## 1. Two Criteria of Theoreticity


### 1.1 Reference to Nonobservable Entities and Properties



As just explained, a theoretical term may simply be understood as an
expression that refers to nonobservable entities or properties.
Theoreticity, on this understanding, is the negation of observability.
This explanation of theoreticity thus rests on an antecedent
understanding of observability. What makes an entity or property
observable? As Carnap (1966: ch. 23) has pointed out, a philosopher
understands the notion of observability in a narrower sense than a
physicist. For a philosopher, a property is observable if it can be
'directly perceived by the senses'. Hence, such properties
as 'blue', 'hard' and 'colder
than' are paradigmatic examples of observable properties in the
philosopher's understanding of observability. The physicist, by
contrast, would also count quantitative magnitudes that can be
measured in a 'relatively simple, direct way' as
observable. Hence, the physicist views such quantities as temperature,
pressure and intensity of electric current as observable.



The notion of direct perception is spelled out by Carnap (1966: ch.
23) by two conditions. Direct perception means, first, perception
unaided by technical instruments and, second, that the perception is
unaided by inferences. These two conditions are obviously not
satisfied for the measurement of quantities like temperature and
pressure. For the philosopher, only spatial positions of liquids and
pointers are observed when these quantities are measured. To an even
higher degree, we are unable to observe electrons, molecules,
gravitational forces and genes on this narrow understanding of
observability. Hence, expressions referring to such entities qualify
as theoretical.



In sum, a property or object is observable--in the
philosopher's sense--if it can be perceived directly, where
directness of observation excludes the use of technical artifacts and
inferences. Notably, Carnap (1936/37: 455; 1966: 226) admits that his
explanation of the distinction is not sufficiently precise to
determine a sharp line between observational and theoretical terms. He
rather views the theory-observation distinction as being introduced
into a 'continuum of degrees of observability' by choice.
Prominent criticisms of the theory-observation distinction will be
discussed in
 Section 2.1.


### 1.2 Semantic Dependence upon a Scientific Theory



The above explanation of theoreticity may be felt unsatisfactory as it
determines the property of being theoretical only via negation of the
property of being observable (Putnam 1962). This explanation does not
indicate any specific connection between the semantics of theoretical
terms and corresponding scientific theories. There is, however, also a
direct characterization of theoreticity that complements the criterion
of non-observability: an expression is theoretical if and only if its
meaning is determined through the axioms of a scientific theory. This
explanation rests on what has come to be referred to as the
*contextual theory of meaning*, which says that the meaning of
a scientific term depends, in some way or other, on how this term is
incorporated into a scientific theory.



Why adopt the contextual theory of meaning for scientific terms?
Suppose the notion of meaning is understood along the lines of the
Fregean notion of sense. The sense of a term be understood as that
what determines its reference (cf. Church 1956: 6n). It is,
furthermore, a reasonable requirement that a semantic theory must
account for our understanding of the sense and, hence, our methods of
determining the extension of scientific terms (cf. Dummett 1991: 340).
For a large number of scientific terms these methods rest upon axioms
of one or more scientific theories. There is no way of determining the
force function in classical mechanics without using some axiom of this
theory. Familiar methods make use of Newton's second law of
motion, Hooke's law, the law of gravitation etc. Likewise,
virtually all methods of measuring temperature rest upon laws of
thermodynamics. Take measurement by a gas thermometer which is based
on the ideal gas law. The laws of scientific theories are thus
essential to our methods of determining the extension of scientific
terms. The contextual theory of meaning, therefore, makes intelligible
how students in a scientific discipline and scientists grasp the
meaning, or sense, of scientific terms. On this account, understanding
the sense of a term means knowing how to determine its referent, or
extension, at least in part.



The contextual theory of meaning can be traced back at least to the
work of Duhem. His demonstration that a scientific hypothesis in
physics cannot be tested in isolation from its theoretical context is
joined with and motivated by semantic considerations, according to
which it is physical theories that give meaning to the specific
concepts of physics (Duhem 1906: 183). Poincare (1902: 90)
literally claims that certain scientific propositions acquire meaning
only by virtue of the adoption of certain conventions. Perhaps the
most prominent and explicit formulation of the contextual theory of
meaning is to be found in Feyerabend's landmark
"Explanation, Reduction, and Empiricism" (1962: 88):



> 
> 
> For just as the meaning of a term is not an intrinsic property but is
> dependent upon the way in which the term has been incorporated into a
> theory, in the very same manner the content of a whole theory (and
> thereby again the meaning of the descriptive terms which it contains)
> depends upon the way in which it is incorporated into both the set of
> its empirical consequences and the set of all the alternatives which
> are being discussed at a given time: once the contextual theory of
> meaning has been adopted, there is no reason to confine its
> application to a single theory, especially as the boundaries of such a
> language or of such a theory are almost never well defined.
> 
> 
> 



The accounts of a contextual theory of meaning in the works of Duhem,
Poincare and Feyerabend are informal insofar as they do not
crystallize into a corresponding formal semantics for scientific
terms. Such a crystallization is brought about by some of the formal
accounts of theoretical terms to be expounded in
 Section 4.



The view that meaning is bestowed upon a theoretical term through the
axioms of a scientific theory implies that only axiomatized or
axiomatizable scientific theories contain theoretical terms. In fact,
all formal accounts of the semantics of theoretical terms are devised
to apply to axiomatic scientific theories. This is due, in part, to
the fact that physics has dominated the philosophy of science for a
long time. One must wonder, therefore, whether there are any
theoretical terms in, for example, evolutionary biology which has as
yet resisted complete axiomatization. Arguably, there are. Even though
evolutionary biology has not yet been axiomatized, we can recognize
general propositions therein that are essential to determining certain
concepts of this theory. Consider the following two propositions. (i)
Two DNA sequences are homologous if and only if they have a common
ancestor sequence. (ii) There is an inverse correlation between the
number of mutations necessary to transform one DNA-sequence \(S\_1\)
into another \(S\_2\) and the likelihood that \(S\_1\) and \(S\_2\) are
homologous. Notably, these two propositions are used to determine,
among other methods, relations of homology in evolutionary biology.
The majority of general propositions in scientific theories other than
those of physics, however, have instances that fail to be true. (Some
philosophers of science have argued that this so even for a large
number of axioms in physics [Cartwright 1983]).


## 2. Criticisms and Refinements of the Theory-Observation Distinction


### 2.1 Criticisms



The very idea of a clear-cut theory-observation distinction has
received much criticism. First, with the help of sophisticated
instruments, such as telescopes and electron microscopes, we are able
to observe more and more entities, which had to be considered
unobservable at a previous stage of scientific and technical
evolution. Electrons and \(\alpha\)-particles which can be observed in
a cloud chamber are a case in point (Achinstein 1965). Second, assume
observability is understood as excluding the use of instruments. On
this understanding, examples drawing on the use of cloud chambers and
electron microscopes, which are adduced to criticize the
theory-observation distinction, can be dealt with. However, we would
then have to conclude that things being perceived with glasses are not
observed either, which is counterintuitive (Maxwell 1962). Third,
there are concepts applying to or being thought to apply to both
macroscopic and submicroscopic particles. A case in point are spatial
and temporal relations and the color concepts that play an important
role in Newton's corpuscle theory of light. Hence, there are
clear-cut instances of observation concepts that apply to unobservable
entities, which does not seem acceptable (cf. Putnam 1962).



These objections to the theory-observation distinction can be answered
in a relatively straightforward manner from a Carnapian perspective.
As explained in
 Section 1.1,
 Carnap (1936/37, 1966) was quite explicit that the
philosopher's sense of observation excludes the use of
instruments. As for an observer wearing glasses, a proponent of the
theory-observation distinction finds enough material in Carnap
(1936/37: 455) to defend her position. Carnap is aware of the fact
that color concepts are not observable ones for a color-blind person.
He is thus prepared to relativize the distinction in question. In
fact, Carnap's most explicit explanation of observability
defines this notion in such a way that it is relativized to an
organism (1936/37: 454n).



Recall, moreover, that Carnap's theory-observation distinction
was not intended to do justice to our overall understanding of these
notions. Hence, certain quotidian and scientific uses of
'observation', such as observation using glasses, may well
be disregarded when this distinction is drawn as long as the
distinction promises to be fruitful in the logical analysis of
scientific theories. A closer look reveals that Carnap (1966: 226)
agrees with critics of the logical empiricists' agenda, such as
Maxwell (1962) and Achinstein (1965), on there being no clear-cut
theory-observation distinction (see also Carnap's early
(1936/37: 455) for a similar statement):



> 
> 
> There is no question here of who [the physicist thinking that
> temperature is observable or the philosopher who disagrees, H. A.] is
> using the term 'observable' in the right or proper way.
> There is a continuum which starts with direct sensory observations and
> proceeds to enormously complex, indirect methods of observation.
> Obviously no sharp line can be drawn across this continuum; it is a
> matter of degree.
> 
> 
> 



A bit more serious is Putnam's (1962) objection drawing on the
application of apparently clear-cut instances of observation concepts
to submicroscopic particles. Here, Carnap would have to distinguish
between color concepts applying to observable entities and related
color concepts applying to unobservable ones. So, the formal language
in which the logical analysis proceeds would have to contain a
predicate 'red\(\_1\)' applying to macroscopic objects and
another one 'red\(\_2\)' applying to submicroscopic ones.
Again, such a move would be in line with the artificial, or ideal
language philosophy that Carnap advocated (see Lutz (2012) for a
sympathetic discussion of artificial language philosophy).



There is another group of criticisms coming from the careful study of
the history of science: Hanson (1958), Feyerabend (1962) and Kuhn
(1962) aimed to show that observation concepts are
*theory-laden* in a manner that makes their meaning
theory-dependent. Feyerabend's (1978: 32), for example, held
that all terms are theoretical. Hanson (1958: 18) thinks that Tycho
and Kepler were (literally) 'seeing' different things when
perceiving the sun rising because their astronomical background
theories were different. Kuhn (1962) was more tentative when
expounding his variant of the theory-ladenness of observation. In a
discussion of the Sneed formalism of the structuralist school, he
favored a theory-observation distinction that is relativized, first to
a theory and second to an application of this theory (Kuhn 1976).



Virtually all formal accounts of theoretical terms in fact assume that
those phenomena which a theory *T* is meant to account for can
be described in terms of expressions whose meaning does not depend on
*T*. The counter-thesis that the meaning of putative
observation terms depends on a quotidian or scientific theory,
therefore, attacks a core doctrine in the logical empiricists'
and subsequent work on theoretical terms. A thorough discussion and
assessment of theory-ladenness of observation in the works of the
great historians of science is beyond the scope of this entry. Bird
(2004), Bogen (2012) and Oberheim and Hoyningen-Huene (2009) are
entries in the present encyclopedia that address, among other things,
this issue. Schurz (2013: ch. 2.9) defines a criterion of the theory
independence of observation in terms of an ostensive learning
experiment, and shows how such a criterion helps answer the challenges
of theory-ladenness of observation.


### 2.2 Refinements



There is a simple, intuitive and influential proposal how to
relativize the theory-observation distinction in a sensible way: a
term *t* is theoretical with respect to a theory *T*, or
for short, a *T*-term if and only if it is introduced by the
theory *T* at a certain stage in the history of science.
O-terms, by contrast, are those that were antecedently available and
understood before *T* was set forth (Lewis 1970; cf. Hempel
1973). This proposal draws the theory-observation distinction in an
apparently sharp way by means of relativizing that distinction to a
particular theory. Needless to say, the proposal is in line with the
contextual theory of meaning.



The distinction between *T*-terms and antecedently available
O-terms has two particular merits. First, it circumvents the view that
any sharp line between theoretical and observational terms is
conventional and arbitrary. Second, it connects the theory-observation
distinction with what seemed to have motivated that distinction in the
first place, viz., an investigation how we come to understand the
meaning of terms that appear to be meaningful in virtue of certain
scientific theories.



A similar proposal of a relativized theory-observation distinction was
made by Sneed in his seminal *The Logical Structure of Mathematical
Physics* (1979: ch. II). Here is a simplified and more syntactic
formulation of Sneed's criterion of *T*-theoreticity:




**Definition 1 (*T*-theoreticity)** 
  

A term *t* is theoretical with respect to the theory
*T*, or for short, *T*-theoretical if and only if any
method of determining the extension of *t*, or some part of
that extension, rests on some axiom of *T*.




It remains to explain what it is for a method *m* of
determining the extension of *t* to rest upon an axiom
\(\phi\). This relation obtains if and only if the use of *m*
depends on \(\phi\) being a true sentence. In other
words, *m* rests upon \(\phi\) if and only if the
hypothetical assumption of \(\phi\) being false or indeterminate would
invalidate the use of *m* in the sense that we would
be lacking the commonly presumed justification for using *m*.
The qualification 'or some part of that
extension' has been introduced in the present definition because
we cannot expect a single measurement method to determine the
extension of a scientific quantity completely.
*T*-non-theoreticity is the negation of
*T*-theoreticity:




**Definition 2 (*T*-non-theoreticity)** 
  

A term *t* is *T*-non-theoretical if and only if it is
not *T*-theoretical.




The concepts of classical particle mechanics (henceforth abbreviated
by CPM) exemplify well the notions of *T*-theoreticity and
*T*-non-theoreticity. As has been indicated above, all methods
of determining the force acting upon a particle make use of some axiom
of classical particle mechanics, such as Newton's laws of motion
or some law about special forces. Hence, force is CPM-theoretical.
Measurement of spatial distances, by contrast, is possible without
using axioms of CPM. Hence, the concept of spatial distance is
CPM-non-theoretical. The concept of mass is less straightforward to
classify as we can measure this concept using classical collision
mechanics (CCM). Still, it was seen to be CPM-theoretical by the
structuralists since CCM appeared reducible to CPM (Balzer et al.
1987: ch. 2).



Suppose for a term *t* once introduced by a scientific theory
\(T\_1\) novel methods of determination become established through
another theory \(T\_2\), where these methods do not depend on any axiom
of \(T\_1\). Then, *t* would neither qualify as
\(T\_1\)-theoretical nor as \(T\_2\)-theoretical. It is preferable, in
this situation, to relativize Definition 1 to theory-nets *N*,
i.e., compounds of several theories. Whether there
are such cases has not yet been settled.



The original exposition of the theoreticity criterion by Sneed (1979)
is a bit more involved as it makes use of set-theoretic predicates and
intended applications, rather technical notions of the
*structuralist approach to scientific theories*. There has been
a lively discussion, mainly but not exclusively within the
structuralist school, how to express the relativized notion of
theoreticity most properly (Balzer 1986; 1996). As noted above, Kuhn
(1976) proposed a twofold relativization of theoreticity, viz., first
to a scientific theory and second to applications of such
theories.



Notably, Sneed's criterion of *T*-theoreticity suggests a
strategy that allows us to regain a global, non-relativized
theory-observation distinction: simply take a term *t* to be
theoretical if and only if it holds, for all methods *m*
of determining its extension, that *m*
rests upon some axiom of some theory *T*. A
term *t* is non-theoretical, or observational, if and only if
there are means of determining its extension, at least in part, that
do not rest upon any axiom of any theory. This criterion is still
relative to our present stage of explicit theories but comes closer to
the original intention of Carnap's theory-observation
distinction, according to which observation is understood in the
narrow sense of unaided perception.


## 3. Two Problems of Theoretical Terms



The problem of theoretical terms is a recurrent theme in the
philosophy of science literature (Achinstein 1965; Sneed 1979: ch. II;
Tuomela 1973: ch. V; Friedman 2011). Different shades of meaning have
been associated with this problem. In its most comprehensive
formulation, the problem of theoretical terms is to give a proper
account of the meaning and reference of theoretical terms. There are
at least two kinds of expression that pose a distinct problem of
theoretical terms, respectively. First, unary predicates referring to
theoretical entities, such as 'electron',
'neutrino' and 'nucleotide'. Second, non-unary
theoretical predicates, such as 'homology' in evolutionary
biology and theoretical function expressions, such as
'force', 'temperature' and 'intensity of
the electromagnetic field' in physics. Sneed's problem of
theoretical terms, as expounded in (1979: ch. II), concerns only the
latter kind of expression. We shall now start surveying problems
concerning the semantics of expressions for theoretical entities and
then move on to expressions for theoretical relations and
functions.


### 3.1 Theoretical Entities



A proper semantics for theoretical terms involves an account of
reference and one of meaning and understanding. Reference fixing needs
to be related to meaning as we want to answer the following question:
how do we come to refer successfully to theoretical entities? This
question calls for different answers depending on what particular
conception of a theoretical entity is adopted. The issue of realism
and its alternatives, therefore, comes into play at this point.



For the realist, theoretical entities exist independently of our
theories about the world. Also, natural kinds that classify these
entities exist independently of our theories (cf. Psillos 1999; Lewis
1984). The instrumentalist picture is commonly reported to account for
theoretical entities in terms of mere fictions. The formalist variant
of instrumentalism denies that theoretical terms have referents in the
first place. Between these two extreme cases there is a number of
intermediate
 positions.[1]



Carnap (1958; 1966: ch. 26) attempted to attain a metaphysically
neutral position so as to avoid any commitment to or denial of
scientific realism. In his account of the theoretical language of
science, theoretical entities were conceived of as mathematical
entities that are related to observable events in certain determinate
ways. An electron, for example, figures as a certain distribution of
charge and mass in a four-dimensional manifold of real numbers, where
charge and mass are real-valued functions. These functions and the
four-dimensional manifold itself are to be related to observable
events by means of universal axioms. Notably, Carnap would not have
accepted a characterization of his view as antirealist or non-realist
since he thought the metaphysical doctrine of realism to be void of
content.



There thus are different metaphysical views of theoretical entities in
science, each of which is consistent with the understanding that a
theoretical entity is inaccessible by means of unaided perception.
First, theoretical entities are mind and language independent. Second,
theoretical entities are mind and language dependent in some way or
other. Third, they are conceived of as mathematical entities that are
related to the observable world in certain determinate ways. We may
thus distinguish between (i) a realist view, (ii) a collection of
non-realist views and (iii) a Pythagorean view of theoretical
entities.



Now, there are three major accounts of reference and meaning that have
been used, implicitly or explicitly, for the semantics of theoretical
terms: (i) the descriptivist picture, (ii) causal and
causal-historical theories and (iii) hybrid ones that combine
descriptivist ideas with causal elements (Reimer 2010). Accounts of
reference and meaning other than these play no significant role in the
philosophy of science. Hence, we need to survey at least nine
combinations consisting, first, of an abstract characterization of the
nature of a theoretical entity (realist, non-realist and Pythagorean),
and, second, a particular account of reference (descriptivist, causal
and hybrid). Some of these combinations are plainly inconsistent and,
hence, can be dealt with very briefly. Let us start with the realist
view of theoretical entities.


#### 3.1.1 The realist view



The descriptivist picture is highly intuitive with regard to our
understanding of expressions referring to theoretical entities on the
realist view. According to this picture, an electron is a
spatiotemporal entity with such and such a mass and such and such a
charge. We detect and recognize electrons when identifying entities
having these properties. The descriptivist explanation of meaning and
reference makes use of theoretical functions, mass and electric charge
in the present example. The semantics of theoretical entities,
therefore, is connected with the semantics of theoretical relations
and functions, which will be dealt with in the next subsection. It
seems to hold, in general, that theoretical entities in the sciences
are to be characterized in terms of theoretical functions and
relations.



The descriptivist account, however, faces two particular problems with
regard to the historic evolution of scientific theories. First, if
descriptions of theoretical entities are constitutive of the meaning
of corresponding unary predicates, one must wonder what the common
core of understanding is that adherents of successive theories share
and whether there is such a core at all. Were Rutherford and Bohr
talking about the same type of entities when using the expression
'electron'? Issues of incommensurability arise with the
descriptivist picture (Psillos 1999: 280). A second problem arises
when elements of the description of an entity in a given theory
*T* are judged wrong from the viewpoint of a successor theory
\(T'\). Then, on a strict reading of the descriptivist account, the
corresponding theoretical term failed to refer in *T*. For if
there is nothing that satisfies a description, the corresponding
expression has no referent. This is a simple consequence of the theory
of description by Russell in his famous "On Denoting"
(1905). Hence, an account of weighting descriptions is needed in order
to circumvent such failures of reference.



As is well known, Kripke (1980) set forth a causal-historical account
of reference as an alternative to the descriptivist picture. This
account begins with an initial baptism that introduces a name and goes
on with causal chains transmitting the reference of the name from
speaker to speaker. In this picture, Aristotle is the man once
baptized so; he might not have been the student of Plato or done any
other thing commonly attributed to him. Kripke thought this picture to
apply both to proper names and general terms. It is hardly indicated,
however, how this picture works for expressions referring to
theoretical entities (cf. Papineau 1996). Kripke's story is
particularly counterintuitive in view of the ahistorical manner of
teaching in the natural sciences. There, up-to-date textbooks and
recent journal articles are more important than the original,
historical introduction of a theoretical term. The Kripkean causal
story may also be read as an account of reference fixing without being
read as a story of grasping the meaning of theoretical terms.
Reference, however, needs to be related to meaning so as to ensure
that scientists know what they are talking about and are able to
identify the entities under investigation. Notably, even for
expressions of everyday language, the charge of not explaining meaning
has been leveled against Krikpe's causal-historical account
(Reimer 2010). The same charge applies to Putnam's (1975) causal
account of reference and meaning, which Putnam himself abandoned in
his (1980).



A purely causal or causal-historical account of reference does not
seem a viable option for theoretical terms. More promising are hybrid
accounts that combine descriptivist intuitions with causal elements.
Such an account has been given by Psillos (1999: 296):


1. A term *t* refers to an entity *x* if and
only if *x* satisfies the core causal description
associated with *t*.
2. Two terms \(t'\) and *t* denote the same entity if and only
if (a) their putative referents play the same causal role with respect
to a network of phenomena; and (b) the core causal description of
\(t'\) takes up the kind-constitutive properties of the core causal
description associated with *t*.



This account has two particular merits. First, it is much closer to
the way scientists understand and use theoretical terms than purely
causal accounts. Because of this, it is not only an account of
reference but also one of meaning for theoretical terms. In purely
causal accounts, by contrast, there is a tendency to abandon the
notion of meaning altogether. Second, it promises to ensure a more
stable notion of reference than in purely descriptivist accounts of
reference and meaning. Notably, the kind of causation that
Psillos's hybrid account refers to is different from the
causal-historical chains that Kripke thought responsible for the
transmission of reference among speakers. No further explanation,
however, is given of what a kind-constitutive property is and how we
are to recognize such a property. Psillos (1999: 288n) merely infers
the existence of such properties from the assumption of there being
natural kinds.


#### 3.1.2 Non-realist views



Non-realist and antirealist semantics for theoretical terms are
motivated by the presumption that the problem of theoretical terms has
no satisfying realist solution. What does a non-realist semantics of
theoretical terms look like? The view that theoretical entities are
mere fictions often figures in realist portrays of antirealism. This
view is hardly seriously maintained by any philosopher of science in
the twentieth century. If one were to devise a formal or informal
semantics for the view that theoretical entities are mere fictions, a
purely descriptive account seems most promising. Such an account could
in particular make heavy use of the Fregean notion of sense. For one
important motivation for introducing this notion is to explain our
understanding of expressions like 'Odysseus' and
'Pegasus'.



Formalist variants of instrumentalism are a more serious alternative
to realist semantics than the fiction view of theoretical entities.
Formalist views in the philosophy of mathematics aim to account for
mathematical concepts and objects in terms of syntactic entities and
operations thereupon within a calculus. Such views have been carried
over to theoretical concepts and objects in the natural sciences, with
the qualification that the observational part of the calculus is
interpreted in such a way that its symbols refer to physical or
phenomenal objects. Cognitive access to theoretical entities is thus
explained in terms of our cognitive access to the symbols and rules of
the calculus in the context of an antecedent understanding of the
observation terms. Formalist ideas were sympathetically entertained by
Hermann Weyl (1949). He was driven towards such ideas by adherence to
Hilbert's distinction between real and ideal elements and the
corresponding distinction between real and ideal propositions (Hilbert
1926). Propositions of the observation language were construed as real
in the sense of this Hilbertian distinction, while theoretical
propositions are construed as ideal. The content of an ideal
proposition is understood in terms of the (syntactic) consistency of
the whole system consisting of ideal and real propositions being
asserted.


#### 3.1.3 The Pythagorean view



There remains to discuss the view that theoretical entities are
mathematical entities which are related to observable events in
certain determinate ways. This theory is clearly of the descriptivist
type, as we shall see more clearly when dealing with the formal
account by Carnap in
 Section 4.
 No causal elements are needed in Carnap's Pythagorean
empiricism.



The Pythagorean view shifts the problem of theoretical terms to the
theory of meaning and reference for mathematical expressions. The
question of how we are able to refer successfully to electrons is
answered by the Pythagorean by pointing out that we are able to refer
successfully to mathematical entities. Moreover, the Pythagorean
explains, it is part of the notion of an electron that corresponding
mathematical entities are connected to observable phenomena by means
of axioms and inference rules. The empirical surplus of theoretical
entities in comparison to "pure" mathematical entities is
thus captured by axioms and inference rules that establish connections
to empirical phenomena. Since mathematical entities do not, by
themselves, have connections to observable phenomena, the question of
truth and falsehood may not be put in a truth-conditional manner for
those axioms that connect mathematical entities with phenomenal events
(cf.
 Section 4.2).
 Carnap (1958), therefore, came to speak of *postulates* when
referring to the axioms of a scientific theory.



How do we come to refer successfully to mathematical entities? This,
of course, is a problem in the philosophy of mathematics. (For a
classical paper that addresses this problem see Benacerraf (1973)).
Carnap has not much to say about meaning and reference of mathematical
expressions in his seminal "The Methodological Character of
Theoretical Concepts" (1956) but discusses these issues in his
"Empiricism, Semantics, and Ontology" (1950). There he
aims at establishing a metaphysically neutral position that avoids a
commitment to Platonist, nominalist or formalist conceptions of
mathematical objects. A proponent of the Pythagorean view other than
Carnap is Hermann Weyl (1949). As for the cognition of mathematical
entities, Weyl largely followed Hilbert's formalism in his later
work. Hence, there is a non-empty intersection between the Pythagorean
view and the formalist view of theoretical entities. Unlike Carnap,
Weyl did not characterize the interpretation of theoretical terms by
means of model-theoretic notions.


### 3.2 Theoretical Functions and Relations



For theoretical functions and relations, a particular problem arises
from the idea that a theoretical term is, by definition, semantically
dependent upon a scientific theory. Let us recall the above
explanation of *T*-theoreticity: a term *t* is
*T*-theoretical if and only if any method of determining the
extension of *t*, or some part of that extension, rests upon
some axiom of *T*. Let \(\phi\) be such an axiom and *m*
be a corresponding method of determination. The
present explanation of *T*-theoreticity, then, means that *m*
is valid only on condition of \(\phi\) being true.
The latter dependency holds because \(\phi\) is used either explicitly
in calculations to determine *t* or in the calibration of
measurement devices. Such devices, then, perform the calculation
implicitly. A case in point is measurement of temperature by a gas
thermometer. Such a device rests upon the law that changes of
temperature result in proportional changes in the volume of gases.



Suppose now *t* is theoretical with respect to a theory
*T*. Then it holds that in order to measure *t*, we need
to assume the truth of some axiom \(\phi\) of *T*. Suppose,
further, that *t* has occurrences in \(\phi\), as is standard
in examples of *T*-theoreticity. From this it follows that, in
standard truth-conditional semantics, the truth value of \(\phi\) is
dependent on the semantic value of *t*. This leads to the
following epistemological problem: on the one hand, we need to know
the extension of *t* in order to find out whether \(\phi\) is
true. On the other hand, it is simply impossible to determine the
extension of *t* without using \(\phi\) or some other axiom of
*T*. This mutual dependency between the semantic values of
\(\phi\) and *t* makes it difficult, if not even impossible, to
have evidence for \(\phi\) being true in any of its applications (cf.
Andreas 2008).



We could, of course, use an alternative measurement method of
*t*, say one resting upon an axiom \(\psi\) of *T*, to
gain evidence for the axiom \(\phi\) being true in some selected
instances. This move, however, only shifts the problem to applications
of another axiom of *T*. For these applications the same type
of difficulty arises, viz., mutual dependency of the semantic values
of \(\psi\) and *t*. We are thus caught either in a vicious
circle or in an infinite regress when attempting to gain evidence for
the propriety of a single measurement of a theoretical term. Sneed
(1979: ch. II) was the first to describe that particular difficulty in
the present manner and termed it *the problem of theoretical
terms*. Measurement of the force function in classical mechanics
exemplifies this problem well. There is no method of measuring force
that does not rest upon some law of classical mechanics. Likewise, it
is impossible to measure temperature without using some law that
depends upon either phenomenological or statistical
thermodynamics.



Although its formulation is primarily epistemological, Sneed's
problem of theoretical terms has a semantic reading. Let the meaning
of a term be identified with the methods of determining its extension,
as suggested in
 Section 1.2.
 Then we can say that our understanding of *T*-theoretical
relations and functions originates from the axioms of the scientific
theory *T*. In standard truth-conditional semantics, by
contrast, one assumes that the truth value of an axiom is determined
by the semantic values of those descriptive symbols which have
occurrences in \(\phi\). Among these symbols, there are theoretical
terms of *T*. Hence, it appears that standard truth-conditional
semantics does not accord with the order of our grasping the meaning
of theoretical terms. In the next section, we will deal with indirect
means of interpreting theoretical terms. These proved to be ways out
of the present problem of theoretical terms.


## 4. Formal Accounts



A few notational conventions and preliminary considerations are
necessary to explain the formal accounts of theoretical terms.
Essential to all of these accounts is the division of the set of
descriptive symbols into a set \(V\_o\) of observational and another
set \(V\_t\) of theoretical terms. (The descriptive symbols of a formal
language are simply the non-logical ones.) A scientific theory thus be
formulated in a language \(L(V\_o,V\_t)\). The division of the
descriptive vocabulary gives rise to a related distinction between
*T*- and *C*-axioms among the axioms of a scientific
theory. The *T*-axioms contain only \(V\_t\) symbols as
descriptive ones, while the *C*-axioms contain both \(V\_o\) and
\(V\_t\) symbols. The latter axioms establish a connection between the
theoretical and the observational terms. *TC* designates the
conjunction of *T*- and *C*-axioms and \(A(\TC)\) the
set of these axioms. Let \(n\_1 ,\ldots ,n\_k\) be the elements of
\(V\_o\) and \(t\_1 ,\ldots ,t\_n\) the elements of \(V\_t\). Then,
*TC* is a proposition of the following type:

 

\[\tag{\(\TC\)} \TC(n\_1 ,\ldots ,n\_k, t\_1 ,\ldots ,t\_n).\]


Ramsey ([1929] 1931) assumes that there is but one domain of
interpretation for all descriptive symbols. Carnap (1956, 1958), by
contrast, distinguishes between a domain of interpretation for
observational terms and another for theoretical terms. Notably, the
latter domain contains exclusively mathematical entities. Ketland
(2004) has emphasized the importance of distinguishing between an
observational and a theoretical domain of interpretation. *TC*
is a first-order sentence in Ramsey's seminal article
"Theories" ([1929] 1931). Carnap (1956; 1958), however,
works with higher-order logic to allow for the formulation of
mathematical propositions and concepts.


### 4.1 The Ramsey Sentence

 

The Ramsey sentence of a theory *TC* in the language \(L(V\_o,V\_t)\) is obtained by the following two transformations of the conjunction of *T*- and *C*-axioms. First, replace all theoretical symbols in this conjunction by higher-order variables of appropriate type. Second, bind these variables by higher-order existential quantifiers. As result one obtains a higher-order sentence of the following form: \[\tag{\(\TC^R\)} \exists X\_1 \ldots \exists X\_n \TC(n\_1 , \ldots ,n\_k, X\_1 , \ldots ,X\_n)\]


where \(X\_1 , \ldots ,X\_n\) are higher-order variables. This sentence
says that there is an extensional interpretation of the theoretical
terms that verifies, together with an antecedently given
interpretation of the observation language \(L(V\_o)\), the axioms
*TC*. The Ramsey sentence expresses an apparently weaker
proposition than *TC*, at least in standard truth-conditional
semantics. If one thinks that the Ramsey sentence expresses the
proposition of a scientific theory more properly than *TC*, one
holds the *Ramsey view* of scientific theories.



Why should one prefer the Ramsey view to the standard one? Ramsey
([1929] 1931: 231) himself seemed to have a contextual theory of
meaning in mind when proposing the replacement of theoretical terms
with appropriate higher-order variables:



> 
> 
> Any additions to the theory, whether in the form of new axioms or
> particular assertions like \(\alpha(0, 3)\) are to be made within the
> scope of the original \(\alpha\), \(\beta\), \(\gamma\). They are not,
> therefore, strictly propositions by themselves just as the different
> sentences in a story beginning 'Once upon a time' have not
> complete meanings and so are not propositions by themselves.
> 
> 
> 



\(\alpha\), \(\beta\), and \(\gamma\) figure in this explanation as
theoretical terms to be replaced by higher-order variables. Ramsey
goes on to suggest that the meaning of a theoretical sentence \(\phi\)
is the difference between


1. \((\TC \wedge A \wedge \phi)^R\)



and


2. \((\TC \wedge A)^R\)



where *A* stands for the set of observation sentences
being asserted and (...)\(^R\) for the operation of Ramsification,
i.e., existentially generalizing on all theoretical terms. This
proposal of expressing theoretical assertions clearly makes such
assertions dependent upon the context of the theory *TC*.
Ramsey ([1929] 1931: 235n) thinks that a theoretical assertion
\(\phi\) is not meaningful if no observational evidence can be found
for either \(\phi\) or its negation. In this case there is no stock
*A* of observation sentences such that (1) and (2)
differ in truth value.



Another important argument in favor of the Ramsey view was given later
by Sneed (1979: ch. III). It is easy to show that Sneed's
problem of theoretical terms (which concerns relations and functions)
does not arise in the first place on the Ramsey view. For \(\TC^R\)
only says that there are extensions of the theoretical terms such that
each axiom of the set \(A(\TC)\) is satisfied in the context of a
given interpretation of the observational language. No claim, however,
is made by \(\TC^R\) as to whether or not the sentences of \(A(\TC)\)
are true. Nonetheless, it can be shown that \(\TC^R\) and *TC*
have the same observational consequences:




**Proposition 1**. For all \(L(V\_o)\) sentences \(\phi\),
\(\TC^R\) \(\vdash \phi\) if and only if \(\TC \vdash \phi\), where
\(\vdash\) designates the relation of logical consequence in classical
logic.




Hence, the Ramsey sentence cannot be true in case the original theory
*TC* is not consistent with the observable facts. For a
discussion of empirical adequacy and Ramsification see Ketland
(2004).



One difficulty, however, remains with the Ramsey view. It concerns the
representation of deductive reasoning, for many logicians the primary
objective of logic. Now, Ramsey ([1929] 1931: 232) thinks that the
'incompleteness' of theoretical assertions does not affect
our reasoning. No formal account, however, is given that relates our
deductive practice, in which abundant use of theoretical terms is
made, to the existentially quantified variables in the Ramsey
sentence. We lack a translation of theoretical sentences (other than
the axioms) that is in keeping with the view that the meaning of a
theoretical sentence \(\phi\) is the difference between \((\TC \wedge
A \wedge \phi)^R\) and \((\TC \wedge A)^R\). As Ramsey observes, it
would not be correct to take \((\TC \wedge A \wedge \phi)^R\) as
translation of a theoretical sentence \(\phi\) since both \((\TC
\wedge A \wedge \phi)^R\) and \((\TC \wedge A \wedge \neg \phi)^R\)
may well be true. Such a translation would not obey the laws of
classical logic. These laws, however, are supposed to govern deductive
reasoning in science. A proper semantics of theoretical terms should
take the semantic peculiarities of these terms into account without
revising the rules of deductive reasoning in classical logic.



There thus remains the challenge of relating the apparent use of
theoretical terms in deductive scientific reasoning to the Ramsey
formulation of scientific theories. Carnap was well aware of this
challenge and addressed it using a sentence that became labeled later
on the *Carnap sentence* of a scientific theory (Carnap 1958;
1966: ch. 23):

 

\[ \tag{\(A\_T\)} \TC^R \rightarrow \TC.\]


This sentence is part of a proposal to draw the analytic-synthetic
distinction at the global level of a scientific theory (as this
distinction proved not to be applicable to single axioms): the
analytic part of the theory is given by its Carnap sentence \(A\_T\),
whereas the synthetic part is identified with the theory's
Ramsey sentence in light of Proposition 1. Carnap (1958) wants \(A\_T\)
to be understood as follows: if the Ramsey sentence is true, then the
theoretical terms be interpreted such that *TC* comes out true
as well. So, on condition of \(\TC^R\) being true, we can recover the
original formulation of the theory in which the theoretical terms
occur as constants. For, obviously, *TC* is derivable from
\(\TC^R\) and \(A\_T\) using Modus Ponens.



From the viewpoint of standard truth-conditional semantics, however,
this instruction to interpret the Carnap sentence appears arbitrary,
if not even misguided. For in standard semantics, the Ramsey sentence
may well be true without *TC* being so (cf. Ketland 2004).
Hence, the Carnap sentence would not count as analytic, as Carnap
intended. Carnap's interpretation of \(A\_T\) receives a sound
foundation in his (1961) proposal to define theoretical terms using
Hilbert's epsilon operator, as we shall see in
 Section 4.3.


### 4.2 Indirect Interpretation



The notion of an indirect interpretation was introduced by Carnap in
his *Foundations of Logic and Mathematics* (1939: ch.
23-24) with the intention of accounting for the semantics of
theoretical terms in physics. It goes without saying that this notion
is understood against the background of the notion of a direct
interpretation. Carnap had the following distinction in mind. The
interpretation of a descriptive symbol is direct if and only if (i) it
is given by an assignment of an extension or an intension, and (ii)
this assignment is made by expressions of the metalanguage. The
interpretation of a descriptive symbol is indirect, by contrast, if
and only if it is specified by one or several sentences of the object
language, which then figure as axioms in the respective calculus. Here
are two simple examples of a direct interpretation:




'*R*' designates the property of being
rational.



'*A*' designates the property of being an
animal.




The predicate '*H*', by contrast, is interpreted in
an indirect manner by a definition in the object language:



\[\forall x(Hx \leftrightarrow Rx \wedge Ax).\]


Interpretation of a symbol by a definition counts as one type of
indirect interpretation. Another type is the interpretation of
theoretical terms by the axioms of a scientific theory. Carnap (1939:
65) remains content with a merely syntactic explanation of indirect
interpretation:



> 
> 
> The calculus is first constructed floating in the air, so to speak;
> the construction begins at the top and then adds lower and lower
> levels. Finally, by the semantical rules, the lowest level is anchored
> at the solid ground of the observable facts. The laws, whether general
> or special, are not directly interpreted, but only the singular
> sentences.
> 
> 
> 



The laws \(A(\TC)\) are thus simply adopted as axioms in the calculus
without assuming any prior interpretation or reference to the world
for theoretical terms. (A sentence \(\phi\) being an axiom of a
calculus *C* means that \(\phi\) can be used in any formal
derivation in *C* without being a member of the premises.) This
account amounts to a formalist understanding of the theoretical
language in science. It has two particular merits. First, it
circumvents Sneed's problem of theoretical terms since the
axioms are not required to be true in the interpretation of the
respective language that represents the facts of the
theory-independent world. The need for assuming such an interpretation
is simply denied. Second, the account is in line with the contextual
theory meaning for theoretical terms as our understanding of such
terms is explained in terms of the axioms of the respective scientific
theories (cf.
 Section 1.2).



There are less formalist accounts of indirect interpretation in terms
of explicit model-theoretic notions by Przelecki (1969: ch. 6) and
Andreas
 (2010).[2]
 The latter account is based on ideas about theoretical terms in
Carnap (1958). It emerged from an investigation into the similarities
and dissimilarities between Carnapian postulates and definitions.
Recall that Carnap viewed the axioms of a scientific theory as
postulates since they contribute to the interpretation of theoretical
terms. When explaining the Carnap sentence \(\TC^R \rightarrow \TC\),
Carnap says that, if the Ramsey sentence is true, the theoretical
terms are to be understood in accordance with some interpretation that
satisfies *TC*. This is the sense in which we can say that
Carnapian postulates contribute to the interpretation of theoretical
terms in a manner akin to the interpretation of a defined term by the
corresponding definition. Postulates and definitions alike impose a
constraint on the admissible, or intended, interpretation of the
complete language \(L(V)\), where *V* contains basic
and indirectly interpreted terms.



Yet, the interpretation of theoretical terms by axioms of a scientific
theory differs in several ways from that of a defined term by a
definition. First, the introduction of theoretical terms may be joined
with the introduction of another, theoretical domain of
interpretation, in addition to the basic domain of interpretation in
which observation terms are interpreted. Second, it must not be
assumed that the interpretation of theoretical terms results in a
unique determination of the extension of these terms. This is an
implication of Carnap's doctrine of partial interpretation
(1958), as will become obvious by the end of this section. Third,
axioms of a scientific theory are not conservative extensions of the
observation language since they enable us to make predictions.
Definitions, by contrast, must be conservative (cf. Gupta 2009).
Taking these differences into account when observing the semantic
similarities between definitions and Carnapian postulates suggests the
following explanation: a set \(A(\TC)\) of axioms that interprets a
set \(V\_t\) of theoretical terms on the basis of a language \(L(V\_o)\)
imposes a constraint on the admissible, or intended, interpretations
of the language \(L(V\_o,V\_t)\). An \(L(V\_o,V\_t)\) structure is
*admissible* if and only if it (i) satisfies the axioms
\(A(\TC)\) and (ii) extends the intended interpretation of \(L(V\_o)\)
to include an interpretation of the theoretical terms.



In more formal terms (Andreas 2010: 373; Przelecki 1969: ch. 6):




**Definition 3 (Set \(\mathcal{S}\) of admissible
structures)**
  

Let \(\mathcal{A}\_o\) designate the intended interpretation of the
observation language. Further, \(\MOD(A(\TC))\) designates the set of
\(L(V\_o,V\_t)\) structures that satisfy the axioms \(A(\TC)\).
\(\EXT(\mathcal{A}\_o,V\_t,D\_t)\) is the set of \(L(V\_o,V\_t)\)
structures that extend \(\mathcal{A}\_o\) to interpret the theoretical
terms, where the domain of interpretation may be extended by a domain
\(D\_t\) of theoretical entities.


1. \(\mathcal{S} := \MOD(A(\TC)) \cap \EXT(\mathcal{A}\_o,V\_t,D\_t)\)
if \(\MOD(A(\TC)) \cap \EXT(\mathcal{A}\_o,V\_t,D\_t) \ne
\varnothing\);
2. \(\mathcal{S} := \EXT(\mathcal{A}\_o,V\_t,D\_t)\) if \(\MOD(A(\TC))
\cap \EXT(\mathcal{A}\_o,V\_t,D\_t) = \varnothing\).




Given there is a range of admissible, i.e., intended structures, the
following semantic rules for theoretical sentences suggest
themselves:




**Definition 4 (Semantics of theoretical sentences)**
  

\(\nu : L(V\_o,V\_t) \rightarrow \{T, F, I\}.\)


1. \(\nu(\phi) := T\) if and only if for all structures \(\mathcal{A}
\in \mathcal{S}, \mathcal{A} \vDash \phi\);
2. \(\nu(\phi) := F\) if and only if for all structures \(\mathcal{A}
\in \mathcal{S}, \mathcal{A} \not\vDash \phi\);
3. \(\nu(\phi) := I\) (indeterminate) if and only if there are
structures \(\mathcal{A}\_1, \mathcal{A}\_2 \in \mathcal{S}\) such that
\(\mathcal{A}\_1 \vDash \phi\) but not \(\mathcal{A}\_2 \vDash
\phi\).




These semantic rules are motivated by supervaluation logic (van
Fraassen 1969; Priest 2001: ch. 7). A sentence is true if and only if
it is true in every admissible structure. It is false, by contrast, if
and only if it is false in every admissible structure. And a sentence
does not have a determinate truth value if and only if it is true in,
at least, one admissible structure and false in, at least, another
structure that is also admissible.



A few properties of the present semantics are noteworthy. First, it
accounts for Carnap's idea that the axioms \(A(\TC)\) have a
twofold function, viz., setting forth empirical claims and determining
the meaning of theoretical terms (Carnap 1958). For, on the one hand,
the truth values of the axioms \(A(\TC)\) depend on empirical,
observable facts. These axioms, on the other hand, determine the
admissible interpretations of the theoretical terms. These two
seemingly contradictory properties are combined by allowing the axioms
\(A(\TC)\) to interpret theoretical terms only on condition of there
being a structure that both extends the given interpretation of the
observation language and that satisfies these axioms. If there is no
such structure, the theoretical terms remain uninterpreted. This
semantics, therefore, can be seen to formally work out the old
contextual theory of meaning for theoretical terms.



Second, Sneed's problem of theoretical terms
 (Section 3.2)
 does not arise in the present semantics since the formulation of this
problem is bound to standard truth-conditional semantics. Third, it is
closely related to the Ramsey view of scientific theories as the
following biconditional holds:




**Proposition 2**  \(\TC^R\) if and only if for all
\(\phi \in A(\TC), \nu(\phi) = T\).




Unlike the Ramsey account, however, the semantics of indirect
interpretation does not dispense with theoretical terms. It can rather
be shown that allowing for a range of admissible interpretations as
opposed to a single interpretation does not affect the validity of
standard deductive reasoning (Andreas 2010). Hence, a distinctive
merit of the indirect interpretation semantics of theoretical terms is
that theoretical terms need not be recovered from the Ramsey sentence
in the first place.



The label *partial interpretation* is more common in the
literature to describe Carnap's view that theoretical terms are
interpreted by the axioms or postulates of a scientific theory (Suppe
1974: 86-95). The partial character of interpretation is
retained in the present account since there is a range of admissible
interpretations of the complete language \(L(V\_o,V\_t)\). This allows
for the interpretation of theoretical terms to be strengthened by
further postulates, just as Carnap demanded in his 1958 and 1961. To
strengthen the interpretation of theoretical terms is to further
constrain the range of admissible interpretations of
\(L(V\_o,V\_t)\).



What happens if the axioms A(TC) are inconsistent or fail to be
empirically adequate? In other words, what happens if there is no
interpretation of the theoretical terms such that this interpretation
satisfies all the axioms and agrees with the antecedent interpretation
of the observation terms? In this case, the axioms of the respective
theory fail to interpret the theoretical terms. The set of admissible
structures, as defined by Definition 3, is empty. At the same time, we
think we have some understanding of the theoretical terms, even if the
respective theory fails to be fully consistent with the empirical
data. For example, we think we have some understanding of the
theretical terms in classical mechanics, even though classical
mechanics is not universally applicable. Moreover, we seem to have
inconsistent, yet non-trivial scientific theories, such as classical
electrodynamics.



Consequently, Andreas (2018) generalized the present semantics so as
to capture scientific reasoning in a paraconsistent setting. The
proposal is based on a preferred-models semantics: satisfaction of a
theory and agreement with empirical data comes in degrees. Some
interpretations score better than others. This idea leads to a strict
partial order of interpretations. And we understand the theoretical
terms in such a manner that the set of instances of all the axioms is
satisfied to a maximal extent in the context of a given interpretation
of the observation terms. This proposal parallels certain adaptive
logics, developed by Batens (2000) and Meheus et al. (2016). Notably,
it allows us to capture scientific reasoning with ceteris paribus
laws, i.e., laws that hold true most of the time but have exceptions.
This could be helpful for the analysis of theoretical terms in
scientific disciplines other than physics.



Relatively little research has been done on the semantics of
theoretical terms in theories from scientific disciplines other than
physics. It has remained an open question whether or not formal
semantics that were primarily developed to capture theories in physics
are applicable to theories in biology, chemistry, psychology,
economics, etc. For the structuralist framework by Balzer et al.
(1987) it could be shown that such applications are feasible (see
Balzer et al. 2000). Rakover (2020) and McClimans (2017) studied
theoretical terms in psychology, albeit without explicit
considerations of their formal semantics.


### 4.3 Direct Interpretation



Both the Ramsey view and the indirect interpretation semantics deviate
from standard truth-conditional semantics at the level of theoretical
terms and theoretical sentences. Such a deviation, however, was not
felt to be necessary by all philosophers that have worked on
theoretical concepts. Tuomela (1973: ch. V) defends a position that he
calls *semantic realism* and that retains standard
truth-conditional semantics. Hence, direct interpretation is assumed
for theoretical terms by Tuomela. Yet, semantic realism for
theoretical terms acknowledges there to be an epistemological
distinction between observational and theoretical terms.
Tuomela's (1973: ch. I) criterion of the theory-observation
distinction largely coincides with Sneed's above expounded
criterion. Since direct interpretation of theoretical terms amounts
just to standard realist truth conditions, there is no need for a
further discussion here.


### 4.4 Defining Theoretical Terms



In Weyl (1949), Carnap (1958), Feyerabend (1962) and a number of other
writings, we find different formulations of the idea that the axioms
of a scientific theory determine the meaning of theoretical terms
without these axioms qualifying as proper definitions of theoretical
terms. Lewis, however, wrote a paper with the title "How to
Define Theoretical Terms" (1970). A closer look at the
literature reveals that the very idea of explicitly defining
theoretical terms goes back to Carnap's (1961) use of
Hilbert's epsilon operator in scientific theories. This operator
is an indefinite description operator that was introduced by Hilbert
to designate some object *x* that satisfies an open
formula \(\phi\). So

 

\[\varepsilon x \phi(x)\]


designates some *x* satisfying \(\phi(x)\), where *x*
is the only free variable of \(\phi\) (cf. Avigad and
Zach 2002). Now, Carnap (1961: 161n) explicitly defines theoretical
terms in two steps:

 

\[\tag{\(A\_{T(0)}\)} \bar{t} = \varepsilon \bar{X} \TC(\bar{X}, n\_1 ,\ldots ,n\_k)\]


where \(\bar{X}\)is a sequence of higher-order variables and
\(\bar{t}\) a corresponding instantiation. So, \(\bar{t}\) designates
some sequence of relations and functions that satisfies *TC* in
the context of an antecedently given interpretation of \(V\_o\). Once
such a sequence has been defined via the epsilon-operator, the second
step of the definition is straightforward:

 

\[ \tag{\(A\_{T(i)}\)} t\_i = \varepsilon x (\exists u\_1 \ldots \exists u\_n (\bar{t} = \langle u\_1 ,\ldots ,u\_n\rangle \wedge x = u\_i)).\]


Carnap showed that these definitions imply the Carnap sentence
\(A\_T\). Hence, they allow for direct recovery of the theoretical
terms for the purpose of deductive reasoning on condition of the
Ramsey sentence being true.



Lewis (1970) introduced a number of modifications concerning both the
language of the Carnap sentence and its interpretation in order to
attain proper definitions of theoretical terms. First, theoretical
terms are considered to refer to individuals as opposed to relations
and functions. This move is made coherent by allowing the basic
language \(L(V\_o)\) to contain relations like '*x*
has property *y*'. The basic,
i.e., non-theoretical language is thus no observation language in this
account. Yet, it serves as the basis for introducing theoretical
terms. The set \(V\_o\) of 'O-terms' is best described as
our antecedently understood vocabulary.



Second, denotationless terms are dealt with along the lines of free
logic by Dana Scott (1967). An improper description, for example,
denotes nothing in the domain of discourse. Atomic sentences
containing denotationless terms are either true or false. An identity
statement that contains a denotationless term on both sides is always
true. If just one side of an identity formula has an occurrence of a
denotationless term, this identity statement is false.



Third, Lewis (1970) insists on a unique interpretation of theoretical
terms, thus rejecting Carnap's doctrine of partial
interpretation. Carnap (1961) is most explicit about the indeterminacy
that this doctrine implies. This indeterminacy of theoretical terms
drives Carnap to using Hilbert's \(\varepsilon\)-operator there,
as just explained. For Lewis, by contrast, a theoretical term is
denotationless if its interpretation is not uniquely determined by the
Ramsey sentence. For a scientific theory to be true, it must have a
unique interpretation.



Using these modifications, Lewis transforms the Carnap sentence into
three Carnap-Lewis postulates, so to speak:

 

\[ \begin{align\*} \tag{CL1} \exists y\_1 &\ldots \exists y\_n \forall x\_1 \ldots \forall x\_n \\ &(\TC(n\_1 ,\ldots ,n\_k, x\_1 ,\ldots ,x\_n) \leftrightarrow y\_1 = x\_1 \wedge \ldots \wedge y\_n = x\_n) \rightarrow \\ & \TC(n\_1 ,\ldots ,n\_k, t\_1 ,\ldots ,t\_k) \\ &\\ \tag{CL2} \,\,\neg \exists x\_1 &\ldots \exists x\_n \TC(n\_1 ,\ldots ,n\_k, x\_1 ,\ldots ,x\_n) \rightarrow \\ & \neg \exists x(x = t\_1) \wedge \ldots \wedge \neg \exists x(x = t\_n) \\ &\\ \tag{CL3} \exists x\_1 &\ldots \exists x\_n \TC(n\_1 ,\ldots ,n\_k, x\_1 ,\ldots ,x\_n) \wedge \mathord{}\\ &\neg \exists y\_1 \ldots \exists y\_n \forall x\_1 \ldots \forall x\_n \\ & (\TC(n\_1 ,\ldots ,n\_k, x\_1 ,\ldots ,x\_n) \leftrightarrow y\_1 = x\_1 \wedge \ldots \wedge y\_n = x\_n) \rightarrow \\ &\neg \exists x(x = t\_1) \wedge \ldots \wedge \neg \exists x(x=t\_n). \end{align\*}\]


These postulates look more difficult than they actually are. CL1 says
that, if *TC* has a unique realization, then it is realized by
the entities named by \(t\_1,\ldots,t\_k\). Realization of a theory
*TC*, in this formulation, means interpretation of the
descriptive terms under which *TC* comes out true, where the
interpretation of the \(V\_o\) terms is antecedently given. So, CL1 is
to be read as saying that the theoretical terms are to be understood
as designating those entities that uniquely realize *TC*, in
the context of an antecedently given interpretation of the \(V\_o\)
terms. CL2 says that, if the Ramsey sentence is false, the theoretical
terms do not designate anything. To see this, recall that \(\neg
\exists x(x=t\_i)\) means, in free logic, that \(t\_i\) is
denotationless. In case the theory *TC* has multiple
realizations, the theoretical terms are denotationless too. This is
expressed by CL3.



CL1-CL3 are equivalent, in free logic, to a set of sentences
that properly define the theoretical terms \(t\_i (1 \le i \le
n)\):

 

\[ \begin{align} \tag{\(D\_i\)} t\_i = \iota y\_i \exists y\_1 &\ldots \exists y\_{i-1} \exists y\_{i+1}\ldots \exists y\_n \forall x\_1 \ldots \forall x\_n \\ &(\TC(n\_1 , \ldots ,n\_k, x\_1 , \ldots ,x\_n) \leftrightarrow \\ & y\_1 = x\_1 \wedge \ldots \wedge y\_i = x\_i \wedge \ldots \wedge y\_n = x\_n). \end{align} \]


\(t\_i\) designates, according to this definition schema, the
*i*-th component in that sequence of entities that uniquely
realizes *TC*. If there is no such sequence, \(t\_i (1 \le i \le
n)\) is denotationless. But even if a theoretical term \(t\_i\) fails
to have a denotation, the definition \(D\_i\) of this term remains true
as long as the complete language \(L(V\_o,V\_t)\) is interpreted in
accordance with the postulates CL1-CL3, thanks to the use of
free logic.



A few further properties of Lewis's definitions of theoretical
terms are noteworthy. First, they specify the interpretation of
theoretical terms uniquely. This property is obvious for the case of
unique realization of *TC* but holds as well for the other
cases since assignment of no denotation counts as interpretation of a
descriptive symbol in free logic. Second, it can be shown that these
definitions do not allow for the derivation of any \(L(V\_o)\)
sentences except logical truths, just as the original Carnap sentence
did. Lewis, therefore, in fact succeeds in *defining*
theoretical terms. He does so without attempting to divide the axioms
\(A(\TC)\) into definitions and synthetic claims about the
spatiotemporal world.



The replacement of theoretical relation and function symbols with
individual terms was judged counterintuitive by Papineau (1996). A
reformulation, however, of Lewis's definitions using second- or
higher-order variables is not difficult to accomplish, as Schurz
(2005) has shown. In this reformulation the problem arises that
theoretical terms are usually not uniquely interpreted since our
observational evidence is most of the time insufficient to determine
the extension of theoretical relation and function symbols completely.
Theoretical functions, such as temperature, pressure, electromagnetic
force etc., are determined only for objects that have been subjected
to appropriate measurements, however indirect. In view of this
problem, Schurz (2005) suggests letting the higher-order quantifiers
range only over those extensions that correspond to *natural
kind* properties. This restriction renders the requirement of
unique interpretation of theoretical terms plausible once again. Such
a reading was also suggested by Psillos (1999: ch. 3) with reference
to Lewis's (1984) discussion of Putnam's (1980)
model-theoretic argument. In that paper, Lewis himself suggests the
restriction of the interpretation of descriptive symbols to extensions
corresponding to natural kind properties.



One final note on indirect interpretation is in order. Both Carnap
(1961) and Lewis (1970) interpret theoretical terms indirectly simply
because any definition is an instance of an indirect interpretation.
For this reason, Sneed's problem of theoretical terms
 (Section 3.2)
 does not arise. Yet, the pattern of Carnap's and Lewis's
proposals conforms to the pattern of a definition in the narrow sense
and not to the peculiar pattern of indirect interpretation that Carnap
(1939) envisioned for the interpretation of theoretical terms. This is
why the indirect interpretation semantics has been separated from the
present discussion of defining theoretical terms.


## 5. Conclusion



The idea that there are scientific terms whose meaning is determined
by a scientific theory goes back to Duhem and Poincare. Such
terms came to be referred to as *theoretical terms* in
twentieth century philosophy of science. Properties and entities that
are observable in the sense of direct, unaided perception did not seem
to depend on scientific theories as forces, electrons and nucleotides
did. Hence, philosophers of science and logicians set out to
investigate the semantics of theoretical terms. Various formal
accounts resulted from these investigations, among which the Ramsey
sentence by Ramsey ([1929] 1931), Carnap's notion of indirect
interpretation (1939; 1958) and Lewis's (1970) proposal of
defining theoretical terms are the most prominent ones. Though not all
philosophers of science understand the notion of a theoretical term in
such way that semantic dependence upon a scientific theory is
essential, this view prevails in the literature.



The theory-observation distinction has been attacked heavily and is
presumably discredited by a large number of philosophers of science.
Still, this distinction continues to permeate a number of important
strands in the philosophy of science, such as scientific realism and
its alternatives and the logical analysis of scientific theories. A
case in point is the recent interest in the Ramsey account of
scientific theories which emerged in the wake of Worral's
structural realism (cf. Ladyman 2009). We have seen, moreover, that
the formal accounts of theoretical terms work well with a
theory-observation distinction that is relativized to a particular
theory. Critics of that distinction, by contrast, have commonly
attacked a global and static division into theoretical and
observational terms (Maxwell 1962; Achinstein 1965). Note finally that
Carnap assigned no ontological significance to the theory-observation
distinction in the sense that entities of the one type would be
existent in a more genuine way than ones of the other.