction, and intervention. For the Syntactic View, what is not (or
cannot be) reconstructed axiomatically is not theoretical, while for
the Semantic View, what is not (or cannot be) modeled mathematically
is not theoretical. In contrast, in part due to its pluralism about
what a scientific theory actually (and possibly) is, and because it
interprets theory structure as distributed in practices, the Pragmatic
View resists the definitional and normative terms set by the other two
views. As a result, the Pragmatic View ultimately reforms the very
concepts of "theory" and "theory structure."




This encyclopedia entry will be organized as follows. After presenting
this piece's two sustained examples, immediately below, the
three views are reviewed in as many substantive sections. Each section
starts with a brief overview before characterizing that
perspective's account of theory structure. Newtonian mechanics
is used as a running example within each section. The interpretation
of theory structure--viz., how theory "hooks up" with
phenomena, experiment, and the world--is also reviewed in each
section. In the final section of this entry, we turn to population
genetics and an analysis of the Hardy-Weinberg Principle (HWP) to
compare and contrast each view. The Conclusion suggests, and remains
non-committal about, three kinds of relations among the views:
*identity*, *combat*, and *complementarity*.
Theory is not a single, static entity that we are seeing from three
different perspectives, as we might represent the Earth using three
distinct mathematical map projections. Rather, theory itself changes
as a consequence of perspective adopted.


### 1.2 Two Examples: Newtonian Mechanics and Population Genetics



Two examples will be used to illustrate differences between the three
views: Newtonian mechanics and population genetics. While relativity
theory is *the* preferred theory of the Syntactic View,
Newtonian mechanics is more straightforward. Somewhat permissively
construed, the theory of Newtonian mechanics employs the basic
conceptual machinery of inertial reference frames, centers of mass,
Newton's laws of motion, etc., to describe the dynamics and
kinematics of, among other phenomena, point masses acting
vis-a-vis gravitational forces (e.g. the solar system) or with
respect to forces involved in collisions (e.g., pool balls on a pool
table; a closed container filled with gas). Newtonian mechanics is
explored in each section.



Population genetics investigates the genetic composition of
populations of natural and domesticated species, including the
dynamics and causes of changes in gene frequencies in such populations
(for overviews, see Lloyd 1994 [1988]; Gould 2002; Pigliucci and
Muller 2010; Okasha 2012). Population genetics emerged as a
discipline with the early 20th century work of R.A. Fisher,
Sewall Wright, and J.B.S. Haldane, who synthesized Darwinian
evolutionary theory and Mendelian genetics. One important part of
population genetic theory is the Hardy-Weinberg Principle. HWP is a
null model mathematically stating that gene frequencies remain
unchanged across generations when there is no selection, migration,
random genetic drift, or other evolutionary forces acting in a given
population. HWP peppers early chapters of many introductory textbooks
(e.g., Crow and Kimura 1970; Hartl and Clark 1989; Bergstrom and
Dugatkin 2012). We return to HWP in Section 5 and here merely state
questions each view might ask about population genetics.



The Syntactic View focuses on questions regarding the highest
axiomatic level of population genetics (e.g., Williams 1970, 1973; Van
Valen 1976; Lewis 1980; Tuomi 1981, 1992). Examples of such queries
are:


1. What would be the most convenient metamathematical axiomatization
of evolutionary processes (e.g., natural selection, drift, migration,
speciation, competition)? In which formal language(s) would and could
such axiomatizations be articulated (e.g., first-order predicate
logic, set theory, and category theory)?
2. Which single grammars could contain a variety of deep evolutionary
principles and concepts, such as HWP, "heritability," and
"competitive exclusion"?
3. Which formal and methodological tools would permit a smooth flow
from the metamathematical axiomatization to the mathematical theory of
population genetics?



Investigations of the axiomatized rational reconstruction of theory
shed light on the power and promises, and weaknesses and
incompleteness, of the highest-level theoretical edifice of population
genetics. 



Secondly, the Semantic View primarily examines questions regarding the
mathematical structure of population genetics (Lewontin 1974, Beatty
1981; Lopez Beltran 1987; Thompson 1989, 2007; Lloyd
1994 [1988]). Very generally, this exploration involves the following
questions:


1. What is the form and content of the directly presented class of
mathematical models of evolutionary theory (e.g., HWP)? How could and
should we organize the cluster of mathematical models (sensu Levins
1966) of population genetics?
2. Which additional models (e.g., diagrammatic, narrative, scale)
might be used to enrich our understanding of evolutionary theory?
3. What are the relations among theoretical mathematical models, data
models, and experimental models? How does theory explain and shape
data? How do the data constrain and confirm theory?



The main subject of investigation is mathematical structure, rather
than metamathematics or even alternative model types or modeling
methods. 



Finally, the Pragmatic View asks about the internal complexity of
population genetic theory, as well as about the development and
context of population genetics. In so doing, it inquires into how
purposes and values have influenced the theoretical structure of
evolutionary theory, selecting and shaping current population genetics
from a wide variety of possible alternative theoretical articulations.
The following questions about the structure of population genetic
theory might be here addressed:


1. What role did R.A. Fisher's interest in animal husbandry,
and his tenure at Rothamsted Experimental Station, play in shaping his
influential methodologies of Analysis of Variance (ANOVA) and
experimental design involving randomization, blocking, and factorial
designs?
2. How did the development of computers and computational practices,
statistical techniques, and the molecularization of genetics, shape
theory and theorizing in population genetics, especially from the
1980s to today?
3. How might normative context surrounding the concept of
"race" impact the way concepts such as
"heritability" and "lineage," or principles
such as HWP, are deployed in population genetics?



As when studying an organism, the structure of theory cannot be
understood independently of its history and function.


## 2. The Syntactic View



According to the Syntactic View, which emerged mainly out of work of
the Vienna Circle and Logical Empiricism (see Coffa 1991; Friedman
1999; Creath 2014; Uebel 2014), philosophy most generally practiced
is, and should be, the study of the logic of natural science, or
*Wissenschaftslogik* (Carnap 1937, 1966; Hempel 1966). Robust
and clear logical languages allow us to axiomatically reconstruct
theories, which--by the Syntacticists' definition--are
*sets of sentences in a given logical domain language* (e.g.,
Campbell 1920, 122; Hempel 1958, 46; cf. Carnap 1967 [1928],
SS156, "Theses about the Constructional System"). Domain languages
include "the language of physics, the language of
anthropology" (Carnap 1966, 58). 



This view has been variously baptized as the Received View (Putnam
1962; Hempel 1970), the Syntactic Approach (van Fraassen 1970, 1989),
the Syntactic View (Wessels 1976), the Standard Conception (Hempel
1970), the Orthodox View (Feigl 1970), the Statement View (Moulines
1976, 2002; Stegmuller 1976), the Axiomatic Approach (van
Fraassen 1989), and the Once Received View (Craver 2002). For
historical reasons, and because of the linguistic trichotomy discussed
above, the "Syntactic View" shall be the name of choice in
this entry.


### 2.1 Theory Structure per the Syntactic View



Some conceptual taxonomy is required in order to understand the
logical framework of the structure of scientific theories for the
Syntactic View. We shall distinguish *terms*,
*sentences*, and *languages* (see Table 1). 


#### 2.1.1 Terms



Building upwards from the bottom, let us start with the three kinds of
terms or vocabularies contained in a scientific language: theoretical,
logical, and observational. Examples of theoretical terms are
"molecule," "atom," "proton," and
"protein," and perhaps even macro-level objects and properties
such as "proletariat" and "aggregate demand."
Theoretical terms or concepts can be classificatory (e.g.,
"cat" or "proton"), comparative (e.g.,
"warmer"), or quantitative (e.g.,
"temperature") (Hempel 1952; Carnap 1966, Chapter 5).
Moreover, theoretical terms are "theoretical constructs"
introduced "jointly" as a "theoretical system"
(Hempel 1952, 32). Logical terms include quantifiers (e.g., \(\forall,
\exists\)) and connectives (e.g., \(\wedge, \rightarrow\)). Predicates
such as "hard," "blue," and "hot,"
and relations such as "to the left of" and "smoother
than," are observational terms. 


#### 2.1.2 Sentences



Terms can be strung together into three kinds of sentences:
theoretical, correspondence, and observational. \(T\_S\) is the set of
theoretical sentences that are the axioms, theorems, and laws of the
theory. Theoretical sentences include the laws of Newtonian mechanics
and of the Kinetic Theory of Gases, all suitably axiomatized (e.g.,
Carnap 1966; Hempel 1966). Primitive theoretical sentences (e.g.,
axioms) can be distinguished from derivative theoretical sentences
(e.g., theorems; see Reichenbach 1969 [1924]; Hempel 1958; Feigl
1970). \(C\_S\) is the set of correspondence sentences tying
theoretical sentences to observable phenomena or "to a
'piece of reality'" (Reichenbach 1969 [1924], 8; cf.
Einstein 1934, 1936 [1936], 351). To simplify, they provide the
theoretical syntax with an interpretation and an application, i.e., a
semantics. Suitably axiomatized version of the following sentences
provide semantics to Boyle's law, \(PV = nRT\): "\(V\) in
Boyle's law is equivalent to the measurable volume \(xyz\) of a
physical container such as a glass cube that is \(x\), \(y\), and
\(z\) centimeters in length, width, and height, and in which the gas
measured is contained" and "\(T\) in Boyle's law is
equivalent to the temperature indicated on a reliable thermometer or
other relevant measuring device properly calibrated, attached to the
physical system, and read." Carnap (1987 [1932], 466) presents
two examples of observational sentences, \(O\_S\): "Here (in a
laboratory on the surface of the earth) is a pendulum of such and such
a kind," and "the length of the pendulum is 245.3
cm." Importantly, theoretical sentences can only contain
theoretical and logical terms; correspondence sentences involve all
three kinds of terms; and observational sentences comprise only
logical and observational terms.


#### 2.1.3 Languages



The total domain language of science consists of two languages: the
theoretical language, \(L\_T\), and the observational language, \(L\_O\)
(e.g., Hempel 1966, Chapter 6; Carnap 1966, Chapter 23; the index
entry for "Language," of Feigl, Scriven, and Maxwell 1958,
548 has three subheadings: "observation,"
"theoretical," and "ordinary"). The
theoretical language includes theoretical vocabulary, while the
observational language involves observational terms. Both languages
contain logical terms. Finally, the theoretical language includes, and
is constrained by, the logical calculus, *Calc*, of the
axiomatic system adopted (e.g., Hempel 1958, 46; Suppe 1977, 50-53).
This calculus specifies sentence grammaticality as well as appropriate
deductive and non-ampliative inference rules (e.g., modus ponens)
pertinent to, especially, theoretical sentences. *Calc* can
itself be written in theoretical sentences.


#### 2.1.4 Theory Structure, in General



Table 1 summarizes the Syntactic View's account of theory
structure:




Table 1|  | ***Theory*** | ***Observation*** |
| Sentence Type | \(T\_S\) | \(C\_S\) | \(O\_S\) |
| Term (or vocabulary) | Theoretical & logical | Theoretical, logical & observational | Observational & logical |
| Language | \(L\_T\) | \(L\_T\) & \(L\_O\) | \(L\_O\) |



The salient divide is between theory and observation. Building on
Table 1, there are three different levels of scientific knowledge,
according to the Syntactic View:



\(\{T\_S\} =\) The uninterpreted syntactic system of the scientific
theory.
  

\(\{T\_S, C\_S\} =\) The scientific theory structure of a particular
domain (e.g., physics, anthropology).
  

\(\{T\_S,C\_S,O\_S\} =\) All of the science of a particular domain. 



Scientific theory is thus taken to be a syntactically formulated set
of theoretical sentences (axioms, theorems, and laws) together with
their interpretation via correspondence sentences. As we have seen,
theoretical sentences and correspondence sentences are cleanly
distinct, even if both are included in the structure of a scientific
theory.



Open questions remain. Is the observation language a sub-language of
the theoretical language, or are they both parts of a fuller language
including all the vocabulary? Can the theoretical vocabulary or
language be eliminated in favor of a purely observational vocabulary
or language? Are there other ways of carving up kinds of languages?
First, a "dialectical opposition" between "logic and
experience," "form and content," "constitutive
principles and empirical laws," and "'from
above'... [and] 'from below'" pervades
the work of the syntacticists (Friedman 1999, 34, 63). Whether
syntacticists believe that a synthesis or unification of this general
opposition between the theoretical (i.e., logic, form) and the
observational (i.e., experience, content) is desirable remains a topic
of ongoing discussion. Regarding the second question, Hempel 1958
deflates what he calls "the theoretician's
dilemma"--i.e., the putative reduction without remainder of
theoretical concepts and sentences to observational concepts and
sentences. Finally, other language divisions are possible, as Carnap
1937 argues (see Friedman 1999, Chapter 7). Returning to the main
thread of this section, the distinction toolkit of theoretical and
observational terms, sentences, and languages (Table 1) permit the
syntacticists to render theoretical structure sharply, thereby aiming
at the reconstructive "logic of science"
(*Wissenschafstlogik*) that they so desire.


### 2.2 A Running Example: Newtonian Mechanics



Reichenbach 1969 [1924] stands as a canonical attempt by a central
developer of the Syntactic View of axiomatizing a physical theory,
viz., relativity theory (cf. Friedman 1983, 1999; see also Reichenbach
1965 [1920]). For the purposes of this encyclopedia entry, it is
preferable to turn to another syntactic axiomatization effort. In
axiomatizing Newtonian mechanics, the mid-20th century
mathematical logician Hans Hermes spent significant energy defining
the concept of mass (Hermes 1938, 1959; Jammer 1961). More precisely,
he defines the theoretical concept of "mass ratio" of two
particles colliding inelastically in an inertial reference frame
\(S\). Here is his full definition of mass ratio (1959, 287):



\[
\text{Mass } \alpha xx\_0 \equiv\_{df} \exists S,t,y,y\_o,v,v\_0 
 (G\_{xy} \wedge G\_{x\_0y\_0} \wedge CSt\_{yy\_0} \wedge Vel\_{Svty} 
 \wedge Vel\_{Sv\_0ty\_0} \wedge \alpha|v| = |v\_0|) \vee
 (G\_{xx\_0} \wedge \alpha = 1)
\]


One paraphrase of this definition is, "'the mass of \(x\)
is a times that of \(x\_0\)' is equivalent to 'there
exists a system \(S\), an instant \(t\), momentary mass points \(y\)
and \(y\_0\), and initial velocities \(v\) and \(v\_0\), such that \(y\)
and \(y\_0\) are genidentical, respectively, with \(x\) and \(x\_0\);
the joined mass points move with a velocity of 0 with respect to frame
\(S\) immediately upon colliding at time \(t\); and \(y\) and \(y\_0\)
have determinate velocities \(v\) and \(v\_0\) before the collision in
the ratio a, which could also be 1 if \(x\) and \(x\_0\) are
themselves genidentical.'" Hermes employs the notion of
"genidentical" to describe the relation between two
temporal sections of a *given* particle's world line
(Jammer 1961, 113). Set aside the worry that two distinct particles
cannot be genidentical per Hermes' definition, though they can
have identical properties. In short, this definition is syntactically
complete and is written in first-order predicate logic, as are the
other axioms and definitions in Hermes (1938, 1959). Correspondence
rules connecting a postulated mass \(x\) with an actual mass were not
articulated by Hermes.


### 2.3 Interpreting Theory Structure per the Syntactic View



The link between theory structure and the world, under the Syntactic
View, is contained in the theory itself: \(C\_S\), the set of
correspondence rules. The term "correspondence rules"
(Margenau 1950; Nagel 1961, 97-105; Carnap 1966, Chapter 24) has
a variety of near-synonyms:


1. Dictionary (Campbell 1920)
2. Operational rules (Bridgman 1927)
3. Coordinative definitions (Reichenbach 1969 [1924], 1938)
4. Reduction sentences (Carnap 1936/1937; Hempel 1952)
5. Correspondence postulates (Carnap 1963)
6. Bridge principles (Hempel 1966; Kitcher 1984)
7. Reduction functions (Schaffner 1969, 1976)
8. Bridge laws (Sarkar 1998)



Important differences among these terms cannot be mapped out here.
However, in order to better understand correspondence rules, two of
their functions will be considered: (i) theory *interpretation*
(Carnap, Hempel) and (ii) theory *reduction* (Nagel,
Schaffner). The dominant perspective on correspondence rules is that
they interpret theoretical terms. Unlike "mathematical
theories," the axiomatic system of physics "cannot
have... a splendid isolation from the world" (Carnap 1966,
237). Instead, scientific theories require observational
interpretation through correspondence rules. Even so, surplus meaning
always remains in the theoretical structure (Hempel 1958, 87; Carnap
1966). Second, correspondence rules are seen as necessary for
inter-theoretic reduction (van Riel and Van Gulick 2014). For
instance, they connect observation terms such as
"temperature" in phenomenological thermodynamics (the
reduced theory) to theoretical concepts such as "mean kinetic
energy" in statistical mechanics (the reducing theory).
Correspondence rules unleash the reducing theory's epistemic
power. Notably, Nagel (1961, Chapter 11; 1979) and Schaffner (1969,
1976, 1993) allow for multiple kinds of correspondence rules, between
terms of either vocabulary, in the reducing and the reduced theory
(cf. Callender 1999; Winther 2009; Dizadji-Bahmani, Frigg, and
Hartmann 2010). Correspondence rules are a core part of the structure
of scientific theories and serve as glue between theory and
observation.



Finally, while they are not part of the theory structure, and although
we saw some examples above, observation sentences are worth briefly
reviewing. Correspondence rules attach to the content of observational
sentences. Observational sentences were analyzed as (i) *protocol
sentences* or *Protokollsatze* (e.g., Schlick 1934;
Carnap 1987 [1932], 1937, cf. 1963; Neurath 1983 [1932]), and as (ii)
*experimental laws* (e.g., Campbell 1920; Nagel 1961; Carnap
1966; cf. Duhem 1954 [1906]). Although constrained by *Calc*,
the grammar of these sentences is determined primarily by the order of
nature, as it were. In general, syntacticists do not consider methods
of data acquisition, experiment, and measurement to be philosophically
interesting. In contrast, the confirmation relation between
(collected) data and theory, especially as developed in inductive
logic (e.g., Reichenbach 1938, 1978; Carnap 1962 [1950], 1952), as
well as questions about the conventionality, grammaticality,
foundationalism, atomism, and content of sense-data and synthetic
statements, are considered philosophically important (e.g., Carnap
1987 [1932], 1937, 1966; Neurath 1983 [1932]; Reichenbach 1951;
Schlick 1925 [1918], 1934; for contemporary commentary, see, e.g.,
Creath 1987, 2014; Rutte 1991; Friedman 1999).


### 2.4 Taking Stock: Syntactic View



To summarize, the Syntactic View holds that there are three kinds of
terms or vocabularies: logical, theoretical, and observational; three
kinds of sentences: \(T\_S\), \(C\_S\), and \(O\_S\); and two languages:
\(L\_T\) and \(L\_O\). Moreover, the structure of scientific theories
could be analyzed using the logical tools of metamathematics. The goal
is to reconstruct the logic of science, viz. to articulate an
axiomatic system.



Interestingly, this perspective has able and active defenders today,
who discuss constitutive and axiomatized principles of the historical
"relativized a priori" (Friedman 2001, cf. 2013), argue
that "the semantic view, if plausible, is syntactic"
(Halvorson 2013), and explore "logicism" for, and in, the
philosophy of science (Demopulous 2003, 2013; van Benthem 2012).
Furthermore, for purposes of the syntactic reconstruction of
scientific theories, some continue espousing--or perhaps plea for
the resurrection of--predicate logic (e.g., Lutz 2012, 2014),
while other contemporary syntacticists (e.g., Halvorson 2012, 2013,
2019) endorse more recently developed metamathematical *and*
mathematical equipment, such as category theory, which "turns
out to be a kind of universal mathematical language like set
theory" (Awodey 2006, 2; see Eilenberg and MacLane 1945).
Importantly, Halvorson (2019) urges that interlocutors adopt
"structured" rather than "flat" views of
theories. For the case of the syntactic view this would mean that
rather than accept the usual formulation that a theory is a
*set* of sentences, "... [we] might say that a
theory consists of both sentences and inferential relations between
those sentences" (Halvorson 2019, 277-8). Classical
syntacticists such as Rudolf Carnap (Friedman 1999, 2011; Carus 2007;
Blatti and Lapointe 2016; Koellner ms. in Other Internet Resources)
and Joseph Henry Woodger (Nicholson and Gawne 2014) have recently
received increasing attention.


## 3. The Semantic View



An overarching theme of the Semantic View is that analyzing theory
structure requires employing mathematical tools rather than predicate
logic. After all, defining scientific concepts within a specific
formal language makes any axiomatizing effort dependent on the choice,
nature, and idiosyncrasies of that narrowly-defined language. For
instance, Suppes understands first-order predicate logic, with its
"linguistic" rather than "set-theoretical"
entities, as "utterly impractical" for the formalization
of "theories with more complicated structures like probability
theory" (Suppes 1957, 232, 248-9; cf. Suppes 2002). Van
Fraassen, another influential defender of the Semantic View, believes
that the logical apparatus of the Syntactic View "had moved us
*mille milles de toute habitation scientifique*, isolated in
our own abstract dreams" (van Fraassen 1989, 225). Indeed, what
would the appropriate logical language for specific mathematical
structures be, especially when such structures could be reconstructed
in a variety of formal languages? Why should we imprison mathematics
and mathematical scientific theory in syntactically defined
language(s) when we could, instead, directly investigate the
mathematical objects, relations, and functions of scientific theory?




Consistent with the combat strategy (discussed in the Conclusion),
here is a list of grievances against the Syntactic View discussed at
length in the work of some semanticists.


1. *First-Order Predicate Logic Objection*. Theoretical
structure is intrinsically and invariably tied to the specific choice
of a language, \(L\_T\), expressed in first-order predicate logic. This
places heavy explanatory and representational responsibility on
relatively inflexible and limited languages.
2. *Theory Individuation Objection*. Since theories are
individuated by their linguistic formulations, every change in
high-level syntactic formulations will bring forth a distinct theory.
This produces a reductio: if \(T\_1 = p \rightarrow q\) and \(T\_2 =
\neg p \vee q\) then \(T\_1\) and \(T\_2\), though logically equivalent,
have different syntactic formulations and would be distinct
theories.
3. *Theoretical/Observational Languages Objection*. Drawing
the theoretical/observational distinction in terms of language is
inappropriate, as observability pertains to entities rather than to
concepts.
4. *Unintended Models Objection*. There is no clear way of
distinguishing between intended and unintended models for
syntactically characterized theories (e.g., the Lowenheim-Skolem
theorem, Bays 2014).
5. *Confused Correspondence Rules Objection*. Correspondence
rules are a confused medley of direct meaning relationships between
terms and world, means of inter-theoretic reduction, causal
relationship claims, and manners of theoretical concept testing.
6. *Trivially True yet Non-Useful Objection*. Presenting
scientific theory in a limited axiomatic system, while clearly
syntactically correct, is neither useful nor honest, since scientific
theories are mathematical structures.
7. *Practice and History Ignored Objection*. Syntactic
approaches do not pay sufficient attention to the actual practice and
history of scientific theorizing and experimenting.



What, then, does the Semantic View propose to put in the Syntactic
View's place?


### 3.1 Theory Structure per the Semantic View



Even a minimal description of the Semantic View must acknowledge two
distinct strategies of characterizing and comprehending theory
structure: the *state-space* and the
*set-/model-theoretic* approaches. 


#### 3.1.1 The State-Space Approach



The state-space approach emphasizes the mathematical models of actual
science, and draws a clear line between mathematics and
metamathematics. The structure of a scientific theory is identified
with the "class," "family" or
"cluster" of mathematical models constituting it, rather
than with any metamathematical axioms "yoked to a particular
syntax" (van Fraassen 1989, 366). Under this analysis,
"the correct tool for philosophy of science is mathematics, not
metamathematics"--this is Suppes' slogan, per van
Fraassen (1989, 221; 1980, 65). In particular, a state space or phase
space is an \(N\)-dimensional space, where each of the relevant
variables of a theory correspond to a single dimension and each point
in that space represents a possible *state* of a real system.
An actual, real system can take on, and change, states according to
different kinds of laws, viz., *laws of succession* determining
possible trajectories through that space (e.g., Newtonian kinematic
laws); *laws of co-existence* specifying the permitted regions
of the total space (e.g., Boyle's law); and *laws of
interaction* combining multiple laws of succession or
co-existence, or both (e.g., population genetic models combining laws
of succession for selection and genetic drift, Wright 1969; Lloyd 1994
[1988]; Rice 2004; Clatterbuck, Sober, and Lewontin 2013). Different
models of a given theory will share some dimensions of their state
space while differing in others. Such models will also partially
overlap in laws (for further discussion of state spaces, laws, and
models pertinent to the Semantic View, see Suppe 1977, 224-8;
Lloyd 1994, Chapter 2; Nolte 2010; Weisberg 2013, 26-9).



Historically, the state-space approach emerged from work by Evert
Beth, John von Neumann, and Hermann Weyl, and has important parallels
with Przelecki (1969) and Dalla Chiara Scabia and Toraldo di
Francia (1973) (on the history of the approach see: Suppe 1977; van
Fraassen 1980, 65-67; Lorenzano 2013; advocates of the approach
include: Beatty 1981; Giere 1988, 2004; Giere, Bickle, and Mauldin
2006; Lloyd 1983, 1994 [1988], 2013 In Press; Suppe 1977, 1989;
Thompson, 1989, 2007; van Fraassen 1980, 1989, 2008; for alternative
early analyses of models see, e.g., Braithwaite 1962; Hesse 1966,
1967). Interestingly, van Fraassen (1967, 1970) provides a potential
reconstruction of state spaces via an analysis of
"semi-interpreted languages." Weisberg (2013), building on
many insights from Giere's work, presents a broad view of
modeling that includes mathematical structures that are
"trajectories in state spaces" (29), but also permits
concrete objects and computational structures such as algorithms to be
deemed models. Lorenzano (2013) calls Giere's (and, by
extension, Weisberg's and even Godfrey-Smith's 2006)
approach "model-based," separating it out from the
state-space approach. A more fine-grained classification of the
state-space approach is desirable, particularly if we wish to
understand important lessons stemming from the Pragmatic View of
Theories, as we shall see below.



As an example of a state-space analysis of modeling, consider a
capsule traveling in outer space. An empirically and dynamically
adequate mathematical model of the capsule's behavior would
capture the *position* of the capsule (i.e., three dimensions
of the formal state space), as well as the *velocity* and
*acceleration* vectors for each of the three standard spatial
dimensions (i.e., six more dimensions in the formal state space). If
the mass were unknown or permitted to vary, we would have to add one
more dimension. Possible and actual trajectories of our capsule, with
known mass, within this abstract 9-dimensional state space could be
inferred via Newtonian dynamical laws of motion (example in Lewontin
1974, 6-8; consult Suppe 1989, 4). Importantly, under the
state-space approach, the interesting philosophical work of
characterizing theory structure (e.g., as classes of models), theory
meaning (e.g., data models mapped to theoretical models), and theory
function (e.g., explaining and predicting) happens at the level of
mathematical models. 


#### 3.1.2 The Set-/Model-Theoretic Approach



Lurking in the background of the state-space conception is the fact
that mathematics actually includes set theory and model
theory--i.e., mathematical logic. Indeed, according to some
interlocutors, "metamathematics is part of mathematics"
(Halvorson 2012, 204). Historically, a set-/model-theoretic approach
emerged from Tarski's work and was extensively articulated by
Suppes and his associates (van Fraassen 1980, 67). Set theory is a
general language for formalizing mathematical structures as
collections--i.e., sets--of abstract objects (which can
themselves be relations or functions; see Krivine 2013 [1971]). Model
theory investigates the relations between, on the one hand, the formal
axioms, theorems, and laws of a particular theory and, on the other
hand, the mathematical structures--the models--that provide
an interpretation of that theory, or put differently, that make the
theory's axioms, theorems, and laws true (Hodges 1997, Chapter
2; Jones 2005). Interestingly, model theory often uses set theory
(e.g., Marker 2002); set theory can, in turn, be extended to link
axiomatic theories and semantic models via "set-theoretical
predicates" (e.g., Suppes 1957, 2002). Finally, there are
certain hybrids of these two branches of mathematical logic, including
"partial structures" (e.g., da Costa and French 1990,
2003; Bueno 1997; French 2017; French and Ladyman 1999, 2003; Vickers
2009; Bueno, French, and Ladyman 2012). Lorenzano (2013) provides a
more complex taxonomy of the intellectual landscape of the Semantic
View, including a discussion of Structuralism, a kind of
set-/model-theoretic perspective. Structuralism involves theses about
"theory-nets," theory-relative theoretical vs.
non-theoretical terms, a diversity of intra- and inter-theoretic laws
with different degrees of generality, a typology of inter-theoretic
relations, and a rich account of correspondence rules in scientific
practice (see Moulines 2002; Pereda 2013; Schmidt 2014; Ladyman 2014).
On the whole, the set-/model-theoretic approach of the Semantic View
insists on the inseparability of metamathematics and mathematics. In
preferring to characterize a theory axiomatically in terms of its
intension rather than its extension, it shares the Syntactic
View's aims of reconstructive axiomatization (e.g., Sneed 1979;
Stegmuller 1979; Frigg and Votsis 2011; Halvorson 2013, 2019;
Lutz 2012, 2014, 2017).



An example will help motivate the relation between theory and model.
Two qualifications are required: (i) we return to a more standard
set-/model-theoretic illustration below, viz., McKinsey, Sugar, and
Suppes' (1953) axiomatization of particle mechanics, and (ii)
this motivational example is not from the heartland of model theory
(see Hodges 2013). Following van Fraassen's intuitive case of
"seven-point geometry" (1980, 41-44; 1989,
218-220), also known as "the Fano plane" we see how
a particular geometric figure, the *model*, interprets and
makes true a set of axioms and theorems, the *theory*. In
topology and geometry there is rich background theory regarding how to
close Euclidean planes and spaces to make finite geometries by, for
instance, eliminating parallel lines. Consider the axioms of a
projective plane:


1. For any two points, exactly one line lies on both.
2. For any two lines, exactly one point lies on both.
3. There exists a set of four points such that no line has more than
two of them.



A figure of a geometric model that makes this theory true is:



![Geometric figure including triangle ACE with interior circle BDF and center point G. Point B is on line segment AC, D is on CE, and F is on AE. G is the center of the circle. Point G is on line segments AD, BE, and CF.](fig1.png)

Figure 1.




This is the smallest geometrical model satisfying the three axioms of
the projective plane theory. Indeed, this example fits van
Fraassen's succinct characterization of the theory-model
relation:



> 
> A model is called a model of a theory exactly if the theory is
> entirely true if considered with respect to this model alone.
> (Figuratively: the theory would be true if this model was the whole
> world.) (1989, 218)
> 



That is, if the entire universe consisted solely of these seven points
and seven lines, the projective plane theory would be true. Of course,
our universe is bigger. Because Euclidean geometry includes parallel
lines, the Fano plane is not a model of Euclidean geometry. Even so,
by drawing the plane, we have shown it to be *isomorphic* to
parts of the Euclidean plane. In other words, the Fano plane has been
*embedded* in a Euclidean plane. Below we return to the
concepts of embedding and isomorphism, but this example shall suffice
for now to indicate how a geometric model can provide a semantics for
the axioms of a theory.



In short, for the Semantic View the structure of a scientific theory
is its class of mathematical models. According to some advocates of
this view, the family of models can itself be axiomatized, with those
very models (or other models) serving as axiom truth-makers. 


### 3.2 A Running Example: Newtonian Mechanics



Returning to our running example, consider Suppes' 1957
model-theoretic articulation of particle mechanics, which builds on
his 1953 article with J.C.C. McKinsey and A.C. Sugar. Under this
analysis, there is a domain of set-theoretic objects of the form \(\{
P, T, s, m, f, g \}\), where \(P\) and \(T\) are themselves sets,
\(s\) and \(g\) are binary functions, \(m\) is a unary and \(f\) a
ternary function. \(P\) is the set of particles; \(T\) is a set of
real numbers measuring elapsed times; \(s(p, t)\) is the position of
particle \(p\) at time \(t\); \(m(p)\) is the mass of particle \(p\);
\(f(p, q, t)\) is the force particle \(q\) exerts on \(p\) at time
\(t\); and \(g(p, t)\) is the total resultant force (by all other
particles) on \(p\) at time \(t\). Suppes and his collaborators
defined seven axioms--three kinematical and four
dynamical--characterizing Newtonian particle mechanics (see also
Simon 1954, 1970). Such axioms include Newton's third law
reconstructed in set-theoretic formulation thus (Suppes 1957,
294):

 

\[\tag{Axiom P5}
\text{For } p,q \text{ in } P \text{ and } t \text{ in }T:
 f(p,q,t) = -f(q,p,t)
\]


Importantly, the set-theoretic objects are found in more than one of
the axioms of the theory, and Newton's calculus is reconstructed
in a novel, set-theoretic form. Set-theoretic predicates such as
"is a binary relation" and "is a function" are
also involved in axiomatizing particle mechanics (Suppes 1957, 249).
Once these axioms are made explicit, their models can be specified and
these can, in turn, be applied to actual systems, thereby providing a
semantics for the axioms (e.g., as described in Section 3.3.1 below).
A particular system satisfying these seven axioms is a particle
mechanics system. (For an example of Newtonian mechanics from the
state-space approach, recall the space capsule of Section 3.1.1.)


### 3.3 Interpreting Theory Structure per the Semantic View



How is the theory structure, described in Section 3.1, applied to
empirical phenomena? How do we connect theory and data via observation
and experimental and measuring techniques? The Semantic View
distinguishes theory individuation from both theory-phenomena and
theory-world relations. Three types of analysis of theory
interpretation are worth investigating: (i) *a hierarchy of
models* (e.g., Suppes; Suppe), (ii) *similarity* (e.g.,
Giere; Weisberg), and (iii) *isomorphism* (e.g., van Fraassen;
French and Ladyman).


#### 3.3.1 A Hierarchy of Models



One way of analyzing theory structure interpretation is through a
series of models falling under the highest-level axiomatizations. This
series has been called "a hierarchy of models," though it
need not be considered a nested hierarchy. These models include models
of theory, models of experiment, and models of data (Suppes 1962,
2002). Here is a summary of important parts of the hierarchy (Suppes
1962, Table 1, 259; cf. Giere 2010, Figure 1, 270): 


1. *Axioms of Theory*. Axioms define set-theoretic
predicates, and constitute the core structure of scientific theories,
as reviewed in Section 3.1.2.
2. *Models of Theory.* "Representation Theorems,"
permit us "to discover if an interesting subset of models for
the theory may be found such that any model for the theory is
isomorphic to some member of this subset" (Suppes 1957, 263).
Representation theorem methodology can be extended (i) down the
hierarchy, both to models of experiment and models of data, and (ii)
from isomorphism to homomorphism (Suppes 2002, p. 57 ff.; Suppe 2000;
Cartwright 2008).
3. *Models of Experiment*. Criteria of experimental design
motivate choices for how to set up and analyze experiments. There are
complex mappings between models of experiment thus specified, and (i)
models of theory, (ii) theories of measurement, and (iii) models of
data.
4. *Models of Data*. In building models of data, phenomena
are organized with respect to statistical goodness-of-fit tests and
parameter estimation, in the context of models of theory. Choices
about which parameters to represent must be made.



The temptation to place *phenomena* at the bottom of the
hierarchy must be resisted because phenomena permeate all levels.
Indeed, the "class of phenomena" pertinent to a scientific
theory is its "intended scope" (Suppe 1977, 223; Weisberg
2013, 40). Furthermore, this temptation raises fundamental questions
about scientific representation: "there is the more profound
issue of the relationship between the lower most representation in the
hierarchy--the data model perhaps--and reality itself, but
of course this is hardly something that the semantic approach alone
can be expected to address" (French and Ladyman 1999, 113; cf.
van Fraassen 2008, 257-258, "The 'link' to
reality"). Borrowing from David Chalmers, the "hard
problem" of philosophy of science remains connecting abstract
structures to concrete phenomena, data, and world. 


#### 3.3.2 Similarity



The similarity analysis of theory interpretation combines semantic and
pragmatic dimensions (Giere 1988, 2004, 2010; Giere, Bickle, and
Mauldin 2006; Weisberg 2013). According to Giere, interpretation is
mediated by theoretical hypotheses positing representational relations
between a model and relevant parts of the world. Such relations may be
stated as follows:

 

\[
S \text{ uses } X \text{ to represent } W \text{ for purposes } P.
\]


Here \(S\) is a scientist, research group or community, \(W\) is a
part of the world, and \(X\) is, broadly speaking, any one of a
variety of models (Giere 2004, 743, 747, 2010). Model-world similarity
judgments are conventional and intentional:



> 
> Note that I am not saying that the model itself represents an aspect
> of the world because it is similar to that aspect. ...Anything is
> similar to anything else in countless respects, but not anything
> represents anything else. It is not the model that is doing the
> representing; it is the scientist using the model who is doing the
> representing. (2004, 747)
> 



Relatedly, Weisberg (2013) draws upon Tversky (1977) to develop a
similarity metric for model interpretation (equation 8.10, 148). This
metric combines (i) model-target semantics (90-97), and (ii) the
pragmatics of "context, conceptualization of the target, and the
theoretical goals of the scientist" (149). Giere and Weisberg
thus endorse an abundance of adequate mapping relations between a
given model and the world. From this diversity, scientists and
scientific communities must select particularly useful similarity
relationships for contextual modeling purposes. Because of semantic
pluralism and irreducible intentionality, this similarity analysis of
theory interpretation cannot be accommodated within a hierarchy of
models approach, interpreted as a neat model nesting based on
pre-given semantic relations among models at different levels.


#### 3.3.3 Isomorphism



The term "isomorphism" is a composite of the Greek words
for "equal" and "shape" or "form."
Indeed, in mathematics, isomorphism is a perfect one-to-one, bijective
mapping between two structures or sets. Figure (2) literally and
figuratively captures the term:



![Script writing of isomorphism with mirror image underneath](fig2.png)

Figure 2.




Especially in set theory, category theory, algebra, and topology,
there are various kinds of "-morphisms," viz., of mapping
relations between two structures or models. Figure
(3) indicates five different kinds of homomorphism, arranged in a Venn
diagram.



![Venn diagram with outer circle Hom and 3 intersecting interior circles: Mon, Epi, and End. The intersection of all 3 is Aut and the intersection of Mon and Epi is Iso.](fig3.png)

Figure 3.




Although philosophers have focused on isomorphism, other morphisms
such as monomorphism (i.e., an injective homomorphism where some
elements in the co-domain remain unmapped from the domain) might also
be interesting to investigate, especially for embedding data (i.e.,
the domain) into rich theoretical structures (i.e., the co-domain). To
complete the visualization above, an epimorphism is a surjective
homomorphism, and an endomorphism is a mapping from a structure to
itself, although it need not be a symmetrical--i.e.,
invertible--mapping, which would be an automorph. 



Perhaps the most avid supporter of isomorphism and embedding as the
way to understand theory interpretation is van Fraassen. In a
nutshell, if we distinguish (i) theoretical models, (ii)
"empirical substructures" (van Fraassen 1980, 64, 1989,
227; alternatively: "surface models" 2008, 168), and (iii)
"observable phenomena" (1989, 227, 2008, 168), then, van
Fraassen argues, theory interpretation is a relation of isomorphism
between observable phenomena and empirical substructures, which are
*themselves* isomorphic with one or more theoretical models.
Moreover, if a relation of isomorphism holds between \(X\) and a
richer \(Y\), we say that we have embedded \(X\) in \(Y\). For
instance, with respect to the seven-point geometry above (Figure 1),
van Fraassen contends that isomorphism gives embeddability, and that
the relation of isomorphism "is important because it is also the
exact relation a phenomenon bears to some model or theory, if that
theory is empirically adequate" (1989, 219-20; this kind
of statement seems to be simultaneously descriptive and prescriptive
about scientific representation, see Section 1.1 above). In *The
Scientific Image* he is even clearer about fleshing out the
empirical adequacy of a theory (with its theoretical models) in terms
of isomorphism between "appearances" (i.e., "the
structures which can be described in experimental and measurement
reports," 1980, 64, italics removed) and empirical
substructures. Speaking metaphorically,



> 
> the phenomena are, from a theoretical point of view, small, arbitrary,
> and chaotic--even nasty, brutish, and short...--but can
> be understood as embeddable in beautifully simple but much larger
> mathematical models. (2008, 247; see also van Fraassen 1981, 666 and
> 1989, 230)
> 



Interestingly, and as a defender of an identity strategy (see
Conclusion), Friedman also appeals to embedding and subsumption
relations between theory and phenomena in his analyses of theory
interpretation (Friedman 1981, 1983).
Bueno, da Costa, French, and Ladyman also employ embedding and
(partial) isomorphism in the empirical interpretation of partial
structures (Bueno 1997; Bueno, French, and Ladyman 2012; da Costa and
French 1990, 2003; French 2017; French and Ladyman 1997, 1999, 2003;
Ladyman 2004). Suarez discusses complexities in van
Fraassen's analyses of scientific representation and theory
interpretation (Suarez 1999, 2011). On the one hand,
representation is structural identity between the theoretical and the
empirical. On the other hand, "There is no representation except
in the sense that some things are used, made, or taken, to represent
some things as thus or so" (van Fraassen 2008, 23, italics
removed). The reader interested in learning how van Fraassen
simultaneously endorses acontextually structural and contextually
pragmatic aspects of representation and interpretation should refer to
van Fraassen's (2008) investigations of maps and "the
essential indexical." [To complement the structure vs. function
distinction, see van Fraassen 2008, 309-311 for a structure
("structural relations") vs. history ("the
intellectual processes that lead to those models") distinction;
cf. Ladyman et al. 2011] In all of this, embedding via isomorphism is
a clear contender for theory interpretation under the Semantic
View.


### 3.4 Taking Stock: Semantic View



In short, committing to either a state-space or a set-/model-theoretic
view on theory structure does not imply any particular perspective on
theory interpretation (e.g., hierarchy of models, similarity,
embedding). Instead, commitments to the former are logically and
actually separable from positions on the latter (e.g., Suppes and
Suppe endorse different accounts of theory structure, but share an
understanding of theory interpretation in terms of a hierarchy of
models). The Semantic View is alive and well as a family of analyses
of theory structure, and continues to be developed in interesting ways
both in its state-space and set-/model-theoretic approaches. 


## 4. The Pragmatic View



The Pragmatic View recognizes that a number of assumptions about
scientific theory seem to be shared by the Syntactic and Semantic
Views. Both perspectives agree, very roughly, that theory is (1)
explicit, (2) mathematical, (3) abstract, (4) systematic, (5) readily
individualizable, (6) distinct from data and experiment, and (7)
highly explanatory and predictive (see Flyvbjerg 2001, 38-39;
cf. Dreyfus 1986). The Pragmatic View imagines the structure of
scientific theories rather differently, arguing for a variety of
theses:


1. *Limitations*. Idealized theory structure might be too
weak to ground the predictive and explanatory work syntacticists and
semanticists expect of it (e.g., Cartwright 1983, 1999a, b, 2019;
Morgan and Morrison 1999; Suarez and Cartwright 2008).
2. *Pluralism*. Theory structure is plural and complex both
in the sense of internal variegation and of existing in many types. In
other words, there is an *internal pluralism* of theory (and
model) components (e.g., mathematical concepts, metaphors, analogies,
ontological assumptions, values, natural kinds and classifications,
distinctions, and policy views, e.g., Kuhn 1970; Boumans 1999), as
well as a broad *external pluralism* of different types of
theory (and models) operative in science (e.g., mechanistic,
historical, and mathematical models, e.g., Hacking 2009, Longino 2013). Indeed, it may be better to speak of the structures of
scientific theories, in the double-plural.
3. *Nonformal aspects.* The internal pluralism of theory
structure (thesis #2) includes many nonformal aspects deserving
attention. That is, many components of theory structure, such as
metaphors, analogies, values, and policy views have a non-mathematical
and "informal" nature, and they lie implicit or hidden
(e.g., Bailer-Jones 2002; Craver 2002; Contessa 2006; Morgan 2012). Interestingly, the common understanding of
"formal," which identifies formalization with
mathematization, may itself be a conceptual straightjacket; the term
could be broadened to include "diagram abstraction" and
"principle extraction" (e.g., Griesemer 2013, who
explicitly endorses what he also calls a "Pragmatic View of
Theories").
4. *Function.* Characterizations of the nature and dynamics
of theory structure should pay attention to the user as well as to
purposes and values (e.g., Apostel 1960; Minsky 1965; Morrison 2007;
Winther 2012a).
5. *Practice*. Theory structure is continuous with practice
and "the experimental life," making it difficult to neatly
dichotomize theory and practice (e.g., Hacking 1983, 2009; Shapin and
Schaffer 1985; Galison 1987, 1988, 1997; Suarez and Cartwright
2008, Cartwright 2019).



These are core commitments of the Pragmatic View.



It is important to note at the outset that the Pragmatic View takes
its name from the linguistic trichotomy discussed above, in the
Introduction. This perspective need not imply commitment to, or
association with, American Pragmatism (e.g. the work of Charles S.
Peirce, William James, or John Dewey; cf. Hookway 2013; Richardson
2002). For instance, Hacking (2007a) distinguishes his pragmatic
attitudes from the school of Pragmatism. He maps out alternative
historical routes of influence, in general and on him,
vis-a-vis fallibilism (via Imre Lakatos, Karl Popper; Hacking
2007a, SS1), historically conditioned truthfulness (via Bernard
Williams; Hacking 2007a, SS3), and realism as intervening (via
Francis Everitt, Melissa Franklin; Hacking 2007a, SS4). To borrow
a term from phylogenetics, the Pragmatic View is
"polyphyletic." The components of its analytical framework
have multiple, independent origins, some of which circumnavigate
American Pragmatism.



With this qualification and the five theses above in mind, let us now
turn to the Pragmatic View's analysis of theory structure and
theory interpretation.


### 4.1 Theory Structure per the Pragmatic View



We should distinguish two strands of the Pragmatic View: the
*Pragmatic View of Models* and a proper *Pragmatic View of
Theories*. 


#### 4.1.1 The Pragmatic View of Models



Nancy Cartwright's *How the Laws of Physics Lie*
crystallized the Pragmatic View of Models. Under Cartwright's
analysis, models are the appropriate level of investigation for
philosophers trying to understand science. She argues for significant
limitations of theory (thesis #1), claiming that laws of nature are
rarely true, and are epistemically weak. Theory as a collection of
laws cannot, therefore, support the many kinds of inferences and
explanations that we have come to expect it to license. Cartwright
urges us to turn to models and modeling, which are central to
scientific practice. Moreover, models
"lie"--figuratively and literally--between
theory and the world (cf. Derman 2011). That is, "to explain a
phenomenon is to find a model that fits it into the basic framework of
the theory and that thus allows us to derive analogues for the messy
and complicated phenomenological laws which are true of it." A
plurality of models exist, and models "serve a variety of
purposes" (Cartwright 1983, 152; cf. Suppes 1978). Cartwright is
interested in the practices and purposes of scientific models, and
asks us to focus on models rather than theories.



Cartwright's insights into model pluralism and model practices
stand as a significant contribution of "The Stanford
School" (cf. Cat 2014), and were further developed by the
"models as mediators" group, with participants at LSE,
University of Amsterdam, and University of Toronto (Morgan and
Morrison 1999; Chang 2011; cf. Martinez 2003). This group
insisted on the internal pluralism of model components (thesis #2).
According to Morgan and Morrison, building a model involves
"fitting together... bits which come from disparate
sources," including "stories" (Morgan and Morrison
1999, 15). Boumans (1999) writes:



> 
> 
> model building is like baking a cake without a recipe. The ingredients
> are theoretical ideas, policy views, mathematisations of the cycle,
> metaphors and empirical facts. (67)
> 
> 
> 
> Mathematical moulding is shaping the ingredients in such a
> mathematical form that integration is possible... (90)
> 
> 
> 



In an instructive diagram, Boumans suggests that a variety of factors
besides theory and data feed into a model: metaphors, analogies,
policy views, stylised facts, mathematical techniques, and
mathematical concepts (93). The full range of components involved in a
model will likely vary according to discipline, and with respect to
explanations and interventions sought (e.g., analogies but not policy
views will be important in theoretical physics). In short, model
building involves a complex variety of internal nonformal aspects,
some of which are implicit (theses #2 and #3).



As one example of a nonformal component of model construction and
model structure, consider metaphors and analogies (e.g., Bailer-Jones
2002). Geary (2011) states the "simplest equation" of
metaphor thus: "\(X = Y\)" (8, following Aristotle:
"Metaphor consists in giving the thing a name that belongs to
something else... ," *Poetics*, 1457b). The line
between metaphor and analogy in science is blurry. Some interlocutors
synonymize them (e.g., Hoffman 1980; Brown 2003), others reduce one to
the other (analogy is a form of metaphor, Geary 2011; metaphor is a
kind of analogy, Gentner 1982, 2003), and yet others bracket one to
focus on the other (e.g., Oppenheimer 1956 sets aside metaphor). One
way to distinguish them is to reserve "analogy" for
concrete comparisons, with clearly identifiable and demarcated source
and target domains, and with specific histories, and use
"metaphor" for much broader and indeterminate comparisons,
with diffuse trajectories across discourses. Analogies include the
"lines of force" of electricity and magnetism (Maxwell and
Faraday), the atom as a planetary system (Rutherford and Bohr), the
benzene ring as a snake biting its own tail (Kekule),
Darwin's "natural selection" and "entangled
bank," and behavioral "drives" (Tinbergen) (e.g.,
Hesse 1966, 1967; Bartha 2010). Examples of metaphor are
genetic information, superorganism, and networks (e.g., Keller 1995).
More could be said about other informal model components, but this
discussion of metaphors and analogies shall suffice to hint at how
models do not merely lie between theory and world. Models express a
rich internal pluralism (see also de Chadarevian and Hopwood 2004;
Morgan 2012). 



Model complexity can also be seen in the external plurality of models
(thesis #2). Not all models are mathematical, or even ideally recast
as mathematical. Non-formalized (i.e., non-state-space,
non-set-/model-theoretic) models such as physical, diagrammatic,
material, historical, "remnant," and fictional models are
ubiquitous across the sciences (e.g., Frigg and Hartmann 2012; for the
biological sciences, see Hull 1975; Beatty 1980; Griesemer 1990, 1991
a, b, 2013; Downes 1992; Richards 1992; Winther 2006a; Leonelli
2008; Weisberg 2013). Moreover, computer simulations differ in
important respects from more standard analytical mathematical models
(e.g., Smith 1996; Winsberg 2010; Weisberg 2013). According to some
(e.g., Griesemer 2013; Downes 1992; Godfrey-Smith 2006; Thomson-Jones
2012), this diversity belies claims by semanticists that models can
always be cast "into set theoretic terms" (Lloyd 2013 In
Press), are "always a mathematical structure" (van
Fraassen 1970, 327), or that "formalisation of a theory is an
abstract representation of the theory expressed in a formal deductive
framework... in first-order predicate logic with identity, in set
theory, in matrix algebra and indeed, any branch of
mathematics..." (Thompson 2007, 485-6). Even so, internal
pluralism has been interpreted as supporting a "deflationary
semantic view," which is minimally committed to the perspective
that "model construction is an important part of scientific
theorizing" (Downes 1992, 151). Given the formal and
mathematical framework of the Semantic View (see above), however, the
broad plurality of kinds of models seems to properly belong under a
Pragmatic View of Models.


#### 4.1.2 The Pragmatic View of Theories



Interestingly, while critiquing the Syntactic and Semantic Views on
most matters, the Pragmatic View of Models construed theory, the
process of theorizing, and the structure of scientific theories,
according to terms set by the two earlier views. For instance,
Cartwright tends to conceive of theory as explicit, mathematical,
abstract, and so forth (see the first paragraph of Section 4). She
always resisted "the traditional syntactic/semantic view of
theory" for its "vending machine" view, in which a
theory is a deductive and automated machine that upon receiving
empirical input "gurgitates" and then "drops out the
sought-for representation" (1999a, 184-5). Rather than
reform Syntactic and Semantic accounts of theory and theory structure,
however, she invites us, as we just saw, to think of science as
modeling, "with theory as one small component"
(Cartwright, Shomar, and Suarez 1995, 138; Suarez and
Cartwright 2008). Many have followed her. Kitcher's predilection
is also to accept the terms of the Syntactic and Semantic Views. For
instance, he defines theories as "axiomatic deductive
systems" (1993, 93). In a strategy complementary to
Cartwright's modeling turn, Kitcher encourages us to focus on
practice, including practices of modeling and even practices of
theorizing. In *The Advancement of Science*, practice is
analyzed as a 7-tuple, with the following highly abbreviated
components: (i) a language; (ii) questions; (iii) statements
(pictures, diagrams); (iv) explanatory patterns; (v) standard
examples; (vi) paradigms of experimentation and observation, plus
instruments and tools; and (vii) methodology (Kitcher 1993, 74).
Scientific practice is also center stage for those singing the praises
of "the experimental life" (e.g., Hacking 1983; Shapin and
Schaffer 1985; Galison 1987), and those highlighting the cognitive
grounds of science (e.g., Giere 1988; Martinez 2014) and
science's social and normative context (e.g., Kitcher 1993,
2001; Longino 1995, 2002; Ziman 2000; cf. Simon 1957). Indeed, the
modeling and practice turns in the philosophy of science were
reasonable reactions to the power of axiomatic reconstructive and
mathematical modeling analyses of the structure of scientific
theories. 



Yet, a Pragmatic View of *Theories* is also afoot, one
resisting orthodox characterizations of theory often embraced, at
least early on, by Pragmatic View philosophers such as Cartwright,
Hacking, Kitcher, and Longino. For instance, Craver (2002) accepts
both the Syntactic and Semantic Views, which he humorously and not
inaccurately calls "the Once Received View" and the
"Model Model View." But he also observes:



> 
> While these analyses have advanced our understanding of some formal
> aspects of theories and their uses, they have neglected or obscured
> those aspects dependent upon *nonformal* patterns in theories.
> Progress can be made in understanding scientific theories by attending
> to their diverse nonformal patterns and by identifying the axes along
> which such patterns might differ from one another. (55)
> 



Craver then turns to mechanistic theory as a third theory type (and a
third philosophical analysis of theory structure) that highlights
nonformal patterns:



> 
> Different types of mechanisms can be distinguished on the basis of
> recurrent patterns in their organization. Mechanisms may be organized
> in series, in parallel, or in cycles. They may contain branches and
> joins, and they often include feedback and feedforward subcomponents.
> (71)
> 



Consistent with theses #2 and #3 of the Pragmatic View, we must
recognize the internal pluralism of theories as including nonformal
components. Some of these are used to represent organizational and
compositional relations of complex systems (Craver 2007; Wimsatt 2007; Winther 2011; Walsh 2015). While mechanistic analyses such as
Craver's may not wish to follow every aspect of the Pragmatic
View of Theories, there are important and deep resonances between the
two.



In a review of da Costa and French (2003), Contessa (2006) writes:



> 
> Philosophers of science are increasingly realizing that the
> differences between the syntactic and the semantic view are less
> significant than semanticists would have it and that, ultimately,
> neither is a suitable framework within which to think about scientific
> theories and models. The crucial divide in philosophy of science, I
> think, is not the one between advocates of the syntactic view and
> advocates of the semantic view, but the one between those who think
> that philosophy of science needs a formal framework or other and those
> who think otherwise. (376)
> 



Again, we are invited to develop a non-formal framework of science and
presumably also of scientific theory. (Halvorson 2012, 203 takes
Contessa 2006 to task for advocating "informal philosophy of
science.") Moreover, in asking "what should the content of
a given theory be taken to be on a given occasion?", Vickers
(2009) answers:



> 
> It seems clear that, in addition to theories being vague objects in
> the way that 'heaps' of sand are, there will be
> fundamentally different ways to put together theoretical assumptions
> depending on the particular investigation one is undertaking. For
> example, sometimes it will be more appropriate to focus on the
> assumptions which were *used* by scientists, rather than the
> ones that were *believed* to be true. (247, footnote
> suppressed)
> 



A Pragmatic View of Theories helps make explicit nonformal internal
components of theory structure. 



Key early defenders of the modeling and practice turns have also
recently begun to envision theory in a way distinct from the terms set
by the Syntactic and Semantic Views. Suarez and Cartwright
(2008) extend and distribute theory by arguing that "What we
know 'theoretically' is recorded in a vast number of
places in a vast number of different ways--not just in words and
formulae but in machines, techniques, experiments and applications as
well" (79). And while her influence lies primarily in the
modeling turn, even in characterizing the "vending
machine" view, Cartwright calls for a "reasonable
philosophical account of theories" that is "much more
textured, and... much more laborious" than that adopted by
the Syntactic and Semantic Views (1999a, 185). The theory-data and
theory-world axes need to be rethought. In her 2019 book on
"artful modeling", Cartwright emphasizes the importance of
know-how and creativity in scientific practice, and "praise[s]
engineers and cooks and inventors, as well as experimental physicists
like Millikan and Melissa Franklin" (Cartwright 2019, 76).
Kitcher wishes to transform talk of theories into discussion of
"significance graphs" (2001, 78 ff.). These are network
diagrams illustrating which (and how) questions are considered
significant in the context of particular scientific communities and
norms (cf. Brown 2010). Consistently with a Pragmatic View of
Theories, Morrison (2007) reconsiders and reforms canonical
conceptualizations of "theory." Finally, Longino (2013)
proposes an archaeology of assumptions behind and under different
research programs and theories of human behavior such as
neurobiological, molecular behavioral genetic, and
social-environmental approaches (e.g., Oyama 2000). For instance, two
shared or recurring assumptions across programs and theories are:



> 
> (1) that the approach in question has methods of measuring both the
> behavioral outcome that is the object of investigation and the factors
> whose association with it are the topic of investigation and (2) that
> the resulting measurements are exportable beyond the confines of the
> approach within which they are made. (Longino 2013, 117)
> 



A Pragmatic View of Theories expands the notion of theory to include
nonformal aspects, which surely must include elements from
Boumans' list above (e.g., metaphors, analogies, policy views),
as well as more standard components such as ontological assumptions
(e.g., Kuhn 1970; Levins and Lewontin 1985; Winther 2006b), natural
kinds (e.g., Hacking 2007b), and conditions of application or scope
(e.g., Longino 2013).



In addition to exploring internal theory diversity and in parallel
with plurality of modeling, a Pragmatic View of Theories could also
explore pluralism of modes of theorizing, and of philosophically
analyzing theoretical structure (thesis #2). Craver (2002) provides a
start in this direction in that he accepts three kinds of scientific
theory and of philosophical analysis of scientific theory. A more
synoptic view of the broader pragmatic context in which theories are
embedded can be found in the literature on different
"styles" of scientific reasoning and theorizing (e.g.,
Crombie 1994, 1996; Vicedo 1995; Pickstone 2000; Davidson 2001;
Hacking 2002, 2009; Winther 2012b; Elwick 2007; Mancosu
2010). While there is no univocal or dominant classification of
styles, two lessons are important. First, a rough consensus exists
that theoretical investigations of especially historical, mechanistic,
and mathematical structures and relations will involve different
styles. Second, each style integrates theoretical products and
theorizing processes in unique ways, thus inviting an irreducible
pragmatic methodological pluralism in our philosophical analysis of
the structure of scientific theories. For instance, the structure of
theories of mechanisms in molecular biology or neuroscience involves
flow charts, and is distinct from the
structure of theories of historical processes and patterns as found in
systematics and phylogenetics, which involves phylogenetic trees. As Crombie suggests, we need a
"comparative historical anthropology of thinking." (1996,
71; see Hacking 2009) Mathematical theory hardly remains regnant. It
gives way to a pluralism of theory forms and theory processes. Indeed,
even mathematical theorizing is a pluralistic motley, as Hacking
(2014) argues. Although a "deflationary" Semantic View
could account for pluralism of theory forms, the Pragmatic View of
Theories, drawing on styles, is required to do justice to the immense
variety of theorizing processes, and of philosophical accounts of
theory and theory structure. 



Finally, outstanding work remains in sorting out the philosophical
utility of a variety of proposed units in addition to styles, such as
Kuhn's (1970) paradigms, Lakatos' (1980) research
programmes, Laudan's (1977) research traditions, and
Holton's (1988) themata. A rational comparative historical
anthropology of both theorizing and philosophical analyses of
theorizing remains mostly unmapped (cf. Matheson and Dallmann 2014).
Such a comparative meta-philosophical analysis should also address
Davidson's (1974) worries about "conceptual schemes"
and Popper's (1996 [1976]) critique of "the myth of the
framework" (see Hacking 2002; Godfrey-Smith 2003).


### 4.2 A Running Example: Newtonian Mechanics



Cartwright has done much to develop a Pragmatic View. Start by
considering Newton's second law:

 

\[\tag{1}
F = ma
\]


Here \(F\) is the resultant force on a mass \(m\), and \(a\) is the
net acceleration of \(m\); both \(F\) and \(a\) are vectors. This law
is considered a "general" (Cartwright 1999a, 187) law
expressed with "abstract quantities" (Cartwright 1999b,
249). Newton's second law can be complemented with other laws,
such as (i) Hooke's law for an ideal spring:

 

\[\tag{2}
F = -kx
\]


Here \(k\) is the force constant of the spring, and \(x\) the distance
along the x-axis from the equilibrium position, and (ii)
Coulomb's law modeling the force between two charged
particles:

 

\[\tag{3}
F = K \frac{qq'}{r^2}
\]


Here \(K\) is Coulomb's electrical constant, \(q\) and \(q'\)
are the charges of the two objects, and \(r\) the distance between the
two objects. The picture Cartwright draws for us is that
Newton's, Hooke's, and Coulomb's laws are abstract,
leaving out many details. They can be used to derive mathematical
models of concrete systems. For instance, by combining (1) and (2),
the law of gravitation (a "fundamental" law, Cartwright
1983, 58-59), other source laws, and various simplifying
assumptions, we might create a model for the orbit of Mars, treating
the Sun and Mars as a 2-body system, ignoring the other planets,
asteroids, and Mars' moons. Indeed, the Solar System is a
powerful "nomological machine" (Cartwright 1999a,
50-53), which "is a fixed (enough) arrangement of
components, or factors, with stable (enough) capacities that in the
right sort of stable (enough) environment will, with repeated
operation, give rise to the kind of regular behaviour that we
represent in our scientific laws" (Cartwright 1999a, 50).
Importantly, most natural systems are complex and irregular, and
cannot be neatly characterized as nomological machines. For these
cases, abstract laws "run out" (Cartwright 1983) and are
rarely smoothly "deidealised" (Suarez 1999). In
general, abstract laws predict and explain only within a given domain
of application, and only under ideal conditions. More concrete laws or
models are not directly deduced from them (e.g., Suarez 1999,
Suarez and Cartwright 2008), and they can rarely be combined to
form effective "super-laws" (Cartwright 1983,
70-73). In short, the move from (1) and (2) or from (1) and (3)
to appropriate phenomenological models, is not fully specified by
either abstract law pairing. Indeed, Cartwright developed her notion
of "capacities" to discuss how "the principles of
physics" "are far better rendered as claims about
capacities, capacities that can be assembled and reassembled in
different nomological machines, unending in their variety, to give
rise to different laws" (1999a, 52). Articulating concrete
models requires integrating a mix of mathematical and nonformal
components. Laws (1), (2), and (3) remain only one component, among
many, of the models useful for, e.g., exploring the behavior of the
Solar System, balls on a pool table, or the behavior of charges in
electrical fields.



Shifting examples but not philosophical research program,
Suarez and Cartwright (2008) explains how analogies such as
superconductors as diamagnets (as opposed to ferromagnets) were an
integral part of the mathematical model of superconductivity developed
by Fritz and Heinz London in the 1930s (63; cf. London and London
1935). Suarez and Cartwright gladly accept that this model
"is uncontroversially grounded in classic electromagnetic
theory" (64). However, contra Semantic View Structuralists such
as Bueno, da Costa, French, and Ladyman, they view nonformal aspects
as essential to practices of scientific modeling and theorizing:
"The analogy [of diamagnets] helps us to understand how the
Londons work with their model... which assumptions they add and
which not... a formal reconstruction of the model on its own
cannot help us to understand that" (69). In short, the running
example of Newtonian mechanics, in conjunction with a glimpse into the
use of analogies in mathematical modeling, illustrates the Pragmatic
View's account of theory syntax: theory is constituted by a
plurality of formal and informal components.


### 4.3 Interpreting Theory Structure per the Pragmatic View



As we have explored throughout this section, models and theories have
informal internal components, and there are distinct modes of modeling
and theorizing. Because of the Pragmatic View's attention to
practice, function, and application, distinguishing *structure*
from *interpretation* is more difficult here than under the
Syntactic and Semantic Views. Any synchronic analysis of the structure
of models and theories must respect intentional diachronic processes
of interpreting and using, as we shall now see.



Regarding the import of function in models and theories (thesis #4),
already the Belgian philosopher of science Apostel defined modeling
thus: "Let then \(R(S,P,M,T)\) indicate the main variables of
the modelling relationship. The subject \(S\) takes, in view of the
purpose \(P\), the entity \(M\) as a model for the prototype
\(T\)" (1960, 128, see also Apostel 1970). Purposes took
center-stage in his article title: "Towards the Formal Study of
Models in the Non-Formal Sciences." MIT Artificial Intelligence
trailblazer Minsky also provided a pragmatic analysis:



> 
> We use the term "model" in the following sense: *To an
> observer \(B\), an object \(A^\*\) is a model of an object \(A\) to the
> extent that \(B\) can use \(A^\*\) to answer questions that interest
> him about \(A\).* The model relation is inherently ternary. Any
> attempt to suppress the role of the intentions of the investigator
> \(B\) leads to circular definitions or to ambiguities about
> "essential features" and the like. (1965, 45)
> 



This account is thoroughly intentionalist and anti-essentialist. That
is, mapping relations between model and world are left open and
overdetermined. Specifying the relevant relations depends on
contextual factors such as questions asked, and the kinds of
similarities and isomorphisms deemed to be of interest. The
appropriate relations are selected from an infinite (or, at least,
near-infinite) variety of possible relations (e.g., Rosenblueth and
Wiener 1945; Lowry 1965).



Regarding practice (thesis #5), in addition to ample work on the
experimental life mentioned above, consider a small example. A full
understanding of the content and structure of the London
brothers' model of superconductivity requires attention to
informal aspects such as analogies. Even London and London (1935)
state in the summary of their paper that "the current ["in
a supraconductor"] is characterized as a kind of diamagnetic
volume current" (88). They too saw the diamagnetic analogy as
central to their theoretical practices. Criteria and practices of
theory confirmation also differ from the ones typical of the Syntactic
and Semantic Views. While predictive and explanatory power as well as
empirical adequacy remain important, the Pragmatic View also insists
on a variety of other justificatory criteria, including pragmatic
virtues (sensu Kuhn 1977; Longino 1995) such as fruitfulness and
utility. In a nutshell, the Pragmatic View argues that scientific
theory structure is deeply shaped and constrained by functions and
practices, and that theory can be interpreted and applied validly
according to many different criteria.


### 4.4 Taking Stock: Pragmatic View



The analytical framework of the Pragmatic View remains under
construction. The emphasis is on internal diversity, and on the
external pluralism of models and theories, of modeling and theorizing,
and of philosophical analyses of scientific theories. The Pragmatic
View acknowledges that scientists use and need different kinds of
theories for a variety of purposes. There is no one-size-fits-all
structure of scientific theories. Notably, although the Pragmatic View
does not necessarily endorse the views of the tradition of American
Pragmatism, it has important resonances with the latter school's
emphasis on truth and knowledge as processual, purposive, pluralist,
and context-dependent, and on the social and cognitive structure of
scientific inquiry.



A further qualification in addition to the one above regarding
American Pragmatism is in order. The Pragmatic View has important
precursors in the historicist or "world view" perspectives
of Feyerabend, Hanson, Kuhn, and Toulmin, which were an influential
set of critiques of the Syntactic View utterly distinct from the
Semantic View. This philosophical tradition focused on themes such as
meaning change and incommensurability of terms across world views
(e.g., paradigms), scientific change (e.g., revolutionary: Kuhn 1970;
evolutionary: Toulmin 1972), the interweaving of context of discovery
and context of justification, and scientific rationality (Preston
2012; Bird 2013; Swoyer 2014). The historicists also opposed the idea
that theories can secure meaning and empirical support from a
theory-neutral and purely observational source, as the Syntactic View
had insisted on with its strong distinction between theoretical and
observational vocabularies (cf. Galison 1988). Kuhn's paradigms
or, more precisely, "disciplinary matrices" even had an
internal anatomy with four components: (i) laws or symbolic
generalizations, (ii) ontological assumptions, (iii) values, and (iv)
exemplars (Kuhn 1970, postscript; Godfrey-Smith 2003; Hacking 2012).
This work was concerned more with theory change than with theory
structure and had fewer conceptual resources from sociology of science
and history of science than contemporary Pragmatic View work.
Moreover, paradigms never quite caught on the way analyses of models
and modeling have. Even so, this work did much to convince later
scholars, including many of the Pragmatic View, of certain weaknesses
in understanding theories as deductive axiomatic structures.


## 5. Population Genetics



As a final way to contrast the three views, we return to population
genetics and, especially, to the Hardy-Weinberg Principle (HWP). Both
Woodger (1937, 1959) and Williams (1970, 1973) provide detailed
axiomatizations of certain parts of biology, especially genetics,
developmental biology, and phylogenetics. For instance, Woodger (1937)
constructs an axiomatic system based on ten logical predicates or
relations, including \(\bP\) (*part of*), \(\bT\) (*before
in time*), \(\bU\) (*reproduced by cell division or cell
fusion*), \(\bm\) (*male gamete*), \(\bff\) (*female
gamete*), and \(\bgenet\) (*genetic property*) (cf.
Nicholson and Gawne 2014). Woodger (1959) elaborates these logical
predicates or relations to produce a careful reconstruction of
Mendelian genetics. Here are two axioms in his system (which are
rewritten in contemporary notation, since Woodger used Russell and
Whitehead's *Principia Mathematica* notation):



\[
(m \wedge f) = \varnothing
\]

 

\[
\forall x,y,z,u,v (DLZxyz \wedge DLZuvz) \rightarrow 
 (x=u) \wedge (y=z)
\]


The first axiom should be read thus: "no gamete is both male and
female" (1959, 416). In the second axiom, given that \(DLZxyz\)
is a primitive relation defined as "\(x\) is a zygote which
develops in the environment \(y\) into the life \(z\)" (1959,
415), the translation is "every life develops in one and only
one environment from one and only one zygote" (416). Woodger
claims that "the whole of Mendel's work can be
expressed..." via this axiomatic system. Woodger briefly
mentions that if one assumes that the entire system or population is
random with respect to gamete fusions, "then the Pearson-Hardy
law is derivable" (1959, 427). This was a reference to HWP. In
her explorations of various axiomatizations of Darwinian lineages and
"subclans," and the process of the "expansion of the
fitter," Williams (1970, 1973) also carefully defines concepts,
and axiomatizes basic biological principles of reproduction, natural
selection, fitness, and so forth. However, she does not address HWP.
Of interest is the lack of axiomatization of HWP or other mathematical
principles of population genetics in Woodger's and
Williams' work. Were such principles considered secondary or
uninteresting by Woodger and Williams? Might Woodger's and
Williams' respective axiomatic systems simply lack the power and
conceptual resources to axiomatically reconstruct a mathematical
edifice actually cast in terms of probability theory? Finally, other
friends of the Syntactic View, such as the early Michael Ruse, do not
provide an axiomatization of HWP (Ruse 1975, 241).



Proponents of the Semantic View claim that their perspective on
scientific theory accurately portrays the theoretical structure of
population genetics. Thompson (2007) provides both set-theoretical and
state-space renditions of Mendelian genetics. The first involves
defining a set-theoretic predicate for the system, viz., \(\{P, A, f,
g\}\), where \(P\) and \(A\) are sets representing, respectively, the
total collection of alleles and loci in the population, while \(f\)
and \(g\) are functions assigning an allele to a specific location in,
respectively, the diploid cells of an individual or the haploid
gametic cells. Axioms in this set-theoretic formalization include
"The sets \(P\) and \(A\) are finite and non empty" (2007,
498). In contrast, the state-space approach of the Semantic View
articulates a phase space with each dimension representing allelic (or
genotypic) frequencies (e.g., cover and Chapter 3 of Lloyd 1994
[1988]). As an example, "for population genetic theory, a
central law of succession is the Hardy-Weinberg law" (Thompson
2007, 499). Mathematically, the diploid version of HWP is written
thus:

 

\[
(p+q)^2\_{\text{par}} = (p^2 + 2pq + q^2)\_{\text{off}}
\]


Here \(p\) and \(q\) are the frequencies of two distinct alleles at a
biallelic locus. The left-hand side represents the allele frequencies
in the parental generation and a random mating pattern, while the
right-hand side captures genotype frequencies in the offspring
generation, as predicted from the parental generation. This is a null
theoretical model--actual genotypic and allelic frequencies of
the offspring generation often deviate from predicted frequencies
(e.g., a lethal homozygote recessive would make the
\(q^2\_{\text{off}}\) term = 0). Indeed, HWP holds strictly only in
abstracted and idealized populations with very specific properties
(e.g., infinitely large, individuals reproduce randomly) and only when
there are *no* evolutionary forces operating in the population
(e.g., no selection, mutation, migration, or drift) (e.g., Hartl and
Clark 1989; Winther et al. 2015). HWP is useful also in the way it
interacts with laws of succession for selection, mutation, and so
forth (e.g., Okasha 2012). This powerful population genetic principle
is central to Semantic View analyses of the mathematical articulation
of the theoretical structure of population genetics (see also
Lorenzano 2014, Ginnobili 2016).



Recall that the Pragmatic View highlights the internal and external
pluralism--as well as the purposiveness--of model and theory
structure. Consider recent uses of population genetic theory to
specify the kinds and amounts of population structure existing in
*Homo sapiens*. In particular, different measures and
mathematical modeling methodologies are employed in investigating
human genomic diversity (e.g., Jobling et al. 2004; Barbujani et al.
2013; Kaplan and Winther 2013). It is possible to
distinguish at least two different research projects, each of which
has a unique pragmatic content (e.g., aims, values, and methods).
*Diversity partitioning* assesses genetic variation within and
among pre-determined groups using Analysis of Variance (also crucial
to estimating heritability, Downes 2014). *Clustering analysis*
uses Bayesian modeling techniques to simultaneously produce clusters
and assign individuals to these "unsupervised" cluster
classifications. The robust result of the first modeling project is
that (approximately) 85% of all genetic variance is found within human
subpopulations (e.g., Han Chinese or Sami), 10% across subpopulations
within a continental region, and only 5% is found across continents
(i.e., "African," "Asian," and
"European" - Lewontin 1972, 1974). (Recall also that
we are all already identical at, on average, 999 out of 1000
nucleotides.) To calculate diversity partitions at these three nested
levels, Lewontin (1972) used a Shannon information-theoretic measure
closely related to Sewall Wright's \(F\)-statistic:



\[
F\_{ST} = \frac{H\_T - \bar{H}\_S}{H\_T}
\]


Here \(H\_T\) is the total heterozygosity of the population assessed,
and \(\bar{H}\_S\) is the heterozygosity of each subpopulation (group)
of the relevant population, averaged across all the subpopulations.
\(F\_{ST}\) is bounded by 0 and 1, and is a measure of population
structure, with higher \(F\_{ST}\) values suggesting more structure,
viz., more group differentiation. HWP appears implicitly in both
\(H\_T\) and \(\bar{H}\_S\), which take heterozygosity (\(2pq\)) to be
equal to the *expected* proportion of heterozygotes under HWP
rather than the *actual* frequency of heterozygotes. \(H\_T\) is
computed by using the grand population average of \(p\) and \(q\),
whereas calculating \(\bar{H}\_S\) involves averaging across the
expected heterozygosities of each subpopulation. If random mating
occurs--and thus HWP applies--across the entire population
without respecting subpopulation borders, then \(H\_T\) and
\(\bar{H}\_S\) will be equal (i.e., \(p\) of the total population and
of each individual subpopulation will be the same; likewise for
\(q\)). If, instead, HWP applies only within subpopulations but not
across the population as a whole, then \(\bar{H}\_S\) will be smaller
than \(H\_T\), and \(F\_{ST}\) will be positive (i.e., there will be
"excess homozygosity" across subpopulations, which is
known as the "Wahlund Principle" in population genetics).
This is one way among many to deploy the population-genetic principle
of HWP. Thus, the Lewontin-style diversity partitioning result that
only roughly 5% of the total genetic variance is among races is
equivalent to saying that \(F\_{ST}\) across the big three continental
populations in Lewontin's three-level model is 0.05 (e.g.,
Barbujani et al. 1997). The basic philosophical tendency is to
associate the diversity partitioning research project's
(approximately) 85%-10%-5% result with an *anti-realist*
interpretation of biological race.



In contrast, clustering analysis (e.g., Pritchard et al. 2000;
Rosenberg et al. 2002; cf. Edwards 2003) can be readily
performed even with the small amount of among-continent genetic
variance in *Homo sapiens*. For instance, when the Bayesian
modeling computer program *STRUCTURE* is asked to produce 5
clusters, continental "races" appear--African,
Amerindian, Asian, European, and Pacific Islanders. Interestingly,
this modeling technique is also intimately linked to HWP: "Our
main modeling assumptions are Hardy-Weinberg equilibrium within
populations and complete linkage equilibrium between loci within
populations" (Pritchard et al. 2000, 946). That is, for a cluster to eventually be robust in the
modeling runs, it should meet HWP expectations. Clustering analysis
has sometimes been interpreted as a justification for a
*realist* stance towards biological race (see discussions in
Hochman 2013; Winther and Kaplan 2013; Edge and Rosenberg 2015; Spencer 2015).



This example of the mathematical modeling of human genomic diversity
teaches that basic and simple formal components can be used in
different ways to develop and apply theory, both inside and outside of
science. In contrast to the Syntactic and Semantic Views, the
Pragmatic View foregrounds tensions vis-a-vis ontological
assumptions and political consequences regarding the existence (or
not) of biological race between diversity partitioning (Lewontin 1972)
and clustering analysis (Pritchard et al. 2000) research packages.
These ontological ruptures can be identified despite the fact that
both research projects assess population structure by examining
departures from HWP (i.e., they measure excess homozygosity), and are
completely consistent (e.g., Winther 2014; Ludwig 2015; Edge and
Rosenberg 2015).



This exploration of how the three views on the structure of scientific
theory address population genetics, and in particular HWP, invites a
certain meta-pluralism. That is, the Syntactic View carefully breaks
down fundamental concepts and principles in genetics and population
genetics, articulating definitions and relations among terms. The
Semantic View insightfully decomposes and interweaves the complex
mathematical edifice of population genetics. The Pragmatic View sheds
light on modeling choices and on distinct interpretations and
applications of the same theory or model, both within and without
science. The three perspectives are hardly mutually exclusive. (N.B.,
the two running examples concern theory structure in Newtonian
mechanics and population genetics, independently considered. While
interesting, debates about "evolutionary forces" are
beyond the scope of the current entry; see, e.g., Hitchcock and
Velasco 2014.)


## 6. Conclusion



The structure of scientific theories is a rich topic. Theorizing and
modeling are core activities across the sciences, whether old (e.g.,
relativity theory, evolutionary theory) or new (e.g., climate
modeling, cognitive science, and systems biology). Furthermore, theory
remains essential to developing multipurpose tools such as statistical
models and procedures (e.g., Bayesian models for data analysis,
agent-based models for simulation, network theory for systems
analysis). Given the strength and relevance of theory and theorizing
to the natural sciences, and even to the social sciences (e.g.,
microeconomics, physical, if not cultural, anthropology),
philosophical attention to the structure of scientific theories could
and should increase. This piece has focused on a comparison of three
major perspectives: Syntactic View, Semantic View, and Pragmatic View.
In order to handle these complex debates effectively, we have
sidestepped certain key philosophical questions, including questions
about scientific realism; scientific explanation and prediction;
theoretical and ontological reductionism; knowledge-production and
epistemic inference; the distinction between science and technology;
and the relationship between science and society. Each of these topics
bears further philosophical investigation in light of the three
perspectives here explored.



A table helps summarize general aspects of the three views'
analyses of the structure of scientific theories:





|  |  |  |  |
| --- | --- | --- | --- |
|  | **Syntactic View** | **Semantic View** | **Pragmatic View** |
| Theory Structure | Uninterpreted axiomatic system | (i) State-space,

(ii) Model-/set-theoretic | Internal and external pluralism |
| Theory Interpretation | Correspondence rules | (i) Hierarchy of models,

(ii) Similarity,

(iii) Isomorphism | (i) Structure already inflected by practice, function, and
application

(ii) Pragmatic virtues |
| Is theory interpretation an aspect of theory structure? | Yes | No | Yes, although the distinction is hard to make. |



Table 2. General aspects of each
view's analysis of the structure of scientific theories.




The Syntactic, Semantic, and Pragmatic views are often taken to be
mutually exclusive and, thus, to be in competition with one another.
They indeed make distinct claims about the anatomy of scientific
theories. But one can also imagine them to be complementary, focusing
on different aspects and questions of the structure of scientific
theories and the process of scientific theorizing. For instance, in
exploring nonformal and implicit components of theory, the Pragmatic
View accepts that scientific theories often include mathematical
parts, but tends to be less interested in these components. Moreover,
there is overlap in questions--e.g., Syntactic and Semantic Views
share an interest in formalizing theory; the Semantic and Pragmatic
Views both exhibit concern for scientific practice.



How are these three views ultimately related? A standard philosophical
move is to generalize and abstract, understanding a situation from a
higher level. One "meta" hypothesis is that a given
philosophical analysis of theory structure tends to be associated with
a perceived relationship among the three views here discussed. The
Syntactic View is inclined to interpret the Semantic View's
formal machinery as continuous with its own generalizing axiomatic
strategy, and hence diagnoses many standard Semantic View critiques
(Section 3) as missing their mark (the strategy of *identity*;
e.g., Friedman 1982; Worrall 1984; Halvorson 2012, 2013, 2019; Lutz
2012, 2017; cf. Chakravartty 2001). The Semantic View explicitly
contrasts its characterization of theory structure with the
"linguistic" or "metamathematical" apparatus
of the Syntactic View (the strategy of *combat*; e.g., Suppe
1977; van Fraassen 1980, 1989; Lloyd 1994 [1988]). Finally, the
Pragmatic View, which did not exist as a perspective until relatively
recently, imagines theory as pluralistic and can thus ground a
holistic philosophical investigation. It envisions a meta-pluralism in
which reconstructive axiomatization and mathematical modeling remain
important, though not necessary for all theories. This third view
endorses a panoply of theoretical structures and theorizing styles,
negotiating continuity both between theorizing and "the
experimental life," and among philosophical analyses of the
structure of scientific theories (the strategy of
*complementarity*; e.g., Hacking 1983, 2009; Galison 1988,
1997; Craver 2002; Suarez and Cartwright 2008; Griesemer 2013). Interestingly, Suarez and Pero
(2019) explicitly concur with the Pragmatic View as described in this
article, but believe that "the semantic conception in its bare
minimal expression" is compatible with, if not sufficient for,
capturing "pragmatic elements and themes involved in a more
flexible and open-ended approach to scientific theory"
(Suarez and Pero 2019, 348). By design, the ecumenical
meta-pluralism sanctioned by the Pragmatic View does not completely
offset identity and combat strategies. Moreover, only "partial
acceptance" of the respective views may ultimately be possible.
Even so, the complementarity strategy might be worth developing
further. Compared to identity and combat meta-perspectives, it
provides broader--or at least different--insights into the
structure of scientific theories. More generally, exploring the
relations among these views is itself a rich topic for future
philosophical work, as is investigating their role in, and
interpretation of, active scientific fields ripe for further
philosophical analysis such as climate change (e.g., Winsberg 2018),
model organisms (e.g., Ankeny and Leonelli 2020), and cartography and
GIS (e.g., Winther 2020).