Balzer, Wolfgang - Theoretical Terms a New Perspective, 1986

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Theoretical Terms: A New PerspectiveAuthor(s): Wolfgang BalzerSource: The Journal of Philosophy, Vol. 83, No. 2 (Feb., 1986), pp. 71-90Published by: Journal of Philosophy, Inc.

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THE JOURNAL OF PHILOSOPHYVOLUME LXXXIII, NO. 2, FEBRUARY 1986

THEORETICAL TERMS: A NEW PERSPECTIVE*

P HILOSOPHICAL reflectionbouttheoreticalerms anbetraced ackat leasttoJohn tuartMill.Lateron, nlogicalempiricism,he ssue reappeared s a question fwhether

certain erms ere efinablena formalanguage. heoreticalermswerecharacterizeds nonobservationalerms,nd observationaltermss terms hose enotationsouldbe "directly"erceived iththe human enses.Observationalerms,n contrasto theoreticalterms, ere elt o have clearmeaning.ogical mpiricistsried o

extend hisclearmeaning o theoreticalerms ydefining hemexplicitly.n this,however,hey id not succeed,mainlyas willbecome lear nsectionii) becausetheirogical ormalismastoonarrownd toostrong.nitiallyhedichotomyetweenheoreticalandnontheoreticalerms asdrawn globally,".e.,with espect othewhole anguage fscience.Refinementsllowed or hierarchyof theoriesnwhichheoreticalerms ere o be introducedtepbystepon the basis of an antecedentlyvailablevocabulary.'n allvariants,owever,heobservationalanguagewastaken o be givenandwellunderstood. ne oftheobjections aised gainst mpiri-cismwas hat his referredositionfobservationalanguagesnot

* Special featuresof thesubject treated n thispaper have been presented nmyHabilitation-lecture t thePhilosophicalFaculty of theUniversityfMunich (May1983) and inmy ontribution o the 7th nternationalCongress ofLogic, Methodol-ogy,and Philosophy fScience in Salzburg (July1983). I am indebted toWolfgangStegmiiller nd UlrichGaehde for lluminating iscussions, nd to Ch. Pinnock forcorrectingmyEnglish.

' See RudolfCarnap, "The MethodologicalCharacter ofTheoreticalConcepts,"in H. Feigl and M. Scriven,eds., Minnesota Studies in thePhilosophy of Science,vol. I (Minneapolis:MinnesotaUP, 1956), pp. 38-76; and Carl G. Hempel, "TheMeaningofTheoreticalTerms: A Critique of the Standard EmpiricistConstrual,"in P. Suppes, L. Henkin, A. Joja, and G. C. Moisil,eds., Logic, Methodology, ndPhilosophy of Science, vol. iv (Amsterdam:North-Holland,1974), pp. 367-378.

0022-362X/86/8302/0071$02.00 ? 1986 The Journal of Philosophy, nc.

71








72 THE JOURNAL OF PHILOSOPHY

justified, nd even"wrong".2The decline oflogicalempiricismwas,accordingly,ccompaniedbyabandonmentof the assumptionof a

pre-givenbservationalanguage and concentration nwhatgoesonin "real-life" mpirical heories.3 n linewith hisdevelopment, .D.Sneed proposed a new criterionftheoreticity4hichdid notrely na given observational anguage.Accordingto his criterion, he dis-tinction etweentheoretical nd nontheoretical erms sdrawnrela-tive to a given theory. he criterion, owever, s formulated n apragmatic evelwhich eavesmuch room forvaguenessand discus-sion. There weremanyreservations,specially mong thoseaccus-tomedto thinkingn thetradition f logical empiricism.

In thepresentpaper we takeSneed's criterion s a point ofde-parture, nd we try obring hediscussionbackto the evel ofclaritycharacteristic flogicalempiricism.We providea nonpragmatic x-planationforSneed's "problem of theoretical erms,"which howshow thisproblem rises s a questionaboutmeaning ather han s aquestion of howtodistinguishheoretical romnontheoreticalerms(section I). Further,weoffer new andprecisedefinitionftheoret-ical termssection II), which pplies toexisting heories, eproduces

existingdistinctions,s in linewiththe intuitions f logicalempiri-cism, nd also throws ome clarifyingight n Sneed's account.Theclaimassociatedwith henewapproach is that topens thedoor foran empirical nvestigationf morecomprehensive artsof thewebof empirical heoriesbyestablishingwhichterms n whichtheoriesare nontheoreticalnd, therefore, ave to be presupposedas givenby other,underlying heories.

I. SNEED'S PROBLEM OF THEORETICAL TERMS

With ome simplificationneed's criterion an be stated as follows.Term t oftheoryT is T-theoreticallf in any determinationf thattermT is presupposedas valid. (The index 1' is used in order todistinguishneed's notionfrom he notion to be treated n sectionIII. Consequently,we willalso speak of theoreticityl.)

Some explanation s necessaryhere;some ofthe technicaldetails,however,willbe used only n thefollowing ections.We consider atheory that s, in our context, n empiricaltheory) s constituted(amongotherthings) ya language L, a classofmodelsM, and a set

of intendedapplications :2Norwood RussellHanson, PatternsofDiscovery New York:Cambridge,1958).3 See Hilary Putnam's complaints on philosophical discussions of theoretical

terms nhis "What TheoriesAre Not," in E. Nagel, P. Suppes, and A. Tarski,eds.,Logic, Methodologyand Philosophyof Science (Stanford,Calif.:University ress,1962), pp. 240-251; especially p. 243.

4The Logical StructureofMathematical Physics Dordrecht:Reidel, 1971), p.31.








THEORETICAL TERMS 73

T= (L,M,I,* **>

The language will have to be many-sorted nd of higherorder in

most ases. Forreasons of simplicity eassume thenonlogicalvocab-ularyto consist of onlyfinitelymanyfunction ymbols r *. , fi;extension o individual onstants nd "proper" predicatesdoes notcreateany difficulties. hese symbolswillbe called the terms f T,and the thterm r the th function fTwill be ustfi. L determinestheclass S of structuresorinterpretations)forin theusualway.5fL isthe anguageoftheory wewill lso saythat tructures orL arestructures or T. If there re, say,k sorts, hen each structure or T

has the form(D1i, ,Dk; fl, fm>

whereD1, * **,Dkare sets of "objects") andf1, **,fmre functions"over" thesesets.6For each i - m, is an interpretationr realiza-tion of the termfi.The class M ofmodels is a subclassof S, and isusually haracterized ya set A ofaxioms formulas fL) such thatthe axiomsare validpreciselyn all members fM.

Intendedapplications are "real systems" o whichscientists in-

tend" toapplythetheorynd which reconceivedof nthe anguageofT. An intended pplicationx of T therefore sgivenbysomerealsystem, hich,naddition,gives rise to a structure orL. We cannotand need notbe verypreciseaboutwhat tmeansto "giverise to" astructure orL; fortunatelyhepresentdiscussiondoes notcruciallydepend on thispoint.A propertreatmentwould lead us deep intothetheory fknowledge.Also, we believe that nthepresentdiscus-sion reference o intendedapplicationscan be completely voided,

but onlyat the priceof much technicalcomplication.An examplewillprovide s much ntuition s is needed in thepresent ontext. fscientists efer o somerealsystems an application i.e.,an intendedapplication nour sense)of classicalparticlemechanics, hey ook atthe system s exhibiting osition, mass,and force functions.They"see' moving particles, nd theyassume or believe thatconstantmasses are attached to theparticles nd that ome forces ct on theparticles.f the samesystema roulettewheel naction,for xample)

5See Joseph R. Shoenfield,Mathematical Logic (Reading,Mass.: Addison-Wes-ley, 1967), p. 18.

6Compare my EmpiricalClaims n ExchangeEconomics," inW. Stegmiiller,W.Balzer, and W. Spohn, eds., Philosophy of Economics (Berlin-Heidelberg-NewYork: Springer-Verlag,1982), pp. 16-40, especiallythe appendix, for a simpledefinition f this"over".

In thefollowingwe willdenote byx-i[f ] theresultofreplacingfi n x byf alwayson the assumptionthatf is of appropriate type).









is called an application of probability heory, cientists eventuallythe same as before) "see" onlypossible outcomes,events, nd rela-

tive frequencies s realizedin the system.In order to understand need's criterion,we have to specify wo

expressions whichare used in the above formulation nd also inSneed's original ersion, amelya determination or ome term' nd'to presupposeT as valid nthe determination or ome term'. t willturn ut that ach of theseexpressions equiresfar-reachingonsid-erations.Therefore,we will say that Sneed's criterion ontains twoimportant eatures:determination nd presupposition.

The intuitionforming he basis of thiscriterion s an intuitionabout scientific ractice, about what scientists ctuallydo and be-lieve. Essentially, his s a matter f pragmatics, ot of ogic. Roughly,the idea is this.There is given a theoryT and a group of scientistsworkingwith T. For some reason or other t becomesnecessary oknowsome function-values f certain of T's functions or certaingiven arguments.This knowledge cannot be acquired by "directobservation," t least not in cases of advanced theories.So somefurther ction is necessary: xperiments nd measurementswillbe

performednorder to determine he desired values. All theseactivi-ties, nsofar s they esult n one ormoreuniquely dentifiable unc-tion-values, epresent eterminations or some term. Of course,allthisneeds further larification.)

By ooking t concrete xamplesof such determinations e recog-nize that, nthe course of a determination,cientistsngeneral i.e.,with heexceptionof a fewvery asicmeasurements) se theoreticalknowledge, ormulas, quations. They perform ertaincalculations

or draw nferences n thebasisofgivenformulasn order to obtainthefunction-valueheywant o know.Usually, his heoretical nowl-edge in a given oncrete ontext s used without urtherjustification:it s (hypothetically)ssumed orpresupposed.All kindsof theoreticalassumptions sed in the course of a determinationre "presupposi-tions." For one "piece of presupposition" one equation, one for-mula)twocases are possible:either t stemsfrom"is part of") sometheoryT' different romT, or it is part of,or identicalwithT (i.e.,withT's axioms). n order to determineor "calculate," as they ay),

forexample,the masses of elementary articles,physicists se therelativisticersion f the aw ofconservation fmomentum,which sthe centralpartof relativisticollisionmechanics; n order to deter-mine theanglesofparticles'pathsafter ollisiontheyuse formulasfrom physical) eometry, hich s a theory ifferent rom ollisionmechanics.









According oSneed, thisdistinctionf whether r not the theoriespresupposed n a determination orsome term re all identicalwith

T is the same (bydefinition)s the distinction etweenT-theoreticaland T-nontheoretical erms.A term i of T is T-theoreticallf in alldeterminations orfJ he pieces of theoryused (presupposed) byscientists orthepurpose of determinationre part of,or identicalwith T. In other words, T-theoretical erms re termsthat can bedetermined nly bymeansof (by using) T.

Withrespectto what willbe said in sections I and III itmust bestressed hatthis riterion ftheoreticity,s pragmatic. hat termf,is T-theoreticals a statement bout how scientists oldingT pro-

ceed, iftheywant to determine i.The criterion istinguishesheo-retical terms by how scientists ct. In contrastto this,the crite-rion-or ratherdefinition-of theoreticityo be presented n sec-tion II willbe purely ogical.

Fromthiscriterion f theoreticitynd fromthehypothesis hatsome of T's terms re T-theoretical, need deduced whathe calledtheproblem ftheoretical, erms.Again wewilldiscussonly simpli-fiedversion.The problemconsistsof a kindof circularity ithre-

spectto thetesting f a theory ymeansofmeasurement. or ifwewant to test T by means of measurementwe have to determine(among otherthings) t least some values of T's theoreticalfunc-tions. But this,by virtueof thecriterion,s possible only f we pre-suppose T as alreadyvalid. So, in order to testT by meansofmea-surement,we have to presuppose thatT already s valid-which isjust whatwewant to find ut by meansoftesting. tillmore briefly:in order to find ut whetherT isvalid wehave topresupposethatTisvalid. f we accept a proposalofWolfgang tegmiiller's,8 amely o

explicate to presuppose' by "to imply ogically" we arrive at thefollowing:o test hevalidityf T bymeans ofmeasurement ogicallyimplies hatT isvalid. This of course seemshardly ompatiblewithcertainview bout testing, amely hattesting theory houldbe anenterprise ndependent f and certainly otpresupposing hetheoryto be tested.On theotherhand, thepresentformulation oes notreveal any strict ogical circulatory,nd it is doubtfulwhetherby

7Some authors have attempted o clarify he issue bymeans of further xplica-tion.See Stegmiiller, p. cit.; Andreas Kamlah, "An Improved Definition f Theo-retical n a Given Theory'," Erkenntnis,x, 3 (October 1976): 349-359; and W.Balzer and Carlos Ulisses Moulines, "On Theoreticity,"Synthese,XLIV, 3 (July1980): 467-494.

8 See The Structure and Dynamics of Theories (New York: Springer-Verlag,1976).









reformulationuch a circle can be constructed.This may be thereasonwhy he problemoftheoretical, ermshas notbeen accepted

as a real problemby many philosophersof science, especiallybythosewho workwith ogical tools.

If we do not bother about details,thepresentdiscussionmaybesummarized s follows.Termfi is T-theoreticallfT is presupposed(used) inanydeterminationorfi.The problemoftheoretical, ermsconsistsof the existenceof T-theoreticall erms i. For if T is pre-supposed in any determination orfi, how can we test T (on theassumption hatthis s possibleonlyon the basis ofsome concretelydetermined alues of T's functionsncluding i)? The characteriza-tion and the resulting roblemof theoretical1 erms re onlyslightvariants f each other;one might aythat he problem f theoretical1termss ust a slight eformulationf the claimthat theory ontainsT-theoreticall erms.The heart of thematter s givenby the obser-vationor claimthat cientists or omefi nfactpresuppose T duringall determinationsorfi (fi being T-theoreticall).

The questionwhether his s so for ome function i of an existingtheory e.g., themass function f classicalmechanics)has been ac-

tively iscussed,but withoutdefiniteresults.The main reason forthismaybe seen in the strongly ragmatic haracter f the originalformulation.9

II. AN EXPLANATION OF THE PROBLEM OF THEORETICAL TERMS

We willnow "derive" (and thereby xplain)the problemof theoreti-cal, termsfrom n assumptionwhichgoes beyondthe mere obser-vationof scientific racticeand whichcan be regardedas a specialinstanceof a philosophicalpositionconcerningmeaning nd refer-ence. Besides being of interest n its own, this xplanationwillmakeclear thattheproblem f theoretical1erms rises from particularview about the meaningof a term n a theoryand thus n fact s aproblemof meaning) and that tis not really onnected to thedis-tinction etween theoretical nd nontheoretical erms. n order to

9 It maybe noted that on Sneed's originalaccount the notion of a theory on-tainedthe distinction etweentheoretical nd nontheoretical erms s a constitutiveelement. But by acceptingsuch a distinctionwe are not forced to accept his crite-

rion of theoreticity,oo. In principle,his theory oncept mightgo togetherwithdifferentriteriaof theoreticity.he two items re separated in thepresentframebecause we do not even insist hat a theory n general should contain a distinctionbetweentheoretical nd nontheoretical erms. t should also be noted thatwe haveincluded language as an explicit lementof a theory, houghtheconceptdescribedabove stillrepresents kindof "nonstatement"view of theories.We do not attachmuch mportance,however, o the distinction-sometimesdiscussedrecently-be-tween "statement" and "nonstatement"views.









state our explanation, some preliminary,larifying onsiderationsare necessary.

Firstwehave to clarify hatwe meanbymeasurement. n general,measurement ims at finding he function-value f some functionfor given rgument . Thisvalue,f(b), esults rom process duringwhichmeasurement akes place. We call this a processof measure-ment. rocessesofmeasurement onsist f realsystems hich hangeover time (zero change as a special case is included). Often, theprocess is initiatedby an experimenter;t mayconsist of what isgoing on in a certainexperimental etup. It also may be given bywhat is happeningin some measuring nstrument. he important

point here is that thesystem ealizedduringa process ofmeasure-ment in fact s a system hatmay be conceptualized and properlydifferentiatedrom ts"environment." fwe assume thatonly omeobjects and functions re realized in such a system,tmaybe con-ceptualized n the formx = KDi,*. . , Dk; fi, *., fm> lready ntro-duced. The functionf, he value ofwhich tb s to be measured, henwillbe amongf, ... ,fm,ayf= f, and b willbe in thedomainoff:bE Dom( f). bmaybe someobject properormaybe constructed ut

of objects and (or) numbers. n mechanics,for nstance, hemass-functionm takesparticles s arguments,n geometry he distancefunction akespairs ofpointsofspace, and inmechanics, gain,theforcefunction akesparticles, eal numbers, nd integersl'; so therespective unction-values ave the following orms:m(p),d(Kb, >),f(p, t, ). The function-valuefb)willbe called themeasuredvalue. Inthefollowing,he thfunctionfoccurringn x willbe denotedbyfxand, consequently, measured value (measured n x) byfX(b).

As longas we are concentrating n the measurement f one single

function-value, e mayassume thattheprocessof measurement ymeans ofwhich his alueisobtained sgovernedby ometheory inthesense thatthesystem ealizedduring hatprocess s an intendedapplicationof T. That is,all the functions f T (moreaccurately: llfunction-symbolsf T) have interpretationsn that ystem, nd sci-entists ntend to apply T to that system.This assumptionseemsratherrestrictive t first ight; t may be objected that n cases offundamentalmeasurement s well as in cases where complicated

apparatus involving ifferent heories s used our assumptionwillnot be satisfied. utincasesoffundamentalmeasurement will ust

10At least in theformulation fJ. C. C. McKinsey,A. C. Sugar, and P. Suppes,"Axiomatic Foundations of Classical Particle Mechanics," Journal of RationalMechanics and Analysis, II (1953): 253-272.









be some theory f extensive ystemsor the like),and in cases ofcomplicated rocedures nvolvingeveral heories he procedurecan

be split up-at least conceptually-into a "chain of measure-ments" so thateach linkofthechain willbe governedby onlyonetheory.n the latter ase we may restrict ur analysis o one of thechain's linkswithout hanging he situation.

Each process of measurementthus constitutes structureforsome theory,whichmoreover s an intended application. Such astructure willbe called a measuringmodelor a measuringmodelforfx(b). The generalcondition characterizingmeasuringmodel x

forfx(b)sthatfx(b) e uniquelydetermined ytheotherfunctions fx and bysome lawlike onnectionbetween those andfiX.Note thatthiskind ofuniqueness s different rom he unique dependencyoffX(b)on b. The latter s expressedby the formula

Vb, E Dom(fx)(b = c f (b) = fix(c))

whereastheformer onditionofuniquenessreads like this see fn6above for notation):

Vf,f'(a(x_j[f])A u(x-i[f']) f(b) =f'(b))

where a-representsthe lawlike connection among the differentfunctions f T.

In a second stepwe have to think bout what it is that s to bemeasured n a measuringmodel.Of course, fx is a measuringmodelforfiX(b)henfiX(b)s measuredbymeans of x. But this s not aninterestingnswer o our question.Usually, fa processof measure-

ment sstarted,we have certain deas aboutwhatwe want o measureevenbefore heprocesshasgiven result.Measurementmakes enseonly fwe can identifyheentityo be measured ndependentlyf theresultof measurement. his is indicatedalready n thegrammar f'to measure'. We always aythatwe measure omething,nd inmorespecific ontexts his something"usuallywillhave a name. In phys-ics,for nstance,we measuremassorweight rforce; nchemistryemeasureatomicweights rvalences; neconomicswe measurepricesor quantities.But wenever"justmeasure."The problem s to iden-

tify heentitywe want to measure.The identificationf what t is thatwe want to measurestraight-

forwardlynvolves hilosophicalreflection,ecause itrequiresus to

" See my "Theory and Measurement," Erkenntnis,XIX, 1 (May 1983): 3-25,p. 15.









take a definitepoint of view about how theory nd evidence arerelatedtoeach other.We know he differentccountsthathave been

givenofthisrelation, anging rom heoperationalist iewthatwhatwewant omeasurebymeansofmeasuringmodelx in fact snothingbutfix(b),o theradical holistic r coherentist iewthatwhatwe wantto measurecan be identifiednlybythetotality f ourknowledge, tthe other end of a spectrum f possible alternatives. or thepur-poses of this paper it is notnecessary o takea particularpositionhere. The crucial point s thatwhatwe wanttomeasure by means ofmeasuringmodel x has to be somehow dentifiedndependentlyfthe concreteprocess of measurementrepresentedby x, and thatthere are different lternative iewsabout how the identificationproceeds (or should proceed).

For our purposes t s sufficiento introduce new abel denotingtheentityo bemeasured ngeneral, .e., regardless ftheparticularphilosophicalpointofviewone may ake. Let us denote theentityobe measured n a measuringmodel for X(b) byf(b). The index 'i'here does not indicate anypositionoffi in some structure; t onlyindicates hatfi softhe sametype sfix.dentityf thetypes ffi nd

fiX seems to be a minimalrequirementwhich will be part of anyidentificationffi. In concrete ases thefunctionfilwayswillhave aname, e.g., 'mass,' 'weight,' 'force,' 'atomic weight,' 'valence,''quantity,' tc. It is notnecessary, owever, o introduce symbolfor such names, because we do not need to talk here about therepresentationffi in the language.

The problemof how toidentifyhefunction ne wants o measurebymeans ofmeasuringmodelx, i.e., theproblemof how to identifyfi, in itspresentformulation eems to havereceived ittle ttention.In anotherformulation-equivalentto thepresentone-the prob-lemhas been at the centerofdiscussionsfor the asthundredyears.In thisother formulationheproblem swhether perationalism rcoherentism r some intermediate osition represents he correctaccountof scientificractice. n thepresent ontext fmeasurementand in the conceptual framehere developed we will speak of theproblem f meaningfulmeasurement.n theterminology evelopedabove itmaybe restated s follows. uppose we want tomeasuref

for omegiven rgument ; i.e.,we wantto determine hevaluefi(b).Some processofmeasurementwillbe initiated uch thatboccurs inthe system orresponding o thatprocess. This systemwillbe gov-erned by some theoryT and willgive rise to a measuringmodelx = KDlx, .. , Dkx;flx, ... frnX>.But what s obtained measured)bymeansof x isf%X(b)nd notfi(b).The problemof meaningfulmea-









surement henconsists fspecifying set of conditions nder whichfX(b)can be identifiedwith i(b).The specification f such conditions

will provide a (partial) dentificationffi.We can now see that measuringf" means applying ome process

of measurement that results in some value fx(b), and then "some-how" passingover fromfx(b) ofi(b). n order to get a clear picture,we have to spell out how precisely his transition rom X(b)tofi(b)works or is justified; that s, we have to providea solution of theproblemof meaningfulmeasurement. here are two "classical" so-lutions: he operationalistnd the coherentist. n a third reliminary

step we have to describe the coherentist olutionof the problem ofmeaningfulmeasurement,nd thismaybest be done by indicatingalso theoperationalist lternative.

The operationalistolution s simple.The operationalist efinesJbythe measuredvalues thatoccurinvariousdistinguished rocessesof measurement.So fi(b) = fx(b) by definition ffi. A bit moreprecisely, i is operationallydefined n the followingway. There isspecified ome method fmeasurement which nourframewe canconceiveof as just consisting f a class ofmeasuringmodels (those

modelswhichresultfrom pplying hespecified method"). Thenfiisdefined s the union of all ith functionsfx ccurringnmeasuringmodelsx of B:f,= U{ fx/xE B}. This is not theplace to discuss themerits nd shortcomings f operationalism. f there is reason tobelieve that differentmeasuringmodels of B willyield identicalvaluesforobjects they ave incommon, .e., if,for ll x,yE B and allbE Dom(fx)n Dom(fy),wehavefx(b)=fy(b),thenthe above defini-tion offi is one possible solution of the problemof meaningful

measurement. his solutionyields n "operationalistwayof measur-ingfi" in the followingway. We choose some measuringmodel xsuch thatx is a member fthe classB thatdefinesfi nd such that heargument at which i is to be determineds in the domainoffix;wethen determine iX(b) ymeans of x. Since, bydefinition ffi, fi(b)= fiX(b),we have determined fi(b) in this way.

The coherentist olutionof theproblemofmeaningfulmeasure-ment s a bit morecomplicatedbecause of the factthat ihere has amoreindependent tatus.We restrict urselvesto the case inwhich

"coherentism" omprises nlyone theoryT. The moregeneralandmoreradically olistic ase perhapscouldbe treated n thesameway.The problem n thegeneralcase is thatwe do not have (yet) clearpicture f theover-all tructure f science whichwecould substituteforour present "theoryT." Roughly, he coherentist olutionpro-ceeds intwosteps. First, iisdefinedbymeansofT, and, second,an









assumption s made fromwhich, n the situationof measurementwiththehelp of a measuringmodel x, it follows hat ,X(b)= fi(b).

First, he definitionffirefers o "all of" T; this s in contrast otheoperationalist efinitionffiwhichrefers nly o somemethodBofmeasurement.According o the coherentist ccount,fi is definedas theunionof all functionsi occurring n models of T which lsoare intendedapplications:

fi= U{ffx/xMfn }

We will ssume n thefollowinghatfi o defined gainisa function,although hisdoes not follow rom he definition. nalysis fexisting

theories howsthat his ssumption in the presentor in some otherequivalentform) s oftenused implicitly,nd maybe regardedas anessentialfeature of empiricaltheories.'2 t should be noted that,eventhoughfisexplicitly efined nterms f theconstituentsf) T,thisdoes not mply hatfi an be formally haracterized, ecause theset of intended pplications ppearing n thedefiniens ffi can befixedonly pragmatically.

Second, giventhisdefinitionffiand given omeprocessof mea-

surement aptured by measuringmodelx for x(b), how can we getfrom iX(b) o fi(b)?The coherentist nsweris: by presupposingTduring the process of measurement. hat is, it is presupposed thatthe axioms of T are valid in the system hat s realizedduringtheprocessofmeasurement. n otherwords: t is presupposed thatthemeasuringmodel given by the process ofmeasurement s a propermodelof T. In otherwords: fthismeasuringmodel is denotedbyxthentopresupposeT during heprocessof measurement iving isetoxjust means thatx E M. It isnotdifficulto see thatpresupposing

Tduring the process of measurementi.e., presupposing hatx E M)in fact mplies hatfix(b) f1(b). or ifx is a measuringmodel thenx isan intended pplicationofT(x E I) byourstipulationsn measuringmodels.This, togetherwith E M, implies hatfxE {ff/y EM n },and,from urassumption hatU{f /yE Mfn } is a functionndbydefinition ffi, we finally btainfX(b)= fi(b)foranyb E Dom(fx).

Accordingly,he coherentistwayof measuring i is this. Choosesome measuringmodel x so thatb occurs in the domain offXand

determinefix(b).Presuppose T as valid during thisprocess (i.e.,presuppose thatx E M). From this nd the general assumption hatfi is a function,tfollows, s was ust shown, hatfix(b) fi(b), nd so

12 In Sneed's workthis ssumption cquires the statusof a componentofempiri-cal theories ngeneral. There, it s givenbya specialkindof Sneedian "constraints,"the so-called "identity onstraints."









we have obtained the desiredvaluefi(b).'3 n this ituation,we will aythatfJs measured n x in thecoherentistay.

It seemshelpfulhere to reflect n whywe have chosen the label'coherentist' orthis econdsolution.Roughly, oherentismsa viewabout meaning and truth.Words acquire their meaningthroughtheirrole and position n a whole language, and truth, orrespond-ingly, lso has to be understood relative o the total anguage.As aconsequence truth annot be checked on the basis of some unshak-able observation entences; tcan be checkedonlyfrom inside", bycoherence of thewholesystem. he coherentistwayofmeasuringfiis ust a very pecial case ofthisgeneralpointof view. n order to findout whethera theoryT is "true" we cannot relyon results ofmeasurements hat re independent fT, because, accordingto thecoherentist ccountofmeasuringfieverymeasurementffi lreadypresupposesT. Butifno measurementsndependent f T are possi-ble, then such measurementsndependentof T ("basic sentences")are not available as a basisfortestingT. Onlythecoherentist,nter-nal wayof assessingT's truth emainsopen. The holisticview thatthemeaning of a termdepends on large portionsof science is not

new, of course. Even withrespectto physical heories t had beenadvanced alreadybyNormanR. Campbell.'4We noware in a positionto state theargument hatexplainsthe

problemof theoretical,terms as a problemabout meaning.Theargumenthas three premises:

(PI) Foranymeasurementhat sperformedn order o test heory itisnecessaryhat omefunctionf T bemeasured uring hepro-cessofmeasurement.

(P2)There redifferentays fmeasuringhefunctionsfT, amongthem heoperationalistnd the oherentistay.

(P3) Thecoherentistay fmeasuringhefunctionsfT isthe orrectway.

From (PI)-(P3) it is concluded that

(C) For nymeasurementhat sperformednorder otest heoryit snecessaryopresuppose asvalid uringheprocess f measure-ment.

"Formally, the assumption thatfi is a function might be weakened or evendropped and replaced by some complicated or conventional "definition" off,.

However, there s no evidence from xisting heoriesthatwouldback sucha move.14 See Foundations ofScience, unalteredrepublication f thefirst ditionof the

workformerly ublished under the titlePhysics: The Elements New York: Doverpublications,1957), chap. ii, especiallypp. 42/3 and 49/50. In spiteof suchsimilar-itieswithrespect to approaches to meaning, here s no proposal to be found n theliterature f a criterion ftheoreticityimilar o thatputforward y Sneed (as far sI know).









Witha fewfurtherteps it can be seen that C), in fact,followsfrom PI)-(P3). First, n (PI) we have to substitute omethingmore

accurateforthe phrase some function f T is measured.'We mayreplacethisby there s some i - msuchthatfi s measured.'By P2)and (P3) anynoncoherentistwayofmeasuring.fis excluded; so in

(PI) wehave to substitutethere s some - m suchthatfi smeasuredin thecoherentist ay'. PI) thenbecomes P'): For anymeasurementwhich s performedn order to test theoryT and which s repre-sented bya measuringmodel x, there s some i - m such that i ismeasured n x in thecoherentistway.But,byour previousexplica-tionof tomeasurefi n x in the coherentist ay,' his mplies hat he

measuringmodelaccording o thecoherentistccount of measuringfi is a propermodel of T, and this n turnmeans thatT has beenpresupposedas validduring he processof measurement. yputtingall thistogetherwe obtain, n fact, C) from PI)-(P3).'5

If we think bout thestatus f premises PI)-(P3) itseemsthat PI)and (P2)have thecharacter f "analytic" entences erving o fix hemeaningof 'test of a theory'and 'measurement,'whereas P3) iS

clearly mpirical n character."6P3) is an empirical tatementbout

how scientists ctuallymeasure f there s already ome theory ov-erningthe process of measurement.P3) mightbe false,and, forinstance,the operationalistwayof measuring i mightbe correct.The pointhere,however, s not to decide between P3) and itspossi-ble alternatives. he point s to show that P3) implies C), given hat

(PI) and (P2)are not questioned.That is,the problemoftheoretical,termss essentially consequence of thecoherentistwayof measur-ing thefunctions f a theory.n otherwords, t follows rom cien-tists'beliefs as expressed n scientific ractice)thattheories-if at

hand-should be takenseriouslyn the contextofmeasurement.To summarize his ection,wemay aythat heproblem ftheoret-

ical, terms s a special case of or followsfrom generalcoherentistviewof meaning nd truth. t arisesbecause scientistseem to checkthe truthof theories by means of measurements n which thosetheories re alreadypresupposed, .e.,in a coherentist ay.The mostinterestingeature four explanation,however, s that tapplies toall termsof a theory: hedistinction etweentheoretical nd non-

15 With some trivial ntermediate xplicativesteps the argumentcan be trans-formed nto a proper logical derivation.However, no further larity s achievedthereby.

16 We accept the thesis thatthere is no sharp distinctionbetween analytic ndsynthetic entences, and this applies to the distinction oncerning the status of(P1)-(P3), too. On theother hand it should be noted that such acceptance does notimplyany rejection of the use of precise (and even formal)concepts in statingtheoretical s well as metatheoretical deas.









theoretical ermswas used neitherntheabove argumentnor in thesubsequentexplications. ince thecriterion ftheoreticity,s only

slight eformulationfthe problemoftheoretical, erms, ur expla-nation casts somedoubt on whether his riterion eallyprovides nadequate distinction etweentheoretical nd nontheoretical erms.Withthese doubts in mind we now turn to a new criterion-orbetter-a new definitionof theoreticitywhich was found in thecourse of researchaiming t clarification f Sneed's account.

III. A NEW DEFINITION OF THEORETICAL TERMS

Sneed's criterion f theoreticitys based on two tems:presupposi-tion and determination. s we ust saw,the problemsofpresupposi-tionessentiallyrise from particular heory fmeaning, nd it isnot clearhow thisfeaturecontributes o the distinctionmongtheterms f a theory. o "determination" eemed tobe a morepromis-ing area. In fact,UlrichGaehde'7 was able to draw a formaldistinc-tionbycharacterizingT-admissible eterminations"s thosewhichare compatiblewith T's invariances, nd thisdistinction ittedwithprevious ntuitions bout theoretical erms.We willpresenthere amodifiedversionof Gaehde's definitionwhich s slightlyweaker"

and muchmoresimple. n order to stress he formal haracter f thedefinitionwe choose to speakabout "definability"nstead of "mea-surement" r "determination."'8Roughly, hen, nd in theconcep-tual frameoutlinedin section , the new definition f theoreticalterms an be statedas follows.Termfiof theoryT is T-theoretical2if fi is weakly nd invariantlyefinable n T.

For further xplanation et us startbynotingthat n first-ordertheories xplicitdefinabilityffi in T is equivalentto thesemanticrequirementhat, n each modelx,fX be uniquelydetermined ytheotherfunctions f x (and byx beinga model):

Vx EM Vf,f'(x4i[f] M A x_f'] E M - f=f)

Weak definabilitys obtained from this if we replace the class ofmodels M bysomesubclassB c M (representingome "subtheory"T' of T) in the above formula:

3Bc M V xE B Vf,f'(x_i[f] E B A xJi[f']E B -f =f)

In "real-life" theories B is givenby so-called "special laws" likeHooke's law in mechanics, he ideal-gas aw in thermodynamics,r

17 T-Theoretizitdt und Holismus (Frankfurt/Main-Bern: eter Lang Verlag,1983). In English,see also his "Formal Conditions of TheoreticalTerms," forth-coming.

18 Compare my"Theoryand Measurement," op. cit., for a similardefinitionntermsof measurement.









the aw ofdiminishingeturnsn economics. But weak definabilitysexpressedby the lastformula s not interestingecause it is trivial:

takeB tobe a singleton.ntuitively, hat smissingnthe formula sof course a characterizationf B beinga law. Fortunately,here s awayoutof thisproblemwhichdoes not depend on an explication flawlikeness.n the present ontext t s sufficiento restricthe classof "T-admissible ubtheories"B by requiring hatB have thesamedegree of invarianceforfi as M. In order to explicatethe lastex-pression, et us look very ormallyt possiblevariations ffXn somemodelx. In general, mongthosefunctionsf hichwecan substitutein x forfX herewillbe functionsf uch thatx4i[f is nota modelofT but only structure or T, and there lso willbe functionsf' uchthatx_[f'] is a modelofT. The classofall thosef' forwhichx_[f']again is a modelof T we call the range of nvarianceoffi in x withrespectto M, denoted by RI(M, i, x):

RI(M, i, x) = {f/x[f] E M}

Thisdefinitionmakes ense andmaybe used also for ubsetsB of M.In concrete heoriesRI(M, i, x) can be exhaustedbysome classrx f

transformationsn the sense thatprecisely llmembersf fRI(M, i,x) can be obtained fromfiX y some transformation0E rx:f = 0(fiX).

It is then aid that heory is invariant ndertransformationsf theformgivenby Tx, and thisis whywe speak of the "range of in-variance." The range of invariance of the position functioninmechanics, or nstance, s describedby Galilean transformationsnthis sense._ Wesaythat subclassB ofM has the amedegreeof nvarianceforfi as T iff, or all x E B, the range of invarianceoffi in x with

respectto B is at least "as great as" (i.e., contains) the range ofinvarianceoff, n x withrespectto M: RI(M, i, x) c RI(B, i, x).Because of B c M this mpliesthatthe tworangesare equal. Intu-itively,fB has thesamedegreeofinvariance or i as T, then ntheaxiomscharacterizing the connection ffiwith heotherfunctionsis not "stronger" than it is in the axioms for M. In first-ordertheories, hismeans thattheaxiomsforB expressfurther equire-mentsonlyfor the function-symbolsifferentrom i. Stilldiffer-

ently:B represents ome special law forT which s invariant nderthe same transformationsoff2) s T. By putting ogether he tworequirementswe obtain the following,more explicitdefinition ftheoreticity2:

Term iis T-theoretical2ff here s someB ' M such hat(i) Vx E B Vf, '(x_i[f E B A x_[f'] E B -f = f)(ii) Vx E B(RI(M, ,x) - RI(B, ,x))









In the verbalformulationwe speak of "definability" ecause of re-quirement i). In somesuitable ubclassofmodels "subtheory"), .e.,

in some special cases,fi can be uniquelydetermined, nd in thissubtheoryi therefore s definable.The requirement n degrees ofinvariance arrows own the class ofadmissible subtheories"whichmaybe used inorderto "define"fi. Without he atter equirementeach termof T wouldbe T-theoretical2.

In quantitative heories hetransformationsescribingRI(M, i, x)usuallywillbe composed of dilations and (eventually) thertrans-formations.f this s so thenrequirements i) and (ii) can never be

jointly atisfied, ecause ifB has the samedegree ofinvariance s Tthen i is determinedn B at mostup to transformationsfscale. Inorderto makeroomforquantitative heoretical erms ccurringntheorieswith nvariances f scale it therefore snecessary oweakentheuniquenesscondition i). Equalityhas to be replacedby"equiva-lence of scale"; i.e., '=' by -', where f - f" means thatf' can beobtainedfromf ysometransformationfscale, .e., sometransfor-mationof theformf'(a) = ff(a) + -y.'9f thisqualifications takenintoaccount,thentheoreticity2eproduces theinformal istinction

betweentheoretical nd nontheoretical erms s drawn n the itera-turebyphilosophers f science. In classicalmechanics,massmandthecomponentforcesfaremechanics-theoretical2;nclassicalcolli-sion mechanicsmassm is theoretical2; nd in exchangeeconomicsequilibrium prices and equilibrium distributionsare theoreti-cal2-all otherfunctions eingnontheoretical2, espectively.20c-cordingto our presentapproach, in contrast o earlierconsidera-tions of such questions,these resultsare now provable, once the

axiomatization s given.The difficultartof such proofs s to showthatsome term s T-nontheoretical2,or thisamounts to showingthat here s no invariantubtheorynwhich he term an be defined.

This remark eveals theclose connectionthattheoreticity2as tothe originalaccount of logical empiricism. he T-nontheoretical2terms repartof theobservationanguage ifwe use this erm n themoresophisticated ersion s referringo theantecedentlyvailablevocabulary),heirmeaninghas tobe already stablishedfwewant ouse them n thecontextof T. For T offers o means to determine

themeven invery pecial cases; i.e., T does not contribute o fixing

19Of course,heref andf' musttaketheirvalues inspaces inwhichmultiplicationand additionmake sense. But this s ust anotherwayofsaying hatfi s a quantity.Compare ibid. fora more explicit treatment f thisfeature.

20 See my"On a New Definitionof Theoreticity,"Dialectica, xxxix, 2 (1985):127-145, for xiomatizations f thesetheories nd forformalproofs bout theore-ticity.









theirmeaning.T-theoretical2 erms, n the otherhand, are thosetermsntroducedby meansof T whichbecome meaningfulhrough

theirrole or position n T. The presentdefinition lso shows whylogical empiricists as well as theirsuccessors) did not succeed indefiningheoretical erms. he notionofa definition s explicated ntermsof first-orderogic simply s too strongto give an adequatepicture of hownew terms re introduced n theframeof compre-hensive mpirical heories. Theoretical" terms re notdefinable nthe sense of first-orderefinability;uch definitionwould deprivethemofany mportance.Butthey re "definable" na weaker ense,namely,weakly nd invariantly,nd by thistheirmeaning s deter-mined at least as far s is necessary orus to use them nconnectionwith hetheory theycome from."

The status fT-nontheoretical2erms s beingnot determinable tall "in T" is also relevantfora deeper understanding f theoreti-city1. he explanationforthe problemoftheoretical, erms iven nsection I does notdependon whether he term nder considerationis T-theoretical, r not. The problem of theoretical1 ermsarisesfromthe coherentistwayof measuringthe functions f T, and it

arisesforT-nontheoreticallerms n thesameway.So, inconnectionwiththisexplanation, natural question to ask is why, hen, therecan be T-nontheoreticallerms t all. For if T is presupposed n alldeterminationsorall terms, hen all terms f T should be T-theo-reticall. But in the light of theoreticity2e can now see whyT-nontheoreticall ermsmayindeed occur. In determinations f T-nontheoretical2 erms tmakes no differencewhetherT is presup-posed or not. For ifa term s T-nontheoretical2,hen, ccordingtothe above definitions,tsdeterminationy meansof T is impossible

(there s no subtheory in whichthe termcan be "defined"); so,even if T is presupposed duringthe course of measurement T-theoreticityl)hisdoes nothelp, does not contribute o finding hemeasuredvalue.Therefore, n thepragmaticevel ofwhat cientistsdo, it will be difficult o detectevidencefortheiractuallypresup-posing T in determinations f T-nontheoretical2erms. n otherwords, T-nontheoretical2 erms to the determination f which Tcannotcontribute re likely o function s T-nontheoreticallerms

on thepragmaticevel. n this ense wemay ay that hedefinitionftheoretical2erms n the lightofwhatwas said in section I aboveexplainswhy here re T-nontheoreticallerms t all (at east n sometheories).

Finally,we wantto stress he mportance f the newdefinition orthe philosophyof science. Up to now no precise and "workable"distinction etweentheoretical nd nontheoretical ermswas avail-









able.2' Butonly f t is available will tbecomepossible to investigatemoreglobalstructures fscience. Moreprecisely,with he newdefi-

nitionof theoreticity2 e obtaina preciseand workablerelationof"presupposition" mongtheories.TheoryTpresupposestheory ' ifsome T-nontheoretical2erm s T'-theoretical2. he idea here is thatT' gives omemeaning o (someway fdeterminationor) heT-non-theoretical2ermwhich annot be providedbyT. On thebasis of thenewdefinition he structure f thispresupposition elationcan beinvestigatedmpirically,.e., bydetermining herelation n the do-mainof empirical heories.By thispossibility heholist'sclaimthatscience andknowledge) orms ne (or at leastonly fewratherbig)inseparableunit(s) gets a newdimension fprecision nd empiricaldecidability.

One of thisJOURNAL'S editorsnoted-quite correctly-that T-nontheoretical2 ermsmayfall ntotwocategories,namely a) thosewhosedeterminations via some othertheoryT*, and (b) those forwhichwedo nothavesuch a T*. In fact, heremaybe T-nontheore-tical2termsthatare not determinedby anyothertheoryT*. Theexistence of such termsraises philosophicalquestionsabout how

science is anchored in human practice and in natural language.These questionsare beyond thescope ofthepresentpaper,butthefactthat our distinctioneads into such deep philosophicalwatercertainly oes not countagainst our approach.

CONCLUSION

Giventwocriteria r definitions f theoreticityhichobviouslynei-therare identical nor lead to identicalresults, t is naturalto askwhich s to be preferred.This, however,turnsout to be difficult,because the twodefinitionsre not reallycompatible. Let us firstconsider some aspectsof detail.

First,theoreticity2as littleto do with"presupposition,"whichwas thecentralfeatureof theoreticityl.WhetherT is or is notpre-supposedduringmeasurements orfi s irrelevant owhetherfis oris not T-theoretical2.n order to apply the newdefinition, o prob-lemofmeaningfulmeasurement eeds to be solved; thequestionofidentifyingunctionfi oes notarise.This, ofcourse, does notmeanthat the latterproblemand question are not interestingn other

respects; tonlymeansthat hey an be separated from hequestionof theoreticity2.n view of the twoaspectsto be found n theoreti-city,-presupposition and determination-wecan saythattheoreti-

21Formal, philosophicallyminded proposals, like,e.g., David Lewis's "How toDefineTheoreticalTerms" have neverbeen applied to real empiricaltheories, ndforgood reasons; for Lewis's paper see this OURNAL, LXVII, 13 (July9, 1970):427-446.









city2 s a more economical concept because it involvesthe secondaspect only.

Second, as concerns the aspect of determination, heoreticity2gives a muchmore detailed picturethan theoreticityl. ven on avery nformal eveltheoreticity2tresses he role ofspecial laws andof invariances. This, togetherwith the conceptual apparatus ofmeasuringmodels,represents good deal ofrefinementver theo-reticity,

Third, theoreticity2oes not imply any problemof theoreticalterms.This is due to an exchangeof quantifiersn the two defini-tions.Whereasthedefinitionftheoreticity,n section I startswithan universal uantifier ver "determinations"or "measurement"),the definition f theoreticity2tartswithan existential uantifier'there s some B ' M. . . .' Intuitively,heproblemof theoretical,terms rises preciselybecause of the universalformof quantifica-tion. fonly he existence fsomepossibilityfdeterminationsomeB) is required, hen there s roomfordeterminations hat ventuallydo notpresuppose22 heoryT-even iftheterm o be determinedsT-theoretical2.n the ight f the discussion f section I this esult s

not very urprising. f the problem of theoretical, erms ssentiallyfollowsfrom the coherentistway of measuringfi, i.e., fromtheaspect of presupposition n theoreticityl,nd if theoreticity2oesnot contain any such aspect,thentheoreticity2illnotyield uch aproblem.

Fourth,theoreticity2an be made formally recise, providedweare willing o talk bout theories f theform utlinedabove. In thisrespect theoreticity2s clearlysuperior to theoreticityl, hich-asalready tressed-is largely pragmatic ffair.

Fifth, heoreticity2rovidesan explanation for the existenceofT-nontheoreticallerms. hat such terms xist snot obvious, t leastifwe lookat theoreticity,rom hemore generalpointof viewdevel-oped in section I.

These five solateditemsall favortheoreticity2,ut they re notsufficiento bridge the gap betweenpragmatic onsiderations ndlogical distinctions. ery broadly, he relationbetween thetwono-tions is this.Theoreticity,s more comprehensive,n its possible

rangeofapplication which ncludesnonformalizedheories) s wellas in tsconceptualapproach because itrelieson or at least scloselyconnectedwith theory fmeaning).Theoreticity2,ncontrast, p-plies only to formalized heoriesand uses only"one half" of theconceptualframeof theoreticityl, amelythe "half" centering n

22 At least if we do not adopt the coherentistpicture.









determination. ut in thisdomain theoreticity2rovides more pre-cise and refined icture. fwe wantto pressthis elation nto a single

expression,perhaps specialization' s best. Theoreticity2 aybe re-garded as a specializationof theoreticityl. his relation does notdevaluate theoreticity2,ecause often it is through"interesting"specializations hat comprehensive heoriesgain empiricalcontentand scientificeputationthink f Newtonianmechanics nd the awof gravitation).

Let us concludeby summarizing ur main results n theformoffour theses: 1) The distinction etweentheoretical nd nontheore-

tical terms fa theory an be drawn n a precise waywithout efer-ence to questionsofmeaning, nd in line with he original deas oflogical empiricism.2) Sneed's pragmatic problem of theoretical,terms"- which rises from he existenceoftheoretical, erms-canbe explained by a coherentist heory fmeaning,which s appliedtothespecial case of themeaningof a function n a theory. 3) If thecriterion f theoreticity,s reformulatedn a way that focusesonquestionsofmeaning, t no longercontributes o thedrawingof adistinction etween T-theoretical nd T-nontheoretical erms. 4)

The new definitionof theoretical2terms explains why-despite(3)-on the pragmatic evel of scientists'behaviortheredo existT-nontheoreticallerms.

WOLFGANG BALZER

Universitait unchen

Documents

Balzer, Wolfgang - Theoretical Terms a New Perspective, 1986