Murray Sidman Equivalência de estímulo e contingência de reforçamento

7/31/2019 Murray Sidman Equivalncia de estmulo e contingncia de reforamento

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128 MURRAY SIDMAN

Fig. 1. Two-, three-, and four-term contingencies.

a direct outcome of reinforcement contin-gencies. In accord with what I have seen hap-pening in the laboratory, I have proposedthat the reinforcement contingency gener-ates the equivalence relation.

The notion is that a reinforcement contin-gency produces at least two types of outcome:analytic units and equivalence relations. Mostfamiliar to us are the units of analysis a con-tingency may give rise to (Sidman, 1986), asdiagrammed in Figure 1. We know the two-term responsereinforcer unit as operant re-

inforcement. A defined response (rsp1) pro-duces a defined reinforcer (rf1); no otherresponse (rspX) does so.

We call the fundamental three-term unitsimple discrimination. Now, the two-term unitcomes under the control of discriminativestimuli. The defined response produces its re-inforcer only in the presence of a defined dis-criminative stimulus (B1); in the presence ofother stimuli (B2, XX), no response of anykind produces the defined consequence.

Conditional discrimination is our name forthe four-term unit. Here, the three-term unitcomes under the control of a conditional

stimulus (Cumming & Berryman, 1965).Now, the defined response may produce itsreinforcer in the presence of either of twodiscriminative stimuli (B1 or B2), dependingon which conditional stimulus is present (A1or A2). Analytic units may, of course, havefive, six, or more termsfor example, sec-

ond-order conditional discrimination, and soon. Figure 1 does not show those larger units.

Less familiar than the standard units ofanalysis is the second type of outcome: Thereinforcement contingency also producesequivalence relations. These consist of or-dered pairs of all positive elements that par-ticipate in the contingency. I shall have moreto say about this definition of the equivalencerelation; it is not as new as it may seem. Fornow, let me just note that it is no more thana concise summary of what we see happening

when our procedures produce the phenom-ena we describe as equivalence relations.

THE DESCRIPTIVE SYSTEMAND THE THEORY

First, however, I will say a few words abouta matter that is really a side issue in the pres-ent context, but I would like to keep that is-sue from obscuring the main thread of thediscussion. My concern is that some may con-fuse the theory that equivalence relationsoriginate in reinforcement contingencies

with the system my collaborators and I have

proposed for describing the behavioral phe-nomena of equivalence (Sidman et al., 1982;Sidman & Tailby, 1982). The descriptive sys-tem does involve one theoretical assumption,namely, the hypothesis that our behavioraldata represent real-world instances of themathematical abstraction that is termed equiv-


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129EQUIVALENCE AND REINFORCEMENT

alence relation. If that simple assumption is cor-rectif our observations are real examples ofthe mathematical abstractionthen mathe-matical set theory gives what I have argued isa consistent, coherent, productive, and par-

simonious way to describe our data (Sidman,1997b). The point I want to make at present,however, is that the descriptive system is in-dependent of any theory of where equiva-lence comes from.

Although it is a mathematical concept,then, the equivalence relation turns out todescribe behavior that we see happening.

Whatever their source, whether it is the re-inforcement contingency (e.g., Sidman,1994) or something more (e.g., Hayes, 1991;Horne & Lowe, 1996), the emergent unitsturn out to be predictable and describable asexamples of the properties that define an

equivalence relation. To propose that equiv-alence originates in the reinforcement con-tingency, however, is to postulate a new out-come of the contingency, an outcome thatdoes go beyond the establishment of the n-term analytic units we have already becomefamiliar with through the work of Skinner(e.g., 1938, 1953) and others. That outcomeis the potential for the emergence of new an-alytic units (Sidman, 1986); for example, thenew conditional discriminations that subjectsperform in our standard tests for equivalencerelations.

CONTINGENCIES

Let us look more closely at the hypothesisthat reinforcement contingencies generateequivalence relations. Although no single di-agram can illustrate all that goes on in a be-havioral episode, Figure 2 can provide at leasta reasonable starting point for illustrating

what happens in four-term unitsin condi-tional discriminations. The conditional dis-crimination procedures illustrated in Figure2 are often called matching to sample. We usu-ally call the conditional stimuli samples andthe discriminative stimuli comparisons. One

problem here is that the illustrated condition-al discriminations entail only two comparisonstimuli per trial. The use of only two compar-isons is dangerous (Carrigan & Sidman, 1992;Green & Saunders, 1998; Johnson & Sidman,1993; Sidman, 1987). For example, we cannottell if the controlling comparison is the one

the subject selects or the one the subject re-jects. The simplification, however, permitsone to use expository diagrams that are sim-pler and less space consuming.

We can see, first, that given a defined con-

ditional stimulus or sample (e.g., A1), two de-fined discriminative stimuli become availableto the subject (Comparisons B1 and B2). Thepositive comparison (B1) sets the occasionfor the defined response to produce the de-fined reinforcer; the negative comparison(B2) does not.

By including the Xs, which denote unde-fined elements, we acknowledge the possibil-ity of uncontrolled variables at each stage ofthe contingencies. For example, the subjectmay do other things (rspX) instead of the de-fined response. The subjects responses mayproduce undefined consequences (rfX); also,

undefined discriminative and conditionalstimuli (XX) may control what the subjectdoes.

This diagram illustrates not just sequencesof stimuli and responses. It also shows contin-gencies, events that are true only under cer-tain conditions: If this, then that; if not this,then not that. For example, in the upper-most conditional discrimination: If the de-fined sample is A1 (not A2), and if Compari-son B1 (not B2 or XX) controls the definedresponse (rsp1 and not rspX), then and onlythen will the defined reinforcer be forthcom-ing; if Comparison B2 controls responding

when A1 is the sample, then no response, de-fined or undefined, will produce the definedreinforcer. And if A2 (not A1) is the sample,then reinforcement becomes subject to a dif-ferent set of conditions. In equivalence exper-iments, we often set up similar contingenciesto teach a second conditional discrimination.

A possibility appears below the first dashedline. Now, the former comparisons, B1 andB2, are samples; new stimuli, C1 and C2, arethe comparisons. Finally (below the seconddashed line), in the presence only of unde-fined stimuli (XX)for example, between tri-alsno response can produce anything but

undefined consequences. (Even during trials,we sometimes say that the subjects attentionwanders.)

The bold lettering shows how we oftenoversimplify diagrams of conditional discrim-inations. In those abbreviated diagrams, weomit all elements that are denoted here in


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130 MURRAY SIDMAN

Fig. 2. Four-term contingencies (conditional discriminations) and the ordered stimulus pairs in the resultingequivalence relation.

plain type. Such bare-bones diagrams do notshow any contingencies. An unfortunate con-sequence is that we come to ignore the pro-cedural complexities and talk about associ-ations, or associated events rather thancontingent events. We come to think linear-lyabout temporal sequences of eventsandnot about simultaneous options. I shall havemore to say about the incomplete under-

standing that can result when we attributecausation in a behavioral episode solely totemporally sequential events.

Given these conditional discriminations asa baseline, we go on to demonstrate theequivalence relation by showing the emer-gence of all of the conditional discrimina-

tions that exemplify the relation. Those con-ditional discriminations are summarizedbelow the diagrams. In this notation, the boldupper case Rdenotes an equivalence relationthat includes the pair of elements on eitherside of the R. In establishing the baselineconditional discriminations, we explicitlyteach a conditional relation between A1 andB1, B1 and C1, A2 and B2, and B2 and C2.

If the conditional relations that our baselineprocedures establish are elements in set the-orys equivalence relation, R, then the de-scription ofRmust also include certain emer-gent conditional discriminations. They areemergent in the sense that we do not explic-itly teach them in the baseline.


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In accord with the descriptive scheme thatset theory provides, these emergent condi-tional relations are classified as demonstra-tions of reflexivity, symmetry, and transitivity:reflexivity, in which we show that the relation

holds between each stimulus and itself (oftencalled identity matching); symmetry, in which

we show that the relation holds when we re-verse the elements of each baseline condi-tional discriminationthe former baselinesamples now serving as comparisons and theformer baseline comparisons now serving assamples; and transitivity, in which we show therelation holding when we test new condition-al discriminations in which samples comefrom one baseline conditional discriminationand comparisons from the other. Here, I wantto emphasize simply that our analytic units,

both baseline and emergent, are the out-comes of contingencies that are much moreintricate than our usual bare-bones diagramsreveal. Let us see now where these complex-ities take us.

The diagram shows that the alternative re-sponses and stimuli involved in the contin-gencies can influence the baseline and theemergent conditional discriminations. Forexample, we might make Comparison B2, orsome undefined stimuli, very similar to B1;or those other stimuli may be more attractiveto the subject than B1 is; or some undefinedresponse may be much easier than Response

1 is for the subject; or some undefined con-sequence may be a more effective reinforcerthan what we have defined as the reinforcer.Such possibilities will weaken the AB condi-tional discrimination and any relation wemight expect to be derived from it.

This kind of nonlinear analysis, as Goldia-mond (1975, 1982) termed it, can be exper-imentally fruitful, but I note it here mainly toillustrate how an examination of the contin-gencies that give rise to equivalence relationsleads immediately to meaningful and experi-mentally answerable questions. Other theo-

ries of equivalence fail to consider the possi-bility that unspecified options in thecontingencies may introduce competingsources of control, dismissing that possibilityas a methodological issue that is supposedlyonly tangential to theory. For me, method-ology and theory are inseparable.

FOUR-TERM CONTINGENCIESAND EQUIVALENCE

Having briefly noted a few examples ofnonlinear contingency analysis, I am going tomove on to other considerations. In the low-est section of Figure 2, I have simply listed allthe sample and comparison stimulus pairs

whose members must turn out to be condi-tionally related if each baseline contingencyhas created both a four-term analytic unit andan equivalence relation. Although this listdoes not classify the stimulus pairs accordingto particular properties of the equivalence re-lation, it contains the same related pairs. Allof these are needed to fulfill our definitionof equivalence. In subsequent figures, I willdescribe the equivalence relation by listingthe baseline and emergent conditional dis-

criminations as event pairs, without designat-ing the particular property of equivalencethat each pair helps to define.

Note that the list includes pairs of condi-tional and discriminative stimuli only: A, B,and C. Yet, I said earlier that the equivalencerelation consists of ordered pairs of all posi-tive elements that participate in the reinforce-ment contingency. The uppermost diagramsshow two other kinds of elements that alsoparticipate in the contingencies and becomecomponents of the analytic units. These arethe defined reinforcers and responses. Whyhave I not included Reinforcer 1 and Re-

sponse 1 in the pairs of elements that makeup the equivalence relation? No other theoryposes this question. It arises as a theoreticalquery only if one considers the reinforce-ment contingency to be the source of equiv-alence, and the relation to include orderedpairs of all positive elements that participatein the contingency.

It turns out that the equivalence relationdoes include all elements of the contingency.To demonstrate this, one has to arrange spe-cial conditions. In the four-term units dia-grammed in Figure 2, Reinforcer 1 and Re-sponse 1 are common to all of the units. If

pairs of events that included those elementswere to remain in the equivalence relation,the contingencies themselves could not work.They could not work because all conditionaland discriminative stimuli in all units that thereinforcement contingencies create would berelated to the same defined reinforcer and


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132 MURRAY SIDMAN

response. These common elements wouldbring all stimuli into one large equivalenceclass. With Stimuli A1, A2, B1, B2, C1, and C2all becoming equivalent to each other, sub-

jects would fail to demonstrate the condition-

al or even the simple discriminations this di-agram calls for. With all of the stimuli beingmembers of a single class, subjects would, forexample, treat the A1B1 and A1B2 pairs asequivalent; would react to B2 and B1 in thesame way; and so on.

But we know, of course, that the contingen-cies do work; subjects do form simple andconditional discriminations even with only asingle defined reinforcer and response. Ourtheory requires us to assume that when thetwo outcomes of the reinforcement contin-gency come into conflict, the analytic unittakes precedence over the equivalence rela-

tion, as it must if we are to learn to reacteffectively to the world around us. In orderfor the common response and reinforcer el-ements to retain their membership in the an-alytic unit, they must selectively drop out ofthe equivalence relation.

Reinforcement contingencies that involvejust one reinforcer and one defined responsemust at first generate one large equivalenceclass. The demands of the contingencies,however, must cause the response and rein-forcer elements to drop out of that class even-tually, making it possible for the smaller clas-ses, A1B1C1 and A2B2C2, to form.

Elsewhere, I have outlined experimentaltests of the proposition that three- and four-term simple and conditional discriminationprocedures generate at first a large equiva-lence class that contains pairs of all contin-gency components, and that eventually, thecommon reinforcer and response elementsdrop out of the class (Sidman, 1994, pp. 411414). That research has yet to be carried out.

When it is, its results will be crucial to thetheory that equivalence arises from reinforce-ment contingencies.

Nevertheless, even in the absence of directexperimental demonstrations that all ele-

ments of the contingency enter into theequivalence relation at least temporarily, ouranalysis provides other reasons for anticipat-ing a confirmation of that theoretical prop-osition. For example, it is quite well estab-lished now that when the contingencies andequivalence relations do not conflict, the

equivalence relation does include orderedpairs of all elements of the analytic unit, in-cluding the reinforcer and the response.

THE EQUIVALENCE RELATIONAND THE REINFORCER

To test whether the reinforcer remains inthe equivalence relation when its presencethere does not conflict with establishment ofthe analytic unit requires only a modestchange in the contingencies we have beenlooking at. The diagrams in Figure 3 are ex-actly the same as before, except that Rein-forcer 2 has replaced Reinforcer 1 in twounits: When A1 is the sample, the defined re-sponse to Comparison B1 still produces Re-inforcer 1, but when A2 is the sample, thedefined response to Comparison B2 produc-

es Reinforcer 2. Similarly, in the second con-ditional discrimination, the conditional B2C2relation now leads to Reinforcer 2.

The addition of Reinforcer 2 provides dif-ferential consequences within each condi-tional discrimination. Without a common re-inforcer to help bring about class union andget in the way of the contingencys establish-ment of the analytic unit, the reinforcers nolonger have to drop out of the equivalenceclasses. Now, tests in which the reinforcingstimuli are used also as sample and compar-ison stimuli should reveal new members ofthe equivalence relation. These are under-

lined in the list of pairs below the diagrams.Fourteen more related pairs should be addedby what has been called performance-specificor class-specific reinforcement, but perhapsshould be called contingency-specific rein-forcement.

Dube, McIlvane, and their collaboratorshave confirmed these expectations (Dube &McIlvane, 1995; Dube, McIlvane, Mackay, &Stoddard, 1987; Dube, McIlvane, Maguire,Mackay, & Stoddard, 1989; McIlvane, Dube,Kledaras, de Rose, & Stoddard, 1992). Bydemonstrating the emergence of many of theexpected new related pairs, each including a

stimulus that is also a reinforcer, they showedthat the reinforcers in the four-term units do

join the conditional and discriminative stim-uli as members of the equivalence classes.These findings support the notion that equiv-alence relations, consisting of ordered pairsof all contingently related elements, includ-


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Fig. 3. Four-term contingencies (AB and BC conditional discriminations) with contingency-specific reinforcers(rf1 and rf2). Underlining indicates the event pairs that the use of contingency-specific reinforcers adds to theequivalence relation.

ing the reinforcers, arise directly from the re-inforcement contingency. No other majortheory of equivalence leads us even to ask

whether the reinforcer belongs to the equiv-

alence class.Let us pursue the inquiry further, with

small changes in our contingency diagram.Figure 4 simply introduces new sample andcomparison stimuli in the second conditionaldiscrimination. Instead of B1 and B2 as sam-ples (as in Figure 3), we now have D1 andD2, stimuli that appear nowhere else in thebaseline contingencies. The comparison stim-uli, C1 and C2, also appear nowhere else.

With these changesand, of course, a newsubjectwhat are we to expect in the way ofemergent conditional discriminations?

The two conditional discriminations (AB

and DC) now have no conditional or discrim-inative stimuli in common. Nevertheless, thelist below the diagram indicates that an equiv-alence relation is still to be expected. The re-lation will include the same 32 event pairsthat had been included before. For example,the subject will still match Stimulus A1 to C1

and A2 to C2 even though none of thosestimuli are now directly related to a commonsample or comparison. Furthermore, also in-cluded in that relation will be emergent con-

ditionally related pairs (underlined) that in-clude the D stimuli as members.

How is all this supposed to happen? Rein-forcer 1 is now a component of two contin-gencies, one including A1 and B1 and theother including D1 and C1. This common re-inforcer element will bring all four of thosestimuli along with it into the same class (classunion). Similarly, Reinforcer 2 will bring A2,B2, C2, and D2 into another class. The equiv-alence relation thus comes to include anoth-er 18 pairs of elements.

This relatively simple experiment is crucial.With careful procedural management, all of

the listed conditional discriminations mustemerge from these baselines. If they do notemerge, the theory that equivalence arises di-rectly from the reinforcement contingencybecomes untenable. I know of no other the-ory of equivalence that offers such a clear op-portunity for disproof.


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134 MURRAY SIDMAN

Fig. 4. Four-term contingencies (AB and DC conditional discriminations) with contingency-specific reinforcers(rf1 and rf2). Underlining indicates the event pairs added to the equivalence relation by the substitution of the DCfor the former BC conditional discriminations.

Although the experiment has not beendone exactly as proposed here, Dube and hiscollaborators, working with human subjects,have again provided decisive data (Dube &McIlvane, 1995; Dube et al., 1989). Theyshowed that stimuli can become members ofthe same equivalence class even when theyhave been related in common to no otherevent than a reinforcer. Figure 5 illustratesone of their procedures (Dube & McIlvane,1995). They first established a baseline of

identity matching, with contingency-specificreinforcement. In one conditional discrimi-nation (uppermost in Figure 5), when A1 wasthe sample, the subjects selection of Com-parison A1 produced Reinforcer 1; with A2 asthe sample, selecting Comparison A2 pro-duced Reinforcer 2. In the other baseline

conditional discrimination (below the firstdashed line), subjects learned to match B1 toitself and B2 to itself, with Reinforcer 1 fol-lowing selections of Comparison B1 in thepresence of Sample B1, and Reinforcer 2 fol-lowing selections of Comparison B2 in thepresence of Sample B2.

The list of emergent conditional discrimi-nations below the diagram describes theequivalence relation that is to be expected.The subject will match Stimuli A1 to B1, A2

to B2, B1 to A1, and B2 to A2, even thoughnone of those have been directly related to acommon sample or comparison. Dube andMcIlvane (1995) reported such test results forseveral of their subjects. The reason to expectthese emergent conditional discriminations isthat Reinforcer 1, a component of the two


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Fig. 5. Four-term contingencies (AA and BB identity matching) with contingency-specific reinforcers (rf1 andrf2), and the ordered event pairs in the resulting equivalence relation (after Dube & McIlvane, 1995).

baseline contingencies that include A1 andB1, brings both of those stimuli along with itinto the same class. Similarly, Reinforcer 2, acomponent of the two contingencies that in-

clude A2 and B2, brings both of those stimulialong with it into another class. Like the ex-periment that was suggested in Figure 4, thisone, too, is critical for the theory that equiv-alence arises directly from the reinforcementcontingency and that the equivalence rela-tion will include all of the positive elementsof the contingency that do not conflict withthe establishment of the analytic unit itself.

Reichmuth (1997), working with sea lions,used yet another experimental procedure toshow that stimuli become members of a func-tional class after having been related only tothe same reinforcer as previously established

members of the class (see the repeated rever-sal procedure, below). No other theory thathas come to my attention has predicted thesefindings. If other theorists should now claimsuch findings to support their theory, they

would also be obliged to show that their the-oretical derivation is sufficiently rigorous to

have required abandonment of their theoryin the eventuality of negative results. Theoriesthat can handle both positive and negativeresults from the same experiment cannot

claim support from either.

THE EQUIVALENCE RELATIONAND THE RES PON SE

If the equivalence relation contains or-dered pairs of all positive components of thereinforcement contingency, what about theresponse? The definition does not exclude re-sponses from the pairs of elements that makeup the equivalence relation. So far, however,

we have used a single defined response(rsp1), which has to drop out of the equiva-lence relation if reinforcement is to create

the four-term unit. A second crucial experi-ment is called for, as outlined in Figure 6.Here, instead of providing contingency-spe-cific reinforcement, we keep the reinforcerconstant (rf1) but require differential re-sponding to the discriminative stimuli. In thepresence of Comparison B1 or C1, Response


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136 MURRAY SIDMAN

Fig. 6. Four-term contingencies (AB and DC conditional discriminations), with contingency-specific defined re-sponses (rsp1 and rsp2). Underlining indicates the event pairs that the use of contingency-specific responses adds tothe equivalence relations.

1 still leads to reinforcement, but in the pres-ence of B2 or C2, Response 2 is now re-quired.

Again, the two baseline conditional dis-criminations, AB and DC, have no condition-al or discriminative stimuli in common. Nev-ertheless, these baseline contingenciesshould generate an equivalence relation thatincludes all of the listed emergent condition-al discriminations. For example, it will in-clude the same AC, DB, and AD event pairsthat had been included before, even thoughnone of those stimuli are related to each oth-

er via a common sample or comparison.Response 1 is now a component of two con-

tingencies, one including A1 and B1 and theother including D1 and C1. This common el-ement brings all four stimuli along with itinto the same class (class union). In a similar

way, Response 2 brings A2, B2, C2, and D2

into another class. The equivalence relationthus comes to include all of the underlinedevent pairs. If all of these component pairsdo not emerge, then again, the theory thatequivalence comes directly from the rein-forcement contingency will become untena-ble.

This experiment awaits doing. It is proce-durally difficult and will take some real in-genuity to arrange tests for emergent condi-tional discriminations in which Response 1 orResponse 2 serves as a sample or comparison.Even without such tests, however, the emer-

gence of new conditional discriminations inwhich A and B stimuli are related to D andC stimuli seems to have no other tenable ex-planation than the common relation of thosestimuli to defined responses.

In this instance, other theories may offer adifferent explanation, called transfer of func-


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tion(e.g., Hayes, 1991, 1992, 1994). The func-tion being transferred is the control of a spe-cific response. The functional control ofResponse 1 by Stimuli B1 and C1 is said notonly to bring about a relation between those

stimuli but also to transfer to other stimulithat are related to B1 and C1, namely, A1 andD1, respectively. Similarly, the functional con-trol of Response 2 is said to transfer to allstimuli that are related to B2 and C2.

The problem here is that transfer of func-tion explains nothing. Function transfer issimply what we observe. Theorists who wouldappeal to transfer of function as an explana-tion that somehow goes beyond description

would be assuming that the transfer of con-trol from one stimulus to another does notitself require explanation. If they are addinga new theoretical process that has no utility

except to account for transfer of function,then they are committing the logical fallacyof naming an observed phenomenon andthen using the name as an explanation. Fur-thermore, such a theory would have to ac-knowledge explicitly that a failure to observetransfer would constitute evidence against thetheory. Theories that appeal to transfer offunction as an explanatory concept, however,have never specified the conditions under

which transfer is and is not to be expected.That omission leaves the theories free toclaim support from observations of functiontransfer and to ignore any failures to observe

such transfer. (The validity of any failure toobserve transfer would, of course, depend onits own procedural integrity; see below for ad-ditional clarification of this point.)

By contrast, the theory that the reinforce-ment contingencies here will create the de-picted equivalence relationthat is to say,

will generate the emergent conditional dis-criminationsis an acknowledged theory. Itis also a most attractively simple and eleganttheory in that it postulates no additional ex-planatory process. Perhaps more important isits empirical testability; it can be disproved.

DIFFERENTIAL RESPONSES ANDDIFFERENTIAL REINFORCERS

It is possible, of course, to set up contin-gencies to require specific responses and re-inforcers. Figure 7 illustrates this, in the con-text once more of the original AB and BC

(rather than DC) conditional discrimina-tions. Now, the events in Class 1 include bothResponse 1 and Reinforcer 1; Class 2 includesResponse 2 and Reinforcer 2. With effectivecontrols for the extra difficulty that teaching

a second response introduces, the use of dif-ferential responses and reinforcers shouldgreatly facilitate both the learning of baselineand the emergence of derived conditionaldiscriminations. Without a common responseand reinforcer to bring about a large equiv-alence class, which then has to break downbefore the analytic units can form, an initialconflict will no longer exist between theequivalence relation and the establishment ofthree- and four-term units. Many students

who ordinarily take a long time to learn con-ditional discriminations, or who fail to learn,or who fail to show equivalence relations,

should improve their performances. The the-ory that reinforcement contingencies gener-ate the equivalence relation requires confir-mation of that prediction also. Althoughsome positive evidence exists (e.g., Lowen-kron, 1984, 1989; McIlvane et al., 1992; Pe-terson, 1984; Reichmuth, 1997; Schenk, 1994;Trapold, 1970), the most definitiveexperiments, with contingency-specific rein-forcers and responses, have yet to be done.

THREE-TERM CONTINGENCIESAND EQUIVALENCE

A simpler paradigm for testing whether re-sponses and reinforcers are included in theequivalence relation is the three-term contin-gency. The theory that the reinforcementcontingency generates the equivalence rela-tion places no lower limit on the complexityof the contingency. Although such a limitmay eventually prove to be necessary, exten-sion of the theory to three-term operant con-tingencies has been well established, sparkedby the work of de Rose and his colleagues (deRose, McIlvane, Dube, Galpin, & Stoddard,1988; de Rose, McIlvane, Dube, & Stoddard,1988) and by Vaughan (1988). Also, Manabe,Kawashima, and Staddon (1995), Braga-Ken-

yon, Andrade, Ahern, and Sidman (2000),1

1 Braga-Kenyon, P., Andrade, M., Ahearn, W. H., & Sid-man, M. (2000). Inclusion of defined responses in equivalencerelations: A systematic replication of Manabe et al. (1995).Manuscript submitted for publication.


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138 MURRAY SIDMAN

Fig. 7. Four-term contingencies (AB and BC conditional discriminations) with contingency-specific responses(rsp1 and rsp2) and reinforcers (rf1 and rf2). Underlining indicates the event pairs added to the equivalence relationby the use of both contingency-specific responses and reinforcers.

and Andrade, Braga-Kenyon, Ahern, and Sid-man (1999)2 have reported several experi-ments in which emergent performances seemto be explainable in no other way than byinclusion of defined responses in equivalenceclasses that three-term contingencies gener-ate. By examining three-term contingenciesmore closely, we can help to make visiblesome additional implications of the theorythat equivalence relations consist of pairs ofall positive elements that are involved in re-

inforcement contingencies.Each of the conditional discriminations wehave been looking at includes one successive

2Andrade, M., Braga-Kenyon, P., Ahearn, W., & Sid-man, M. (1999, May). Equivalence classes and three-term con-tingencies. Poster session presented at the annual meetingof the Association for Behavior Analysis, Chicago.

and two simultaneous simple discriminations.For example, in the context of Sample A1 or

A2 (the presentation of A1 or A2 separatelyrequires a successive discrimination betweenthem), we see a simultaneous presentation ofComparisons B1 and B2; a defined responseto only one of these will produce the definedreinforcer. This, of course, requires a simul-taneous discrimination between B1 and B2;both are present at the same time. Figure 8,

which isolates one of the simple discrimina-

tions that was involved in our four-term con-tingency, shows this three-term contingency.In the presence of B1, Response 1 will pro-duce Reinforcer 1; in the presence of otherstimuli, no response will produce the definedreinforcer.

Our theory tells us that even while the con-


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Fig. 8. A three-term contingency, and the resultingequivalence relation that can actually not be measured(see text).

Fig. 9. One set of three-term contingencies with con-tingency-specific defined responses and reinforcers, andthe event pairs that make up the resulting equivalencerelation.

tingency establishes this three-term unitasimple discriminationStimulus B1, Re-sponse 1, and Reinforcer 1 will become mem-

bers of the same equivalence class. To estab-lish the class requires no more than to set upthe depicted contingency. In practice, how-ever, none of the event pairs that are listed asmembers of the equivalence relation are ac-tually demonstrable without some proceduralchanges. This is because the defined ele-ments, B1, rsp1, and rf1, have no defined op-tions. For example, in an identity-matchingtest for the B1B1 pair, what stimulus wouldserve as the incorrect comparison?

Figure 9 illustrates one way to solve thisproblem. The simple discrimination now in-

volves contingency-specific responses and re-

inforcers. In the presence of Stimulus B1, Re-sponse 1 produces Reinforcer 1, andResponse 2 produces no defined conse-quence. In the presence of B2, however, Re-sponse 2 produces Reinforcer 2, and Re-sponse 1 produces no defined consequence.

Again, conditional discrimination tests aredifficult to design when responses have tofunction as samples or comparisons, but theother element pairs in the equivalence rela-tion can be tested relatively easily. No one has

yet carried out such tests after having estab-lished only this relatively simple baseline. Giv-en procedural integrity, their results will once

more cause the theory to stand or fall.Manabe et al. (1995), using budgerigars as

subjects, introduced a technique that, al-though indirect, helps to get around an in-herent difficulty in attempting to determine

whether defined responses are equivalenceclass members. To verify that an equivalence

class includes a response, one must show thatthe response and not its controlling stimulusis the essential element (Sidman, 1994, p.377). Although some aspects of the Manabeet al. procedures require clarification, and ad-ditional controls might have made their con-clusions more definitive (see, e.g., K. Saun-ders & Williams, 1998; Sidman, 1994, p. 471),a systematic replication by Braga-Kenyon etal. (2000),1 with human subjects, confirmedthe utility of their innovative procedure. Theresults obtained by Braga-Kenyon et al. sup-

ported the hypothesis that the event pairsmaking up an equivalence relation includethe three-term contingencys defined re-sponses. Figure 10 illustrates their technique,a slight modification of the Manabe et al. pro-cedure.

In Phase 1, they taught their subjects a sim-ple discrimination, placing Responses 1 and2 under the discriminative control of StimuliB1 and B2, respectively. Then, in Phase 2,came a test of whether the three-term contin-gency had generated an equivalence relationin which one class consisted of Stimulus B1and Response 1 and the other consisted of

Stimulus B2 and Response 2. The critical testhad the subjects learning a conditional dis-crimination in which the discriminative stim-uli from Phase 1 (B1 and B2) served as com-parisons, with new stimuli (A1 and A2)serving as samples. The ingenious aspects ofthis procedure were (a) requiring a defined


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140 MURRAY SIDMAN

Fig. 10. The inclusion of defined responses in equivalence classes established by three-term contingencies (afterManabe et al., 1995). See text for details.

response to the sample before the compari-sons could appear; and (b) leaving the de-fined response somewhat flexibleon any tri-al, regardless of the sample, either Response1 or Response 2 (but no other) would pro-duce the comparisons. Then, once the com-parisons appeared, the subjects could pro-

duce the reinforcer not with Responses 1 or2 but with a new response (Response 3). Themain experimental question was whethereach of the sample stimuli (A1 and A2) wouldcome to control the same response that itsrelated comparison (B1 or B2) had con-trolled in the previous simple discrimination.

Several of the human subjects (Braga-Ken-yon et al., 20001) came to respond consis-tently to the sample stimuli. On trials with A1as the sample, they produced the compari-sons by means of Response 1; with A2 as thesample, they typically emitted Response 2.Even though the contingencies did not re-

quire the subjects to respond differentially toSamples A1 and A2, they did so.

These Phase 2 results are essential if ourtheory is to survive. If equivalence arises di-rectly from the reinforcement contingency,

with the equivalence class consisting of or-dered pairs of all positive elements that par-

ticipate in the contingency, then the three-term contingency in Phase 1 must haveestablished an equivalence relation that in-cluded the event pairs (B1, rsp1) and (B2,rsp2). Then, in Phase 2, the expanded con-tingency must have brought the event pairs(A1, B1) and (A2, B2) into an equivalence

relation. With the common B stimuli bring-ing about class union, each response, along

with its directly related B stimulus from Phase1 and its indirectly related A stimulus fromPhase 2, became members of expandedequivalence classes. Only if Responses 1 and2 had become members of equivalence clas-ses because of their involvement in three-term contingencies could we predict thatthose responses would come under the con-trol of Stimuli A1 and A2.

Once again, to explain these results by ap-pealing to transfer of function (transfer ofcontrol over the defined responses from the

B to the A stimuli) would add nothing to thestory except a postulated process that itself

would need explanation. Furthermore, I havenot yet seen any theorists admit that a failureto produce such transfer would negate theirtheory. By contrast, our theory holds that theobserved transfer had to come about because


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Fig. 11. The inclusion of defined responses in equivalence classes established by four-term contingencies.

three-term contingencies generate equiva-lence relations, with defined responses in-cluded in the equivalence classes along withthe stimuli. Failure of transfer under the con-ditions of this experiment would have negat-ed the theory that is being elucidated here.

The procedure that Figure 10 summarizescan be more generally useful to test for theinclusion of defined responses in equivalenceclasses. It provides a way around the problem,noted in conjunction with Figure 6, of howto arrange conditional discrimination tests in

which responses have to serve as samples orcomparisons. Although Phase 1 in Figure 10used a simple discrimination with contingen-cy-specific responses, it might just as well haveused a conditional discrimination, as in thePhase 1 section of Figure 11. Then, in Phase2, Stimuli A1 and A2 from Phase 1 wouldserve as comparisons; new stimuli, C1 and C2,

would serve as samples. Again, a defined re-sponse to the sample is required on eachPhase 2 trial before comparisons can appear,and the defined response can be either Re-sponse 1 or Response 2, regardless of thesample. Once more, we would expect to ob-serve subjects responding differentially to the

new samples, with each one coming to con-trol the same response that its related A stim-ulus had controlled in Phase 1. This exten-sion of the Manabe et al. (1995) procedurehas yet to be carried out, but when it is, it willprovide yet another opportunity to support

or to disprove the theory under consider-ation.

More tests of the theory in the context ofthree-term contingencies become possible if

we teach a subject additional simple discrim-inations. Figure 12 adds just one. Now, Stim-uli B1 and C1 will become members of oneclass because both are involved in contingen-cies that include Response 1 and Reinforcer1 as common elements. Stimuli B2 and C2

will become members of another class be-cause they are components of contingenciesthat include Response 2 and Reinforcer 2.The equivalence relation will now include all

of the element pairs that the C stimuli belongto. This prediction, too, which has not yetbeen tested, requires confirmation if the the-ory being proposed is to stand up.

The use of two or more simple discrimi-nations opens up another way to find out

whether three-term contingencies are suffi-


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Fig. 12. Two sets of three-term contingencies withcontingency-specific defined responses and reinforcers,and the event pairs that make up the resulting equiva-lence relation.

Fig. 13. Three-term contingencies in a repeated re-versal procedure, and the event pairs in the resultingequivalence relation. There are no contingency-specificresponses or reinforcers.

cient to establish equivalence relations. Thisis the discrimination-reversal procedure that

Vaughan pioneered with pigeons as subjects(Vaughan, 1988; see also Dube, Callahan, &McIlvane, 1993). Figure 13 illustrates a sim-plified version of Vaughans procedure. The

diagrams above the double dashed line illus-trate the baseline contingencies that teach asubject to respond only in the presence of aset of positive stimuli, B1 and C1. Once thesubject has learned that discrimination, Stim-uli B1 and C1 are made negative, and theother stimuli, B2 and C2, become positive, asillustrated below the double dashed line. Thediscrimination continues to be reversed eachtime the subject learns to respond only tomembers of the current positive set (B1 andC1 or B2 and C2). The establishment of anequivalence class is suggested when the sub-

ject experiences a reversed contingency with

only one of the newly positive stimuli, andthen shifts responding immediately whennext encountering the other member of thatset.

Such a result, obtained by Vaughan (1988)with pigeons, was replicated and extended tosea lions by Reichmuth (1997) and to human

subjects by Sidman, Wynne, Maguire, andBarnes (1989). As Vaughan indicated, this ex-periment brought equivalence classes andfunctional classes under the same rubric. Wecan now also see that its results are not onlyto be expected but are actually required ifequivalence relations do come directly fromreinforcement contingencies, including

three-term contingencies.Vaughans (1988) original repeated rever-

sal procedure did not permit any conclusionsto be drawn with respect to the inclusion ofdefined responses or reinforcers in equiva-lence classes. Because all of the contingenciesincluded Response 1 and Reinforcer 1, thoseelements had to drop out of the equivalencerelation if subjects were to discriminate the Band C stimuli. Once the common responseand reinforcer had dropped out, the eventpairs listed below the diagrams in Figure 13

would have defined the equivalence relation.Both Reichmuth (1997) and Sidman et al.

(1989) confirmed the emergence of thoselisted stimulus pairs that did not involve iden-tity matching: (B1, C1), (C1, B1), (B2, C2),and (C2, B2).

Modifying the repeated discrimination re-versal procedure by using contingency-specif-ic defined responses and reinforcers would


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Fig. 14. Three-term contingencies in a repeated re-versal procedure with contingency-specific responses andreinforcers. Underlining indicates the event pairs addedto the equivalence relation by the use of contingency-specific responses and reinforcers.

permit more extensive tests of our theory.

Figure 14 illustrates such a modification. Asbefore, the original contingencies (above thedouble dashed lines) set up B1 and C1 as pos-itive discriminative stimuli; either one sets theoccasion for Response 1 to produce Reinforc-er 1. With the contingencies reversed (belowthe double dashed lines), B2 and C2 becomepositive; now, either of these stimuli sets theoccasion for Response 2 to produce Reinforc-er 2. One effect of using contingency-specificresponses and reinforcers will be an enlarge-ment of the equivalence relation. With ap-propriate testing, all 24 of the underlinedevent pairs in the list below the diagrams

should emerge, in addition to the eight pairsthat were to be expected before.

The use of differential responses and re-inforcers should facilitate both the learningof the original three-term contingencies andthe successive reversals. As with four-termcontingencies, this facilitation should come

about because without a common responseand reinforcer to bring about an initial largeequivalence class, no conflict would exist be-tween the equivalence relation and the for-mation of the three-term units. This predic-

tion has not yet been fully tested, butReichmuth (1997) did find that the use ofcontingency-specific reinforcers facilitatedlearning of contingency reversals by sea lions.

Reichmuth (1997) also used the repeateddiscrimination-reversal procedure to find out

whether the equivalence classes that three-and four-term contingencies give rise to in-clude the reinforcers. She first used condition-al discriminations, with contingency-specificreinforcers, to establish equivalence relations.Then, using entirely different stimuli, she setup two equivalence classes by means of the re-peated reversal procedure illustrated in Figure14 (but without requiring differential respons-es). Then, she added stimuli from the condi-tional discriminations to the two sets of stimuliin the simple discrimination-reversal proce-dure, maintaining the contingency-specific re-inforcers. The new stimuli took their places inthe classes that the reversal procedure had es-tablished, even though they had been relatedto the stimuli in those classes in no way exceptthrough their shared reinforcement contin-gencies.

No other current theory that might claimto accommodate such findings has been putforth in a way that makes them crucial. Ifstimuli that had been unrelated except for

shared reinforcers had failed to transfer fromclasses established by means of conditionaldiscriminations to classes established bymeans of simple discrimination reversals, oth-er theories would have considered such a fail-ure to be without significance. That is to say,no other theorists would have paid attentionto negative results. By contrast, the theorythat a reinforcement contingency gives risedirectly to an equivalence relation, with therelation consisting of all possible pairs ofevents that the contingency specifies (includ-ing reinforcers and responses), not only pre-dicts the findings noted above without need-

ing any theoretical modifications or additionsbut also requires those findings as necessaryfor its survival.

SOME FINAL GENERALITIESTo help make it evident that the rather pro-

saic theory that equivalence arises directly


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from reinforcement contingencies is produc-tive, I have outlined many of the experimentsthat the theory generates. Some of the mostexciting of those experiments have beendone, with results that support the theory,

but many remain to be performed.Every theory, of course, has gray areas thatmust eventually be looked into. The theoryunder consideration here is no exception. Forexample, how do we deal with those instancesin which a reinforcement contingency fails togenerate an equivalence relation? First, ofcourse, procedural factors in the relevant ex-periments must be ruled out (e.g., Carrigan &Sidman, 1992; Harrison & Green, 1990; John-son & Sidman, 1993; Kelly, Green, & Sidman,1998; Sidman, 1994, pp. 259263, 406414,511512, 524525; Stikeleather & Sidman,1990). That is to say, the experiments must be

procedurally valid. Second, other variables,not specified in the description of the rein-forcement contingency itselffor example, re-inforcement and stimulus control variables, orneurological variablesare highly likely to berelevant. Explicating their relevance will in-crease the theorys breadth without discredit-ing it.

Third, members of some species may notshow equivalence relations even when oper-ant reinforcement contingencies do producethree- and four-term units of analysis. (It mustbe noted, however, that so far, there has beenno definitive demonstration that any species

sensitive to reinforcement contingencies is in-capable of equivalence relations.) Variabilitymay also exist within a species, including thehuman, when factors like developmental re-tardation, acquired brain damage, sensorydeficiencies, or genetic abnormalities may befound to bear on the production of equiva-lence relations by reinforcement contingen-cies (e.g., Devany, Hayes, & Nelson, 1986;Green, 1990; Sidman, 1994, pp. 266271). Asformulated, however, the present theory isneutral with respect to the relevance of neu-rological structure and function, genetic fac-tors, or developmental processes. Those stud-

ies that have shown equivalence in somenonhumans (Reichmuth, 1997; Schusterman& Kastak, 1993, 1998) or can be interpretedas having done so (e.g., Manabe et al., 1995;Schusterman, Reichmuth, & Kastak, 2000;Urcuioli & DeMarse, 1997; Vaughan, 1988;Zentall, 1998; Zentall & Urcuioli, 1993) are

sufficient to fulfill the demands of the theory.If a demonstration is forthcoming, however,that a particular species is incapable of equiv-alence relations even though it is sensitive toreinforcement contingencies, such a demon-

stration will not require abandonment of thetheory offered here. To set limiting condi-tions will not negate the theory that equiva-lence relations arise from reinforcement con-tingencies. Instead, the identification of suchlimits will establish opportunities for researchto identify their sources. Accurate predictionof which nonhumansor even humanscanand cannot show equivalence relations, farfrom diminishing the theory, will increase itsbreadth. Indeed, such prediction is likely torequire theoretical considerations and empir-ical support that only sciences other than be-havior analysis can provide.

In addition to the specific experiments thatit generates, the theory also gives rise to a wayof looking at equivalence relations that dif-fers somewhat from our usual conceptualiza-tions of operant conditioning. For example,by including defined responses in the equiv-alence classes that a contingency generates,

we remove the distinction between stimuliand responses when considering classes (Sid-man, 1994, pp. 384386). The unidirectionalarrows of time and causality, which are so im-portant when we are talking about condition-ing, are not relevant to relations betweenclass members. This gives rise to the bag

analogy (Sidman, 1994, p. 381): An equiva-lence relation can be thought of as a bag thatcontains ordered pairs of all events that thecontingency specifies; the bag can be shakenand the elements mixed without regard toany spatial or temporal relations amongthem. To document the relation, all we haveto do is reach into the bag and pull out itsmember pairs.

One consequence of this analogy is that wehave to look critically at the notion that struc-tural or linear-associative variables can differ-entiate the members of equivalence classes.One such variable is directionality or sam-

ple as node versus comparison as node (seeR. Saunders & Green, 1999, for a review ofthis issue). Another has been called nodaldistance (Fields, Adams, & Verhave, 1993),or the least number of nodes that must beinvolved for a particular stimulus pair in a setof conditional discriminations to be included


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in an equivalence relation. As I have pointedout elsewhere, however (Sidman, 1994, p.543), the notion that members of a class dif-fer from each other with respect to the cri-terion for class membership contradicts the

very concept of classes. I therefore suggestedexperimental tests of certain procedural var-iables that might change our interpretationsof the relevant experiments (Sidman, 1994,pp. 273279, 525528, 537549). R. Saundersand Green also have suggested a number ofprocedural factors that might account for theresults of experiments that have given rise tonotions of directionality and nodal distance.Once again, methodology rears its ugly head,raising questions that have priority over the-ory.

One more issue: Is naming a critical deter-miner of the emergent performances that de-fine equivalence relations (Dugdale & Lowe,1990; Horne & Lowe, 1996)? Inclusion of thedefined response component of the contin-gency in the equivalence class should some-

what defuse the naming controversy. Anyname we apply to stimuli is a defined discrim-inative response. Our theory states explicitlythat any defined response components of thecontingencies have a status that is equal in ev-ery way to the stimulus and reinforcer mem-bers of the classes. Although just as important,responses require no separate treatment. Ithink this provides a simple but satisfactory

resolution to the naming controversy.In brief summary, then, equivalence rela-tions require no new concepts beyond the no-tion that the reinforcement contingency es-tablishes them in addition to the familiaranalytic units, and that equivalence relationsconsist of ordered pairs of all positive ele-ments that participate in the contingency.That notion, however simple it may seem,gives rise to some new ways of thinking aboutbehavior. Perhaps more important are thenew experiments it leads toexperimentsthat, up to this time, no other theory has mo-tivated. Although the war of words is likely to

continue, with other theories being adaptedto unpredicted outcomes (Clayton & Hayes,1999), the very simplicity of the theory beingexplicated here makes each of the suggestedexperiments crucial. This conception ofequivalence relations cannot ignore failuresto observe the predicted results.

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Received November 12, 1999Final acceptance April 27, 2000

Murray Sidman Equivalência de estímulo e contingência de reforçamento

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