PhilosophicalFoundationsofPhysics

byRudolphCarnap(1966)

CHAPTER23TheoriesandNonobservablesCHAPTER24CorrespondenceRulesCHAPTER25HowNewEmpiricalLawsAreDerivedfromTheoreticalLawsCHAPTER26TheRamseySentence

CHAPTER23TheoriesandNonobservables

ONEOFTHEmost important distinctions between two types of laws inscienceisthedistinctionbetweenwhatmaybecalled(thereisnogenerallyaccepted terminology for them) empirical laws and theoretical laws.Empirical laws are laws that can be confirmed directly by empiricalobservations.Theterm"observable"isoftenusedforanyphenomenonthatcan be directly observed, so it can be said that empirical laws are lawsaboutobservable.

Here, a warning must be issued. Philosophers and scientists have quitedifferentwaysofusingtheterms"observable"and"nonobservable".Toaphilosopher, "observable" has a very narrowmeaning. It applies to suchpropertiesas"blue","hard","hot".Thesearepropertiesdirectlyperceivedby thesenses.To thephysicist, thewordhasamuchbroadermeaning. Itincludes any quantitativemagnitude that can bemeasured in a relativelysimple, direct way. A philosopher would not consider a temperature of,perhaps, 80 degrees centigrade, or aweight of 93 pounds, an observablebecause there is no direct sensory perception of such magnitudes. To a

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physicist, both are observables because they can be measured in anextremely simpleway. The object to beweighed is placed on a balancescale. The temperature is measured with a thermometer. The physicistwould not say that the mass of a molecule, let alone the mass of anelectron, is something observable, because here the procedures ofmeasurementaremuchmorecomplicatedandindirect.Butmagnitudesthatcanbeestablishedbyrelativelysimpleprocedures-lengthwitharuler,timewith a clock, or frequency of lightwaveswith a spectrometer-are calledobservables.

Aphilosophermight object that the intensity of an electric current is notreally observed.Only a pointer positionwas observed.An ammeterwasattachedto thecircuitand itwasnoted that thepointerpointed toamarklabelled 5.3. Certainly the current's intensity was not observed. It wasinferredfromwhatwasobserved.

Thephysicistwouldreplythatthiswastrueenough,buttheinferencewasnotverycomplicated.Theprocedureofmeasurementissosimple,sowellestablished, that it could not be doubted that the ammeterwould give anaccuratemeasurementofcurrentintensity.Therefore,itisincludedamongwhatarecalledobservables.

Thereisnoquestionhereofwhoisusingtheterm"observable"inarightor proper way. There is a continuum which starts with direct sensoryobservations and proceeds to enormously complex, indirect methods ofobservation.Obviouslynosharplinecanbedrawnacrossthiscontinuum;itis amatter of degree. A philosopher is sure that the sound of his wife'svoice, coming from across the room, is an observable. But suppose helistens to her on the telephone. Is her voice an observable or isn't it? Aphysicistwouldcertainlysaythatwhenhelooksatsomethingthroughanordinarymicroscope,heisobservingitdirectly.Isthisalsothecasewhenhe looks into an electron microscope? Does he observe the path of aparticlewhenheseesthetrackitmakesinabubblechamber?Ingeneral,thephysicistspeaksofobservablesinaverywidesensecomparedwiththenarrow sense of the philosopher, but, in both cases, the line separatingobservablefromnonobservableishighlyarbitrary.Itiswelltokeepthisinmindwheneverthesetermsareencounteredinabookbyaphilosopherorscientist.Individualauthorswilldrawthelinewhereitismostconvenient,

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dependingontheirpointsofview,andthereisnoreasonwhytheyshouldnothavethisprivilege.

Empirical laws, in my terminology, are laws containing terms eitherdirectly observable by the senses or measurable by relatively simpletechniques.Sometimessuchlawsarecalledempiricalgeneralisations,asareminder that they have been obtained by generalising results found byobservationsandmeasurements.They includenotonly simplequalitativelaws(suchas,"Allravensareblack")butalsoquantitativelawsthatarisefrom simple measurements. The laws relating pressure, volume, andtemperature of gases are of this type.Ohm's law, connecting the electricpotentialdifference,resistance,andintensityofcurrent,isanotherfamiliarexample. The scientist makes repeated measurements, finds certainregularities,andexpressestheminalaw.Thesearetheempiricallaws.Asindicated in earlier chapters, they are used for explaining observed factsandforpredictingfutureobservableevents.

Thereisnocommonlyacceptedtermforthesecondkindoflaws,whichIcall theoretical laws. Sometimes they are called abstract or hypotheticallaws. "Hypothetical" is perhaps not suitable because it suggests that thedistinctionbetweenthetwotypesoflawsisbasedonthedegreetowhichthelawsareconfirmed.Butanempiricallaw,ifitisatentativehypothesis,confirmedonlytoalowdegree,wouldstillbeanempiricallawalthoughitmightbesaidthatitwasratherhypothetical.Atheoreticallawisnottobedistinguished from an empirical law by the fact that it is not wellestablished,butby the fact that it contains termsof adifferentkind.Theterms of a theoretical law do not refer to observables even when thephysicist'swidemeaning forwhat can be observed is adopted. They arelaws about such entities as molecules, atoms, electrons, protons,electromagneticfields,andothersthatcannotbemeasuredinsimple,directways.

Ifthereisastaticfieldoflargedimensions,whichdoesnotvaryfrompointtopoint,physicists call it anobservable fieldbecause it canbemeasuredwithasimpleapparatus.Butifthefieldchangesfrompointtopointinverysmalldistances,orvariesveryquicklyintime,perhapschangingbillionsoftimes each second, then it cannot be directly measured by simpletechniques.Physicistswouldnotcallsuchafieldanobservable.Sometimes

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aphysicistwilldistinguishbetweenobservablesandnonobservablesinjustthisway. If themagnitude remains the samewithin large enough spatialdistances, or large enough time intervals, so that an apparatus can beapplied for a directmeasurement of themagnitude, it is called amacro-event. If themagnitude changeswithin such extremely small intervals ofspaceandtimethatitcannotbedirectlymeasuredbysimpleapparatus,itisa micro-event. (Earlier authors used the terms "microscopic" and"macroscopic", but today many authors have shortened these terms to"micro" and "macro". ) A micro-process is simply a process involvingextremelysmallintervalsofspaceandtime.Forexample,theoscillationofanelectromagneticwaveofvisiblelightisamicro-process.Noinstrumentcan directly measure how its intensity varies. The distinction betweenmacro-andmicro-conceptsissometimestakentobeparalleltoobservableand nonobservable. It is not exactly the same, but it is roughly so.Theoretical lawsconcernnonobservables,andveryoften thesearemicro-processes.Ifso, thelawsaresometimescalledmicro-laws.Iusetheterm"theoreticallaws"inawidersensethanthis,toincludeallthoselawsthatcontainnonobservables, regardlessofwhether theyaremicro-conceptsormacro-concepts.

It is true, as shown earlier, that the concepts "observable" and"nonobservable" cannot be sharply defined because they lie on acontinuum. In actual practice, however, the difference is usually greatenoughsothereisnotlikelytobedebate.Allphysicistswouldagreethatthelawsrelatingpressure,volume,andtemperatureofagas,forexample,are empirical laws. Here the amount of gas is large enough so that themagnitudes to be measured remain constant over a sufficiently largevolumeofspaceandperiodoftimetopermitdirect,simplemeasurementswhich can then be generalised into laws.All physicistswould agree thatlaws about the behaviour of single molecules are theoretical. Such lawsconcern amicro-process aboutwhich generalisations cannot be based onsimple,directmeasurements.

Theoretical laws are, of course, more general than empirical laws. It isimportanttounderstand,however,thattheoreticallawscannotbearrivedatsimplybytakingtheempiricallaws,thengeneralisingafewstepsfurther.Howdoesaphysicistarriveatanempiricallaw?Heobservescertainevents

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in nature.He notices a certain regularity.He describes this regularity bymakinganinductivegeneralisation.Itmightbesupposedthathecouldnowputtogetheragroupofempiricallaws,obsenesomesortofpattern,makeawiderinductivegeneralisation,andarriveatatheoreticallaw.Suchisnotthecase.

Tomake this clear, suppose it has been observed that a certain iron barexpandswhenheated.Aftertheexperimenthasbeenrepeatedmanytimes,alwayswiththesameresult,theregularityisgeneralisedbysayingthatthisbarexpandswhenheated.Anempiricallawhasbeenstated,eventhoughithasanarrowrangeandappliesonlytooneparticularironbar.Nowfurthertestsaremadeofother ironobjectswith theensuingdiscovery thateverytimeanironobjectisheateditexpands.Thispermitsamoregenerallawtobe formulated, namely that all bodies of iron expand when heated. Insimilarfashion,thestillmoregenerallaws"Allmetals...",then"Allsolidbodies...",aredeveloped.Theseareallsimplegeneralisations,eachabitmoregeneralthanthepreviousone,buttheyareallempiricallaws.Why?Because ineachcase, theobjectsdealtwithareobservable (iron,copper,metal, solidbodies); in eachcase the increases in temperature and lengtharemeasurablebysimple,directtechniques.

In contrast, a theoretical law relating to this process would refer to thebehaviourofmoleculesintheironbar.Inwhatwayisthebehaviourofthemoleculesconnectedwiththeexpansionofthebarwhenheated?Youseeatoncethatwearenowspeakingofnonobservables.Wemustintroduceatheory-theatomictheoryofmatter-andwearequicklyplungedintoatomiclawsinvolvingconceptsradicallydifferentfromthosewehadbefore.Itistrue that these theoretical concepts differ from concepts of length andtemperature only in the degree to which they are directly or indirectlyobservable,but thedifferenceissogreat that there isnodebateabout theradicallydifferentnatureofthelawsthatmustbeformulated.

Theoretical laws are related to empirical laws in a way somewhatanalogous to the way empirical laws are related to single facts. Anempiricallawhelpstoexplainafactthathasbeenobservedandtopredictafactnotyetobserved.Insimilarfashion,thetheoreticallawhelpstoexplainempirical laws already formulated, and to permit the derivation of newempiricallaws.Justasthesingle,separatefactsfallintoplaceinanorderly

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pattern when they are generalised in an empirical law, the single andseparateempiricallawsfitintotheorderlypatternofatheoreticallaw.Thisraisesoneof themainproblems in themethodologyofscience.Howcanthekindofknowledgethatwilljustifytheassertionofatheoreticallawbeobtained? An empirical lawmay be justified by making observations ofsingle facts. But to justify a theoretical law, comparable observationscannot be made because the entities referred to in theoretical laws arenonobservables.

Before takingup this problem, some remarksmade in an earlier chapter,abouttheuseoftheword"fact",shouldberepeated.Itisimportantinthepresent context to be extremely careful in the use of this word becausesomeauthors,especiallyscientists,use"fact"or"empiricalfact"forsomepropositions which I would call empirical laws. For example, manyphysicistswill refer to the 'fact" that thespecificheatofcopper is .090. Iwould call this a law because in its full formulation it is seen to be auniversalconditionalstatement:"Foranyx,andanytimet, ifxisasolidbodyof copper, then the specific heat of x at t is .090."Somephysicistsmayevenspeakofthelawofthermalexpansion,Ohm'slaw,andothers,asfacts.Ofcourse, theycanthensaythat theoretical lawshelpexplainsuchfacts.Thissoundslikemystatementthatempiricallawsexplainfacts,buttheword"fact"isbeingusedhereintwodifferentways.Irestrictthewordto particular, concrete facts that can be spatiotemporally specified, notthermalexpansion ingeneral,but theexpansionof this ironbarobservedthismorningat teno'clockwhen itwasheated. It is important tobear inmindtherestrictedwayinwhichIspeakoffacts.Iftheword"fact"isusedin an ambiguous manner, the important difference between the ways inwhichempiricalandtheoreticallawsserveforexplanationwillbeentirelyblurred.

Howcantheoreticallawsbediscovered?Wecannotsay:"Let'sjustcollectmore andmore data, then generalise beyond the empirical laws untilwereach theoretical ones."No theoretical lawwas ever found thatway.Weobserve stones and trees and flowers, noting various regularities anddescribing them by empirical laws. But no matter how long or howcarefully we observe such things, we never reach a point at which weobserve a molecule. The term "molecule" never arises as a result of

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observations. For this reason, no amount of generalisation fromobservations will ever produce a theory of molecular processes. Such atheorymustariseinanotherway.Itisstatednotasageneralisationoffactsbutasahypothesis.Thehypothesisisthentestedinamanneranalogousincertainwaystothetestingofanempiricallaw.Fromthehypothesis,certainempirical lawsarederived,and theseempirical lawsare tested in turnbyobservationoffacts.Perhapstheempiricallawsderivedfromthetheoryarealreadyknownandwellconfirmed. (Such lawsmayevenhavemotivatedtheformulationof the theoretical law.)Regardlessofwhether thederivedempirical laws are known and confirmed, or whether they are new lawsconfirmed by new observations, the confirmation of such derived lawsprovidesindirectconfirmationofthetheoreticallaw.

The point to be made clear is this. A scientist does not start with oneempiricallaw,perhapsBoyle'slawforgases,andthenseekatheoryaboutmolecules from which this law can be derived. The scientist tries toformulate amuchmoregeneral theory fromwhichavarietyof empiricallaws can be derived. The more such laws, the greater their variety andapparent lack of connection with one another, the stronger will be thetheory that explains them. Some of these derived laws may have beenknown before, but the theory may also make it possible to derive newempiricallawswhichcanbeconfirmedbynewtests.Ifthisisthecase,itcanbesaidthatthetheorymadeitpossibletopredictnewempiricallaws.The prediction is understood in a hypothetical way. If the theory holds,certainempirical lawswill alsohold.Thepredictedempirical lawspeaksabout relations between observables, so it is now possible to makeexperiments to see if the empirical law holds. If the empirical law isconfirmed, it provides indirect confirmation of the theory. Everyconfirmationofa law,empiricalor theoretical, is,ofcourse,onlypartial,nevercompleteandabsolute.Butinthecaseofempiricallaws,itisamoredirect confirmation. The confirmation of a theoretical law is indirect,because it takes place only through the confirmation of empirical lawsderivedfromthetheory.

Thesupremevalueofanew theory is itspower topredictnewempiricallaws. It is true that it alsohasvalue inexplainingknownempirical laws,butthis isaminorvalue.Ifascientistproposesanewtheoreticalsystem,

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fromwhichnonewlawscanbederived, thenit is logicallyequivalent tothe set of all known empirical laws. The theory may have a certainelegance, and itmay simplify to some degree the set of all known laws,althoughitisnotlikelythattherewouldbeanessentialsimplification.Onthe other hand, every new theory in physics that has led to a great leapforward has been a theory from which new empirical laws could bederived.IfEinsteinhaddonenomorethanproposehistheoryofrelativityasanelegantnewtheorythatwouldembracecertainknownlaws-perhapsalsosimplifythemtoacertaindegree-thenhistheorywouldnothavehadsucharevolutionaryeffect.

Of course it was quite otherwise. The theory of relativity led to newempirical lawswhich explained for the first time such phenomena as themovementoftheperihelionofMercury,andthebendingoflightraysintheneighbourhoodofthesun.Thesepredictionsshowedthatrelativitytheorywasmorethanjustanewwayofexpressingtheoldlaws.Indeed,itwasatheory of great predictive power. The consequences that can be derivedfrom Einstein's theory are far from being exhausted. These areconsequences that could not have been derived from earlier theories.Usually a theory of such power does have an elegance, and a unifyingeffectonknownlaws.Itissimplerthanthetotalcollectionofknownlaws.Butthegreatvalueofthetheoryliesinitspowertosuggestnewlawsthatcanbeconfirmedbyempiricalmeans.

CHAPTER24CorrespondenceRules

AN IMPORTANT qualificationmust now be added to the discussion oftheoretical laws and terms given in the last chapter. The statement thatempiricallawsarederivedfromtheoreticallawsisanoversimplification.Itis not possible to derive themdirectly because a theoretical law containstheoreticalterms,whereasanempiricallawcontainsonlyobservableterms.

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Thispreventsanydirectdeductionofanempirical lawfroma theoreticalone.

To understand this, imagine that we are back in the nineteenth century,preparingtostateforthefirsttimesometheoreticallawsaboutmoleculesin a gas. These laws are to describe the number of molecules per unitvolume of the gas, the molecular velocities, and so forth. To simplifymatters,weassumethatallthemoleculeshavethesamevelocity.(Thiswasindeed the original assumption; later it was abandoned in favour of acertainprobabilitydistributionofvelocities.)Furtherassumptionsmustbemade about what happenswhenmolecules collide.We do not know theexactshapeofmolecules,soletussupposethattheyaretinyspheres.Howdo spheres collide? There are laws about colliding spheres, but theyconcernlargebodies.Sincewecannot

directly observe molecules, we assume their collisions are analogous tothoseof largebodies;perhaps theybehave likeperfectbilliardballsonafrictionless table. These are, of course, only assumptions; guessessuggestedbyanalogieswithknownmacro-laws.

Butnowwecomeupagainstadifficultproblem.Ourtheoreticallawsdealexclusivelywiththebehaviourofmolecules,whichcannotbeseen.How,therefore,canwededucefromsuchlawsalawaboutobservablepropertiessuchasthepressureortemperatureofagasorpropertiesofsoundwavesthat pass through the gas? The theoretical laws contain only theoreticalterms. What we seek are empirical laws containing observable terms.Obviously, such laws cannot be derived without having something elsegiveninadditiontothetheoreticallaws.

Thesomethingelsethatmustbegivenisthis:asetofrulesconnectingthetheoreticaltermswiththeobservableterms.Scientistsandphilosophersofscience have long recognised the need for such a set of rules, and theirnaturehasbeenoftendiscussed.Anexampleofsucharuleis:"Ifthereisan electromagnetic oscillation of a specified frequency, then there is avisiblegreenish-bluecolourofacertainhue."Heresomethingobservableisconnectedwithanonobservablemicro-process.

Another example is: "The temperature (measured by a thermometer and,

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therefore, an observable in thewider sense explained earlier) of a gas isproportional to the mean kinetic energy of its molecules." This ruleconnects a nonobservable in molecular theory, the kinetic energy ofmolecules,withanobservable,thetemperatureofthegas.Ifstatementsofthiskinddidnotexist, therewouldbenowayofderivingempirical lawsaboutobservablesfromtheoreticallawsaboutnonobservables.

Different writers have different names for these rules. I call them"correspondence rules". P. W. Bridgman calls them operational rules.NormanR.Campbellspeaksof themas the"Dictionary".'Since therulesconnectaterminoneterminologywithaterminanotherterminology,theuse of the rules is analogous to the use of a French-English dictionary.What does the French word "cheval" mean? You look it up in thedictionaryandfindthatitmeans"horse".Itisnotreallythatsimplewhenaset of rules is used for connecting nonobservables with observables;nevertheless,thereisananalogyherethatmakesCampbell's"Dictionary"asuggestivenameforthesetofrules.

Thereisatemptationattimestothinkthatthesetofrulesprovidesameansfor defining theoretical terms,whereas just the opposite is really true.Atheoreticaltermcanneverbeexplicitlydefinedonthebasisofobservableterms, although sometimes an observable can be defined in theoreticalterms. For example, "iron" can be defined as a substance consisting ofsmall crystalline parts, each having a certain arrangement of atoms andeachatombeingaconfigurationofparticlesofacertaintype.Intheoreticaltermsthen, it ispossible toexpresswhat ismeantbytheobservable term"iron",butthereverseisnottrue.

Thereisnoanswertothequestion:"Exactlywhatisanelectron?"Laterweshallcomebacktothisquestion,becauseitisthekindthatphilosophersarealwaysaskingscientists.Theywantthephysicisttotellthemjustwhathemeans by "electricity", "magnetism", "gravity", "a molecule". If thephysicist explains them in theoretical terms, the philosopher may bedisappointed."ThatisnotwhatImeantatall",hewillsay."Iwantyoutotell me, in ordinary language, what those terms mean." Sometimes thephilosopherwrites a book inwhich he talks about the greatmysteries ofnature."Noone",hewrites,"hasbeenablesofar,andperhapsnooneeverwillbeable, togiveusastraightforwardanswer to thequestion: 'What is

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electricity?' And so electricity remains forever one of the great,unfathomablemysteriesoftheuniverse."

There is no special mystery here. There is only an improperly phrasedquestion.Definitionsthatcannot,inthenatureofthecase,begiven,shouldnotbedemanded.Ifachilddoesnotknowwhatanelephantis,wecantellhimitisahugeanimalwithbigearsandalongtrunk.Wecanshowhimapicture of an elephant. It serves admirably to define an elephant inobservable terms that a child can understand. By analogy, there is atemptationtobelievethat,whenascientistintroducestheoreticalterms,heshould also be able to define them in familiar terms. But this is notpossible.Thereisnowayaphysicistcanshowusapictureofelectricityinthewayhecanshowhischildapictureofanelephant.Eventhecellofanorganism, although it cannot be seen with the unaided eye, can berepresented by a picture because the cell can be seenwhen it is viewedthroughamicroscope.Butwedonotpossessapictureoftheelectron.Wecannot say how it looks or how it feels, because it cannot be seen ortouched.Thebestwecandoistosaythatitisanextremelysmallbodythatbehaves in a certain manner. This may seem to be analogous to ourdescriptionofanelephant.Wecandescribeanelephantasa largeanimalthatbehavesinacertainmanner.Whynotdothesamewithanelectron?

The answer is that a physicist can describe the behaviour of an electrononly by stating theoretical laws, and these laws contain only theoreticalterms.Theydescribethefieldproducedbyanelectron, thereactionofanelectrontoafield,andsoon.Ifanelectron is inanelectrostaticfield, itsvelocity will accelerate in a certain way. Unfortunately, the electron'saccelerationisanunobservable.It isnot liketheaccelerationofabilliardball,which can be studied by direct observation. There is noway that atheoretical concept can be defined in terms of observables. We must,therefore,resignourselvestothefactthatdefinitionsofthekindthatcanbesuppliedforobservabletermscannotbeformulatedfortheoreticalterms.

Itistruethatsomeauthors,includingBridgman,havespokenoftherulesas"operationaldefinitions".Bridgmanhadacertain justification,becausehe used his rules in a somewhat different way, I believe, than mostphysicistsusethem.Hewasagreatphysicistandwascertainlyawareofhisdeparturefromtheusualuseofrules,buthewaswillingtoacceptcertain

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formsofspeechthatarenotcustomary,andthisexplainshisdeparture.Itwaspointedout inapreviouschapter thatBridgmanpreferred tosay thatthere is not just one concept of intensity of electric current, but a dozenconcepts.Eachprocedurebywhichamagnitudecanbemeasuredprovidesan operational definition for that magnitude. Since there are differentproceduresformeasuringcurrent,therearedifferentconcepts.Forthesakeofconvenience,thephysicistspeaksofjustoneconceptofcurrent.Strictlyspeaking,Bridgmanbelieved,heshouldrecognisemanydifferentconcepts,eachdefinedbyadifferentoperationalprocedureofmeasurement.

Wearefacedherewithachoicebetweentwodifferentphysicallanguages.If the customary procedure among physicists is followed, the variousconceptsofcurrentwillbereplacedbyoneconcept.Thismeans,however,thatyouplacetheconceptinyourtheoreticallaws,becausetheoperationalrules are just correspondence rules, as I call them, which connect thetheoretical terms with the empirical ones. Any claim to possessing adefinition-that is,anoperationaldefinitionofthetheoreticalconceptmustbegivenup.Bridgmancouldspeakofhavingoperationaldefinitionsforhistheoretical termsonlybecausehewasnot speakingof ageneral concept.Hewasspeakingofpartialconcepts,eachdefinedbyadifferentempiricalprocedure.

Even in Bridgman's terminology, the question of whether his partialconcepts can be adequately defined by operational rules is problematic.Reichenbachspeaksoftenofwhathecalls"correlativedefinitions".(InhisGerman publications, he calls them Zuordnungsdefinitionen, fromzuordnen,whichmeans to correlate.) Perhaps correlation is a better termthan definition for what Bridgman's rules actually do. In geometry, forinstance, Reichenbach points out that the axiom system of geometry, asdeveloped by David Hilbert, for example, is an uninterpreted axiomsystem.Thebasicconceptsofpoint, line,andplanecould justaswellbecalled "class alpha", "class beta", and "class gamma". We must not beseduced by the sound of familiarwords, such as "point" and "line", intothinkingtheymustbetakenintheirordinarymeaning.Intheaxiomsystem,they are uninterpreted terms. But when geometry is applied to physics,these termsmustbeconnectedwithsomething in thephysicalworld.Wecansay,forexample,thatthelinesofthegeometryareexemplifiedbyrays

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of light in a vacuum or by stretched cords. In order to connect theuninterpreted terms with observable physical phenomena, we must haverulesforestablishingtheconnection.

Whatwecall theserulesis,ofcourse,onlyaterminologicalquestion;weshould be cautious and not speak of them as definitions. They are notdefinitionsinanystrictsense.Wecannotgiveareallyadequatedefinitionof the geometrical concept of "line" by referring to anything in nature.Light rays, stretched strings, and so on are only approximately straight;moreover,theyarenotlines,butonlysegmentsoflines.Ingeometry,alineisinfiniteinlengthandabsolutelystraight.Neitherpropertyisexhibitedbyany phenomenon in nature. For that reason, it is not possible to give anoperational definition, in the strict sense of the word, of concepts intheoreticalgeometry.Thesameistrueofalltheothertheoreticalconceptsofphysics.Strictlyspeaking,thereareno"definitions"ofsuchconcepts.Iprefer not to speak of "operational definitions" or even to useReichenbach's term "correlative definitions". Inmy publications (only inrecentyearshaveIwrittenaboutthisquestion),Ihavecalledthem"rulesofcorrespondence"or,moresimply,"correspondencerules".

Campbellandotherauthorsoftenspeakoftheentitiesintheoreticalphysicsasmathematicalentities.Theymeanbythisthattheentitiesarerelatedtoeachother inways that canbeexpressedbymathematical functions.Butthey are notmathematical entities of the sort that canbedefined in puremathematics.Inpuremathematics,itispossibletodefinevariouskindsofnumbers,thefunctionoflogarithm,theexponentialfunction,andsoforth.It is not possible, however, to define such terms as "electron" and"temperature"bypuremathematics.Physicaltermscanbeintroducedonlywiththehelpofnon-logicalconstants,basedonobservationsoftheactualworld.Herewehaveanessentialdifferencebetweenanaxiomaticsysteminmathematicsandanaxiomaticsysteminphysics.

If we wish to give an interpretation to a term in a mathematical axiomsystem,wecandoitbygivingadefinitioninlogic.Consider,forexample,theterm"number"asitisusedinPeano'saxiomsystem.Wecandefineitinlogicalterms,bytheFrege-Russellmethod,forexample.Inthiswaytheconceptof"number"acquiresacomplete,explicitdefinitiononthebasisofpurelogic.Thereisnoneedtoestablishaconnectionbetweenthenumber5

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and such observables as "blue" and "hot".The terms have only a logicalinterpretation; no connectionwith the actualworld is needed.Sometimesanaxiomsysteminmathematicsiscalledatheory.Mathematiciansspeakof set theory, group theory, matrix theory, probability theory. Here theword "theory" is used in a purely analytic way. It denotes a deductivesystemthatmakesnoreferencetotheactualworld.Wemustalwaysbearinmindthatsuchauseoftheword"theory"isentirelydifferentfromitsuseinreferencetoempiricaltheoriessuchasrelativitytheory,quantumtheory,psychoanalyticaltheory,andKeynesianeconomictheory.

Apostulatesysteminphysicscannothave,asmathematicaltheorieshave,asplendidisolationfromtheworld.Itsaxiomaticterms-"electron","field",and so on-must be interpreted by correspondence rules that connect theterms with observable phenomena. This interpretation is necessarilyincomplete. Because it is always incomplete, the system is left open tomakeitpossible toaddnewrulesofcorrespondence. Indeed, this iswhatcontinuallyhappens in thehistoryofphysics. Iamnot thinkingnowofarevolutioninphysics,inwhichanentirelynewtheoryisdeveloped,butofless radical changes that modify existing theories. Nineteenth-centuryphysics provides a good example, because classical mechanics andelectromagnetics had been established, and, formany decades, therewasrelatively littlechangeinfundamental laws.Thebasic theoriesofphysicsremained unchanged. There was, however, a steady addition of newcorrespondence rules, because new procedures were continually beingdevelopedformeasuringthisorthatmagnitude.

Of course, physicists always face the danger that they may developcorrespondencerulesthatwillbeincompatiblewitheachotherorwiththetheoreticallaws.Aslongassuchincompatibilitydoesnotoccur,however,they are free to add new correspondence rules. The procedure is never-ending. There is always the possibility of adding new rules, therebyincreasingtheamountof interpretationspecifiedfor the theoretical terms;butnomatterhowmuchthisisincreased,theinterpretationisneverfinal.Inamathematicalsystem,itisotherwise.Therealogicalinterpretationofanaxiomatictermiscomplete.Herewefindanotherreasonforreluctancein speaking of theoretical terms as "defined" by correspondence rules. Ittends to blur the important distinction between the nature of an axiom

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systeminpuremathematicsandoneintheoreticalphysics.

Isitnotpossibletointerpretatheoreticaltermbycorrespondencerulessocompletely that no further interpretation would be possible? Perhaps theactualworldislimitedinitsstructureandlaws.Eventuallyapointmaybereached beyond which there will be no room for strengthening theinterpretationofatermbynewcorrespondencerules.Wouldnottherulesthenprovideafinal,explicitdefinitionfortheterm?Yes,butthenthetermwould no longer be theoretical. Itwould become part of the observationlanguage. The history of physics has not yet indicated that physics willbecome complete; there has been only a steady addition of newcorrespondencerulesandacontinualmodificationintheinterpretationsoftheoretical terms.There is nowayof knowingwhether this is an infiniteprocessorwhetheritwilleventuallycometosomesortofend.

Itmay be looked at thisway. There is no prohibition in physics againstmakingthecorrespondencerulesforatermsostrongthatthetermbecomesexplicitly defined and therefore ceases to be theoretical. Neither is thereany basis for assuming that it will always be possible to add newcorrespondence rules. Because the history of physics has shown such asteady, unceasing modification of theoretical concepts, most physicistswouldadviseagainstcorrespondencerulessostrongthatatheoreticaltermbecomes explicitly defined. Moreover, it is a wholly unnecessaryprocedure.Nothingisgainedbyit.Itmayevenhavetheadverseeffectofblockingprogress.

Of course, here again we must recognise that the distinction betweenobservables and nonobservables is amatter of degree.Wemight give anexplicit definition, by empirical procedures, to a concept such as length,becauseitissoeasilyanddirectlymeasured,andisunlikelytobemodifiedby new observations. But it would be rash to seek such strongcorrespondence rules that "electron" would be explicitly defined. Theconcept"electron"issofarremovedfromsimple,directobservationsthatitisbesttokeepittheoretical,opentomodificationsbynewobservations.

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CHAPTER25HowNewEmpiricalLawsAreDerivedfrom

TheoreticalLaws

IN CHAPTER 24, the discussion concerned the ways in whichcorrespondence rules are used for linking the nonobservable terms of atheory with the observable terms of empirical laws. This can be madeclearer by a few examples of the manner in which empirical laws haveactuallybeenderivedfromthelawsofatheory.

The first example concerns the kinetic theory of gases. Its model, orschematicpicture,isoneofsmallparticlescalledmolecules,allinconstantagitation. In its original form, the theory regarded these particles as littleballs, all having the samemass and, when the temperature of the gas isconstant, the sameconstant velocity.Later itwasdiscovered that thegaswouldnotbeinastablestateifeachparticlehadthesamevelocity;itwasnecessarytofindacertainprobabilitydistributionofvelocitiesthatwouldremain stable. This was called the Boltzmann-Maxwell distribution.According to this distribution, there was a certain probability that anymoleculewouldbewithinacertainrangeonthevelocityscale.

When the kinetic theory was first developed, many of the magnitudesoccurringinthelawsofthetheorywerenotknown.Nooneknewthemassof a molecule, or how many molecules a cubic centimetre of gas at acertain temperature and pressure would contain. These magnitudes wereexpressedbycertainparameterswritten into the laws.After theequationswereformulated,adictionaryofcorrespondenceruleswasprepared.Thesecorrespondence rules connected the theoretical terms with observablephenomenainawaythatmadeitpossibletodetermineindirectlythevaluesoftheparametersintheequations.This,inturn,madeitpossibletoderiveempiricallaws.Onecorrespondencerulestatesthatthetemperatureofthegas corresponds to the mean kinetic energy of the molecules. Anothercorrespondence rule connects the pressure of the gas with the impact ofmolecules on the confining wall of a vessel. Although this is adiscontinuousprocessinvolvingdiscretemolecules,thetotaleffectcanberegarded as a constant force pressing on the wall. Thus, by means of

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correspondence rules, thepressure that ismeasuredmacroscopicallybyamanometer (pressure gauge) can be expressed in terms of the statisticalmechanicsofmolecules.

Whatisthedensityofthegas?Densityismassperunitvolume,buthowdowemeasure themassofamolecule?Againourdictionary-averysimpledictionary-suppliesthecorrespondencerule.ThetotalmassMofthegasisthe sum of themassesm of themolecules.M is observable (we simplyweighthegas),butmistheoretical.Thedictionaryofcorrespondencerulesgives the connection between the two concepts. With the aid of thisdictionary, empirical tests of various laws derived from our theory arepossible.On the basis of the theory, it is possible to calculatewhatwillhappentothepressureofthegaswhenitsvolumeremainsconstantanditstemperature is increased.We can calculate what will happen to a soundwaveproducedbystrikingthesideof thevessel,andwhatwillhappenifonly part of the gas is heated. These theoretical laws are worked out intermsofvariousparameters thatoccurwithintheequationsof thetheory.The dictionary of correspondence rules enables us to express theseequations as empirical laws, in which concepts are measurable, so thatempiricalprocedurescansupplyvaluesfortheparameters.Iftheempiricallaws can be confirmed, this provides indirect confirmation of the theory.Many of the empirical laws for gaseswere known, of course, before thekinetic theory was developed. For these laws, the theory provided anexplanation. In addition, the theory led to previously unknown empiricallaws.

The power of a theory to predict new empirical laws is strikinglyexemplifiedbythetheoryofelectromagnetism,whichwasdevelopedabout1860 by two great English physicists,Michael Faraday and JamesClerkMaxwell. (Faraday didmost of the experimentalwork, andMaxwell didmostofthemathematicalwork.)Thetheorydealtwithelectricchargesandhow they behaved in electrical and magnetic fields. The concept of theelectron-a tiny particle with an elementary electric charge-was notformulated until the very end of the century. Maxwell's famous set ofdifferential equations, for describing electromagnetic fields, presupposedonly small discrete bodies of unknown nature, capable of carrying anelectric charge or amagnetic pole.What happenswhen a currentmoves

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along a copper wire? The theory's dictionary made this observablephenomenon correspond to the actual movement along the wire of littlecharged bodies. From Maxwell's theoretical model, it became possible(with the help of correspondence rules, of course) to derivemany of theknownlawsofelectricityandmagnetism.

Themodel didmuchmore than this. Therewas a certain parameter c inMaxwell's equations. According to his model, a disturbance in anelectromagneticfieldwouldbepropagatedbywaveshavingthevelocityc.Electricalexperimentsshowedthevalueofctobeapproximately3x1010centimetrespersecond.Thiswasthesameastheknownvalueforthespeedof light, and it seemed unlikely that it was an accident. Is it possible,physicists asked themselves, that light is simply a special case of thepropagation of an electromagnetic oscillation? It was not long beforeMaxwell's equations were providing explanations for all sorts of opticallaws, including refraction, the velocity of light in different media, andmanyothers.

Physicistswouldhavebeenpleasedenough to find thatMaxwell'smodelexplainedknownelectricalandmagneticlaws;buttheyreceivedadoublebounty.Thetheoryalsoexplainedopticallaws!Finally,thegreatstrengthof the new model was revealed in its power to predict, to formulateempiricallawsthathadnotbeenpreviouslyknown.

The first instancewasprovidedbyHeinrichHertz, theGermanphysicist.About 1890, he began his famous experiments to see whetherelectromagneticwavesoflowfrequencycouldbeproducedanddetectedinthe laboratory.Light is an electromagnetic oscillation andpropagationofwaves at very high frequency. But Maxwell's laws made it possible forsuch waves to have any frequency. Hertz's experiments resulted in hisdiscovery ofwhat at firstwere calledHertzwaves. They are now calledradio waves. At first, Hertz was able to transmit these waves from oneoscillatortoanotheroveronlyasmalldistance-firstafewcentimetres,thenameterormore.Todayaradiobroadcastingstationsendsitswavesmanythousandsofmiles.

Thediscoveryofradiowaveswasonlythebeginningofthederivationofnew laws fromMaxwell's theoreticalmodel.X rayswerediscovered and

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were thought at first tobeparticlesof enormousvelocity andpenetrativepower.Thenitoccurredtophysiciststhat,likelightandradiowaves,thesemight be electromagneticwaves, but of extremely high frequency,muchhigher than the frequencyof visible light.This alsowas later confirmed,and laws dealingwithX rayswere derived fromMaxwell's fundamentalfield equations. X rays proved to bewaves of a certain frequency rangewithinthemuchbroaderfrequencybandofgammarays.TheXraysusedtodayinmedicinearesimplygammaraysofcertainfrequency.Allthiswaslargelypredictableon thebasisofMaxwell'smodel.His theoretical laws,togetherwiththecorrespondencerules,ledtoanenormousvarietyofnewempiricallaws.

Thegreatvarietyof fields inwhichexperimentalconfirmationwasfoundcontributed especially to the strong overall confirmation of Maxwell'stheory. The various branches of physics had originally developed forpracticalreasons; inmostcases, thedivisionswerebasedonourdifferentsense organs. Because the eyes perceive light and colour, we call suchphenomena optics; because our ears hear sounds, we call a branch ofphysics acoustics; and because our bodies feel heat,we have a theory ofheat. We find it useful to construct simple machines based on themovementsofbodies,andwecallitmechanics.Otherphenomena,suchaselectricity and magnetism, cannot be directly perceived, but theirconsequencescanbeobserved.

Inthehistoryofphysics,itisalwaysabigstepforwardwhenonebranchofphysicscanbeexplainedbyanother.Acoustics,forinstance,wasfoundtobe only a part of mechanics, because sound waves are simply elasticitywaves in solids, liquids, and gases.We have already spoken of how thelaws of gases were explained by the mechanics of moving molecules.Maxwell'stheorywasanothergreatleapforwardtowardtheunificationofphysics.Opticswas found tobeapartof electromagnetic theory.Slowlythenotiongrewthatthewholeofphysicsmightsomedaybeunifiedbyonegreat theory. At present there is an enormous gap betweenelectromagnetism on the one side and gravitation on the other. Einsteinmadeseveralattemptstodevelopaunifiedfieldtheorythatmightclosethisgap;morerecently,Heisenbergandothershavemadesimilarattempts.Sofar,however,notheoryhasbeendevisedthatisentirelysatisfactoryorthat

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providesnewempiricallawscapableofbeingconfirmed.

Physics originally began as a descriptive macrophysics, containing anenormousnumberof empirical lawswithnoapparent connections. In thebeginning of a science, scientistsmay be very proud to have discoveredhundredsof laws.But,as thelawsproliferate, theybecomeunhappywiththisstateofaffairs;theybegintosearchforunderlying,unifyingprinciples.In the nineteenth century, there was considerable controversy over thequestion of underlying principles. Some felt that sciencemust find suchprinciples, because otherwise it would be nomore than a description ofnature, not a real explanation. Others thought that that was the wrongapproach,thatunderlyingprinciplesbelongonlytometaphysics.Theyfeltthat the scientist's task is merely to describe, to find out how naturalphenomenaoccur,notwhy.

Todaywe smile abit about thegreat controversyoverdescriptionversusexplanation.Wecanseethattherewassomethingtobesaidforbothsides,but that their way of debating the question was futile. There is no realoppositionbetweenexplanationanddescription.Ofcourse,ifdescriptionistakeninthenarrowestsense,asmerelydescribingwhatacertainscientistdid on a certain day with certain materials, then the opponents of meredescriptionwerequiterightinaskingformore,forarealexplanation.Buttoday we see that description in the broader sense, that of placingphenomenainthecontextofmoregenerallaws,providestheonlytypeofexplanationthatcanbegivenforphenomena.Similarly, if theproponentsofexplanationmeanametaphysicalexplanation,notgroundedinempiricalprocedures, then their opponents were correct in insisting that scienceshould be concerned only with description. Each side had a valid point.Bothdescriptionandexplanation,rightlyunderstood,areessentialaspectsofscience.

The first efforts at explanation, those of the Ionian natural philosophers,werecertainlypartlymetaphysical;theworldisallfire,orallwater,orallchange.Thoseearlyeffortsatscientificexplanationcanbeviewedintwodifferent ways.We can say: "This is not science, but pure metaphysics.There is no possibility of confirmation, no correspondence rules forconnectingthetheorywithobservablephenomena."Ontheotherhand,wecansay:"TheseIoniantheoriesarecertainlynotscientific,butatleastthey

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arepictorialvisionsof theories.Theyare thefirstprimitivebeginningsofscience."

It must not be forgotten that, both in the history of science and in thepsychological history of a creative scientist, a theory has often firstappearedasakindofvisualisation,avisionthatcomesasaninspirationtoa scientist long before he has discovered correspondence rules that mayhelp in confirming his theory. When Democritus said that everythingconsists of atoms, he certainly had not the slightest confirmation for thistheory.Nevertheless,itwasastrokeofgenius,aprofoundinsight,becausetwo thousand years later his vision was confirmed. We should not,therefore,rejecttoorashlyanyanticipatoryvisionofatheory,provideditisone that may be tested at some future time. We are on solid ground,however, if we issue the warning that no hypothesis can claim to bescientificunlessthereisthepossibilitythatitcanbetested.Itdoesnothavetobeconfirmedtobeahypothesis,buttheremustbecorrespondencerulesthatwillpermit, inprinciple,ameansofconfirmingordisconfirming thetheory.Itmaybeenormouslydifficulttothinkofexperimentsthatcantestthe theory; this is the case today with various unified field theories thathavebeenproposed.But ifsuchtestsarepossible inprinciple, the theorycanbecalledascientificone.Whenatheoryisfirstproposed,weshouldnotdemandmorethanthis.

Thedevelopmentofsciencefromearlyphilosophywasagradual,step-by-stepprocess.TheIonianphilosophershadonlythemostprimitivetheories.Incontrast, the thinkingofAristotlewasmuchclearerandonmoresolidscientific ground.Hemade experiments, and he knew the importance ofexperiments, although inother respects hewas an apriorist.Thiswas thebeginningofscience.ButitwasnotuntilthetimeofGalileoGalilei,about1600,thatareallygreatemphasiswasplacedontheexperimentalmethodin preference to aprioristic reasoning about nature.Even thoughmanyofGalileo's concepts had previously been stated as theoretical concepts, hewas the first to place theoretical physicson a solid empirical foundation.CertainlyNewton'sphysics(about1670)exhibits thefirstcomprehensive,systematic theory, containing unobservables as theoretical concepts: theuniversal force of gravitation, a general concept of mass, theoreticalpropertiesoflightrays,andsoon.Histheoryofgravitywasoneofgreat

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generality. Between any two particles, small or large, there is a forceproportional to the square of the distance between them.BeforeNewtonadvancedthistheory,scienceprovidednoexplanationthatappliedtoboththefallofastoneandthemovementsofplanetsaroundthesun.

It is very easy for us today to remark how strange it was that it neveroccurred to anyone before Newton that the same force might cause theappletodropandthemoontogoaroundtheearth.Infact,thiswasnotathoughtlikelytooccurtoanyone.Itisnotthattheanswerwassodifficulttogive; it is thatnobodyhadaskedthequestion.This isavitalpoint.Noone had asked: "What is the relation between the forces that heavenlybodiesexertuponeachotherandterrestrialforcesthatcauseobjectstofalltotheground?"Eventospeakinsuchtermsas"terrestrial"and"heavenly"is to make a bipartition, to cut nature into two fundamentally differentregions.ItwasNewton'sgreatinsighttobreakawayfromthisdivision,toassertthatthereisnosuchfundamentalcleavage.Thereisonenature,oneworld.Newton's universal law of gravitationwas the theoretical law thatexplainedforthefirst timeboththefallofanappleandKepler's lawsforthe movements of planets. In Newton's day, it was a psychologicallydifficult,extremelydaringadventuretothinkinsuchgeneralterms.

Later, of course, bymeans of correspondence rules, scientists discoveredhowtodeterminethemassesofastronomicalbodies.Newton'stheoryalsosaidthattwoapples,sidebysideonatable,attracteachother.Theydonotmovetowardeachotherbecausetheattractingforceisextremelysmallandthe friction on the table very large. Physicists eventually succeeded inactually measuring the gravitational forces between two bodies in thelaboratory.Theyusedatorsionbalanceconsistingofabarwithametalballon each end, suspended at its center by a long wire attached to a highceiling. (The longer and thinner thewire, themore easily the barwouldturn.)Actually,thebarnevercametoanabsoluterestbutalwaysoscillatedabit.Butthemeanpointofthebar'soscillationcouldbeestablished.Aftertheexactpositionof themeanpointwasdetermined, a largepileof leadbrickswas constructed near the bar. (Leadwas used because of its greatspecificgravity.Goldhasanevenhigherspecificgravity,butgoldbricksareexpensive.)Itwasfoundthatthemeanoftheoscillatingbarhadshiftedatinyamounttobringoneoftheballsontheendofthebarnearertothe

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leadpile.Theshiftwasonlyafractionofamillimetre,butitwasenoughtoprovidethefirstobservationofagravitationaleffectbetweentwobodiesina laboratory-an effect that had been predicted by Newton's theory ofgravitation.

IthadbeenknownbeforeNewtonthatapplesfalltothegroundandthatthemoon moves around the earth. Nobody before Newton could havepredicted the outcome of the experimentwith the torsion balance. It is aclassicinstanceofthepowerofatheorytopredictanewphenomenonnotpreviouslyobserved.

CHAPTER26TheRamseySentence

[IuseEforthemathematicallogicsymbolmeaning"thereexists"]

SCIENTIFIC THEORY, in the sense in which we are using the term-theoretical postulates combined with correspondence rules that jointheoretical and observational terms- has in recent years been intenselyanalysedanddiscussedbyphilosophersofscience.Muchofthisdiscussionis so new that it has not yet been published. In this chapter, we willintroduceanimportantnewapproachtothetopic,onethatgoesbacktoalittle known paper by the Cambridge logician and economist, FrankPlumptonRamsey.

Ramseydiedin1930attheageoftwenty-six.Hedidnotlivetocompleteabook,butafterhisdeathacollectionofhispaperswaseditedbyRichardBevan Braithwaite and published in 1931 as The Foundations ofMathematics.Ashortpaperentitled"Theories"appearsinthisbook.Inmyopinion, this paper deservesmuchmore recognition than it has received.Perhaps the book's title attracted only readers interested in the logicalfoundations of mathematics, so that other important papers in the book,

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suchasthepaperontheories,tendedtobeoverlooked.

Ramseywas puzzled by the fact that the theoretical terms- terms for theobjects, properties, forces, and events described in a theory- are notmeaningfulinthesamewaythatobservationalterms-"ironrod","hot",and"red"-aremeaningful.How,then,doesatheoreticaltermacquiremeaning?Everyoneagreesthatitderivesitsmeaningfromthecontextofthetheory."Gene"derives itsmeaning fromgenetic theory. "Electron" is interpretedby the postulates of particle physics. But we are faced with manyconfusing, disturbing questions. How can the empirical meaning of atheoreticaltermbedetermined?Whatdoesagiventheorytellusabouttheactualworld?Doesitdescribethestructureoftherealworld,orisitjustanabstract, artificial device for bringing order into the large mass ofexperiencesinsomewhatthesamewaythatasystemofaccountingmakesitpossibletokeeporderlyrecordsofafirm'sfinancialdealings?Canitbesaidthatanelectron"exists"inthesamesensethatanironrodexists?

There are procedures that measure a rod's properties in a simple, directmanner.Itsvolumeandweightcanbedeterminedwithgreataccuracy.Wecanmeasure thewave lengthsof light emittedby the surfaceof aheatedironrodandpreciselydefinewhatwemeanwhenwesaythattheironrodis"red".Butwhenwedealwiththepropertiesoftheoreticalentities,suchasthe"spin"ofanelementaryparticle,thereareonlycomplicated,indirectprocedures for giving the term an empirical meaning. First we mustintroduce "spin" in the context of an elaborate theory of quantummechanics, and then the theory must be connected with laboratoryobservablesbyanothercomplexsetofpostulates-thecorrespondencerules.Clearly,spinisnotempiricallygroundedinthesimple,directmannerthattherednessofaheatedironrodisgrounded.Exactlywhatisitscognitivestatus?Howcantheoreticalterms,whichmustinsomewaybeconnectedwith the actual world and subject to empirical testing, be distinguishedfrom those metaphysical terms so often encountered in traditionalphilosophy-termsthathavenoempiricalmeaning?Howcantherightofascientist to speakof theoreticalconceptsbe justified,withoutat the sametimejustifyingtherightofaphilosophertousemetaphysicalterms?

In seeking answers to these puzzling questions, Ramsey made a novel,startlingsuggestion.Heproposed that thecombinedsystemof theoretical

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and correspondence postulates of a theory be replaced by what is todaycalledthe"Ramseysentenceofthetheory".IntheRamseysentence,whichisequivalenttothetheory'spostulates,theoreticaltermsdonotoccuratall.In other words, the puzzling questions are neatly side-stepped by theeliminationoftheverytermsaboutwhichthequestionsareraised.

In seeking answers to these puzzling questions, Ramsey made a novel,startlingsuggestion.Heproposed that thecombinedsystemof theoreticaland correspondence postulates of a theory be replaced by what is todaycalledthe"Ramseysentenceofthetheory".IntheRamseysentence,whichisequivalenttothetheory'spostulates,theoreticaltermsdonotoccuratall.In other words, the puzzling questions are neatly side-stepped by theeliminationoftheverytermsaboutwhichthequestionsareraised.

Suppose we are concerned with a theory containing n theoretical terms:''T1'',"T2","T3"..."Tn".Thesetermsareintroducedbythepostulatesofthe theory. They are connected with directly observable terms by thetheory's correspondence rules. In these correspondence rules occur mobservational terms: ''O1'', "O2", "O3" . . . "Om". The theory itself is aconjunction of all the theoretical postulates together with all thecorrespondence postulates.A full statement of the theory, therefore,willcontain the combined sets ofT- andO-terms: ''T1'', "T2", "T3" . . . "Tn";''O1'',"O2","O3"..."On".Ramseyproposedthat,inthissentence,thefullstatement of the theory, all the theoretical terms are to be replaced bycorrespondingvariables:''U1'',"U2","U3"..."Un",andthatwhatlogicianscall"existentialquantifiers"-'(EU1)', '(EU2)',..., '(Eun)'-beaddedtothisformula. It is this new sentence,with itsU-variables and their existentialquantifiers,thatiscalledthe"Ramseysentence".

Toseeexactlyhowthisdevelops,considerthefollowingexample.Takethesymbol"Mol" for theclassofmolecules. Insteadofcallingsomething"amolecule",callit"anelementofMol".Similarly,"Hymol"standsfor"theclassofhydrogenmolecules",and"ahydrogenmolecule"is"anelementofHymol".Itisassumedthataspace-timecoordinatesystemhasbeenfixed,sothataspace-timepointcanberepresentedbyitsfourcoordinates:x,y,z,t. Adopt the symbol "Temp" for the concept of temperature. Then, "the(absolute) temperature of the body b, at time t, is 500" can be written,

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"Temp(b,t)=500".Temperatureisthusexpressedasarelationinvolvingabody,atimepoint,andanumber."Thepressureofabodyb,attimet",canbewritten,"Press(b,t)".Theconceptofmassisrepresentedbythesymbol"Mass".For"themassofthebodyb(ingrams)is150"write,"Mass(b)=150".Massisarelationbetweenabodyandanumber.Let"Vel"standforthevelocityofabody(itmaybeamacro-oramicro-body).Forexample,"Vel(b,r)=(rl,r2,r3)",wheretherightsideoftheequationreferstoatripleofrealnumbers,namely,thecomponentsofthevelocityinthedirectionsofx,y,andz.Velisthusarelationconcerningabody,atimecoordinate,andatripleofrealnumbers.

Generallyspeaking,thetheoreticallanguagecontains"classterms"(suchasterms for macro-bodies, micro-bodies, and events) and "relation terms"(suchastermsforvariousphysicalmagnitudes).

Consider theoryTC. (The "T" stands for the theoretical postulates of thetheory,and"C'standsforthepostulatesthatgivethecorrespondencerules.)Thepostulatesofthistheoryincludesomelawsfromthekinetictheoryofgases, laws concerning the motions of molecules, their velocities,collisions,andsoon.Therearegeneral lawsaboutanygas,andtherearespecial lawsabouthydrogen.Inaddition, therearemacro-gas-theorylawsabout the temperature, pressure, and total mass of a (macro-) gas body.Suppose that the theoreticalpostulatesof theoryTCcontainall the termsmentionedabove.Forthesakeofbrevity,insteadofwritingoutinfullallthe T-postulates, write only the theoretical terms, and indicate theconnectingsymbolismbydots:

(T)...Mol...Hymol...Temp...Press...Mass#..Vel...

TocompletethesymbolisationoftheoryTC,thecorrespondencepostulatesfor some, but not necessarily all, of the theoretical terms must beconsidered. These C-postulates may be operational rules for themeasurement of temperature and pressure (that is, a description of theconstructionofathermometerandamanometerandrulesfordeterminingthevaluesoftemperatureandpressurefromthenumbersreadonthescalesof the instruments). The C-postulates will contain the theoretical terms"Temp"and"Press"aswellasanumberofobservationalterms:''O1'',"O2",

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"O3" . . . "Om". Thus, the C-postulates can be expressed in a brief,abbreviatedwaybywriting:

(C)...Temp...O1,...O2,...O3..Press...O4...Om..

Theentiretheorycannowbeindicatedinthefollowingform:

(TC)...Mol...Hymol...Temp...Press...Mass...Vel...;...Temp...O1,...O2,...O3.....Press.....O4...Om..

To transform this theory TC into its Ramsey sentence, two steps arerequired. First, replace all the theoretical terms (class terms and relationterms)witharbitrarilychosenclassandrelationvariables.Wherever"Mol"occurs in the theory, substitute the variable ''C1'', for example.Wherever"Hymol"occursinthetheoryreplaceitbyanotherclassvariable,suchas"C2".Therelationterm"Temp"isreplacedeverywhere(bothintheTandCportionsofthetheory)byarelationvariable,suchas ''R1''.Inthesameway, "Press", "Mass", and "Vel" are replaced by three other relationvariables,"R2","R3",and"R4"respectively,forexample.Thefinalresultmaybeindicatedinthisway:

..C1...C2...R1...R2...R3...R4...;...R1...O1...O2...O3...R2...O4...Om...

This result (which shouldbe thought of as completelywrittenout, ratherthanabbreviatedasitisherewiththehelpofdots)isnolongerasentence(asT,C,andTCare).Itisanopensentenceformulaor,asitissometimescalled,asentenceformorasentencefunction.

The second step, which transforms the open sentence formula into theRamseysentence,RTC,consistsofwritinginfrontofthesentenceformulasixexistentialquantifiers,oneforeachofthesixvariables:

(RTC)(EC1)(EC2)(ER1)(ER2)(ER3)(ER4)[...C1...C2...R1...R2....R3...R4...;...R1...O1...O2..O3...R2....O4...Om.--]

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Aformulaprecededbyanexistentialquantifierassertsthatthereisatleastone entity (of the type to which it refers) that satisfies the conditionexpressedbytheformula.Thus,theRamseysentenceindicatedabovesays(roughly speaking) that there is (at least) one classC1, one classC2, onerelationR1,oneR2,oneR3andoneR4,suchthat:

(1) these six classes and relations are connected with one another in aspecifiedway(namely,asspecifiedinthefirstorTpartoftheformula),

(2)thetworelations,R1andR2~~areconnectedwiththemobservationalentities,O1,...,Ominacertainway(namely,asspecifiedinthesecondorCpartoftheformula).

Theimportant thingtonoteis that intheRamseysentencethetheoreticaltermshavedisappeared.Intheirplacearevariables.Thevariable"C1"doesnotrefertoanyparticularclass.Theassertionisonlythatthereisatleastone class that satisfies certain conditions. The meaning of the Ramseysentenceisnotchangedinanywayifthevariablesarearbitrarilychanged.Forexample, thesymbols ''C1'' and"C2" can be interchangedor replacedwithotherarbitraryvariables, suchas 'X1 and"X2".Themeaningof thesentenceremainsthesame.

It may appear that the Ramsey sentence is no more than just an othersomewhat roundabout way of expressing the original theory. In a sense,thisistrue.Itiseasytoshowthatanystatementabouttherealworldthatdoes not contain theoretical terms-that is, any statement capable ofempiricalconfirmation-that followsfromthe theorywillalso followfromtheRamsey sentence. In otherwords, theRamsey sentence has preciselythe same explanatory and predictive power as the original system ofpostulates. Ramseywas the first to see this. It was an important insight,although few of his colleagues gave it much attention. One of theexceptionswasBraithwaite,whowasRamsey'sfriendandwhoeditedhispapers. In his book, Scientific Explanation (1953), Braithwaite discussesRamsey'sinsight,emphasisingitsimportance.

The important fact is that we can now avoid all the troublesomemetaphysicalquestionsthatplaguetheoriginalformulationoftheoriesand

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canintroduceasimplificationintotheformulationoftheories.Before,wehadtheoreticalterms,suchas"electron",ofdubious"reality"becausetheyweresofarremovedfromtheobservableworld.Whateverpartialempiricalmeaningcouldbegiventothesetermscouldbegivenonlybytheindirectprocedureofstatingasystemoftheoreticalpostulatesandconnectingthosepostulateswithempiricalobservationsbymeansofcorrespondence rules.In Ramsey's way of talking about the external world, a term such as"electron"vanishes.Thisdoesnotinanywayimplythatelectronsvanish,or, more precisely, that whatever it is in the external world that issymbolised by the word "electron" vanishes. The Ramsey sentencecontinues to assert, through its existential quantifiers, that there issomethingintheexternalworldthathasallthosepropertiesthatphysicistsassigntotheelectron.Itdoesnotquestiontheexistence-the"reality"-ofthissomething. It merely proposes a different way of talking about thatsomething.Thetroublesomequestionitavoidsisnot,"Doelectronsexist?"but,"Whatistheexactmeaningoftheterm'electron'?"InRamsey'swayofspeaking about the world, this question does not arise. It is no longernecessary to inquire about the meaning of "electron", because the termitselfdoesnotappearinRamsey'slanguage.

Itisimportanttounderstand-andthispointwasnotsufficientlystressedbyRamsey-thatRamsey's approach cannot be said to bring theories into theobservationlanguageif"observationlanguage"means(asisoftenthecase)alanguagecontainingonlyobservationaltermsandthetermsofelementarylogic andmathematics.Modern physics demands extremely complicated,high-level mathematics. Relativity theory, for instance, calls for non-Euclideangeometryandtensorcalculus,andquantummechanicscallsforequally sophisticatedmathematical concepts. It cannot be said, therefore,thataphysical theory,expressedasaRamseysentence, isasentenceinasimple observational language. It requires an extended observationallanguage,which isobservationalbecause it containsno theoretical terms,but has been extended to include an advanced, complicated logic,embracingvirtuallythewholeofmathematics.

Supposethat,inthelogicalpartofthisextendedobservationlanguage,weprovide for a seriesD0,D1,D2, . . . of domains ofmathematical entitiessuchthat:

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(1)ThedomainDocontainsthenaturalnumbers(0,1,2,..)

(2)ForanydomainDnsthedomainDn+1containsallclassesofelementsofDn.

The extended language contains variables for all these kinds of entities,togetherwithsuitablelogicalrulesforusingthem.Itismyopinionthatthislanguage is sufficient, not only for formulating all present theories ofphysics,butalsoforallfuturetheories,atleastforalongtimetocome.Ofcourse, it is not possible to foresee the kinds of particles, fields,interactions, or other concepts that physicists may introduce in futurecenturies.However, Ibelieve that such theoreticalconcepts, regardlessofhowbizarreandcomplextheymaybe,can-bymeansofRamsey'sdevice-beformulatedinessentiallythesameextendedobservationlanguagethatisnow available, which contains the observational terms combined withadvancedlogicandmathematics.

On the other hand, Ramsey certainly did not mean-and no one hassuggested-thatphysicistsshouldabandon theoretical terms in theirspeechandwriting.Todo sowould require enormously complicated statements.For example, it is easy to say in the customary language that a certainobject has a mass of five grams. In the symbolic notation of a theory,beforeitischangedtoaRamseysentence,onecansaythatacertainobjectNo.17hasamassoffivegramsbywriting,"Mass(17)=5".InRamsey'slanguage,however, the theoretical term"Mass"doesnotappear.There isonlythevariable(asinthepreviousexample)"R3".Howcanthesentence"Mass (17) = 5" be translated into Ramsey's language? "R3 (17) = 5"obviously will not do; it is not even a sentence. The formula must besupplemented by the assumptions concerning the relation R3 that arespecified in theRamseysentence.Moreover, itwouldnotbesufficient topick out only those postulateformulas containing "R3".An the postulatesare needed.Therefore, the translation of even this brief sentence into theRamsey language demands an immensely long sentence, which containsthe formulas corresponding to all the theoretical postulates, all thecorrespondencepostulates,andtheirexistentialquantifiers.Evenwhentheabbreviatedformusedearlierisadopted,thetranslationisratherlong:

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(ECl)(EC2)...(ER3)(ER4)[...C1...C2...R1...R2...R3...R4...;...R1..Ol...O2...O3...R2...O4...Om...andR3(17)=5].

ItisevidentthatitwouldbeinconvenienttosubstitutetheRamseywayofspeaking for the ordinary discourse of physics inwhich theoretical termsare used. Ramsey merely meant to make clear that it was possible toformulateanytheoryinalanguagethatdidnotrequiretheoreticaltermsbutthatsaidthesamethingastheconventionallanguage.

On the other hand, Ramsey certainly did not mean-and no one hassuggested-thatphysicistsshouldabandon theoretical terms in theirspeechandwriting.Todo sowould require enormously complicated statements.For example, it is easy to say in the customary language that a certainobject has a mass of five grams. In the symbolic notation of a theory,beforeitischangedtoaRamseysentence,onecansaythatacertainobjectNo.17hasamassoffivegramsbywriting,"Mass(17)=5".InRamsey'slanguage,however, the theoretical term"Mass"doesnotappear.There isonlythevariable(asinthepreviousexample)"R3".Howcanthesentence"Mass (17) = 5" be translated into Ramsey's language? "R3 (17) = 5"obviously will not do; it is not even a sentence. The formula must besupplemented by the assumptions concerning the relation R3 that arespecified in theRamseysentence.Moreover, itwouldnotbesufficient topick out only those postulateformulas containing "R3".An the postulatesare needed.Therefore, the translation of even this brief sentence into theRamsey language demands an immensely long sentence, which containsthe formulas corresponding to all the theoretical postulates, all thecorrespondencepostulates,andtheirexistentialquantifiers.Evenwhentheabbreviatedformusedearlierisadopted,thetranslationisratherlong:

(ECl)(EC2)...(ER3)(ER4)[...C1...C2...R1...R2...R3...R4...;...R1..Ol...O2...O3...R2...O4...Om...andR3(17)=5].

ItisevidentthatitwouldbeinconvenienttosubstitutetheRamseywayofspeaking for the ordinary discourse of physics inwhich theoretical termsare used. Ramsey merely meant to make clear that it was possible toformulateanytheoryinalanguagethatdidnotrequiretheoreticaltermsbutthatsaidthesamethingastheconventionallanguage.

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When we say it "says the same thing", we mean this only so far as allobservableconsequencesareconcerned.Itdoesnot,ofcourse,sayexactlythe same thing. The former language presupposes that theoretical terms,such as "electron" and "mass", point to something that is somehowmorethanwhatissuppliedbythecontextofthetheoryitself.Somewritershavecalledthisthe"surplusmeaning"ofaterm.Whenthissurplusmeaningistaken into account, the two languages are certainly not equivalent. TheRamsey sentence represents the full observational content of a theory. Itwas Ramsey's great insight that this observational content is all that isneededforthetheorytofunctionastheory,thatis,toexplainknownfactsandpredictnewones.

It is true that physicists find it vastly more convenient to talk in theshorthand language that includes theoretical terms, such as "proton","electron",and"neutron".Butiftheyareaskedwhetherelectrons"really"exist, theymayrespond indifferentways.Somephysicistsarecontent tothinkabout such termsas "electron" in theRamseyway.Theyevade thequestionaboutexistencebystatingthattherearecertainobservableevents,in bubble chambers and so on, that can be described by certainmathematical functions, within the framework of a certain theoreticalsystem.Beyond that theywillassertnothing.Toaskwhether there reallyareelectronsisthesame-fromtheRamseypointofview-asaskingwhetherquantum physics is true. The answer is that, to the extent that quantumphysics has been confirmed by tests, it is justifiable to say that there areinstancesofcertainkindsofeventsthat,inthelanguageofthetheory,arecalled"electrons".

This point of view is sometimes called the "instrumentalist" view oftheories. It is close to the position defended by Charles Peirce, JohnDewey, and other pragmatists, aswell as bymany other philosophers ofscience.Fromthispointofview,theoriesarenotabout"reality".Theyaresimply language tools for organising the observational phenomena ofexperience into some sort of pattern that will function efficiently inpredictingnewobservables.Thetheoreticaltermsareconvenientsymbols.The postulates containing them are adopted because they are useful, notbecausetheyare"true".Theyhavenosurplusmeaningbeyondthewayinwhichtheyfunctioninthesystem.Itismeaninglesstotalkaboutthe"real"

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electronorthe"real"electromagneticfield.

Opposed to this view is the "descriptive" or "realist" view of theories.(Sometimesthesetwoaredistinguished,butitisnotnecessarytodelveintothese subtle differences.) Advocates of this approach find it bothconvenientandpsychologicallycomfortingtothinkofelectrons,magneticfields, and gravitational waves as actual entities about which science issteadilylearningmore.Theypointoutthatthereisnosharplineseparatinganobservable,suchasanapple,fromanunobservable,suchasaneutron.Anamoebaisnotobservablebythenakedeye,butitisobservablethrougha light microscope. A virus is not observable even through a lightmicroscope, but its structure can be seen quite distinctly through anelectronmicroscope.Aprotoncannotbeobservedinthisdirectway,butitstrackthroughabubblechambercanbeobserved.Ifitispermissibletosaythattheamoebais"real",thereisnoreasonwhyitisnotpermissibletosaythat the proton is equally real. The changing view about the structure ofelectrons,genes,andotherthingsdoesnotmeanthatthereisnotsomething"there",behindeachobservablephenomenon;itmerelyindicatesthatmoreandmoreisbeinglearnedaboutthestructureofthoseentities.

Proponents of the descriptive view remind us that unobservable entitieshave a habit of passingover into theobservable realmasmorepowerfulinstruments of observation are developed. At one time, "virus" was atheoretical term. The same is true of "molecule". Ernst Mach was soopposedtothinkingofamoleculeasanexisting"thing"thatheoncecalledit a "valueless image". Today, even atoms in a crystal lattice can bephotographed by bombarding themwith elementary particles; in a sense,theatomitselfhasbecomeanobservable.Defendersofthisviewarguethatitisasreasonabletosaythatanatom"exists"asitistosaythatadistantstar,observableonlyasafaintspotoflightonalong-exposedphotographicplate,exists.Thereis,ofcourse,nocomparablewaytoobserveanelectron.But that is no reason for refusing to say it exists. Today, little is knownabout its structure; tomorrowagreatdealmaybeknown. It isascorrect,saytheadvocatesofthedescriptiveapproach,tospeakofanelectronasanexistingthingasitistospeakofapplesandtablesandgalaxiesasexistingthings.

It is obvious that there is a difference between the meanings of the

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instrumentalist and the realist ways of speaking.My own view, which Ishallnotelaboratehere,isthattheconflictbetweenthetwoapproachesisessentially linguistic. It is a question of whichway of speaking is to bepreferred under a given set of circumstances. To say that a theory is areliable instrument-that is, that thepredictionsofobservableevents that ityieldswillbeconfirmed-isessentiallythesameassayingthatthetheoryistrue and that the theoretical, unobservable entities it speaks about exist.Thus, there isno incompatibilitybetween the thesisof the instrumentalistand thatof the realist.At least, there isno incompatibility so longas theformer avoids such negative assertions as, ". . . but the theory does notconsistofsentenceswhichareeithertrueorfalse,andtheatoms,electrons,andthelikedonotreallyexist".

Source:PhilosophicalFoundationsofPhysics(1966)publ.BasicBooksInc.Chapters23to26reproducedhere.

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