A simple explanation of theoreticity says that a term is theoreticalif and only if it refers to nonobservational entities. Paradigmaticexamples of such entities are electrons, neutrinos, gravitationalforces, genes etc. There is yet another explanation of theoreticity: atheoretical term is one whose meaning becomes determined through theaxioms of a scientific theory. The meaning of the term‘force’, for example, is seen to be determined byNewton’s laws of motion and further laws about special forces,such as the law of gravitation. Theoreticity is a property that iscommonly applied to both expressions in the language of science, andreferents and concepts of such expressions. Objects, relations andfunctions as well as concepts thereof may thus qualify as theoreticalin a derived sense.

Several semantics have been devised that aim to explain how ascientific theory contributes to the interpretation of its theoreticalterms and as such determines what they mean and how they areunderstood. All of these semantics assume the respective theory to begiven in an axiomatic fashion. Yet, theoretical terms are alsorecognizable in scientific theories which have as yet resisted asatisfying axiomatization.

Theoretical terms pertain to a number of topics in the philosophy ofscience. A fully fledged semantics of such terms commonly involves astatement about scientific realism and its alternatives. Such asemantics, moreover, may involve an account of how observation isrelated to theory in science. All formal accounts of theoretical termsdeny the analytic-synthetic distinction to be applicable to the axiomsof a scientific theory. The recognition of theoretical terms in thelanguage of science by Carnap thus amounts to a rejection of anessential tenet of early logical empiricism and positivism, viz., thedemonstration that all empirically significant sentences aretranslatable into an observation language. The present articleexplains the principal distinction between observational andtheoretical terms, discusses important criticisms and refinements ofthis distinction and investigates two problems concerning thesemantics of theoretical terms. Finally, the major formal accounts ofthis semantics are expounded.

• 1. Two Criteria of Theoreticity
  • 1.1 Reference to Nonobservable Entities and Properties
  • 1.2 Semantic Dependence upon a Scientific Theory
• 2. Criticisms and Refinements of the Theory-Observation Distinction
  • 2.1 Criticisms
  • 2.2 Refinements
• 3. Two Problems of Theoretical Terms
  • 3.1 Theoretical Entities
    • 3.1.1 The realist view
    • 3.1.2 Non-realist views
    • 3.1.3 The Pythagorean view
  • 3.2 Theoretical Functions and Relations
• 4. Formal Accounts
  • 4.1 The Ramsey Sentence
  • 4.2 Indirect Interpretation
  • 4.3 Direct Interpretation
  • 4.4 Defining Theoretical Terms
• 5. Conclusion
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Two Criteria of Theoreticity

1.1 Reference to Nonobservable Entities and Properties

As just explained, a theoretical term may simply be understood as anexpression that refers to nonobservable entities or properties.Theoreticity, on this understanding, is the negation of observability.This explanation of theoreticity thus rests on an antecedentunderstanding of observability. What makes an entity or propertyobservable? As Carnap (1966: ch. 23) has pointed out, a philosopherunderstands the notion of observability in a narrower sense than aphysicist. For a philosopher, a property is observable if it can be‘directly perceived by the senses’. Hence, such propertiesas ‘blue’, ‘hard’ and ‘colderthan’ are paradigmatic examples of observable properties in thephilosopher’s understanding of observability. The physicist, bycontrast, would also count quantitative magnitudes that can bemeasured in a ‘relatively simple, direct way’ asobservable. 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, perceptionunaided by technical instruments and, second, that the perception isunaided by inferences. These two conditions are obviously notsatisfied for the measurement of quantities like temperature andpressure. For the philosopher, only spatial positions of liquids andpointers are observed when these quantities are measured. To an evenhigher degree, we are unable to observe electrons, molecules,gravitational forces and genes on this narrow understanding ofobservability. Hence, expressions referring to such entities qualifyas theoretical.

In sum, a property or object is observable—in thephilosopher’s sense—if it can be perceived directly, wheredirectness of observation excludes the use of technical artifacts andinferences. Notably, Carnap (1936/37: 455; 1966: 226) admits that hisexplanation of the distinction is not sufficiently precise todetermine a sharp line between observational and theoretical terms. Herather views the theory-observation distinction as being introducedinto a ‘continuum of degrees of observability’ by choice.Prominent criticisms of the theory-observation distinction will bediscussed inSection 2.1.

1.2 Semantic Dependence upon a Scientific Theory

The above explanation of theoreticity may be felt unsatisfactory as itdetermines the property of being theoretical only via negation of theproperty of being observable (Putnam 1962). This explanation does notindicate any specific connection between the semantics of theoreticalterms and corresponding scientific theories. There is, however, also adirect characterization of theoreticity that complements the criterionof non-observability: an expression is theoretical if and only if itsmeaning is determined through the axioms of a scientific theory. Thisexplanation rests on what has come to be referred to as thecontextual theory of meaning, which says that the meaning ofa scientific term depends, in some way or other, on how this term isincorporated 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 theFregean notion of sense. The sense of a term be understood as thatwhat determines its reference (cf. Church 1956: 6n). It is,furthermore, a reasonable requirement that a semantic theory mustaccount for our understanding of the sense and, hence, our methods ofdetermining the extension of scientific terms (cf. Dummett 1991: 340).For a large number of scientific terms these methods rest upon axiomsof one or more scientific theories. There is no way of determining theforce function in classical mechanics without using some axiom of thistheory. Familiar methods make use of Newton’s second law ofmotion, Hooke’s law, the law of gravitation etc. Likewise,virtually all methods of measuring temperature rest upon laws ofthermodynamics. Take measurement by a gas thermometer which is basedon the ideal gas law. The laws of scientific theories are thusessential to our methods of determining the extension of scientificterms. The contextual theory of meaning, therefore, makes intelligiblehow students in a scientific discipline and scientists grasp themeaning, or sense, of scientific terms. On this account, understandingthe sense of a term means knowing how to determine its referent, orextension, at least in part.

The contextual theory of meaning can be traced back at least to thework of Duhem. His demonstration that a scientific hypothesis inphysics cannot be tested in isolation from its theoretical context isjoined with and motivated by semantic considerations, according towhich it is physical theories that give meaning to the specificconcepts of physics (Duhem 1906: 183). Poincaré (1902: 90)literally claims that certain scientific propositions acquire meaningonly by virtue of the adoption of certain conventions. Perhaps themost prominent and explicit formulation of the contextual theory ofmeaning 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 isdependent upon the way in which the term has been incorporated into atheory, in the very same manner the content of a whole theory (andthereby again the meaning of the descriptive terms which it contains)depends upon the way in which it is incorporated into both the set ofits empirical consequences and the set of all the alternatives whichare being discussed at a given time: once the contextual theory ofmeaning has been adopted, there is no reason to confine itsapplication to a single theory, especially as the boundaries of such alanguage or of such a theory are almost never well defined.

The accounts of a contextual theory of meaning in the works of Duhem,Poincaré and Feyerabend are informal insofar as they do notcrystallize into a corresponding formal semantics for scientificterms. Such a crystallization is brought about by some of the formalaccounts of theoretical terms to be expounded inSection 4.

The view that meaning is bestowed upon a theoretical term through theaxioms of a scientific theory implies that only axiomatized oraxiomatizable scientific theories contain theoretical terms. In fact,all formal accounts of the semantics of theoretical terms are devisedto apply to axiomatic scientific theories. This is due, in part, tothe fact that physics has dominated the philosophy of science for along time. One must wonder, therefore, whether there are anytheoretical terms in, for example, evolutionary biology which has asyet resisted complete axiomatization. Arguably, there are. Even thoughevolutionary biology has not yet been axiomatized, we can recognizegeneral propositions therein that are essential to determining certainconcepts of this theory. Consider the following two propositions. (i)Two DNA sequences are homologous if and only if they have a commonancestor sequence. (ii) There is an inverse correlation between thenumber of mutations necessary to transform one DNA-sequence \(S_1\)into another \(S_2\) and the likelihood that \(S_1\) and \(S_2\) arehomologous. 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 thanthose of physics, however, have instances that fail to be true. (Somephilosophers of science have argued that this so even for a largenumber 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 hasreceived much criticism. First, with the help of sophisticatedinstruments, such as telescopes and electron microscopes, we are ableto observe more and more entities, which had to be consideredunobservable at a previous stage of scientific and technicalevolution. Electrons and \(\alpha\)-particles which can be observed ina cloud chamber are a case in point (Achinstein 1965). Second, assumeobservability is understood as excluding the use of instruments. Onthis understanding, examples drawing on the use of cloud chambers andelectron microscopes, which are adduced to criticize thetheory-observation distinction, can be dealt with. However, we wouldthen have to conclude that things being perceived with glasses are notobserved either, which is counterintuitive (Maxwell 1962). Third,there are concepts applying to or being thought to apply to bothmacroscopic and submicroscopic particles. A case in point are spatialand temporal relations and the color concepts that play an importantrole in Newton’s corpuscle theory of light. Hence, there areclear-cut instances of observation concepts that apply to unobservableentities, which does not seem acceptable (cf. Putnam 1962).

These objections to the theory-observation distinction can be answeredin a relatively straightforward manner from a Carnapian perspective.As explained inSection 1.1, Carnap (1936/37, 1966) was quite explicit that thephilosopher’s sense of observation excludes the use ofinstruments. As for an observer wearing glasses, a proponent of thetheory-observation distinction finds enough material in Carnap(1936/37: 455) to defend her position. Carnap is aware of the factthat color concepts are not observable ones for a color-blind person.He is thus prepared to relativize the distinction in question. Infact, Carnap’s most explicit explanation of observabilitydefines this notion in such a way that it is relativized to anorganism (1936/37: 454n).

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

There is no question here of who [the physicist thinking thattemperature is observable or the philosopher who disagrees, H. A.] isusing the term ‘observable’ in the right or proper way.There is a continuum which starts with direct sensory observations andproceeds to enormously complex, indirect methods of observation.Obviously no sharp line can be drawn across this continuum; it is amatter of degree.

A bit more serious is Putnam’s (1962) objection drawing on theapplication of apparently clear-cut instances of observation conceptsto submicroscopic particles. Here, Carnap would have to distinguishbetween color concepts applying to observable entities and relatedcolor concepts applying to unobservable ones. So, the formal languagein which the logical analysis proceeds would have to contain apredicate ‘red\(_1\)’ applying to macroscopic objects andanother one ‘red\(_2\)’ applying to submicroscopic ones.Again, such a move would be in line with the artificial, or ideallanguage philosophy that Carnap advocated (see Lutz (2012) for asympathetic discussion of artificial language philosophy).

There is another group of criticisms coming from the careful study ofthe history of science: Hanson (1958), Feyerabend (1962) and Kuhn(1962) aimed to show that observation concepts aretheory-laden in a manner that makes their meaningtheory-dependent. Feyerabend’s (1978: 32), for example, heldthat all terms are theoretical. Hanson (1958: 18) thinks that Tychoand Kepler were (literally) ‘seeing’ different things whenperceiving the sun rising because their astronomical backgroundtheories were different. Kuhn (1962) was more tentative whenexpounding his variant of the theory-ladenness of observation. In adiscussion of the Sneed formalism of the structuralist school, hefavored a theory-observation distinction that is relativized, first toa theory and second to an application of this theory (Kuhn 1976).

Virtually all formal accounts of theoretical terms in fact assume thatthose phenomena which a theoryT is meant to account for canbe described in terms of expressions whose meaning does not depend onT. The counter-thesis that the meaning of putativeobservation 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 andassessment of theory-ladenness of observation in the works of thegreat historians of science is beyond the scope of this entry. Bird(2004), Bogen (2012) and Oberheim and Hoyningen-Huene (2009) areentries in the present encyclopedia that address, among other things,this issue. Schurz (2013: ch. 2.9) defines a criterion of the theoryindependence of observation in terms of an ostensive learningexperiment, and shows how such a criterion helps answer the challengesof theory-ladenness of observation.

2.2 Refinements

There is a simple, intuitive and influential proposal how torelativize the theory-observation distinction in a sensible way: atermt is theoretical with respect to a theoryT, orfor short, aT-term if and only if it is introduced by thetheoryT at a certain stage in the history of science.O-terms, by contrast, are those that were antecedently available andunderstood beforeT was set forth (Lewis 1970; cf. Hempel1973). This proposal draws the theory-observation distinction in anapparently sharp way by means of relativizing that distinction to aparticular theory. Needless to say, the proposal is in line with thecontextual theory of meaning.

The distinction betweenT-terms and antecedently availableO-terms has two particular merits. First, it circumvents the view thatany sharp line between theoretical and observational terms isconventional and arbitrary. Second, it connects the theory-observationdistinction with what seemed to have motivated that distinction in thefirst place, viz., an investigation how we come to understand themeaning of terms that appear to be meaningful in virtue of certainscientific theories.

A similar proposal of a relativized theory-observation distinction wasmade by Sneed in his seminalThe Logical Structure of MathematicalPhysics (1979: ch. II). Here is a simplified and more syntacticformulation of Sneed’s criterion ofT-theoreticity:

It remains to explain what it is for a methodm ofdetermining the extension oft to rest upon an axiom\(\phi\). This relation obtains if and only if the use ofmdepends on \(\phi\) being a true sentence. In otherwords,m rests upon \(\phi\) if and only if thehypothetical assumption of \(\phi\) being false or indeterminate wouldinvalidate the use ofm in the sense that we wouldbe lacking the commonly presumed justification for usingm.The qualification ‘or some part of thatextension’ has been introduced in the present definition becausewe cannot expect a single measurement method to determine theextension of a scientific quantity completely.T-non-theoreticity is the negation ofT-theoreticity:

The concepts of classical particle mechanics (henceforth abbreviatedby CPM) exemplify well the notions ofT-theoreticity andT-non-theoreticity. As has been indicated above, all methodsof determining the force acting upon a particle make use of some axiomof classical particle mechanics, such as Newton’s laws of motionor some law about special forces. Hence, force is CPM-theoretical.Measurement of spatial distances, by contrast, is possible withoutusing axioms of CPM. Hence, the concept of spatial distance isCPM-non-theoretical. The concept of mass is less straightforward toclassify as we can measure this concept using classical collisionmechanics (CCM). Still, it was seen to be CPM-theoretical by thestructuralists since CCM appeared reducible to CPM (Balzer et al.1987: ch. 2).

Suppose for a termt once introduced by a scientific theory\(T_1\) novel methods of determination become established throughanother theory \(T_2\), where these methods do not depend on any axiomof \(T_1\). Then,t would neither qualify as\(T_1\)-theoretical nor as \(T_2\)-theoretical. It is preferable, inthis situation, to relativize Definition 1 to theory-netsN,i.e., compounds of several theories. Whether thereare 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 andintended applications, rather technical notions of thestructuralist approach to scientific theories. There has beena lively discussion, mainly but not exclusively within thestructuralist school, how to express the relativized notion oftheoreticity most properly (Balzer 1986; 1996). As noted above, Kuhn(1976) proposed a twofold relativization of theoreticity, viz., firstto a scientific theory and second to applications of suchtheories.

Notably, Sneed’s criterion ofT-theoreticity suggests astrategy that allows us to regain a global, non-relativizedtheory-observation distinction: simply take a termt to betheoretical if and only if it holds, for all methodsmof determining its extension, thatmrests upon some axiom of some theoryT. Atermt is non-theoretical, or observational, if and only ifthere are means of determining its extension, at least in part, thatdo not rest upon any axiom of any theory. This criterion is stillrelative to our present stage of explicit theories but comes closer tothe original intention of Carnap’s theory-observationdistinction, according to which observation is understood in thenarrow sense of unaided perception.

3. Two Problems of Theoretical Terms

The problem of theoretical terms is a recurrent theme in thephilosophy of science literature (Achinstein 1965; Sneed 1979: ch. II;Tuomela 1973: ch. V; Friedman 2011). Different shades of meaning havebeen associated with this problem. In its most comprehensiveformulation, the problem of theoretical terms is to give a properaccount of the meaning and reference of theoretical terms. There areat least two kinds of expression that pose a distinct problem oftheoretical terms, respectively. First, unary predicates referring totheoretical entities, such as ‘electron’,‘neutrino’ and ‘nucleotide’. Second, non-unarytheoretical predicates, such as ‘homology’ in evolutionarybiology and theoretical function expressions, such as‘force’, ‘temperature’ and ‘intensity ofthe electromagnetic field’ in physics. Sneed’s problem oftheoretical terms, as expounded in (1979: ch. II), concerns only thelatter kind of expression. We shall now start surveying problemsconcerning the semantics of expressions for theoretical entities andthen move on to expressions for theoretical relations andfunctions.

3.1 Theoretical Entities

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

For the realist, theoretical entities exist independently of ourtheories about the world. Also, natural kinds that classify theseentities exist independently of our theories (cf. Psillos 1999; Lewis1984). The instrumentalist picture is commonly reported to account fortheoretical entities in terms of mere fictions. The formalist variantof instrumentalism denies that theoretical terms have referents in thefirst place. Between these two extreme cases there is a number ofintermediate positions.^[1]

Carnap (1958; 1966: ch. 26) attempted to attain a metaphysicallyneutral position so as to avoid any commitment to or denial ofscientific realism. In his account of the theoretical language ofscience, theoretical entities were conceived of as mathematicalentities that are related to observable events in certain determinateways. An electron, for example, figures as a certain distribution ofcharge and mass in a four-dimensional manifold of real numbers, wherecharge and mass are real-valued functions. These functions and thefour-dimensional manifold itself are to be related to observableevents by means of universal axioms. Notably, Carnap would not haveaccepted a characterization of his view as antirealist or non-realistsince he thought the metaphysical doctrine of realism to be void ofcontent.

There thus are different metaphysical views of theoretical entities inscience, each of which is consistent with the understanding that atheoretical 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 orother. Third, they are conceived of as mathematical entities that arerelated to the observable world in certain determinate ways. We maythus distinguish between (i) a realist view, (ii) a collection ofnon-realist views and (iii) a Pythagorean view of theoreticalentities.

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

3.1.1 The realist view

The descriptivist picture is highly intuitive with regard to ourunderstanding of expressions referring to theoretical entities on therealist view. According to this picture, an electron is aspatiotemporal entity with such and such a mass and such and such acharge. We detect and recognize electrons when identifying entitieshaving these properties. The descriptivist explanation of meaning andreference makes use of theoretical functions, mass and electric chargein the present example. The semantics of theoretical entities,therefore, is connected with the semantics of theoretical relationsand functions, which will be dealt with in the next subsection. Itseems to hold, in general, that theoretical entities in the sciencesare to be characterized in terms of theoretical functions andrelations.

The descriptivist account, however, faces two particular problems withregard to the historic evolution of scientific theories. First, ifdescriptions of theoretical entities are constitutive of the meaningof corresponding unary predicates, one must wonder what the commoncore of understanding is that adherents of successive theories shareand whether there is such a core at all. Were Rutherford and Bohrtalking about the same type of entities when using the expression‘electron’? Issues of incommensurability arise with thedescriptivist picture (Psillos 1999: 280). A second problem ariseswhen elements of the description of an entity in a given theoryT are judged wrong from the viewpoint of a successor theory\(T'\). Then, on a strict reading of the descriptivist account, thecorresponding theoretical term failed to refer inT. For ifthere is nothing that satisfies a description, the correspondingexpression has no referent. This is a simple consequence of the theoryof description by Russell in his famous “On Denoting”(1905). Hence, an account of weighting descriptions is needed in orderto circumvent such failures of reference.

As is well known, Kripke (1980) set forth a causal-historical accountof reference as an alternative to the descriptivist picture. Thisaccount begins with an initial baptism that introduces a name and goeson with causal chains transmitting the reference of the name fromspeaker to speaker. In this picture, Aristotle is the man oncebaptized so; he might not have been the student of Plato or done anyother thing commonly attributed to him. Kripke thought this picture toapply both to proper names and general terms. It is hardly indicated,however, how this picture works for expressions referring totheoretical entities (cf. Papineau 1996). Kripke’s story isparticularly counterintuitive in view of the ahistorical manner ofteaching in the natural sciences. There, up-to-date textbooks andrecent journal articles are more important than the original,historical introduction of a theoretical term. The Kripkean causalstory may also be read as an account of reference fixing without beingread as a story of grasping the meaning of theoretical terms.Reference, however, needs to be related to meaning so as to ensurethat scientists know what they are talking about and are able toidentify the entities under investigation. Notably, even forexpressions of everyday language, the charge of not explaining meaninghas been leveled against Krikpe’s causal-historical account(Reimer 2010). The same charge applies to Putnam’s (1975) causalaccount of reference and meaning, which Putnam himself abandoned inhis (1980).

A purely causal or causal-historical account of reference does notseem a viable option for theoretical terms. More promising are hybridaccounts that combine descriptivist intuitions with causal elements.Such an account has been given by Psillos (1999: 296):

1. A termt refers to an entityx if andonly ifx satisfies the core causal descriptionassociated witht.
2. Two terms \(t'\) andt denote the same entity if and onlyif (a) their putative referents play the same causal role with respectto a network of phenomena; and (b) the core causal description of\(t'\) takes up the kind-constitutive properties of the core causaldescription associated witht.

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

3.1.2 Non-realist views

Non-realist and antirealist semantics for theoretical terms aremotivated by the presumption that the problem of theoretical terms hasno satisfying realist solution. What does a non-realist semantics oftheoretical terms look like? The view that theoretical entities aremere fictions often figures in realist portrays of antirealism. Thisview is hardly seriously maintained by any philosopher of science inthe twentieth century. If one were to devise a formal or informalsemantics for the view that theoretical entities are mere fictions, apurely descriptive account seems most promising. Such an account couldin particular make heavy use of the Fregean notion of sense. For oneimportant motivation for introducing this notion is to explain ourunderstanding of expressions like ‘Odysseus’ and‘Pegasus’.

Formalist variants of instrumentalism are a more serious alternativeto realist semantics than the fiction view of theoretical entities.Formalist views in the philosophy of mathematics aim to account formathematical concepts and objects in terms of syntactic entities andoperations thereupon within a calculus. Such views have been carriedover to theoretical concepts and objects in the natural sciences, withthe qualification that the observational part of the calculus isinterpreted in such a way that its symbols refer to physical orphenomenal objects. Cognitive access to theoretical entities is thusexplained in terms of our cognitive access to the symbols and rules ofthe calculus in the context of an antecedent understanding of theobservation terms. Formalist ideas were sympathetically entertained byHermann Weyl (1949). He was driven towards such ideas by adherence toHilbert’s distinction between real and ideal elements and thecorresponding distinction between real and ideal propositions (Hilbert1926). Propositions of the observation language were construed as realin the sense of this Hilbertian distinction, while theoreticalpropositions are construed as ideal. The content of an idealproposition is understood in terms of the (syntactic) consistency ofthe whole system consisting of ideal and real propositions beingasserted.

3.1.3 The Pythagorean view

There remains to discuss the view that theoretical entities aremathematical entities which are related to observable events incertain determinate ways. This theory is clearly of the descriptivisttype, as we shall see more clearly when dealing with the formalaccount by Carnap inSection 4. No causal elements are needed in Carnap’s Pythagoreanempiricism.

The Pythagorean view shifts the problem of theoretical terms to thetheory of meaning and reference for mathematical expressions. Thequestion of how we are able to refer successfully to electrons isanswered by the Pythagorean by pointing out that we are able to refersuccessfully to mathematical entities. Moreover, the Pythagoreanexplains, it is part of the notion of an electron that correspondingmathematical entities are connected to observable phenomena by meansof axioms and inference rules. The empirical surplus of theoreticalentities in comparison to “pure” mathematical entities isthus captured by axioms and inference rules that establish connectionsto empirical phenomena. Since mathematical entities do not, bythemselves, have connections to observable phenomena, the question oftruth and falsehood may not be put in a truth-conditional manner forthose axioms that connect mathematical entities with phenomenal events(cf.Section 4.2). Carnap (1958), therefore, came to speak ofpostulates whenreferring 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 aclassical paper that addresses this problem see Benacerraf (1973)).Carnap has not much to say about meaning and reference of mathematicalexpressions in his seminal “The Methodological Character ofTheoretical Concepts” (1956) but discusses these issues in his“Empiricism, Semantics, and Ontology” (1950). There heaims at establishing a metaphysically neutral position that avoids acommitment to Platonist, nominalist or formalist conceptions ofmathematical objects. A proponent of the Pythagorean view other thanCarnap is Hermann Weyl (1949). As for the cognition of mathematicalentities, Weyl largely followed Hilbert’s formalism in his laterwork. Hence, there is a non-empty intersection between the Pythagoreanview and the formalist view of theoretical entities. Unlike Carnap,Weyl did not characterize the interpretation of theoretical terms bymeans of model-theoretic notions.

3.2 Theoretical Functions and Relations

For theoretical functions and relations, a particular problem arisesfrom the idea that a theoretical term is, by definition, semanticallydependent upon a scientific theory. Let us recall the aboveexplanation ofT-theoreticity: a termt isT-theoretical if and only if any method of determining theextension oft, or some part of that extension, rests uponsome axiom ofT. Let \(\phi\) be such an axiom andmbe a corresponding method of determination. Thepresent explanation ofT-theoreticity, then, means thatmis valid only on condition of \(\phi\) being true.The latter dependency holds because \(\phi\) is used either explicitlyin calculations to determinet or in the calibration ofmeasurement devices. Such devices, then, perform the calculationimplicitly. A case in point is measurement of temperature by a gasthermometer. Such a device rests upon the law that changes oftemperature result in proportional changes in the volume of gases.

Suppose nowt is theoretical with respect to a theoryT. Then it holds that in order to measuret, we needto assume the truth of some axiom \(\phi\) ofT. Suppose,further, thatt has occurrences in \(\phi\), as is standardin examples ofT-theoreticity. From this it follows that, instandard truth-conditional semantics, the truth value of \(\phi\) isdependent on the semantic value oft. This leads to thefollowing epistemological problem: on the one hand, we need to knowthe extension oft in order to find out whether \(\phi\) istrue. On the other hand, it is simply impossible to determine theextension oft without using \(\phi\) or some other axiom ofT. This mutual dependency between the semantic values of\(\phi\) andt makes it difficult, if not even impossible, tohave evidence for \(\phi\) being true in any of its applications (cf.Andreas 2008).

We could, of course, use an alternative measurement method oft, say one resting upon an axiom \(\psi\) ofT, togain evidence for the axiom \(\phi\) being true in some selectedinstances. This move, however, only shifts the problem to applicationsof another axiom ofT. For these applications the same typeof difficulty arises, viz., mutual dependency of the semantic valuesof \(\psi\) andt. We are thus caught either in a viciouscircle or in an infinite regress when attempting to gain evidence forthe propriety of a single measurement of a theoretical term. Sneed(1979: ch. II) was the first to describe that particular difficulty inthe present manner and termed itthe problem of theoreticalterms. Measurement of the force function in classical mechanicsexemplifies this problem well. There is no method of measuring forcethat does not rest upon some law of classical mechanics. Likewise, itis impossible to measure temperature without using some law thatdepends upon either phenomenological or statisticalthermodynamics.

Although its formulation is primarily epistemological, Sneed’sproblem of theoretical terms has a semantic reading. Let the meaningof a term be identified with the methods of determining its extension,as suggested inSection 1.2. Then we can say that our understanding ofT-theoreticalrelations and functions originates from the axioms of the scientifictheoryT. In standard truth-conditional semantics, bycontrast, one assumes that the truth value of an axiom is determinedby the semantic values of those descriptive symbols which haveoccurrences in \(\phi\). Among these symbols, there are theoreticalterms ofT. Hence, it appears that standard truth-conditionalsemantics does not accord with the order of our grasping the meaningof theoretical terms. In the next section, we will deal with indirectmeans of interpreting theoretical terms. These proved to be ways outof the present problem of theoretical terms.

4. Formal Accounts

A few notational conventions and preliminary considerations arenecessary to explain the formal accounts of theoretical terms.Essential to all of these accounts is the division of the set ofdescriptive symbols into a set \(V_o\) of observational and anotherset \(V_t\) of theoretical terms. (The descriptive symbols of a formallanguage are simply the non-logical ones.) A scientific theory thus beformulated in a language \(L(V_o,V_t)\). The division of thedescriptive vocabulary gives rise to a related distinction betweenT- andC-axioms among the axioms of a scientifictheory. TheT-axioms contain only \(V_t\) symbols asdescriptive ones, while theC-axioms contain both \(V_o\) and\(V_t\) symbols. The latter axioms establish a connection between thetheoretical and the observational terms.TC designates theconjunction ofT- andC-axioms and \(A(\TC)\) theset 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:

Ramsey ([1929] 1931) assumes that there is but one domain ofinterpretation for all descriptive symbols. Carnap (1956, 1958), bycontrast, distinguishes between a domain of interpretation forobservational terms and another for theoretical terms. Notably, thelatter domain contains exclusively mathematical entities. Ketland(2004) has emphasized the importance of distinguishing between anobservational and a theoretical domain of interpretation.TCis 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 ofmathematical propositions and concepts.

4.1 The Ramsey Sentence

where \(X_1 , \ldots ,X_n\) are higher-order variables. This sentencesays that there is an extensional interpretation of the theoreticalterms that verifies, together with an antecedently giveninterpretation of the observation language \(L(V_o)\), the axiomsTC. The Ramsey sentence expresses an apparently weakerproposition thanTC, at least in standard truth-conditionalsemantics. If one thinks that the Ramsey sentence expresses theproposition of a scientific theory more properly thanTC, oneholds theRamsey 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 ofmeaning in mind when proposing the replacement of theoretical termswith appropriate higher-order variables:

Any additions to the theory, whether in the form of new axioms orparticular assertions like \(\alpha(0, 3)\) are to be made within thescope of the original \(\alpha\), \(\beta\), \(\gamma\). They are not,therefore, strictly propositions by themselves just as the differentsentences in a story beginning ‘Once upon a time’ have notcomplete meanings and so are not propositions by themselves.

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

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

and

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

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

Another important argument in favor of the Ramsey view was given laterby Sneed (1979: ch. III). It is easy to show that Sneed’sproblem 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 thateach axiom of the set \(A(\TC)\) is satisfied in the context of agiven 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 \(\TC^R\) andTChave the same observational consequences:

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

One difficulty, however, remains with the Ramsey view. It concerns therepresentation of deductive reasoning, for many logicians the primaryobjective of logic. Now, Ramsey ([1929] 1931: 232) thinks that the‘incompleteness’ of theoretical assertions does not affectour reasoning. No formal account, however, is given that relates ourdeductive practice, in which abundant use of theoretical terms ismade, to the existentially quantified variables in the Ramseysentence. We lack a translation of theoretical sentences (other thanthe axioms) that is in keeping with the view that the meaning of atheoretical sentence \(\phi\) is the difference between \((\TC \wedgeA \wedge \phi)^R\) and \((\TC \wedge A)^R\). As Ramsey observes, itwould not be correct to take \((\TC \wedge A \wedge \phi)^R\) astranslation 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 ofclassical logic. These laws, however, are supposed to govern deductivereasoning in science. A proper semantics of theoretical terms shouldtake the semantic peculiarities of these terms into account withoutrevising the rules of deductive reasoning in classical logic.

There thus remains the challenge of relating the apparent use oftheoretical terms in deductive scientific reasoning to the Ramseyformulation of scientific theories. Carnap was well aware of thischallenge and addressed it using a sentence that became labeled lateron theCarnap sentence of a scientific theory (Carnap 1958;1966: ch. 23):

This sentence is part of a proposal to draw the analytic-syntheticdistinction at the global level of a scientific theory (as thisdistinction proved not to be applicable to single axioms): theanalytic part of the theory is given by its Carnap sentence \(A_T\),whereas the synthetic part is identified with the theory’sRamsey sentence in light of Proposition 1. Carnap (1958) wants \(A_T\)to be understood as follows: if the Ramsey sentence is true, then thetheoretical terms be interpreted such thatTC comes out trueas well. So, on condition of \(\TC^R\) being true, we can recover theoriginal formulation of the theory in which the theoretical termsoccur 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 sentencemay well be true withoutTC being so (cf. Ketland 2004).Hence, the Carnap sentence would not count as analytic, as Carnapintended. Carnap’s interpretation of \(A_T\) receives a soundfoundation in his (1961) proposal to define theoretical terms usingHilbert’s epsilon operator, as we shall see inSection 4.3.

4.2 Indirect Interpretation

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

The predicate ‘H’, by contrast, is interpreted inan indirect manner by a definition in the object language:

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

The calculus is first constructed floating in the air, so to speak;the construction begins at the top and then adds lower and lowerlevels. Finally, by the semantical rules, the lowest level is anchoredat the solid ground of the observable facts. The laws, whether generalor special, are not directly interpreted, but only the singularsentences.

The laws \(A(\TC)\) are thus simply adopted as axioms in the calculuswithout assuming any prior interpretation or reference to the worldfor theoretical terms. (A sentence \(\phi\) being an axiom of acalculusC means that \(\phi\) can be used in any formalderivation inC without being a member of the premises.) Thisaccount amounts to a formalist understanding of the theoreticallanguage in science. It has two particular merits. First, itcircumvents Sneed’s problem of theoretical terms since theaxioms are not required to be true in the interpretation of therespective language that represents the facts of thetheory-independent world. The need for assuming such an interpretationis simply denied. Second, the account is in line with the contextualtheory meaning for theoretical terms as our understanding of suchterms is explained in terms of the axioms of the respective scientifictheories (cf.Section 1.2).

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

Yet, the interpretation of theoretical terms by axioms of a scientifictheory differs in several ways from that of a defined term by adefinition. First, the introduction of theoretical terms may be joinedwith the introduction of another, theoretical domain ofinterpretation, in addition to the basic domain of interpretation inwhich observation terms are interpreted. Second, it must not beassumed that the interpretation of theoretical terms results in aunique determination of the extension of these terms. This is animplication 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 theobservation 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 semanticsimilarities between definitions and Carnapian postulates suggests thefollowing explanation: a set \(A(\TC)\) of axioms that interprets aset \(V_t\) of theoretical terms on the basis of a language \(L(V_o)\)imposes a constraint on the admissible, or intended, interpretationsof the language \(L(V_o,V_t)\). An \(L(V_o,V_t)\) structure isadmissible 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):

Given there is a range of admissible, i.e., intended structures, thefollowing semantic rules for theoretical sentences suggestthemselves:

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

A few properties of the present semantics are noteworthy. First, itaccounts for Carnap’s idea that the axioms \(A(\TC)\) have atwofold function, viz., setting forth empirical claims and determiningthe 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 theadmissible interpretations of the theoretical terms. These twoseemingly contradictory properties are combined by allowing the axioms\(A(\TC)\) to interpret theoretical terms only on condition of therebeing a structure that both extends the given interpretation of theobservation language and that satisfies these axioms. If there is nosuch structure, the theoretical terms remain uninterpreted. Thissemantics, therefore, can be seen to formally work out the oldcontextual 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 thisproblem is bound to standard truth-conditional semantics. Third, it isclosely related to the Ramsey view of scientific theories as thefollowing biconditional holds:

Unlike the Ramsey account, however, the semantics of indirectinterpretation does not dispense with theoretical terms. It can ratherbe shown that allowing for a range of admissible interpretations asopposed to a single interpretation does not affect the validity ofstandard deductive reasoning (Andreas 2010). Hence, a distinctivemerit of the indirect interpretation semantics of theoretical terms isthat theoretical terms need not be recovered from the Ramsey sentencein the first place.

The labelpartial interpretation is more common in theliterature to describe Carnap’s view that theoretical terms areinterpreted by the axioms or postulates of a scientific theory (Suppe1974: 86–95). The partial character of interpretation isretained in the present account since there is a range of admissibleinterpretations of the complete language \(L(V_o,V_t)\). This allowsfor the interpretation of theoretical terms to be strengthened byfurther postulates, just as Carnap demanded in his 1958 and 1961. Tostrengthen the interpretation of theoretical terms is to furtherconstrain the range of admissible interpretations of\(L(V_o,V_t)\).

What happens if the axioms A(TC) are inconsistent or fail to beempirically adequate? In other words, what happens if there is nointerpretation of the theoretical terms such that this interpretationsatisfies all the axioms and agrees with the antecedent interpretationof the observation terms? In this case, the axioms of the respectivetheory fail to interpret the theoretical terms. The set of admissiblestructures, as defined by Definition 3, is empty. At the same time, wethink we have some understanding of the theoretical terms, even if therespective theory fails to be fully consistent with the empiricaldata. For example, we think we have some understanding of thetheretical terms in classical mechanics, even though classicalmechanics is not universally applicable. Moreover, we seem to haveinconsistent, yet non-trivial scientific theories, such as classicalelectrodynamics.

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

Relatively little research has been done on the semantics oftheoretical terms in theories from scientific disciplines other thanphysics. It has remained an open question whether or not formalsemantics that were primarily developed to capture theories in physicsare 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 (seeBalzer et al. 2000). Rakover (2020) and McClimans (2017) studiedtheoretical terms in psychology, albeit without explicitconsiderations of their formal semantics.

4.3 Direct Interpretation

Both the Ramsey view and the indirect interpretation semantics deviatefrom standard truth-conditional semantics at the level of theoreticalterms and theoretical sentences. Such a deviation, however, was notfelt to be necessary by all philosophers that have worked ontheoretical concepts. Tuomela (1973: ch. V) defends a position that hecallssemantic realism and that retains standardtruth-conditional semantics. Hence, direct interpretation is assumedfor theoretical terms by Tuomela. Yet, semantic realism fortheoretical terms acknowledges there to be an epistemologicaldistinction between observational and theoretical terms.Tuomela’s (1973: ch. I) criterion of the theory-observationdistinction largely coincides with Sneed’s above expoundedcriterion. Since direct interpretation of theoretical terms amountsjust to standard realist truth conditions, there is no need for afurther discussion here.

4.4 Defining Theoretical Terms

In Weyl (1949), Carnap (1958), Feyerabend (1962) and a number of otherwritings, we find different formulations of the idea that the axiomsof a scientific theory determine the meaning of theoretical termswithout these axioms qualifying as proper definitions of theoreticalterms. Lewis, however, wrote a paper with the title “How toDefine Theoretical Terms” (1970). A closer look at theliterature reveals that the very idea of explicitly definingtheoretical terms goes back to Carnap’s (1961) use ofHilbert’s epsilon operator in scientific theories. This operatoris an indefinite description operator that was introduced by Hilbertto designate some objectx that satisfies an openformula \(\phi\). So

designates somex satisfying \(\phi(x)\), wherexis the only free variable of \(\phi\) (cf. Avigad andZach 2002). Now, Carnap (1961: 161n) explicitly defines theoreticalterms in two steps:

where \(\bar{X}\)is a sequence of higher-order variables and\(\bar{t}\) a corresponding instantiation. So, \(\bar{t}\) designatessome sequence of relations and functions that satisfiesTC inthe context of an antecedently given interpretation of \(V_o\). Oncesuch a sequence has been defined via the epsilon-operator, the secondstep of the definition is straightforward:

Carnap showed that these definitions imply the Carnap sentence\(A_T\). Hence, they allow for direct recovery of the theoreticalterms for the purpose of deductive reasoning on condition of theRamsey sentence being true.

Lewis (1970) introduced a number of modifications concerning both thelanguage of the Carnap sentence and its interpretation in order toattain proper definitions of theoretical terms. First, theoreticalterms are considered to refer to individuals as opposed to relationsand functions. This move is made coherent by allowing the basiclanguage \(L(V_o)\) to contain relations like ‘xhas propertyy’. The basic,i.e., non-theoretical language is thus no observation language in thisaccount. Yet, it serves as the basis for introducing theoreticalterms. The set \(V_o\) of ‘O-terms’ is best described asour antecedently understood vocabulary.

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

Third, Lewis (1970) insists on a unique interpretation of theoreticalterms, thus rejecting Carnap’s doctrine of partialinterpretation. Carnap (1961) is most explicit about the indeterminacythat this doctrine implies. This indeterminacy of theoretical termsdrives Carnap to using Hilbert’s \(\varepsilon\)-operator there,as just explained. For Lewis, by contrast, a theoretical term isdenotationless if its interpretation is not uniquely determined by theRamsey sentence. For a scientific theory to be true, it must have aunique interpretation.

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

These postulates look more difficult than they actually are. CL1 saysthat, ifTC has a unique realization, then it is realized bythe entities named by \(t_1,\ldots,t_k\). Realization of a theoryTC, in this formulation, means interpretation of thedescriptive terms under whichTC comes out true, where theinterpretation of the \(V_o\) terms is antecedently given. So, CL1 isto be read as saying that the theoretical terms are to be understoodas designating those entities that uniquely realizeTC, inthe context of an antecedently given interpretation of the \(V_o\)terms. CL2 says that, if the Ramsey sentence is false, the theoreticalterms do not designate anything. To see this, recall that \(\neg\exists x(x=t_i)\) means, in free logic, that \(t_i\) isdenotationless. In case the theoryTC has multiplerealizations, the theoretical terms are denotationless too. This isexpressed by CL3.

CL1–CL3 are equivalent, in free logic, to a set of sentencesthat properly define the theoretical terms \(t_i (1 \le i \len)\):

\(t_i\) designates, according to this definition schema, thei-th component in that sequence of entities that uniquelyrealizesTC. If there is no such sequence, \(t_i (1 \le i \len)\) is denotationless. But even if a theoretical term \(t_i\) failsto have a denotation, the definition \(D_i\) of this term remains trueas long as the complete language \(L(V_o,V_t)\) is interpreted inaccordance with the postulates CL1–CL3, thanks to the use offree logic.

A few further properties of Lewis’s definitions of theoreticalterms are noteworthy. First, they specify the interpretation oftheoretical terms uniquely. This property is obvious for the case ofunique realization ofTC but holds as well for the othercases since assignment of no denotation counts as interpretation of adescriptive symbol in free logic. Second, it can be shown that thesedefinitions do not allow for the derivation of any \(L(V_o)\)sentences except logical truths, just as the original Carnap sentencedid. Lewis, therefore, in fact succeeds indefiningtheoretical terms. He does so without attempting to divide the axioms\(A(\TC)\) into definitions and synthetic claims about thespatiotemporal world.

The replacement of theoretical relation and function symbols withindividual terms was judged counterintuitive by Papineau (1996). Areformulation, however, of Lewis’s definitions using second- orhigher-order variables is not difficult to accomplish, as Schurz(2005) has shown. In this reformulation the problem arises thattheoretical terms are usually not uniquely interpreted since ourobservational evidence is most of the time insufficient to determinethe extension of theoretical relation and function symbols completely.Theoretical functions, such as temperature, pressure, electromagneticforce etc., are determined only for objects that have been subjectedto appropriate measurements, however indirect. In view of thisproblem, Schurz (2005) suggests letting the higher-order quantifiersrange only over those extensions that correspond tonaturalkind properties. This restriction renders the requirement ofunique interpretation of theoretical terms plausible once again. Sucha reading was also suggested by Psillos (1999: ch. 3) with referenceto Lewis’s (1984) discussion of Putnam’s (1980)model-theoretic argument. In that paper, Lewis himself suggests therestriction of the interpretation of descriptive symbols to extensionscorresponding to natural kind properties.

One final note on indirect interpretation is in order. Both Carnap(1961) and Lewis (1970) interpret theoretical terms indirectly simplybecause 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’sproposals conforms to the pattern of a definition in the narrow senseand not to the peculiar pattern of indirect interpretation that Carnap(1939) envisioned for the interpretation of theoretical terms. This iswhy the indirect interpretation semantics has been separated from thepresent discussion of defining theoretical terms.

5. Conclusion

The idea that there are scientific terms whose meaning is determinedby a scientific theory goes back to Duhem and Poincaré. Suchterms came to be referred to astheoretical terms intwentieth century philosophy of science. Properties and entities thatare observable in the sense of direct, unaided perception did not seemto depend on scientific theories as forces, electrons and nucleotidesdid. Hence, philosophers of science and logicians set out toinvestigate the semantics of theoretical terms. Various formalaccounts resulted from these investigations, among which the Ramseysentence by Ramsey ([1929] 1931), Carnap’s notion of indirectinterpretation (1939; 1958) and Lewis’s (1970) proposal ofdefining theoretical terms are the most prominent ones. Though not allphilosophers of science understand the notion of a theoretical term insuch way that semantic dependence upon a scientific theory isessential, this view prevails in the literature.

The theory-observation distinction has been attacked heavily and ispresumably discredited by a large number of philosophers of science.Still, this distinction continues to permeate a number of importantstrands in the philosophy of science, such as scientific realism andits alternatives and the logical analysis of scientific theories. Acase in point is the recent interest in the Ramsey account ofscientific theories which emerged in the wake of Worral’sstructural realism (cf. Ladyman 2009). We have seen, moreover, thatthe formal accounts of theoretical terms work well with atheory-observation distinction that is relativized to a particulartheory. Critics of that distinction, by contrast, have commonlyattacked a global and static division into theoretical andobservational terms (Maxwell 1962; Achinstein 1965). Note finally thatCarnap assigned no ontological significance to the theory-observationdistinction in the sense that entities of the one type would beexistent in a more genuine way than ones of the other.

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