JOURNAL OF THE EXPERIMENTAL ANALYSIS OF BEHAVIOR

STIMULUS EQUIVALENCE AND TRANSITIVE ASSOCIATIONS:A METHODOLOGICAL ANALYSIS

LANNY FIELDS, THOM VERHAVE, AND STEPHEN FATHTHE COLLEGE OF STATEN ISLAND/CUNY, AND QUEENS COLLEGE/CUNY

When a number of two-stimulus relations are established through training within a set ofstimuli, other two-stimulus relations often emerge in the same set without direct training.These, termed "transitive stimulus relations," have been demonstrated with a variety ofvisual and auditory stimuli. The phenomenon has served as a behavioral model for ex-plaining the emergence of rudimentary comprehension and reading skills, and thedevelopment of generative syntactic repertoires. This article considers the range of rela-tions that can arise between a given number of stimuli in a class, the number of directlyestablished two-stimulus relations necessary for the emergence of transitive relations, theforms that training sets of stimuli can take, and the number of transitive two-stimulus rela-tions that can be induced without direct training. The procedures needed to establish andassess transitive stimulus control, the possible interactions between the training and testingprocedures, and the constraints these interactions place upon the analysis of transitivestimulus control are also examined. The present analysis indicates that in a transitivitytest, choice among such stimuli may be controlled by (1) the relation between the sampleand the positive comparison stimulus (transitive stimulus control), (2) the relation betweenthe sample and the negative comparison stimulus (S- rule control), and (3) possiblediscriminative properties that may inadvertently be established in the positive andnegative comparison stimuli (valence control). Methods are described for distinguishingthese three forms of stimulus control.

Key words: stimulus equivalence, transitive stimulus control, complimentarity, mediatedassociation, nodal training clusters, associative distance, concept formation, cognition,language acquisition

In the early part of the present century,some associationists held the view that in-direct formation of associations betweenideas was an essential characteristic of think-ing (Warren, 1921). Such indirect associa-tions were thought to be mediated by thelinkage of each idea with a common thirdidea or event. For example, if A, B, and Crepresent "ideas," after linking A and B(A-B), as well as A and C (A-C), ideas Band C should be associated without furtherado (B - C). Such relations were called

This research was partially supported by grantsfrom the Research Foundation of the City Universityof New York. We wish to thank Peter Harzem for hisinvaluable editorial assistance in preparing thismanuscript for publication. Reprints may be obtainedfrom Lanny Fields or Stephen Fath, Department ofPsychology, College of Staten Island/CUNY, 130Stuyvesant Place, Staten Island, New York 10301, orfrom Thom Verhave, Department of Psychology,Queens College/CUNY, Kissena Boulevard,Flushing, New York 11367.

mediated associations. Psychologists beganto study mediated association experimen-tally by substituting stimuli for ideas (Peters,1935), and have variously termed thisphenomenon "mediated generalization,""mediated transfer," "equivalent stimuli,""transitive relations," and "derived stimulusrelations." With the exception of the lasttwo, each term implies a mechanism respon-sible for the observed behavior. Because ourgoal is to explore the procedures responsiblefor establishing control of behavior by aspecific set of stimulus relations, and not toexplore any linkage mechanisms, we will usethe terms "transitive stimulus control" and"transitivity" to refer to the phenomenon.This choice is based on the frameworkpresented by Sidman and Tailby (1982),who proposed that for a set of stimuli to beequivalent the stimuli must be reflexive,symmetrical, and transitive. Each of theseproperties can be defined logically, indepen-

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LANNY FIELDS, THOM VERHA VE, and STEPHEN FA TH

dent of behavior that may be empiricallyobserved. Such logical definitions do not, ofcourse, imply that the properties chosen bythe experimenter in fact control the subjects'behavior. Discovering whether that is thecase depends on empirical evidence concern-ing the relations between the experimenter-defined stimulus property and behavior. Inorder to demonstrate transitive stimuluscontrol, the stipulated behavior change mustbe observed under conditions of nonrein-forcement so it cannot be attributed to directreinforcement, and it must be observed inthe presence of novel discriminative stimulithat have the property under consideration.In this paper, we consider how such tran-sitive control may be generated in discrim-inative stimulus sets, so that stimuli in suchsets that have not previously been related tothe responses under study nevertheless con-trol those responses.

In addition to transitivity, symmetry andreflexivity are central concepts that comprisethe notion of stimulus equivalence. Sym-metry has been closely intertwined with theanalysis of transitivity (Jenkins, 1963, 1965;Lazar, 1977; Mackay & Sidman, 1977; Sid-man, 1971; Sidman & Cresson, 1973; Sprad-lin & Dixon, 1976). We do not, however,consider in this article the many complexrelationships between symmetry and tran-sitivity. The control of behavior by the sym-metrical properties of stimuli, their interac-tion with control exerted by the transitiveproperties of stimuli, and theoretical issuesregarding their independence or interde-pendence are extensive and merit the atten-tion of a separate article.The following example illustrates our

usage of terms and the general approachused to study transitive stimulus control ex-perimentally. Let us assume that transitivityis to be established among a picture of a catand the English and Spanish names for cats- and likewise for dogs. We have two stimu-lus classes, one for cats and the other fordogs. Each class contains three stimuluselements. For dog, these are a picture of adog and the words "DOG" and "PERRO."In each trial, a sample stimulus and two

comparison stimuli are presented, only oneof the comparison stimuli being related tothe sample. A response to that comparisonstimulus is reinforced; a response to theunrelated comparison stimulus goes unrein-forced. If a picture of a dog is presented asthe sample with the words "DOG" and"CAT" as comparison stimuli, the choice of"DOG" is reinforced. In this fashion, a two-term relation-a picture of a dog and theword "DOG"-is directly established bytraining. Before assessment of transitivestimulus control, a second two-term relationmust be established. For that purpose, a pic-ture of a dog is presented as a sample withthe words "GATO" and "PERRO" as com-parison stimuli. Reinforcing the response to"PERRO" directly establishes a second two-term relation. Although a picture of a dogand "PERRO," and a picture of a dog and"DOG" were directly linked by training,"DOG" and "PERRO" were not. Whetherthe theoretical transitive relation between"DOG" and "PERRO" actually controlsbehavior can be assessed by presenting"DOG" as sample with "PERRO" and"GATO" as comparison stimuli. The choiceof "PERRO" would demonstrate transitivestimulus control. If, in fact, reflexive andsymmetrical stimulus control were alsodemonstrated, the words and pictures wouldnow function as a class, the members ofwhich are behaviorally equivalent (Sidman& Cresson, 1973).

Transitive stimulus control was firstdemonstrated by Peters (1935) using visuallypresented nonsense syllables as pairedassociates and later byJenkins (1963, 1965),Kjeldergaard and Horton (1960), and Goss(1961). In these studies, however, groupdesigns were used. After conditioning A- Band B-C, A-C was conditioned. Rate oflearning A-C was compared with the rateof learning A- C for a control group of sub-jects who had not previously learned themediated pairs. More rapid acquisition ofA- C by the experimental group than by thecontrol group indicated the "presence ofmediational associates."More recently, transitivity has been dem-

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onstrated directly in single subjects, as il-lustrated in the preceding example, usingconditional discrimination or matching-to-sample methods (Lazar, 1977; Premack,1971, 1976; Sidman, 1971; Sidman & Cres-son, 1973; Sidman, Cresson, & Willson-Morris, 1974; Sidman & Tailby, 1982;Spradlin, Cotter, & Baxley, 1973; Spradlin& Dixon, 1976; Wetherby, Karlan, & Sprad-lin, 1983). The subjects used in these andother studies of transitivity included normaladults and children (Lazar, 1977; Lazar &Kotlarchyk, 1980; Sidman, Rauzin, Lazar,Cunningham, Tailby, & Carrigan, 1982),retarded individuals and global aphasics(Dixon & Spradlin, 1976; Glass, Gazzaniga,& Premack, 1973; Lazar, 1977; Mackay& Sidman, 1977; Sidman, 1971; Sidmanet al., 1974), and primates (Premack, 1971;Salmon & D'Amato, 1983). In these ex-

periments a wide range of visual andauditory stimuli were used, and in some,classes consisting of visual stimuli only were

established. Sidman and Tailby (1982) usedupper- and lower-case Greek alphabeticsymbols; Lazar (1977) used complex visualstimuli consisting of sets of four triangles ar-

rayed in different orientations in one set ofexperiments, and upper- and lower-caseGreek alphabetic symbols in conjunctionwith colored stimulus patches in another.Others have established classes consisting ofboth visual and auditory stimuli. Sidman(1971) and Sidman and Cresson (1973) es-

tablished transitive stimulus control usingpictures of common objects and animals,their written names, and their dictatednames. Spradlin and his colleagues(Spradlin et al., 1973; Spradlin & Dixon,1976) established transitive stimulus controlusing visual nonsense shapes and auditorynonsense words.

QUANTITATIVE ANALYSIS: THERELATIONSHIP BETWEEN TRANSI-TIVITY AND STIMULUS CLASS SIZE

In the above example, the stimulus classreferring to canines consisted of threeelements, a picture of a dog and the words

"DOG" and "PERRO." A stimulus class can,

of course, consist of more than threeelements. For example, adding the spokenword "dog" and the sound of a barking dogwould increase the stimulus class of "canine"to five elements. Indeed, the number ofelements per class for any particular class isindefinite.As stimulus classes become larger, the

number of potential transitive relationsamong the elements within that class also in-creases. Several important research ques-

tions may be asked concerning stimulusclasses of varying sizes. How many two-termrelations exist? What is the minimumnumber that must be established by trainingin order to link all elements? How many

ways are there of linking all elements? Howmany and which two-term stimulus relationsare transitive? How can transitive stimuluscontrol be assessed? In the present articlethese issues are systematically examined.

Number of Two-Term Relations in a

Stimulus ClassConsider a stimulus class consisting of five

elements, A, B, C, D, and E. All of the two-term relations in the class can be identifiedby establishing a matrix with the elementslisted as column and row headings. The in-tersections of columns and rows containingdifferent letters define all of the two-termrelations that exist within the stimulus class.Table 1 designates all possible two-termrelations for a class containing five elements.Each two-term relation is generic and doesnot specify the function that may be servedby each stimulus.

Similar matrices can be generated forstimulus classes of any size. The number of

Table 1Stimulus-combination matrix showing the set of two-term relations for a stimulus class containing fiveelements.

A B C D E

A AB AC AD AEB BC BD BEC CD CED DE

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LANNY FIELDS, THOM VERHA VE, and STEPHEN FA TH

two-term relations that exist in a stimulusclass of a given size is specified by the follow-ing formula: (N- 1)N/2, where N is thenumber of stimuli in the class.

Number of Training Pairs Needed to Create aStimulus Class

Theoretically, the minimal conditionneeded to link all stimuli within a class en-tails the establishment of (N- 1) two-termrelations by direct training, provided eachelement in the stimulus class is used at leastonce. All of the remaining two-term rela-tions should be transitive, because allelements are indirectly linked by trainingwith common intermediary elements. Forexample, if AB, AC, AD, and AE were usedfor training, the transitive relation, EB,could be established because of the commonrelations of E and B with A. A transitiverelation may also be developed in a differentway, through serial intermediate linkages.For example, if AB, BC, CD, and DE wereused for training, the transitive relation, EB,could have been established because of therelation of E and B through serial in-termediate linkages with C and D.

Nodal Training ClustersAlthough (N- 1) two-term relations are

needed to establish a stimulus class, manydifferent sets of two-term relations may beused for that purpose. Each will be called a"nodal training cluster." For a stimulus classof N elements, there are (N- 2) possiblenodal training clusters that can be used tolink all stimuli in a set.The particular two-term relations that

comprise each nodal training cluster can bespecified and systematically interrelated byusing the concept of a "node." A node is astimulus that is related to more than oneother stimulus used in training. A stimulusrelated to only one other stimulus is called a"single." The specific combinations of two-term relations that comprise each trainingcluster are determined by listing all of thenumerical combinations of nodes and singlesthat make up the total number of stimuliwithin a class, with the restriction that at

least two singles must occur in any combina-tion. For example, with a stimulus class of 6elements, training pairs can be chosen thatcontain 4 nodes and 2 singles, 3 nodes and 3singles, 2 nodes and 4 singles, or 1 node and5 singles. The training clusters that could beused to link all elements in a class are showngraphically in Table 2 as letter-line arrays,for stimulus classes containing 3, 4, 5, and 6elements.

Although combinations of individualstimuli in nodes could be different fromthose listed in Table 2, the table shows allpossible combinations of nodes and singlesin a class. For example, a 6-element classcan be established with the cluster AB, BC,BF, CD, and DE, which has 3 nodes, orwith the cluster of AB, AC, CD, DE, andDF, which also has 3 nodes; the latter isshown in Table 2. Both clusters have 3nodes- one linked to 3 stimuli and two linkedto 2 stimuli each. The fact that the letterscorresponding to nodes differ for eachtraining cluster is of no consequence becausethey are assigned to actual stimuli in an ar-bitrary manner. Inasmuch as the same argu-ment can be made for all other variants ofany set of nodal clusters, (N- 2) nodalclusters specify all possible training com-binations that can be used to establish astimulus class.

Associative Distance and TransitivityIn addition to the different ways in which

the stimuli within a class can be linked bytraining, the number of training nodes mayalso be an important variable that influencesthe degree of transitive stimulus control. Forexample, with five stimuli in a class, A, B,C, D, and E, two of the ways in which stim-uli can be linked are (1) by training with a1-node cluster AB, AC, AD, and AE, or (2)by training with a 3-node cluster AB, BC,CD, and DE. The number of intermediatestimuli that link a transitive pair differs aftereach procedure. Thus, after training withthe 1-node cluster, the transitive pair BE isrelated by one intermediate link, A, whereasafter training with the 3-node cluster, BE isrelated by two intermediate links, C and D.

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Table 2

147

Number of nodal training clusters for stimulus classes containing three to six elements.Capital letters represent stimuli. The end-points of each horizontal line indicate the stimulilinked in training. Nodes (indicated by asterisks) are the stimuli that have at least two lineend-points beneath them. Singles are stimuli with only one line end-point beneath them.

No. Training No. No.Stimuli Clusters Nodes Singles *A B C

3 1 1 2

*A B C2

2

3

3

2 2

4

D

A *B *C D

*A B C D E

2

3

1

2

3

*A B *C D5 3

5

6

6

6

3

4

4

4

3

2

5

4

3

E

A *B *C *D E

*A B C D E F

*A B *C D E F

*A B *C *D E F

A *B *C *D *E F6 4

4

4

5

I

4 2

LANNY FIELDS, THOM VERHA VE, and STEPHEN FA TH

The term "associative distance" is definedhere concretely in terms of the number of in-termediate links needed to connect thestimuli comprising a transitive pair. Ifassociative distance influences transitivity,fewer transitive relations should emerge

after training with clusters containing more

nodes. If the same degree of transitivity isfound regardless of training cluster, we can

conclude that transitivity is not influencedby nodes or associative distance.The nodes and singles used in training

will influence the specific stimulus pairs usedto test for transitive stimulus control. For ex-

ample, if a 1-node training cluster is used toestablish a 4-element stimulus class, AB,AC, and AD would be used for training, andBC, CD, and BD would be used to test tran-sitive stimulus control. Alternatively, if a

2-node cluster is used to establish the class,AB, BC, and CD would be used for train-ing, and AC, BD, and AD would be used totest transitive stimulus control. Otherpossibilities can be deduced by examiningTable 2.

The Class Size/Transitivity FunctionThe maximum number of two-term tran-

sitive relations in a particular stimulus classis specified by the following formula:(N- 2) (N- 1)/2, where N is the number ofstimuli in a class.The minimum numbers of training pairs

required to establish a stimulus class and thenumbers of potential transitive relations forstimulus classes ranging from 3 to 16 ele-ments are shown in Table 3. In a stimulusclass of three elements, the minimum numberof two-term relations that must be trained togenerate transitive relations is greater than

the number of transitive relations that are

generated. In a stimulus class of fourelements, these numbers are equal; in largerclasses, the number of transitive relations isalways greater. As stimulus class size in-creases, the number of transitive relationsincreases at an ever expanding rate. Thetheoretical relationship could be consideredmetaphorically as the behavioral analog of a

"cognitive breeder reactor," because thenumber of transitive relations created far ex-

ceeds the number of relations that must beestablished by training. Does this theoreticalrelationship correctly describe the empiricalrelationship induced in subjects exposed toappropriate training? A review of the ex-

isting literature indicates that transitive stim-ulus control has been demonstrated usingfrom 2 to 20 stimulus classes with from threeto eight stimuli per class (Dixon & Spradlin,1976; Lazar, 1977; Lazar & Kotlarchyk,1980; Mackay & Sidman, 1977; Sidman &Cresson, 1973; Sidman et al., 1974; Sid-man & Tailby, 1982; Spradlin et al., 1973;Spradlin & Dixon, 1976; Stromer & Os-borne, 1982; Wetherby et al., 1983). Un-fortunately, the functional relation betweenstimulus class size and the degree of tran-sitive stimulus control cannot be derivedfrom the existing literature, because dif-ferent training and testing procedures were

used in each study. These studies were notdesigned to explore the relationship betweenstimulus class size and number of transitivestimulus relations per se, and all of the possi-ble test configurations for transitive stimuluscontrol were not used. The remainder of thisarticle delineates the procedures that can beused to explore this relationship and iden-tifies the factors that must be considered

Table 3The class size/transitivity function. Numbers of training pairs needed to establish a stimulus classand the numbers of transitive relations that should emerge are shown for stimulus classes con-taining different numbers of elements.

Class Size3 4 5 6 7 8 9 10 11 12 13 14 15 16

No. TrainingPairs 2 3 4 5 6 7 8 9 10 11 12 13 14 15

No. TransitiveRelations 1 3 6 10 15 21 28 36 45 55 66 78 91 105

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when assessing transitive stimulus control.

Interaction of Class Size and NodesClass size is not, of course, the only

variable that influences degree of transi-tivity. As indicated in the previous section,the configuration of nodes used for trainingmay also influence the degree of transitivity;that is, increasing the number of nodes maydecrease the degree of transitive stimuluscontrol. If both nodality and class size in-fluence transitive stimulus control, thenumbers of transitive relations predicted inthis section specify the upper limits of tran-sitivity that can be expected when nodaltraining clusters are manipulated.

EMPIRICAL ANALYSIS:ESTABLISHING TRANSITIVE

STIMULUS CONTROL

So far, the formal and quantitative aspectsof transitive stimulus control have beendiscussed. In this section we consider theprocedures that must be used to establishtransitive stimulus control, including con-straints on training conditions, interactionsthat may exist between training and testingconditions, methods of analyzing the choicesmade by subjects during transitivity testtrials, and possible alternative explanationsof results that appear to reflect control bytransitive relations. To facilitate this discus-sion, a simple notational convention isneeded for identifying stimuli and their classmembership.A single stimulus class was used to il-

lustrate the logical relations that existbetween the stimuli within a class; however,at least two stimulus classes and many com-binations of stimuli must be presented dur-ing training and testing when establishingtransitive stimulus control. Thus, we shalldesignate each stimulus within a class by aletter, and each class by a number, with theresult that each stimulus has a letter andnumber designation (e.g., C2 would refer toa particular element in class 2).

Trial Contingencies: Training Trials andTesting TrialsThe establishment and assessment of tran-

sitive stimulus control have been exploredusing a conditional discrimination para-digm. Typically, training and testing areconducted on a discrete-trial basis. Threestimuli are presented during each trial: asample (Sa), a positive comparison stimulus(Co+), and a negative comparison stimulus(Co-). The terms "sample" and "compari-son" are used descriptively and do not implyany process engaged in when responding tothe stimuli.Each training trial begins with the presen-

tation of a sample. Responding to it producesthe Co+ and Co-. The Co+ is anothermember of the stimulus class from which thesample was drawn, and the Co- is one of thestimuli from the other class. For example, ifAI is presented as a sample, then Cl could bepresented as the Co+ with any one of thestimuli in the other class, X2, as Co-. Aresponse to Co+ is reinforced and a responseto Co- goes unreinforced. Either responseterminates the stimuli. All trials remain ac-tive until a response is made. After an inter-trial interval (ITI), another trial is presented.

Test trials have the same format as train-ing trials. They differ in terms of the specificstimuli presented as samples and compar-ison stimuli, and by the fact that no rein-forcement occurs following any response.

If during training some of the stimuli in aclass are always used as samples and neveras comparison stimuli, and some other stim-uli in the class are used as comparisonstimuli and never as samples, training is saidto be unidirectional. If, however, the stimuliare used both as samples and comparisonstimuli, training is said to be bidirectional.The following analysis assumes the use ofbidirectional training and testing.

A Specific ExampleTo study transitive stimulus control, some

central issues regarding training and assess-ment procedures must be addressed. We willdo so by considering as an example the con-

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LANNY FIELDS, THOM VERHA VE, and STEPHEN FATH

Table 4All possible bidirectional training trials for two stimulus classes with four stimuli per class.

Class 1Sa Co+ Co- Sa Co+ Co- Sa Co+ Co- Sa Co+ Co- Sa Co+ Co- Sa Co+ Co-

1. Al B1 A2 Bl Al A2 Al Ci A2 Cl Al A2 Al DI A2 DI Al A22. Al Bl B2 Bl Al B2 Al Cl B2 Cl Al B2 Al DI B2 DI Al B23. Al B1 C2 Bl Al C2 Al Cl C2 Cl Al C2 Al DI C2 DI Al C24. Al B1 D2 Bl Al D2 Al Ci D2 Cl Al D2 Al Dl D2 DI Al D2

Class 2Sa Co+ Co- Sa Co+ Co- Sa Co+ Co- Sa Co+ Co- Sa Co+ Co- Sa Co+ Co-

5. A2 B2 Al B2 A2 Al A2 C2 Al C2 A2 Al A2 D2 Al D2 A2 Al6. A2 B2 BI B2 A2 Bl A2 C2 Bi C2 A2 Bi A2 D2 Bi D2 A2 BI7. A2 B2 Cl B2 A2 Cl A2 C2 Cl C2 A2 Cl A2 D2 Cl D2 A2 Cl8. A2 B2 Dl B2 A2 Dl A2 C2 Dl C2 A2 DI A2 D2 Dl D2 A2 DI

ditions needed to establish and assess tran-sitive stimulus control using two stimulusclasses with four stimuli per class. Table 4shows all possible training trial configura-tions when a training set containing. onenode and three singles is used. Stimuluspairs AB, AC, and AD are used for training,responding to the Co+ being reinforced inthe context of all available negative com-parison stimuli drawn from the other class.All trials on a given row share a commonCo-. For adjacent trial pairs, the stimuliused as sample and positive comparison arereversed.

Because AB, AC, and AD were used intraining, in test trials stimulus pairs BD,CD, and BC are presented. Each pair isused bidirectionally and in the context of allnegative comparison stimuli drawn from theother stimulus class. Table 5, which has thesame format as Table 4, presents all of thepossible configurations of stimuli that can beused in test trials to assess transitive stimuluscontrol.

Training Options and Constraints on TestingIf during bidirectional training all the

combinations in Table 4 are presented, tran-sitive stimulus control could be assessed bypresenting the test configurations shown inTable 5 and recording the subject's choices.If, for example, responses were made ex-clusively to Co+, one might conclude thatthe transitive relations controlled respond-ing. Such a conclusion, however, would beerroneous. Close analysis of the matrices

presented in Tables 4 and 5 indicates thatconsistent responding to Co+ can be ex-plained by reference to an entirely differentsource of control. During training, respond-ing could have come under the control ofSa/Co- combinations because in the pres-ence of those combinations, responding to"not Co-" was reinforced. This has beencalled "S- rule control of behavior" (Carter& Werner, 1978). Each Sa/Co- used intraining as shown in Table 4 also appears inthe test matrix shown in Table 5. For exam-ple, B1 Al C2 is used as a training con-figuration, and Bi DI C2 is used as a testconfiguration. It is possible, therefore, thatchoices of the Co+ during test trials couldreflect the S- rule, rather than transitivestimulus control. For this reason, using ex-haustive bidirectional training to establishcontrol by transitive relations precludes theunambiguous assessment of transitive stimu-lus control.

Partial Bidirectional Training andComplementarity

In light of the above discussion, it appearsthat if transitive stimulus control is to beassessed unambiguously, training must beconducted with a subset of the trial con-figurations. Such a subset must link togetherall of the stimuli in a class and do so for eachstimulus class. However, not every stimulusin the opposing class can be used as a Co- intraining. These points can be illustrated byreferring to Tables 4 and 5. Each line of thetraining matrix defines a training subset that

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Table 5All possible transitivity tests trial configurations after training with two stimulus classescontaining four elements each.

Class ISa Co+ Co- Sa Co+ Co- Sa Co+ Co- Sa Co+ Co- Sa Co+ Co- Sa Co+ Co-

1. B Cl A2 C1 BI A2 Bi DI A2 DI Bi A2 C1 DI A2 Dl C1 A22. B1 Cl B2 Ci BI B2 Bi DI B2 DI Bi B2 Ci DI B2 Dl Ci B23. B1 Cl C2 Ci Bi C2 Bi DI C2 DI Bi C2 C1 DI C2 DI Ci C24. B1 Cl D2 Ci B1 D2 Bi Dl D2 DI Bi D2 Cl DI D2 DI Ci D2

Class 2Sa Co+ Co- Sa Co+ Co- Sa Co+ Co- Sa Co+ Co- Sa Co+ Co- Sa Co+ Co-

5. B2 C2 Al C2 B2 Al B2 D2 Al D2 B2 Al C2 D2 Al D2 C2 Al6. B2 C2 BI C2 B2 BI B2 D2 BI D2 B2 Bi C2 D2 BI D2 C2 BI7. B2 C2 Cl C2 B2 Cl B2 D2 Cl D2 B2 Cl C2 D2 Cl D2 C2 Cl8. B2 C2 DI C2 B2 Dl B2 D2 Dl D2 B2 Dl C2 D2 Dl D2 C2 Dl

uses one Co- and links all stimuli in a class.If the trials listed on lines 1, 2, 5, and 6 ofTable 4 are used for training, trials listed onlines 3, 4, 7, and 8 of Table 5 can be used totest transitive stimulus control. In that case,the negative comparison stimuli used in thepresence of specific samples are different inthe training and testing configurations.Thus, choices of Co+ stimuli in test trialscould not be attributed to control of behaviorby the S- rule.

In general, there is complementary rela-tion between the trials used for the trainingand the testing of transitive stimulus control.As the number of training trials increases,the number of test trials decreases cor-respondingly. By the same token, the use ofspecific training configurations eliminatesthe possibility of using those particular con-figurations in test trials. This relationshipwill be referred to as the complementarityprinciple, which is illustrated in detail byreference to Tables 4 and 5. Each line oftrials in both Tables is numbered. The train-ing trials listed on each line of Table 4 linkall stimuli in a class in the context of a singleCo-. Each line of training trials has a cor-responding line of test trials (Table 5) thatuses the same Co-. Using the trials listed ina given line of Table 4 precludes the assess-ment of transitive stimulus control with testtrials listed on the corresponding line ofTable 5. Transitive stimulus control can,however, be established and assessed usingtrials designated by the combinations of the

lines in Tables 4 and 5. These combinationsare listed in Table 6.When training is minimal, a broad range

of transitivity testing can be conducted. Asthe subset of training configurations in-creases in number, the variety and numberof usable test configurations decrease. Thus,the generality of transitivity testing is in-versely related to the extent of training.Because this analysis is valid regardless ofstimulus class size and the number ofstimulus classes used in training and testing,the complementarity principle points, to anunavoidable procedural constraint that mustbe considered when attempting to establishand assess transitive stimulus control.

Valence: Control by Comparison Stimuli OnlyConsider an experiment in which partial

training and complementary transitivitytests are conducted. If subjects responduniformly to Co+ during testing, the S- rulecannot -be used to explain the outcome. Toattribute the results to transitive stimuluscontrol, however, control by yet anotheraspect of the test stimuli must be ruled out.This is discriminative control by the com-parison stimuli only. On different trials intraining, each stimulus serves as Co+ andCo-. When used as a Co+, it functions as adiscriminative stimulus (SD); when used as aCo-, it functions, conversely, as an SA.Therefore, each comparison stimulus mayacquire discriminative properties related tothe frequency with which it occasioned rein-

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LANNY FIELDS, THOM VERHA VE, and STEPHEN FATH

Table 6Possible complementary combinations for training and testing. The line numbers refer tothose shown in Tables 4 and 5.

Stimulus Class 1 2 1 2 1 2

Train with Lines 1,2,3 5,6,7 1,2 5,6 1 5Test with Lines 4 8 3,4 7,8 2,3,4 6,7,8No. Train. Trials 18 18 12 12 6 6No. Test Trials 6 6 12 12 18 18

forcement and nonreinforcement. On eachtransitivity test trial, then, the choice of a

given comparison stimulus could be affectedby the discriminative control directly estab-lished during training. In other words, testperformance could have been controlled bythe Co+/Co- relation rather than the tran-sitive Sa/Co+ relation. To discover whetherand to what extent this was the case, the fre-quency with which each comparison stimu-lus was presented in training as a functionalSD and SA can be established. These two fre-quencies may, in turn, be expressed as a

derived measure called "valence," which issimply the result of subtracting SA frequencyfrom the SD frequency. Thus, if a positivecorrelation exists between the choice of Co+and relative valences of the comparisonstimuli, these choices could be explained bythe simple disciminative properties of thecomparison stimuli rather than the transitiverelations between samples and comparisonstimuli. Alternatively, if there is no correla-tion between the relative valences of thecomparison stimuli and the responses madeto the Co+, the evidence strongly indicatescontrol by transitive stimulus relations.To illustrate the above analysis, assume

that training is conducted using the trialslisted on lines 1, 2, 5, and 6 of Table 4. Eachtime a stimulus appears as Co+ it is given an

increment of +1 because it sets the occasionfor the reinforcement of a response to thestimulus. Likewise, each time a stimulus ap-

pears as Co-, it is given a decrement of -1because it sets the occasion for nonreinforce-ment of a response to that stimulus. Thevalence accumulated in this fashion by eachstimulus can be estimated by summing thepositive and negative values.

Table 7 lists each stimulus, the number oftimes it was used as Co+ and Co-, and itsvalence. These values were determined byassuming that training was conducted usingthe trials listed on lines 1, 2, 5, and 6 inTable 4.Once the valence of each stimulus has

been estimated, the valences of the com-

parison stimuli used in each transitivity testconfiguration can be compared. Table 8 liststhe comparison stimuli used in each tran-sitivity test configuration and their valences.These configurations were obtained fromlines 3, 4, 7, and 8 of Table 5, which showthe complementary set of stimuli for the setused in training. Although valences differed,valence disparities were zero. Therefore,because the choice of any Co+ cannot be ex-

plained in terms of the discriminative prop-

erties of the stimuli, all of these configura-tions can be used in testing for transitivestimulus control.

Valence Disparities and Assessing TransitivityThree possible types of valence disparity

can exist in a test trial. A "neutral" test con-

figuration is one in which the valences of theCo+ and Co- are equal. A "strong" test con-

Table 7Valence values of comparison stimuli after trainingwith lines 1, 2, 5, and 6 of Table 4.

Stimulus Co+ Co- Valence

Al +6 -6 0B1 +2 --6 -4C1 +2 0 +2DI +2 0 +2A2 +6 -6 0B2 +2 -6 -4C2 +2 0 +2D2 +2 0 +2

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EQUIVALENCE AND TRANSITIVITY

Table 8Valence disparities for comparison stimuli in transitivetest trials after training with lines 1, 2, 5, and 6 ofTable 4.

Comparison ValenceStimuli Valence Disparity Test

Co+ Co- Co+ Co- Co+ less Co- Type

Bi B2 -4 -4 0 NeutralAl A2 0 0 0 NeutralCl C2 +2 +2 0 NeutralDl D2 +2 +2 0 NeutralCl D2 +2 +2 0 NeutralBi C2 +2 +2 0 NeutralBi Dl +2 +2 0 NeutralBi D2 +2 +2 0 NeutralDl Cl +2 +2 0 NeutralB2 Dl +2 +2 0 NeutralB2 Cl +2 +2 0 Neutral

figuration is one in which the valence of theCo- is more positive than the valence of theCo+ (e.g., Co+ = +4 and Co- = +7, or

Co+ = -4 and Co- = -2). Finally, an "in-adequate" test configuration is one in whichthe valence of the Co+ is more positive thanthe valence of the Co-(e.g., Co+ = +4 andCo- = +2, or Co+ = -4 and Co- =-7, or

Co+ = +4 and Co- = -2).The relative valences of the comparison

stimuli indicate whether the choice made on

a test trial can be attributed unequivocally totransitive stimuli control. With a neutral testconfiguration, the responses are unlikely tobe influenced by the valences of the com-

parison stimuli. The choice of Co+,therefore, would indicate control by tran-sitive relations. With a strong test configura-tion, the valences of the comparison stimuliwould suggest a bias in favor of a response toCo-. Thus, responding to Co+ would in-dicate strong control exerted by transitivestimulus relations. Finally, an inadequatetest configuration would not be helpful indistinguishing between discriminative stim-ulus control and transitive stimulus control ifresponding occurred to Co+. Responding toCo-, however, would provide strongevidence for discriminative control andagainst transitive stimulus control.

If, in training, different nodal trainingclusters are used for each stimulus class, it ispossible to have transitive test configurations

that are inadequate, neutral, and strong allwithin the same experiment. It is also possi-ble for each of the strong test configurationsto have different valence disparities betweenthe Co+ and Co- stimuli. Some examplesare shown in Table 9. The figures given inthat table are not actual data but are intendedas examples of the kind of data that may beobtained. If for each configuration the per-centage of the occasions when a response oc-curs to Co+ is assessed as a function of thedisparity between the valences of Co+ andCo- (see Columns 2 and 4 of Table 9), thatfunctional relation may reflect the relativedegree of control exerted by the transitiverelations between stimuli on the one hand, orthe discriminative properties of the choicestimuli on the other. On that function, the50% point would define the condition underwhich both kinds of control are equally likely.

It is also possible that Co+ stimuli will bechosen on all test trials regardless of valencedisparities. Such a result would demonstratethat responding is controlled predominantlyby transitive stimulus relations. Under suchconditions discriminative control might stillbe manifested by influencing some dimen-sional property of responding. For example,latency of responding might be systemati-cally related to valence disparity. Columns2, 5, and 6 of Table 9 provide a hypotheticalexample of such a possibility.

Testing with Novel and/or Familiar NegativeComparison Stimuli

In our discussion so far, all asssessment oftransitive stimulus control has been accom-plished with test configurations composed ofstimuli also used in training. It is possible,however, that exhaustive bidirectional train-ing may be necessary to establish transitivestimulus control. In that case, due to com-plementarity, it would not be possible toassess transitive stimulus control using onlythe stimuli that were part of the trainingset -a necessary consequence of the S- rule.This constraint can be circumvented byusing "novel stimuli" as negative comparisonstimuli during transitivity testing. Novelstimuli are those that have not previously

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LANNY FIELDS, THOM VERHA VE, and STEPHEN FA TH

Table 9Possible valence disparities on transitive test trials, and potential outcomes.

(1) (2) (3) (4) (5) (6)Valence Valence Percent Probability LatencyValue Disparity Test of Co+ of Co+ of Co+

Co+ Co- Co+ less Co- Type Choice Choice Choice

+4 +4 0 Neutral+4 +7 -3 Strong 90. 1.0 0.8+4 +6 -2 Strong 70. 1.0 2.5+4 +5 -1 Strong 50. 1.0 4.3

been used in that experiment with that sub-ject. For example, if exhaustive bidirectionaltraining is conducted using AB, AC, andAD for two stimulus classes, each containingfour elements, transitivity testing would beconducted using BC, BD, and CD combina-tions where novel stimuli would be used asCo- stimuli. Thus, Bi DI N3 and BI DIN4 would be two configurations used fortransitivity testing. Because the constraintsof complimentarity are now lifted, transitivestimulus control can be measured using allof the test configurations presented in Table5, provided each is modified by substitutinga novel stimulus for the Co- listed in thattable.

Positive results obtained when using novelCo- stimuli would not only constituteevidence of transitive stimulus control, butalso would demonstrate that transitive stim-ulus control is not context dependent and isnot inhibited by the presence of new stimuli.In contrast, negative results obtained whenusing novel Co- stimuli could be due to (1)the absence of transitive stimulus control, (2)novel stimuli having suppressed transitivestimulus control, or (3) transitivity beingcontext specific.

If negative tests results are obtained whennovel Co- stimuli are used during transi-tivity testing, whether this was due to thesuppressive effects of novel stimulation canbe experimentally assessed by using familiarrather than novel Co- stimuli. Familiarstimuli are those that have not been used intransitivity training, but have been usedpreviously in separate discrimination train-ing. Thus, these familiar stimuli are knownto be discriminable. Positive results obtained

with familiar Co- stimuli would constituteevidence of transitive stimulus control. Incontrast, negative results obtained withfamiliar Co- stimuli would rule out thepossibility of accounting for the negativeresults in terms of the inhibition of transitivi-ty by stimulus novelty. Rather, such resultswould imply that transitive control eitherhad not developed or was context depen-dent.

APPLICATIONS OF PRINCIPLES TOTHE ANALYSIS OF RESEARCH

In this final section we illustrate how theprinciples presented here may be used toanalyze actual experimental research. Forthis purpose, we will consider the experi-ment reported by Wetherby et al. (1983).Four stimuli per class and two stimulusclasses were used to study transitive andsymmetrical stimulus control. These stimuliare represented symbolically and pictoriallyin Table 10. Training was conducted usingAC, BC, and AD pairs in a unidirectionalfashion. This comprised a 2-node trainingcluster that differed from the 2-node clustershown in Table 2. It can be transformed,however, to the 2-node cluster of Table 2 byswitching the position of the A stimulus asshown in Table 11. These transformationsdo not make any assumptions about thebehavioral functions served by the stimuli.

Table 12 presents the exhaustive bidirec-tional training configurations that can beused to establish two stimulus classes, eachcontaining four stimuli, where two nodes areused. The training configurations used byWetherby et al. are shown in italics.

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EQUIVALENCE AND TRANSITIVITY 155

Table 10Stimulus classes and the stimuli used in the experimentby Wetherby, Karlan, and Spradlin, 1983.

Stimuli within ClassStimulus A B C DClass Symb. Pict. Symb. Pict. Symb. Pict. Symb. Pict.

1 Al BDl ci t Dl 7

2 A2/\ 92 /E\ C2 -A D2

The Sa/Co- combinations used in train-ing comprise the stimulus combinations thateliminate transitivity testing configurationsbecause of confounding by the S- rule. Inthe present case, these combinations wereA1/C2, B1/C2, and A1/D2 for Class 1 andA2/C1, A2/B1, and A2/D1 for Class 2.

Table 1 1Transformation of nodal training clusters in Wetherbyet al., 1983.

Wetherby et al.A B C D

Transformed to the 2-node training cluster in Table 2.B C A D

Table 13Valence values of the comparison stimuli used byWetherby et al., 1983.

Stimulus Co+ Co- ValenceAl 0 0 0B1 0 0 ' 0C1 +2 -2 0DI +1 -1 0A2 0 0 0B2 0 0 0C2 +2 -2 0D2 +2 -2 0

The valences of the stimuli that were usedin training are shown in Table 13. Becausethe valences were all zero, all combinationsof choice stimuli in transitivity tests wereneutral; therefore, all possible test con-figurations could be used with the exceptionof those ruled out by the S- rule. Testing oftransitive stimulus control was done bidirec-tionally using the BD, AB, and CD con-figurations listed in Table 14.Table 15 illustrates all of the bidirectional

Table 14Transitivity test configurations used by Wetherby etal., 1983.

Type Class 1 Class 2

BD Bi DI D2 B2 D2 DIDB DI Bi B2 D2 B2 BiAB Al Bi B2 A2 B2 BIBA BI Al A2 B2 A2 AlCD Cl Dl D2 C2 D2 DIDC DI Cl C2 D2 C2 Cl

Table 12Bidirectional training trials: four stimuli per class, two stimulus classes, 2-node trainingcluster. The configurations used by Wetherby et al. (1983) are in italics and marked byasterisks.

Class ISa Co+ Co- Sa Co+ Co- Sa Co+ Co- Sa Co+ Co- Sa Co+ Co- Sa Co+ Co-

l. Al Ci A2 Cl Al A2 B1 C1 A2 Cl BI A2 Al Dl A2 DI Al A22. A1 C1 B2 Cl Al B2 Bl C1 B2 C1 BI B2 Al Dl B2 DI Al B23. Al Cl C2* Cl Al C2 BJ Cl C2* Ci BI C2 Al DI C2 DI Al C24. A1 C1 D2 Cl Al D2 Bi Cl D2 C1 BI D2 Al DI D2* DI Al D2

Class 2Sa Co+ Co- Sa Co+ Co- Sa CO+ Co- Sa Co+ Co- Sa Co+ Co- Sa Co+ Co-

5. A2 C2 Al C2 A2 Al B2 C2 Al C2 B2 Al A2 D2 Al D2 A2 Al6. A2 C2 BI C2 A2 BI B2 C2 Bi C2 B2 BI A2 D2 BI D2 A2 BI7. A2 C2 CG* C2 A2 C1 B2 C2 Cl* C2 B2 C1 A2 D2 Cl D2 A2 Cl8. A2 C2 DI C2 A2 Dl B2 C2 Dl C2 B2 Dl A2 D2 DI* D2 A2 Dl

156 LANNY FIELDS, THOM VERHA VE, and STEPHEN FATH

Table 15Transitivity tests trials after training with two classes, four stimuli per class, and a 2-nodetraining cluster. The configuration used by Wetherby et al. (1983) are in italics andmarked by asterisks. The configurations that would permit confounding by the S- rule arein bold letters.

Class 1Sa Co+ Co- Sa Co+ Co- Sa Co+ Co- Sa Co+ Co- Sa Co+ Co- Sa Co+ Co-

1. Bl Al A2* Al Bi A2 BI DI A2 DI BI A2 Cl DI A2 DI C1 A22. B1 A1 B2 Al BJ B2* BI DI B2 DI BJ B2* Cl Dl B2 DI C1 B23. B1 A1 C2 Al BI C2 B1 DI C2 Dl Bi C2 Cl DI C2 DI Cl C2*4. Bl Al D2 A1 B1 D2 B1 DI D2* DI Bl D2 CG DI D2* D1 C1 D2

Class 2Sa Co+ Co- Sa Co+ Co- Sa Co+ Co- Sa Co+ Co- Sa Co+ Co- Sa Co+ Co-

5. B2 A2 Al* A2 B2 Al B2 D2 Al D2 B2 Al C2 D2 Al D2 C2 Al6. B2 A2 Bi A2 B2 B1* B2 D2 Bi D2 B2 Bl *C2 D2 BI D2 C2 B17.B2 A2 C1 A2 B2 C1 B2 D2 C1 D2 B2 Cl C2 D2 Cl D2 C2 CG*8.B2 A2 Dl A2 B2 Dl B2 D2 DI* D2 B2 Dl C2 D2 Dl* D2 C2 Dl

transitivity test configurations that can beused after training with four classes, twostimuli per class, and with 2-node trainingclusters. All test configurations that were infact used are italicized; all configurationsthat could not have been used because of theS- rule are presented in bold letters.

Eight test configurations were not usablebecause of the S- rule. Although all of theremaining configurations could have beenused for assessing transitive stimulus con-trol, only some of them were actually used.Those used were appropriately selectedbecause any responding to Co+ could not beexplained by the S- rule or by valencedisparity. Although all of the transitiveSa/Co+ configurations were assessed, eachtest occurred only in the context of thespecific Co+/Co- pairs used in training(i.e., Xl Cl C2). On the basis of thisevidence, it is possible to assert that tran-sitivity occurs only in the context of thespecific sets of positive and negative com-parisons used in training; that is, transitivestimulus control may be context dependentand may or may not occur in the presence ofnew comparison configurations (i.e.,Xl Cl not-C2). Such possible contextdependency could have been evaluated byusing those test configurations shown inTable 15 that were not used in this experi-ment, because those configurations differfrom the configurations actually used only in

that dimension of context specificity.Demonstrating transitive stimulus control inthe presence of the remaining configurationswould have increased the generality of thefinding.These analyses should be of assistance in

the choice of stimulus configurations forboth the training and testing phases of ex-periments concerned with stimulus tran-sitivity. The variables stipulated here alsocomprise a behavioral framework for inter-preting phenomena that have traditionallybeen analyzed using cognitive and/orassociative constructs.

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Received December 8, 1983Final acceptance May 24, 1984