The Model in Theory Construction

8/12/2019 The Model in Theory Construction

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Psychological eviewVol . 67 No. 2 1960

THE MODEL IN THEORY CONSTRUCTIONR OY L A C H M A N *

New York UniversityThe scientif ic enterprise as here con-ceived consis ts of two related but dis-cr imina t ively di f ferent act ivi t ies . On eis theobservation ofobjectsan devents,both experimentally an d less formally.The other involves the use of math-ematical and natural l inguis t ic sym-bols, along with the rules for their

manipulat ion, to represent these sen-sory experiences, to organize what isobserved into some comprehensibleorder, and by proper symbolic manip-ulation to arr ive at a representat ion ofwhat has not yet been observed. Thecommentary that accompanies theselat ter theoret ical undertakings fre-quently includes the term "model,"themean ing of which can , at best , beimperfectly ascertained from the con-text. It has been claimed, not in-frequently, that while some distinctreferent can be delineated in physics ,the term model has lost all sem-blance of mean ing in psychology (e.g.,Underwood, 1957, p. 25 7). Actual ly ,the physical scientist has permi t tedhimself wide lati tude in the use of theterm "model" and the psychologisthas followed suit.The plan of this paper is to analyzecertain meanings and func t ions of theconcept of a model ; d i s t inc t ions madewill then be exemplified in some actualtheoriesfrom contemporary psychologyan d classical physic s. Im plication s ofthe properties of models for theoryconstruct ion will then be examined ina systematicfashion. The consequences

iThe author, no w a t Hilo Campus , Uni-versi ty of Hawaii , is indebted to H. H.Kendler, M. Tatsuoka, M.K. Mun i tz , E. D.Neimark, Linda Weingar t en ,and H . Jagodafo r reading th e manusc r ip t .

of this analysis for a sample of method-ological problems currently at issuewill be explored.Two properties of the analysis tofollow should be noted. First, al-though the profus ion of ideas referredto by the term model is often in ter-related and compounded, it will be

necessary for analyt ic purposes to sep-arate the various meanings and func-t ions into discrete categories. W ith thisexception, the explication of the con-cept of the model will follow as closelyas possible the actual scientificusage.Secondly, while the symbolic form ofconveying the ideas of a model maytake the form of a word, sentence,diagram, or mathematical calculus, thedis t inct ions to be made concern themeanings of the concept of model an dits functional properties within a theo-retical system.Since model has been used inter-changeably with or in reference tovarious parts of a theory, it is neces-sary to dis t in guish such theoret icals t ruc tures , if only crudely . For ourpresent purpose, it will suffice to differ-entiate among the parts of certaintheories: (a) the principles, postulates,or hypothetical ideas, including therelat ions among them; ( f t ) the sen-tences, equations, or theorems, derivedtherefrom; (c ) thecoordinating defini-t ions relat ing theoret ical terms to theobservational sentences. The textualcommentary accompanying the type oftheory with which we are concernedm ay not explicitly designate all theparts of such a structure. It is partof the task of methodological analysisto isolate this struc ture. Finally, i twill be useful to d is t inguish the formal

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114 R O Y L A C H M A Ntheory from the separate system whichis the model. This meaning of modelm ay refer to, but need not necessarilyrefer to, what is frequently called theanalogy. A model as an order struc-turally independent of the theory isthe promin ent not ion . Equally im-portant , more than on e model gen-erally funct ions for a theory. Themodel, consisting of a separate system,brings to bear an external organizationof ideas, laws, or relationships uponthe hypothetical propositions of atheory or the phenomena it encom-passes. This external organizat ion ormodel contributes to the construction,application, an d interpretat ion of thetheory. Precisely how this is accom-plished can be clarified by analyzingthe meanings an d funct ions of themodel concept.

F U N C T I O N S O F M O D E L S/. Models Providing Modes ofRepresentation

Empirical elements an d relation-ships which constitute the phenomenato beorganized by a theory are kn ownto us by the symbolic system or lan-guage that designates these data. Byintroducing a model constituting aseparately organized system, we areproviding an addi t ional system ofrepresentat ion for the phenomena anda suitable way ofspeaking about them.An essential characteris t ic of modu-lar 2representat ion con st i tutes the fu r -n i sh ing of new ways of regarding orthinking about the empirical objectsan d events . Attr ibu tes an d meaningsof the 'model are t rans ferred from theirinitial context of usage to the newsetting. This application of unprec-edented modes of representat ion canbe executed on two levels. Objec tsand events formulated directly from2Hereinaf ter the word "modular" is in -tended to perta in to a "model," not a mode.

observation are thought about in then ew an d u n u s u a l fashion prescribedby the model.At another level, a model m ay pro-vide novel modes for conceiv ing thehypothetical ideas an d postulates of atheory. Consider the con di t ion ingmode l : applied to perceptual phe-nomena, it i llustrat es the first or em -pirical level of applicat ion. If percep-tion is regarded as though it were aconditioned response, to some deter-minable degree perc eptual behaviorshould be capable of t rea tmen t in ac-cordance w i th the lawsofc ond i t ion ing :frequency, stimulation interval, etc.;i f so, the relationships an d languageof condi t ion ing may be frui tfully ap -plied. Howes an d Solomon's (1951)findings maybe soregarded. In thisexample, some such analogy with con-di t ioning is said to provide an em-pirical or pretheoretical model (Koch,1954). In contrast, the fractionalant icipatory response mechanism ratheory(Amsel , 1958; Kendler &Levine, 1951; Mol tz , 1957; Spence,195la, 195Ib; etc.) illustrates the sec-ond level of representative funct ion ingfor the condi t ion ing model . Here, thepostulated n onobs ervable fract ion al an-tedat ing response is conceived of asoperating according to condi t ion ingprinciples to -whatever degree desir-able. W e assign an y ind iv idual orcombinat ion of properties from themodel to the theoret ical constructguided by pragmatic considerat ions ofpredictive fertili ty an d consistency ofusage required for the integration ofdiverse areas. To reiterate this im -portant point, a model may be appliedto directly accessible (empirical) ob-jects and events or to inaccessible an dimagin ary conceptsof atheory. Thoughthe levels of modular applicat ion con-sidered are not discrete but representextremes of a con t i nuum, an impor tan tdistinction should not be overlooked:


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THE MODEL IN THEORY C O N S T R U C T I O N 115while theoretical entities can be said tobe any th ing w e want them to be, thisis not so for categories close to ob-servat ion. The latter are spoken ofonly in an as if fashion. By invokingthe condi t ioning model , percept ion,mean ing , an d verbal behavior are notsaid to be cond i t ioned re sponses ; it isonly claimed that if represented andspoken of in such a fashion, impor tan tempir ical consequences can be ob-tained. This point is well il lustratedbyToulmin's analysis of the rectil inearpropagation model of geometrical op-tics, We do not find l ight atomizedinto indiv idualrays: w erepresent it asconsist ing of such rays" (T ou lm i n ,1953, p. 29).//. Models Functioning asRulesofInference

Giventhea s sumpt ion sofsome theoryalong with the exper imental arrange-ments , a tra in of reasoning leads tothe predic t ion sentences . From whatsource do the rules for such in ferencearise?In the language used to talk abouta theory, the term model may referdirectly to a mathemat ical calculus orto a system of relationships from someother source such as electronic com-puters , the classical laws of motion,the laws of con dit ion ing , etc . Any oneof these separate systems in c on junc -t ion wi th the in i t ia l condi t ions of agiven experiment m ay permit the pre-dict ion of what is to be observed.Here, model signifies the rules bywhich the theoretical symbols (some,at least, having been empir ical ly co-ordinated) are manipulated to arriveat new relations; the rules by whichon e sentence is inferred from anotheror the theorems are der ived. Theprobability calculus utilized in stat is-tical learni n g theory (Est es, 1959)exemplifies such am athemat ical mod el,while some of the relations holding

among electronic communication sys-tems play a role in providing thein fe rence rules or calculus for informa-tion theory (Gran t, 1954). Anotherexample is the laws of classical con-dit ionin g whic h, alone or supplemented,are the means by which inferences aredrawn in appl icat ion of rg theory to agiven experimen t (Kendler & Levine,1951; Moltz & Maddi , 1956).Calculational rules of mathemat icalsystems and the semantic pr inciples ofna tura l language are themselves rulesof inference for the symbolism in -volved. In contras t to this mode ofder ivat ion, M u n i t z (1957a, p. 42)recognizes then ecess i tyfor dis t inguish-ing a more specific set of inferencerules provided by cer tain equationsan d supplementary textual commen-tary which prescr ibe the precise man-ner in which the theoretical terms an dsymbols are to be connec ted ; theseoccasionally are termed "implicit defi-nit ions . Although some models m ayimply how assumptions and impl ic i tdefini t ions are to be formula ted , theterm s mathem atical model an d con-di t ioning model apparent ly refer tothe general inference pr inc iples em-ployed in the f irs t sense rather than tothe implicit definitions. This is illus-trated in rg theory by the relation orimplici t definit ion rg sg, for there isnoth ing in the laws of classical con-di t ioning that requires the hypotheticalresponse to have a hypothetical re -sponse-produced cue. But i t is pre-cisely the laws of classical condit ioningthat func t ion as the basis for the rulesby which we arr ive at the consequencesof the several implici t defin i t ions . Anadditional illustration is provided bystatistical learningtheory's operatorordif ference equat ions , the specific choiceof which is in no way dictated by therules of mathematics, al though theoperators may be suggested by somesecond model. While the operator


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116 R O Y L A C H M A Nequat ions de termine the form of thelearning func t ion to be derived, it isonly by the mathematical rules thatthe derivation itself is executed. Itimmedia te ly follows that in the ex-press ions mathemat ica l model an d con-d i t ion ing model , the adject ives preced-ing the term model descr ibe the sourceof the general inference pr inc iples . Inboth cases, this is a separate systemexternal to the s tructure of the theory.The major source of s ta t i s t ica l learn-in g theory ' s in ference ru les are to bef o un d in thetexts on probabi l i ty theoryand the types ofm athemat i c s i nvo lved .The rules of inference for rg can befound in the exper imenta l l i te ra tureon classical condit ioning. For pre-cision in der iva t ion , it is obvious thatthe rules of mathemat i c s are to bepreferred . Fur thermore , if i t is notyet apparent, la ter analysis of specifictheories will show that these modesof in fe rence d rawing do not follow thelogic-book deduction of which somescient is ts and philosophers are sofond .III. Interpretational Function ojModels

The inference ru les or calculus of r fftheory is f o rmu la t e d by amend ing an dadapt ing the rules contained in thecondi t ion ing m o d e l ; the model servesthe addit ional func t ion of in te rpre t ingthe calculus . This is not the case fortheories wi th in ference ru les pr imari lyadopted from a mathemat i ca l ca l cu lus ;the mathemat ica l model may be sup-plementedwith one or more addi t ionalmodels which serve to interpret andmake intell igiblet he in fe rence rulesem -ployed. Estes' stat is t ical learning the-ory (1959), for exam ple, m akes u se ofa s t imulus-sampling m o d e l ; th is secondmodel provides a mode of representa-t ion fun ct ion ing as a coherent way ofspeaking about the theory. M oreover,i t func t i ons as one poss ib le in terpre ta-

t ionof the theore t ica l terms. Form ulaeand sym bols are ren dere d intell igible bythe modular in terpre ta t ion which a lsoshowshow toapply thetheory an d sug-ges ts procedures for extend ing the useof ind iv idua l pa ramete r s and the theoryas a whole . Thus , the s t imulus - sam-pl ing model would tend to suggest thatin e x t e n d i n g the scope of stat is t icallearn ing theory to include mot iva t ionalphenom ena, deprev ia t ional s tates w ouldbe concep tua l ized as p ro d u c i n g di f fer-en t ia l con f igura t ionsof the s t imu lu ssetavailable for sampl ing ; th i s is preciselywhat Estes (1958) has done.

Someapplications of the condi t ion ingmodel suggest , in add i t ion to in fe rencepr inc ip les , the rules for their applica-t ion to the phenomena invo lved , thein te rpre ta t ion of the in fe rence p r in -ciples. Here , the modular in terpre ta-t ion produces propos i t ions suchas: thes t rength ofrg increases to some asymp-tote as a func t ion of the quant i ty , t ime,and magn i tude o f consummatory ex-perience in the goal box. In psychologysuch symptom rela t ions are general lyte rm ed coord ina ting de f in i t ions wh ichdesignate the class of propos i t ions ty in gtogether theore t ica l terms wi th the irassociated empirical sentences. Themodel as an interpretat ion of a theoryguides the fo rmula t i ng of coord ina t ingdef in i t ions. Hutten (1956,p. 87) addssal ient ly, The m odel so becomes a linkbetween theory an d e x p e r ime n t . W eexplain an d tes t the theory in te rmsofthe model.IV. Models Providing PictorialVisualisation

Probably the most common meaningemployed an d serv ice rendered by amodel consis ts of the reproduct ion ofthe theoret ical prototype in terms ofmental p ic tures or images . Agentsmedia t ing th is func t ion range from arigorously integrated separate system toa loose analogy with fami l iar sensory


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TH E M O D E L IN T H E O R Y C O N S T R U C T I O N 117experience. The popular i ty of thismodular activity follows from the ubiq-uitous desire for an in tu i t iv ely sa t i s fy-ing account of any theory. Preoccupa-tion of classical physicists with modelbui ld ing of this l imited pictor ial typeinitiated the counter-view that modelconstruction was a disreputable enter-prise. Duhem (1954) , for one, con-sidered model construction to be bothsuperf luous and the refuge of weakmin ds. Residual form sof this positionare in evidence today ; Carnap (1955,p. 209), while recogn izing esthetic,didact ic, an d heur is t ic value, f indsmodels nonessential for successful ap-plication of theory. Although this m aybet rueof some theoretical activity, as ageneralization it is deficient on severalgrounds and i t neglects cer tain essentialaspects of the behavior of a good num-ber of scientists . In mod ern physics,al though visual representation is nolonger a prerequis i te for the acceptanceof a theory, the pictorial model m ayserve decisively in the ini t ial phase oftheory con struc tion . While contempo-rary quantum theory cannot providecoherent visual izat ion, it did developfrom Bohr's a tom which w as asso-ciated w ith a pic torial mod el. More-over, some curren t theoretical effortsconcerning the atomic nucleus util izethe shell an d water drop models. Ax-iomatizationof a theory, or the at temptsto do so, does not compromise thisargument . Construc t ion of an axio-matic system, apparently, is not at-tempted until theory construction iswell under way, usually with the aid ofsome model. It isnecessary, therefore,to carefully dist in guish simple pictor ialfunctions from the less trivial enter-prises described in the previous sec-t ions. A model providing sa t i s fy ingin tu i t ive pictures m ay also serve one ormoreof thefunc t ions enumerated above,with influence upon theory constructionand application that cannot easily be

overrated. Conversely, a model pro-vid ingonly visual representation serves,at best, d idact ica l ly ; even then, it prob-ably does more to mislead the s tuden tthan to teach him . Consider Lew in's(1951) topological an d field-theoretic almodels. Followin g Braithw aite (1953,p.366n) w emust agree that theessenceof a calculus is not its symbolism butits inference rules, whereas Lewin'smodel func t ions as a calculus thatdoesn't calculate. In contras t , the con-di t ion ing model, while possibly provid-in g some S-R theorists with all thebeaut i ful imagery associated with sali-vating dogs, also contr ibutes the essen-t ial inference rules. The w ork ofStaatsan d Staats (1957, 1958) prov ides anadmirable il lustration by their demon-strations that verbal meaning and at-t i tudes may be treated as analogousto a conventional condit ioned responsewhose laws furn i sh the calculus em -ployed.

I L L U S T R A T I O N S O F M O D E L SAlthough the func t ions enumeratedhere have been illustrated by examplesfrom contemporary psychological re -search an d theory, for some readers thed is t inc t ions made m ay still be rathervague. The situation may be remediedby analyzing d i f feren t models and not-

ing their func t ions in specific theorieswithin cultivated areas of scien ce. Cer-tain theories of classical physics providean excellent opportunity for compara-t ive analysis with current theoreticalefforts inpsychology. Add ition al func-t ions will be served by such an analysisof at once demonstrating the ini t ial con-ten t ion that models are util ized in anessentially similar fashion in both clas-sical physics an d psychology. At thesame time, some light may be cast upona question that has eliciteda good dealof polemic : the problem of the emula-tion of physics by psychologists. Therf theory, statistical learning theory,


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118 R O Y L A C H M A Nand the classical kineti c theory ofgaseshave been selected to illustrate th e roleof the model in a theory. The rigorousmethodological analysis to which thelatter theory has previously been sub-jected is the basis for its selection. Thelanguage to be employed in this illus-tration will be accessible to almostevery student of psychology; we tu rn ,now, to that task.The Kinetic Theory of Gases

Let us first observe the pictorialmodel of the kinetic theory in i ts mostl imited form. A gas is represented bythis initial model as consisting of min-ute invisible particles (molecules ora toms) in incessant mot ion , collidingwith one another and with the walls ofany restricting con ta iner ; at a giveninstant , most of the particles are sepa-rated by vacan t space. Consider, n ow ,the interpretation by means of the pic-torial model of the known properties ofa gas residing in a container composedof rigid walls, one of which is movablein the manner of a pis ton . The ad-justable wall may be moved i n w a r d ;this is interpreted as a reduct ion in theavailable space between the particlesan d accounts for the property of com-pressibili ty of a gas. While pus hin gin the wall w e experience resistanceagainst our efforts. As we cont inu e toreduce the space, the resistance in-creases ever more. The model attrib-utes this to an increase in the numberof collisions following from a reductionin volume an d resulting in an increasein the velocity of the gas particles.Boyle's Law, which states the relation-shipbetween volumeand pressure, thusis attributed to an increase in the rateof impact between the particles and thewall of the container, which is part ofthe general picture of increment in thenumber ofcollisions am ong the particlesproduced by the decrement in the spacebetween them. But the diminut ion in

volume has increased not only the pres-sure but also the temperature. This isaccounted for by the increased velocityof the particles adding to their kineticenergy. Thus, the law relating tem-perature to pressure is acknowledged.While the presentation, thus far, is notdissimilar to D. Bernoulli 's original hy-pothesis (D'Abro, 1951, p. 381) wehave proceeded somewhat beyond theinitial model which essentially involvesa picture ofmass-poin t sin motion func-tioning as a representation for and wayof talking about the hypothetical ideasand postulates of the initial theory.Introducing additional models or theamendmen t of the original provides theinference rules. The first additi on in-volves the well developed laws govern-ing a separate phenomenon which areapplied to the ideas of the pictorialmodel. This may be termed the dy-namical ModelII8for thetheory und erconsideration. Here , the gas particlesare conceived as behaving analogouslyto medium sized bodies such as billiardballs, rocks, or plane t s : the laws ofclassical mechanics governing the mo-t ions of the latter and the principleso fdynamics describing their energy statesare ascribed, approximately, to the mo-t ionsand energy statesof the imaginarygasparticlespictured by the Model IV .The laws provided by the dynamicalmodelare in mathematicalform and arethemselves supplemented by an addi-t ional Model II, the calculus of prob-abilities. Thereb y, the rules of rea-soning or calculus of the theory areformulated, by means of which conse-quences are derived from the init ialtheoretical terms. But the combiningofModelsII canno t , itself, begiven anyrules, for the essential ingredient is thecreative imagination of the scientist.Following Campbell's (1920) de-

8A Roman numeral designating the spe-cif ic type of model wil l be affixed wherethis may be in doubt.


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THE M O D E L IN T H E O R Y C O N S T R U C T I O N 119tailed analysis, let us schematically de-scr ibe one of the more elementary der-ivat ions in a sui tably ideal ized andsimplified fashion. The postulates ofthe theory may be represented by termswhich inc ludethe symbols:t,m,v, I ,n,and (x, y, s). The postulates, whichwe need not reproduce, consist of sev-eral implici t definit ions in the form ofequations relating these theoretical var-iables an d associated textual commen-tary. Now, {x, y , z ) m a y b epic turedby the Model IV as rec tangular co-ordinates for the posi t ion of each of theimaginary par t i c les : m their mass , vtheir mean velocity, their number , tas time, and / as the length of an edgeof the hypothetical container . The im-plicit def init io ns relating the pos tulatedvariables are approximate in form toth e laws which would govern the mo-tions of such an isolated system ofentities pictured by the model. Fromthe ini t ial terms and their postulatedinterrelationships, the expression,

nmy*~ ~ [1]is der ived in accordance with the math-ematical rules involved.4 For some of

*The term p is a compact way ofwr i t i ng a more complex a rrangement ofthe above symbols involved in the postulates.The actual derivation is frequent ly per-formed in the Gibbs ian manner of treatinga gas particle's three components of positionand three of momentum as the coordinates ofa s ingle "phase point located in a "phasespace" of six d imens ions or 6N d imens ionsfor J V particles (Jeans, 1940, p. 17 ff). Forthe present considerations, this i l lus tra testhe not infrequent s i tuat ion of ident icalconsequences being derived f rom somewhatdifferent assumptions.The dis t inc t ion usually made between anideal gas and a real gas wi l l be avoided.Ins tead, the proposi t ions concerning a realgas will be amended wi th a s ta tement con-cerning their range of validi ty . This alter-nat ive method will beemployedi n accordancewi th the view that a scientific la w cannotbe regarded as a universal proposi t ion both

the individual theoretical terms an dseveral combinations of them, coordi-na t ing defin i t ions are stated relatingthem to the appropriate empir icalterm s.The coordinat ing proposi t ions are sug-gested by the pictorial model in itsinterpretative func t ion . Thus, coordi-na t ing / wi th the length of a cubicalcontainer holding a real gas, it followsthat /3 is the volume V of the gas.Likewise, p is coordinated wi th thepressure P on a wall of the vessel, an dm v*/a wi ththeabsolute temperature Tof the gas.B Subs t i t u t i ng the empir icalte rms P, V, T in the theoretically de-derived Express ion 1, the term an/3remains , which is a constant k, an dwe have,

PV=kT, [2]the famil iar gas law relating tempera-ture, pressure, and volume, which isapplicable over an empirically deter-mined range for a given type of gas.Re turn ing to the der ivat ion of Ex-pression 1, application of the inferencerules provided by the laws of dynamicsrequires cer tain assumptions concern-in g properties of the modular par t ic les ;nam ely, they are in f in i tesim al, so thattheir diameters and orientation in spacemay be ignored. In addit ion, they areassumed to be perfectly elastic. Non -elastic particles would suf fer a reduc-t ion in velocity with each impact uponthe container wall an d eventually theirmotionwould cease entirely, precludingthe derivationo f consequencesin agree-m e n t with observation. The cr i t ics ofcer tain theoretical assumptions in psy-chology would do well to ponder theon empir ica l an d logical grounds. Withrespect to the hypotheticalideas of a theory,as here viewed, these are always ideal si tua-t ions, since they are creat ions of the im-agination direc ted toward executing deriva-tions in agreement with experiment an dth e achievment of mathem atical simplicity.6For a definit ion an d analysis of theterm a, consult the work of Campbell(1920, p. 127).


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120 ROY LACHMANimplications of the assumptions of aninfinitesimal and perfectly elastic par-ticle, properties that it is imp oss ible toassign to observable objects.Campbell (1920) discriminates tw oprocesses for exten din g a theory. I twill be recalled that only some of thepostulated variables were coordinatedwith empir ical terms. The form ulat ionof addit ional coordinating defini t ionsfor the uncoordinated terms il lustratesthe first methodo ftheoretical extension.A second method oftheoretical develop-ment involves making amendments tothe postulated terms and the equationsexpressingtheir interrelation ships . Thissecond method proceeds as follows: theprevious assumptions are modified oramended and new theoretical terms areintroduced into the postulates corre-sponding to the addi t ional assumptions .The pictor ial model may prescr ibe thisextension. Thus, viscosity relations ofa gas may be der ived by al ter ing theprevious postulates with a term sug-gested by the pictor ial-dynam ical modelcorresponding to a diameter for thehypothetical particles. Considerationofthe space occupied by the modular par-t icles and their mutual at tractions leadstootherderivations. Thisaccumulationof assumptions following from the ex-tension of ideas contained in the modelhas been termed mod ular d eploymen tby Toulmin (1953). In the presenttheory, experimental agreement breaksdown when the model is deployed toinclude in ternal rota t ions of the im-agi n ary particles. Gen eral failure ofobservational agreement also occurswith the attempted extension of thetheoretical scope to extreme ranges oftemperature an d pressure. In both in -stances,quantum mechanicssupersedesthe classical kinetic theory.ra Theory

Turning to r, theory, let us firstexamine the pictorial model involved.

A rat is placed in a T maze, one armof which isbaited with food ; the secondis empty. A noncorrec t ion procedureo f training is employed. On the trialsduring which the correct goal box isentered, the animal executes uncondi-t ioned responses (UR) of approachmovements , chewing, salivating, etc., tothe uncond i t ioned s t imulus (US), food.On subsequent t r ia l s , s t imulus objec tsan d pat terns of the choice-point an ds ta r t ing arm are encountered at apoin tin space pr iorto the food, or U S. Sincethese choice-point stimuli are locatedspatially antecedent to the U S, they are,therefore, encountered at some tem-poral interval prior to the food or U S,depending on theanimal'srateof loco-motion. The antecedent s t im ulus pat-terns come to el ic i t imaginary or in-visible condit ioned responses (CR),such as chewing and sal ivating, whichlogically cons i s t of components of theempirical U R to food. These invisibleantedating condit ioned responses (rg)are acquired in a fashion exactly anal-ogous to Pavlovian condit ioning of ob-servable responses. The unobservablerg, in turn, produces a hypotheticalproprioceptive response-produced st im-ulus (sg), which likewise functions inthe fashion of an overt an d observables t imulus .

Cues at the choice-point m ay consistof di f ferent visual texture or brightnesspat terns for each of the maze arms, aswell as proprioceptive consequences oflocomotion an d receptor or ientation to-ward the left or r ight en d box. On eand onlyone of these choice-point s t im-uluspat terns is consistently followed bythe U S, food. Thus, only this specificset of s t imul i will selectively begin toact as a CS and elicit the hypotheticalcondit ioned response,rg. Evocation ofrg will occur with increased frequencyan d ampl i tude, the greater the spatialproximity between the set ofcues an dthe goal object, or the greater the s im-


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TH E M O D E L IN T H E O R Y C O N S T R U C T I O N 121ilarity between the choice-point cuesand the goal box s t imul i . This, ofcourse, follows from classical condit ion-ing laws deal ing wi th CS-US in tervalan d general ization. It is possible toimagine a sequenceof events occurr ingdur ing a trial early in t r a in ing which isterminated by the rewarded or correctresponse: 5 locomotes to the choice-point where VT E behavior occurs,af terwhich 5 orients it s muscu la ture an dreceptors toward the cues of the re-w arded end box. Since only a mod-erate amount of training has preceded,the correct cues el ic i t rg wi th lo w f re-quency and min imum amplitude. Ori-ent ing responses are followed by pene-tration into the correct end box whichtermina tes wi th the consummatory orgoalresponse. Notice w hat is imaginedto o c c u r : since the correct choice-pointst imuli have been followed by the U Sof food, these choice-point cues haveincreased their strength of evocationofthe condit ioned response, (rg). At thesame t ime sg, which is available onlywhen rg occurs, would be selectivelypresent pr ior to the performance of thecorrect ins trumental response. Withprolonged training, the s trength an dprobability of rg elicitation by the cor-rec t cues are con t inuous ly enhanced viaclassical condit ioning. A concomitantinc rement in frequency of occur renceofs ff takes place. Likew ise, by means ofS-R cont igui ty ,sg elicits the appropriateselective-learning response wi thgreaterfrequency. In this fashion , the fre-quency of the correct response ap-proaches some asymptote , which forthe present experimental condit ions isunity.A basicform of selective-learning hasbeen accounted for by the model with-out appealing to some special mech-anism of re in fo rcemen t ; the changes inbehavior are mediated solely by clas-sical condi t ion ing and S-R cont igui ty .It might be ques t i oned : why not ex-

plain the behavior by molar S-R con-t igui ty aloneand n ot in troduc e invis ibleeven ts? As we shall see, this non-observable system of events mediatesother predictions for several differentin i t ial condi t ions , mostof which cannotbe derived and integrated by molarprinciples alone. An exactly analogoussituation exists be tween thermodynam-ic s and the kinet icgas theory.Deployment of the model has madethe picture more complex : a class ofevents has been in t roduced in to thenegative en d box. Frus tra t ion an davoidance responses are assumed alsoto"moveforward bymeansofclassicalcondi t ioning to the incorrect cues at thechoice-point. To pursu e this phase ofmodular deployment is not necessaryfor ourpurposes . Let us ins tead exam-in e that which has already been pre-sented. W e have proceeded beyond thepictorial model and descr ibed the in-ference techniques of the theory. Thepictorial m odel itself essentiallyinvolvesthe ideao r p ic tureofobjects and eventsoccurr ing at the terminal por t ion ofa behavioral chain inf luenc in g pr iorevents. Terminal goalbox events func-t ion as US's whose appropriate CR's move forward to de termine and me-diate events occurr ing earlier in thebehavioral sequence. Serving as arepresentation for and m a n n e r of talk-in g about the ideas of ra theory, thepictorial model provides a seeminglynecessary service.The model of classical condit ioningalso provides the inference rules orcalculusofra theory,which for the mostpart is the p r oduc t ofSpence's (19Sla,195Ib) creative theorizing. Nonob-servable rg is conceived as behavinganalogously to observable condi t ioning:the acquis i t ion and extinct ion laws ofthe latter are ascr ibed, approximately,to the form er. In conjunc t ion with theinit ialcondi t ions of a given exper imen t ,the condi t ioning laws mediate the der-


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122 ROY L A C H M A Nivation of specific predic t ions , as hasbeen demon strated. Util izing one ofthe many examples available (Amsel ,1958; Kendler &Levine, 1951; Moltz ,1957; Spence, 1956),le tu sagainexam-ine the manner in wh ic h the laws ofcond i t ion ing serve as the model andprovide the in ferenc e rules. Mo ltz(1957) has demonstrated that the ob-servat ions concerning la tent or nonre-sponse extinction may be derived bythe calculus of rg. Modif ied onlyslightly, Moltz's derivation proceeds asfollows. Placing an animal in a pre-viously rewarded but now empty goalboxresults in therepeated evocation ofrg by the cues that were in closeprox-imi ty to the now absent goal object.Since the CS (goal box cues) elicitingrg is no longer followed by the US( f o o d ) , the extinction laws of classicalcondi t ion ing would require that the hy-pothetical CR (rg) should suffer areduction in strength. Invoki ng thegeneralization of extinction laws, it canbe reasoned that the f requency of rgevocation at the choice-point is reducedand that the reduction is a func t ion ofthe degree of similarity between choice-point an d goal box cues. Since a con-comitant reduction in sg occurs , it fol-lows that a decrement in the probabil i tyof the observable choice behavior mustoccur. Such a reduction in choice be-havior as a consequence of latent ex-t inc t ion had been previously observed.But once again util izing the inferencerules provided by the model, conse-quences not yet observed were derived.Moltz selected two condit ioning laws asi n ference ru les : (a) the rate of evoca-tion of an observable condit ioned re -sponse covaries with relevant dr ivedur ing ext inc t ion, an d (b ) the reduc-tion inresponsestrength duringextinc-tion is a monotonic increas ing func t ionof the number of response evocations intheabsence of the U S. Notice that thelogical status of the two conditioning

laws has now changed from empir icalpropositions to inference rules. Thein i t ialcon d i t ionsof an exper iment mustnow be stated in order to der ive thepred ic t ion sentences . Train in g mayfirs tbe adminis tered in a T maze. Theanimals must then be d iv ided in to an u mb e r of groups wi th di f ferent driveschedules and placed in the empty goalbox. In accordance w ith the dr iv e law( a ) , the animals wi th the highest moti-vat ionshould produc e the greatest emis-sion rate of the imaginary response, rg.These sub jec ts , according to the ext inc-t ion law ( b ) , should show the greatestreduction in the strength ofrg. Thus,the decrement in choice behavior onsubsequent test tr ials should covarywith the dr ive level present dur ingplacement in the empty goal box. Ex-per imenta l veri f ica t ion for this predic-t ion was obtained (Mol tz & Madd i ,1956).

H av ing demonstra ted that the condi-t ioning model of rg theory provides inaddi t ion to pictor ial and representa-t ional func t ion the rules of reasoning,w e need only consider the model 's in-terpretative func t ion . Although coordi-nat ing def in i t ions were sys temat ical lyformulated dur ing the derivation ofobservable consequences, they were notexplicitly denoted. Examin ing f i r s t theempirical antecedent var iables, coordi-na t ing defini t ions may beformu la tedbyimagining that rg is an ordinary condi-t ioned response. This is an al ternateway of stating that the cond i t ion ingmodel is being util ized. Thus, thestrength ofrg is a negatively acceleratedincreas ing func t ion of the number oftr ials that the el ic i t ing s t imulus of rg isfollowed by the reward or U S. Thestrength of rg is a decreasing functionof th e temporal interval between it seliciting s t imulus and the US. In thisfashion, the var ious rela t ionships be-tween an observable CR and the var-iables of which the CR is a funct ion


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TH E MODEL IN THEORY CONSTRUCTION 123define the empirical coordinates of rg.The term define in this last statementmust be qualified. It will be recalledthat in theanalysisof thekinetic theoryof gases, one method of developing thetheory involved the fo rmula t ion and re-vi s iono fcoordinating definitions. Thus,coordinating propositions are to beunderstood as hypotheses rather thanconventions: those that prove fru i t fu lareretained, theuseless coordinatesarediscarded. Finally, considering theconsequent empirical variables, theprobabi l i ty of the correct response inselective learning is coordinated withtheavailabilityof sg at thechoice-point.Having demonstrated the role of themodel in rg theory, an evaluationof thesystem mightbe inorder. Aside fromits apparent promising potential, theprimitive state of the theory precludesanydefinitivejudgment. However,theconditioning model of rg theory doesappear to func t ion in a fashion quitesimilar to the mechanical model oftheprequantitative (Bernoulli's) kinetictheory. Accordingly, on e wonders ifthe conditioning model is destined toplay the same role in psychology thatthemechanical model served inphysics.Statistical Learning Theory

W e shift now to anexamination of amultiple model system. Estes' prob-abili ty learning theory (1959) of ferssome interesting contrasts to rg theory.A s we shall see, thepictorial modelo fEstes' theory invokes nonobservablestimulus occurrences while rg theoryappealed essentially to nonobservableresponse events.If a T-mazeexperimentwith a cor-rection procedure is employed for thepurpose of illustration, then an analysismayberenderedthatissuitablysimpli-fied for thepresent purpose of studyingth e func t ions of the several types ofscientific models. Let us, therefore,consider such an experiment. Over a

large number of trials the reward israndomly placed in the left arm 70of the time; on the remaining 30% ofthe trials, the reward is in the rightarm. When an animal selects a non-rewarded arm, it is permitted to re-trace. In this fash ion , each trial termi-nates with reinforcement. The initialright or left turn is defined as theresponse for that trial. These tworesponse classesare mutually exclusiveand exhaustive.

The pictorial model associated withEstes' theory permits us to imaginethe external environment of the mazeand the internal environment of theanimal as consisting of a verylargebutf ini te number of invisible stimuluselements. On anygiventrial,asampleof these stimulus elements isavailable.Each element has an equal probabilityof being included in the sample (thisis assumed for the present type ofexperimental situation). At any spe-cific moment, agivenstimulus elementis in an associative state with (con-ditioned to) one and only one re-sponse. In addition,theprobability ofagiven response isdependent upon thenumber of stimulus elements condi-tioned to it. The sum of the prob-abilities of the available, mutually ex-clusive,andexhaustive response classesequals unity:one of them must occur.

During the initial trials in the Tmaze, the two alternative responsesmay sometimes occur with equal fre-quency and it is possible to imaginethat an equivalent number of e lemen t sof the stimulus set are conditioned tothe two response classes at the startof training. With continued training,the ratio of stimulus elements con-ditioned to each of the response classesundergoes a change. When the re-warded arm is ultimately selected oneach trial, all the stimulus elements inthe sample available during that trialareconditioned to the reinforced alter-


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124 ROY LACKMANnative. It will be recalled that a re-sponse to the left arm was ass igned are inforcement schedule of 70 . R e-gardless of the init ial response ex-ecuted, and as a result of retracing,70% of alarge series of t raining trialswill terminate with a response to theleft arm . Likewise, 30% of the trialsterminate in the right en d box. Thus,as training progresses, the s t imulusset tends to approach a com posi t ion of70% of the elements condi t ioned tothe left response and 30% associatedwith the alternative respon se. Sinceth e conten t of the subset of st imuluselements sampled on any trial ap-proximates the composi t ion of thetotal set, an init ial response to the leftarm will approach a probabil i ty of .7with continued training. Thus, 70%of the ini t ial responses to the left endbox will be observed over a longseriesof trials.So much for the pictorial modelwhich represents the experimentalsituation as a series of imaginary s t im-ulusevents,the compositeset of whichundergoes a cumula t ive change inassociat ive con nect ion. Un avoidably,other matters such as the assumpt ionsor postulates of the theory were in -c luded. To these and to the rules ofin ference or mathematical model wen ow direct our a t ten t ion .Following the thorough presenta-t ion of Estes (1959), an d Estes an dStraughn(1954)and introducing somemodif icat ions for exposi tory purposes,le t us part ial ly describe the derivat ionsfor the previously portrayed T-mazecondi t ions . The theoret ical postulatesinvolve combinat ions of such symbolsas ,F(n ), 6',v,AI,A2, n, etc.,and therelated textual com men tary . In terpre t -ing these symbols in astrict theoreticallanguage by means of the pictorialmodel, AI and A2 may be imaginedas the two theoretical response classesavailable, n as the number of theo-

retical trials, I T as the probabili ty ofthe hypothetical reinforcing event fol-lowing the occurrence of AI. In ad-di t ion, 0' may be conceived of as theprobabil i ty of any single s t im ulus ele-me n tbein g included in thes ample avail-able on a given theoret ical t r ial , an dF(n ) as the probabil i ty that a givens t imulus element is connected to im-aginary response AI af ter t r ia l n .A few examples f rom Estes' postu-late set will be helpful. On e axiompreviously indicated in the discussionof th e pictorial model is expressed bythe following propos i t ion :

Fw=(l-0 )F [3]when response AZ is reinforced ont r ial n . Interpreted by the pictorialmodel the equat ion s t a t e s : after a re-in forcement of response A2 on t r ia l n,the probabil i ty or proport ion F(n ) ofthe e lemen ts con di t ioned to responseAi equals the proport ion of s t imuluselements not sampled (1 6') on tr ialn that were already condi t ioned to theAI response (F(n_1)) at the end ofthe previous trial. This is a conse-quence of the terminal response A ont r ia l M dur ing which all the elementssampled were condi t ioned toA2. Thepostulate for the consequences of rein-forcement of response AI during theseimaginary events is given by theequat ion,

F(n)= ( -


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THE M O D E L IN T H E O R Y C O N S T R U C T I O N 125preted by the pictorial model, theproposition reads: after a reinforce-men t on trial n, the probabili ty orproport ion of the elements F(n)con-dit ioned to A I after t r ia l n is equal tothat proportionof theelements sampled(0') which were n ot previously con-ditioned (1 F(B-D), plus thatpro-portion of stimulus elements (F(n_i))already condi t ioned to response A\.In this fashion, the postulates an dhypothetical ideas of the theory maybe interpreted by means of thepic-torial model .Starting with the theory's set ofpostulates, parts of which are ex-pressed in the difference equations 3and 4, derivations are made by meansof the inference rules or mathematicalmodels which include the separatesystems of the calculusof probabili ties,the rules of algebra, and the methodsof mathematical induct ion. Among thetheorems derived (Estes, 1959; Estes& Straughn, 1954)is the func t ion ,F(n)=7r' - (7 r ' -F ( 1 ))( l-0 ')-1[6]which expresses the acquisition curvefor the imaginary events .Once again guided by the pictorialmodel, coordinat ing defini t ions areformulated. For theempirical counter-part of the T-maze situation previouslydescribed, F(B) is coordinated withP(B), the average probability of anA\response for a group of subjects . TheA I response is coordinated with thatclass of empirical responses involvingan initial selection of the left en d box.Coordinating w wi th - a , the latter termis defined as the empirical ratio ofreinforcement for the response co-ordinated with AI. In the T-mazesituation exem plified, the reinforc e-ment schedule for the left response is70% or .7 ; in this si tuation = .7 bydefinition. Finally, n is coordinatedwith N, the n u m b e r of empirical trials,one of which is defined as the sequence

o f events ini t ia t ing in the start boxand terminating in the rewarded en db o x . The concept '$',' let us simplysay, is coordinated with the empiricalacquisition rate 6. Substit i tutin g theseterms in equation (6), w e have theempir ical learning funct ion for the T-maze situation previously portrayed,P()=- ( T - P ( I > ) (I-*)*'1[7]which is stated in terms of the datalanguage. When N is large, the ex-t reme right hand term approacheszero, since (1 0) is less than unity.The function, therefore, reduces toP ( N ) -t. This agrees with the ob-servation that asymptotic response fre-quency for a correction procedureT-maze habit approaches the rein-forcement rat io when frequency ofreinforcement is set at i r and (1 ir)for the two alternative responses. Theempirical scope of this law has notbeen determined. That is, the sets ofin i t ial condi t ions (experimental ar-rangements) within which the law iscorrect await empirical demonstration.The several meanings and functionsof scien tific models have been dem on-strated in statistical learning theory.A characteristic feature ofEstes' theoryobviously isthat the rules of inferenceemployed are entirely mathematical inform. As recently as the late19thcentury, a good number of experi-mental physicists were openly express-in g their distaste for mathematics intheir science. Cons equently, it wouldbe surprising not to hear the samesort of thing from some contemporarypsychologists. Perspectivegained withpassage of t ime as well as the fantast icsuccessofmathematicalphysicssilencedthe critics. A good case is yet to bemade against the expectation that thesame might occur in psychology. Bethat as it may, a cogent point is oftenm a d e : fancy mathematics is nosub-sti tute for creative im agin ation . It


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126 ROY LACKMANwould appear, however , that Estes'theory conta ins a good deal of both.C R I T E R IA F O R E V A L U A T I N G A M O D E L

The modes of operation of severalmodels within their respective theorieshave been demonstrated. It becomesnecessary now to formulate cr i ter ia forevaluating a given m odel. FollowingToulmin's (1953) analysis of science,which w as influenced by Wit tgenstein 's(1922) views, several grounds forjudging a model may be stated.Deployability

The degree to which the te rms inthe model in their primary context ofusage can successfully be brought tothenewsetting. In rg theory, deploy-ability means the extent to which thenonobservable rg can be assigned theproperties of i ts observable counter-part,the condi t ioned response. As ad-ditional properties and laws of theseparate system are assigned to rg, thecondi t ioning model is being deployed.Scope

The range of phenomena to whichth e model is applicable. This meansthe n u m b e r orextent of facts and datathat may be derived by use of themodel. According to this view, atheory does not contain formal limits;instead, the scope of applicability isempirically determined.Delineation of the scope of a theoryprovides the guides and incent ives forsubsequent theorizing. This m ay takethe form of changes in the ini t ialtheory or the construction of a morecomprehensive system. In the lat tercase, the original theory frequentlybecomes a first approximation or aspecial instance of the general theory :the equations of the comprehensivetheory reduce to the equations of thel imited system for the special casesinvolved. The efforts of Estes and

other mathematical model buildersmake it possible to envisage this futurestate of affairs in psychology.While "scope" refers to the empiri-cal derivatives of a theory which aregenerated with the aid of a model,"de-ployabili ty is an attribu te of the setof relationships or meanings conta inedin the model that are employed informula t ing the logical proposi t ions ofthe theory. Gen erally, the more amodel is deployed the greater will bethe scope of the t he o ry ; the reverse,however, is not true.Precision

For psychology, this in the presentpaper refers to the degree to whichthe consequences of a theory are un-equivocally derived through applica-t ion of the inference rules provided bythe m odel. This feature is of specialimpor tance since the prequant i tat ivecharacter of most modelsinpsychologyoften permits contradictory derivat ions.Several im plica tion s of the view-point that generated these criteriameri t specific considerat ion. A funda -mental point concerns the cri ter ion oft ruth. Although the terms t rue an dfalse are applicable to empirical sen-tences, these adjectives are devoid ofme a n in g when applied to a model orthe theorythe model serves. A modelhas lesser or greater deployabili ty an dscope and a certain degree of preci-sion, but not t ru th or fals i ty .A n ad junc t to the three judgmenta lcriteria is the dichotomy betweenmodels provid ing c alcul i, coord ina t in gdefini t ions, an d related functions an dthose yielding only pictorial imagery.The latter, in psychology, have takenth e form of cogn itive m aps, S-Rswitchboards , behavioral f ields, and avariety of peculiar things in the nerv-ous system. When m istaken for moreser ious ins t ruments , the exclusivelypictorial model can generate a good


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TH E M O D E L IN T H E O R Y C O N S T R U C T I O N 127deal of useless experimental labor, apoint eloquently presented by Kendler(1952). Where the inferent ia l orin terpreta t ive funct ion is sacrificed forthe best intuitive picture such as in amodel reflectin g, the existential rich-ness of human l i f e . . ." (Allport,1955, p. 11), the sc ient is t usurps thepoet's func t ion .One consequence of confus ing mod-ular te rms wi th empir i ca l sen tences isthe variety of cases cont inuously madeagainst one or ano ther m odel . Forexample, Harlow(1953)contends thatsimple behavior has explanatory valueonly for simpler phenom ena and pro-vides n o in form at ion concern ing com-plex behavior. This view loses allurgency when one seeks the basis forthe d is t inc t ion between s imple andcomplex. Occasionally, the distinctionis held to be self-evident. How ever,this cannot be seriously regarded, asthat which is considered self-evidentfo r on e generation of science is fre-quently bel ieved to be absurd by someother generation. More often, thedist inction between simple and com-plex phenomena is predicated upon themoreextensiveknowledge accumulatedconcerning the f o r m e r : what is rela-tively well understood is simple; thatwhich is less well known is complex.Harlow (1953) an d Asch (1952)have been most cr i t ical of the sim-plicity involved in the condi t ioningmodel. We may look forward to theday when the research being done bythese scientists transforms "complex"incentive-motivational, exploratory an dsocial behaviors in to sim ple phe-nomena .

THE ROLE OF A N A L O G YThere are no sufficient a priorigrounds for determining how wellsome separate system will func t ion asa theoretical model for another realmof phenomena. The successful theo-

rist 's imagination does, initially, graspan analogy between the data of in-terest and the separate system. Asthe basis for selecting some model,recognition of a critical analogy hassomet imes produced the most incred-ible accompl i shments : de Broglie'swave mechanics an d Ein stein 's rela-t iv i ty are two ins tances . Successfulapprehension of analogy is not limitedto the most i l lustr ious examples, foras Muni tz (1957b) has demonstrated,analogy has served a core func t ion inman' s interpretation of his environ-men t from Baby lonian mythology tom odern science. Consequently, thelogician has exerted some effort inthe a t tempt to tor ture the notion ofanalogy into a neat classif icatoryscheme. With the exception of therecent heroic effor ts of Braithwaite(1953), the basis for the analogicalrelationship between a model's sepa-rate system and the ideas of the theoryit serves has not been explicated.

M O D U L A R THEORIES A N DM O D U L A R C O N S T R U C T S

One corollary of the methodologicalposition outlined is the required dis-t inct ion between construc ts in troducedby a model and those defined im-mediately in terms of observables.Stimulus elements and rg are theo-retical or modular constructs, whereasdrive, defined in terms of the knownlaws concerning the energizing andfacilitative effects of deprivation, mightbetermedan empirical construct. Anessentially similar distinction has re-cently been made by Ginsberg (1954),so we need not pu rsue i t fur th er exceptfor on e addit ional consideration. D i-chotomizing constructs along the linessuggested might provide one basis forcategorizing scientif ic theories. Bor-rowing the classif ication of empiricalconstruct theory an d axiomatic-modeltheory from Spence (1957), w ewould


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128 ROY L A C H M A Nsuggest that whereas the former classof theory employs empir ical cons truc ts ,the latter uti l izes modular or theo-retical construc ts . Examples of axio-matic-model theories have been ana-lyzed here; there are no examples ofrelatively pure empirical constructtheory in psychology. Interestin glyenough, Spence's (1956) theory mighthe considered a mixed type, as heemploys empir ical constructs such asdrive an d habit an d theoretical con-structs such as rg and sg.In the course of the last century,theories of the axiomatic-model typewere considered extremely objection-able to a number of scientists an dphilosophers such as Mach, Duhem,an d Ostw ald. Character ist ic objec-t ions concerned the employment ofconcepts at another level of discourse,that is, the appeal to suprasensibleoccurrences. The great successof theaxiomatic-m odel type of theory inclassical physics eventually silenced theposit ivist ic cr i t ics. In contemporarypsychology, however, this viewpointhas sired the school of radical em -pir ic ism. Although the ingenui ty an dimagination of Skin ner 's experimen ta-tion are probably unequaled, his viewson theory (Skin ne r, 1950) should beseverely quest ioned.

TH E M O D E L AN D R E A L I T YO n pragmatic grounds, most scien-t is ts find i t necessary to acknowledgean external world independent of theobserver. A s a consequence, discourseis generated concerning the relation-ship between the models an d con-structs of science and the phenomenathese represent. Since a successfulmodel enables us to predict the fu tureoutcome of sets of events, the model issomet imes assumed to be a literaldescr iption of reality. For example,i t might be argued that the kinetictheory predicts the behavior of gases

and since a gas must be composed ofsomething, why not the moleculespic tured by the m ode l? Such an as-sumption in no way in ter feres wi ththe scientists ' activity and may be dis-regarded. However, on e consequenceof that view is not so harmless . Occa-sionally it is argued that if there is areality, then one and only on e modelcan provide the best description of it.This last proposit ion and the assump-tion of real i ty correspondence uponwhich it is based are seriously dis-credited by theverycontent ofphysicsand psychology. Nu m erous examplesare available of several models servingthe same class of even ts. Con versely,a single model can func t ion in behalfof independent classes of phenomena .Whittaker provides an admirable il lus-tration of the latter state of a f fa i r s :. . . it happens very often that di fferentphysical systems are represented by identicalmathematical descr iption. For example, thevibrations of a membrane which has theshape of an ellipse can becalculated bymeansof a differential equation known as Mathieu'se qua t ion : but this same equation is also ar-rived at when we study the dynamics of ac ircus pe r former ,w ho holds an assis tant bal-anced on a pole while he himself s tands ona spheric al ball rolling on the ground. Ifw e now imagine an observer who discoversthat the fu ture course of a cer tain phenome-non can be predicted by Mathieu's equation,but who is unable for some reason to per-ceive the system which generates the phe-nomenon, then evidently he would be unableto te l l whether the system in question is anell ipt ic memb rane or a var ie ty art i s te (Whi t -taker , 1942, p. 17).

S U M M A R YThe functions of models in theoryconstruction were analytical ly catego-r ized as (a) representational , ( b ) in -

ferent ia l , (c ) interpretational , and (d )pictor ial . Dist inctions introduced wereexemplified in the kinetic theory ofgases, rg theory, an d statistical learn-in g theory. Implications of the anal-


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THE MODEL IN THEORY C O N S T R U C T I O N 129ysis for current methodological prob-lems in psychology were examined.

R F R N SALLPORT, G. W. Becoming: Basic con-siderations for a psychology of personality.New Haven: Yale Univer. Press, 1955.AMSEL, A. The role of f ru s t r a t i ve non-reward in noncontinuous reward situa-tions. Psychol. Bull., 1958, 55, 102-119.ASCH, S. E. Social psychology. NewYork: Prentice-Hall, 1952.BRAITHWAITE, R. B. Scientific explanation.N ew York: Cambridge Univer. Press,1953.CAMPBELL, N. R. Physics; th e elements.

N ew York: Cambridge Univer. Press,1920.CARNAP, R . Foundations of logic an dmathema t i c s . In Int. Encyc. Unif. Sci.Vol. 1. C hicago: Univer. Chicago Press,1955.D'ABRO, A. The rise of the neiv physics.Vol. 1. New York: Dover Publications,1951.DUHEM, P. The aim and structure ofphysical theory. Princeton: PrincetonUniver. Press, 1954.ESTES, W. K. The statistical approach tolearning theory. In S. Koch ed.),

Psychology: A study o f a science. Vol. 2.N ew York: McGraw-Hill, 1959.ESTES, W. K. Stimulus-response theory ofdrive. In M. R. Jones (ed.), Nebraskasymposium on motivation: 1958. L i n c o l n :Univer. Nebraska Press, 1958.ESTES, W . K., & STRAUGHN, J. H. Anal-ysis of a verbal conditioning situation interms of statistical learning theory. /.exp. Psychol., 1954,47, 225-234.GINSBERG, A . Hypothetical constructs an dintervening variables. Psychol. Rev.,1954, 61, 119-131.GRANT, D. A. The discrimination of se-quences in stimulus events and the trans-miss ion of information. Amer. Psy-chologist 1954,9, 62-68.HARLOW, H. F. Mice, monkeys, men, andmotives. Psychol. Rev., 1953, 60, 23-32.HOWES, D. H., & SOLOMON, R. L. Visualduration threshold as a func t ion of word-probabil i ty. /. exp. Psychol., 1951, 41,

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Received June 26, 1959)

The Model in Theory Construction

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