Equivalence Relations and Behavior: An Introductory Tutorial

Murray Sidman, Sarasota, Florida

With an emphasis on procedural fundamentals, the original behavior-analytic equivalence experimentsand the equivalence paradigm are described briefly. A few of the subsequent developments andimplications are noted, with special reference to the possible significance of the findings with respect tolanguage and cognition.

Key words: equivalence relations, stimulus control, conditional discrimination, matching to sample

I was asked to do a brief introduction toequivalence relations in behavior and itsimplications for language research and ap-plication, particularly for the benefit ofyoung readers of this journal who arebecoming more interested in the topic. Thisintroduction to research on equivalencerelations is therefore going to be extremelybasic, starting at the very beginning andemphasizing methodology. Expectations of abig theoretical discussion will only result indisappointed readers. There is still much tobe done in equivalence research and in itsapplications that is independent of any parti-cular theory. We are, of course, interested inwhat any data signify, but there are manykinds of significance besides theoretical. Iwill note some exciting possibilities that I seein the topic of equivalence relations withrespect both to the science of behavior and tomore general intellectual and practical con-cerns. What remains to be done is at least asstimulating as what has been done already.Even after an exciting research program thathas now lasted more than 35 years, I ameager to see others expand on the basics.

For those who want to follow up in moredetail, two references that give my own slanton the field of equivalence relations are myequivalence book (Sidman, 1994) and apaper that expands on some of the materialin the book (Sidman, 2000). I will start herewith a description of our first experiment(Sidman, 1971). Although many investiga-tors have since done more sophisticated andmore revealing studies, the first ones havecertain virtues as an introduction. Even the

very first had features that are still relevant towhat is being done today. For those whowant to go more deeply into the initial data, amore fully controlled replication was pub-lished 2 years later (Sidman & Cresson,1973).

Our basic procedure was matching tosample. That term, matching to sample, re-ferred originally to what experimentersthought of as identity matching, in whichsubjects have to match stimuli that, to us, arephysically the same. In most of our experi-ments, although not all, the stimuli to bematched bore no physical resemblance to eachother. Because the matching criteria werearbitrary, I prefer the procedural name,conditional discrimination. If you are givenStimulus A1, then you match it to Stimulus B1and not to B2, B3, or B4. If you are givenStimulus A2, however, then you match it to B2and not any of the others. If A3, then B3, andso on. The experimenter or teacher determineswhich stimuli are to be related, and thematching is done regardless of any lack ofresemblance between the matched stimuli.

The involvement of arbitrary matchingbrings up what many consider to be the mostinteresting aspect of equivalence relations.The emergence of equivalence relations pro-vides a way to study experimentally whatmight be thought of as a kind of stimulusgeneralization, an elusive kind in which sub-jects come to match stimuli that share nophysical properties and that have never beenpaired with or directly related to each other.

Here is an outline of the experimentalsetup we started with. On any given condi-tional discrimination trial, the subject was tocompare several stimuli (called comparisonstimuli) to a sample and to select one of thosecomparison stimuli by touching it; the choice

Address correspondence to Murray Sidman,3435 Fox Run Road #347, Sarasota, Florida34231 (e-mail: murraysidman@comcast.net).

The Analysis of Verbal Behavior 2009, 25, 517

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that we scored correct was conditional onwhich stimulus was the sample on that parti-cular occasion. In our laboratory, subjects satbefore a matrix of nine keys onto whichstimuli could be projected, one in the centersurrounded by eight others. As an example,Figure 1 shows two trials. At the left, withthe sample word CAR located in the center ofa circle of eight keys, the subject is to selectthe picture of a car from among the com-parisons located in the outer keys. At theright, with the sample word cup, the pictureof a cup is to be selected. On other trials,with different samples (cat, box, cow, etc.),different selections from among those samecomparisons will be correct. The locationsof the comparison pictures change from trialto trial. Because the conditional discrimina-tion terminology is somewhat cumbersome,we often still talk about matching to sampleeven though we are studying nonidentitymatching.

Skinner (1950, pp. 213214) found that hecould not easily get pigeons to do matchingto sample unless he taught them first to peckthe sample key to gain access to comparisonstimuli. This procedure was probably effec-tive because pigeons usually look at whatev-er they peck, so pecking the sample mayhave helped ensure that they observed thesample. Based on Skinners findingalongwith human subject replications, particularlywith normal and handicapped childrentodays standard matching-to-sample proce-dure requires even human subjects to respond

to the sample before the comparisons canappear. Requiring human subjects to touchthe sample key does not, however, guaranteethat they will observe the stimulus on thekey. I suspect that such failures of observa-tion are responsible for the seeming inabilityof some subjects to learn a particularmatching-to-sample taskthey just do notlook at the stimulus on the sample key. Evenwhen they are performing a conditionaldiscrimination perfectly, the stimulus aspectsthat control their behavior may not be thesame as those specified by the experimentalcontingencies (e.g., Carrigan & Sidman,1992; Iversen, Sidman, & Carrigan, 1986;Johnson & Sidman, 1993; McIlvane & Dube,2003). Observing behavior in matching tosample is a ripe area for investigation (Dubeet al., 2006). How can it be measured, taught,and modified?

The first phase of our experiment was tocheck whether our subjects could matchprinted word samples to picture comparisons,as summarized in Figure 1 and in the left sideof Figure 2. This is an instance of what wecall visual-visual word-to-picture matching.We also tested another example of noniden-tity matching to samplethe reverse, orsymmetric version of what we have just beenlooking at (Figure 2, right). Now, the sampleis a picture, and the comparisons are printedwords. This is an example of what we callvisual-visual picture-to-word matching.

These kinds of stimuli interested us be-cause people who can match printed words to

Figure 1. Stimulus displays from two trials of visual-visual word-to-picture matching. The center key onthe left contains the sample word car, and on the right, the sample word cup. Eight comparison stimulisurround each sample.

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the appropriate pictures are said to under-stand the words, to exhibit a simple form ofreading comprehension. For reasons thatneed not concern us here, we wanted to findout whether elementary reading comprehen-sion, tested this way via word-to-picture andpicture-to-word matching, could developwithout being directly taught. Could we getstudents to do these matching tasks withoutour ever having provided them with anyreinforcing consequences for doing so?

We had been working with a group ofinstitutionalized teenaged boys with severemental retardation, boys who were unable todo the two tasks illustrated in Figure 2. Theycould not match printed words to their corre-sponding pictures; they had never learned toread. Indeed, before we could get them to dothe complex matching to sample that thisexperiment required, we had to teach thembasics like sitting quietly, pointing at specificobjects, discriminating simple forms likelines of different orientations and curvatures,and telling circles, squares, and other stan-dard forms from each other. Before we couldexpect them to discriminate words, we had toteach them to discriminate the individualletters, and before that, the forms that makeup the letters. Finally, we had brought themto the point where we could teach them to dowhat was called identity matchingto matchwords to themselves and pictures to them-selvesso they had become familiar withour matching-to-sample procedures. Because

they had shown no evidence of readingcomprehension, they seemed ideally suitedto help answer our question about how toteach it.

We first taught the boys to match dictatedword samples to picture comparisons (Fig-ure 3). Instead of presenting visual sampleson the center key, which remained blank, wepresented auditory samples, dictated words.On the particular trial shown at the left, wedictated, car, car, car, , repeating theword until the end of the trial so that the boywould not have to remember it. On the rightside is an example of another trial, this onewith the dictated word cup as the sample.Before the comparison stimuli could appear,the boy had to touch the blank sample keyand thereby produce comparison pictures onthe outer keys, and we required at least onesounding of the sample word before a touchto the blank key would work. The reason wedid that was to decrease the likelihood thata boy would impulsively press the blanksample and a comparison key without havinga chance to listen to the spoken word. If hedid not listen, he would not have an opportu-nity to learn anything about the dictatedsamples.

Then, having produced comparison stimulion the outer keys, the boy could produce areinforcer by touching the comparison pic-ture that matched the dictated sample. Ontrials other than those shown in Figure 3, wedictated other names. The subjects eventually

Figure 2. Stimulus displays from pretests of visual-visual word-to-picture matching (left) and visual-visual picture-to-word matching (right).

EQUIVALENCE RELATIONS: TUTORIAL 7

learned to match 20 dictated names to corre-sponding pictures. We also used several varia-tions of each picture, so that the boy would notjust observe some irrelevant aspect of apicture. Let us call this auditory-visual word-to-picture matching.

The next step was to teach them to matchthe same dictated words not to pictures butto printed words. The right side of Figure 4

shows this. Again, on the illustrated trial, werepeatedly dictated the word car. On othertrials, we dictated other words. The boycould now procure a reinforcer by touchingthe corresponding printed word rather than apicture. This task, which we call auditory-visual word-to-word matching, was extreme-ly difficult to teach to our first subjects, butthey eventually learned to match the 20

Figure 3. Comparison-stimulus displays from two teaching trials of auditory-visual word-to-picturematching. The center keys are blank and the dictated sample words car and cup are indicated above thecomparison displays (here only, but not to the subjects).

Figure 4. Comparison stimulus displays from teaching trials of auditory-visual word-to-picture matching(left) and auditory-visual word-to-word matching (right).

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dictated names with the corresponding print-ed names.

At this point, some might be tempted tosay that the boys had learned to read text(i.e., printed words). Indeed, we found laterthat learning these auditory-visual matchingtasks often, although not always, did makesubjects able to name the printed words (i.e.,to read them aloud). But we could not say yetthat the boys understood the words, that theywere reading with comprehension. For ex-ample, I can match many words spoken tome in German with their printed counterparts

and I can also read many German wordsaloud with something resembling the Ger-man pronunciationall this, however, with-out having the slightest idea what thosewords mean. We had not yet shown that theboys grasped the relation between printedwords and pictures, which would have indi-cated at least a simple understanding of thewords. To find out if they could now readwith comprehension, we repeated the originalvisual-visual word-to-picture and picture-to-word matching tests that we saw earlier inour procedure illustrations. In Figure 5, the

Figure 5. Procedural summary of the first experiments. The two upper segments illustrate the auditory-visual word-to-picture and word-to-word teaching trials that were shown in Figure 4, and the two lowersegments illustrate subsequent posttest trials of visual-visual word-to-picture and picture-to-wordmatching.

EQUIVALENCE RELATIONS: TUTORIAL 9

upper section repeats the teaching phase.That phase had involved only auditory-visualmatching: dictated words to pictures and toprinted words. The lower section of Figure 5illustrates one trial from each subsequent test,which involved only visual stimuli and noauditory samples. Given a printed word asthe sample (shown at the lower left, in thecenter key), would the boy now select theappropriate picture? And given a picturesample (shown in the lower right), would henow select the appropriate printed word? Theboys had never been able to do these kindsof visual-to-visual matching tasks before. Ifthey could now do them, we would be ableto assert that learning to match dictatedwords both to pictures and to printed wordshad given them the ability to comprehendthe printed words, to match them to theircorresponding pictures.

That is exactly what happened. Althoughour students had never been taught explicitlyto relate text and pictures, they now accu-rately matched nearly every one of the 20printed-word samples to its picture and eachof the 20 picture samples to its correspond-ing printed word. After they had learnedthe original 40 auditory-visual relations viadirect teaching with reinforcement, 40 newvisual-to-visual relations literally emergedin full bloom, so to speak. They could nowread with comprehension without their doingso ever having been reinforced.

This was not the usual transfer-of-trainingphenomenon. It was not that the auditory-visual experience permitted the boys to learnthe visual-visual matching faster than theyotherwise would have. They matched thepictures and printed words perfectly on thevery first posttest trials; they showed readingcomprehension immediately. Nor was thisthe usual stimulus generalization phenome-non. It could not be said that the visual-visualrelations between printed words and picturesemerged because of any physical resem-blances between related stimuli. In everydaylanguage, we could say that the printed wordshad become symbols for the pictures.

The first time I saw this happen was a bigevent in my life. For me, it was an experiencecomparable to the first time I shaped a ratsbar pressing, and then to the first time I con-ditioned avoidance behavior with the free-operant procedure. My excitement also

matched what I felt the first time I taught adifficult circle-ellipse discrimination error-lessly by means of a stimulus-fading proce-dure, and then, to the first time I found thatsome patients who had suffered strokes andhad lost the ability to express themselvesvocally could nevertheless understand wordswhen they were tested nonvocally, that isto say, when they were tested with exactlythe same matching-to-sample procedures wehave been looking at here. Although theycould not say the names, some of them couldstill match the printed words and pictures.

I am convinced that the best way to getones feet wet in equivalence, to experiencethe same excitement I did during our firstequivalence experiment, would be to do thisexperiment oneself. One can do it easily withfewer stimuli and with table-top procedures.Just seeing it happen would be more likely tostimulate interest than would any exposureto published papers, lectures, or theoreticalcontroversies, especially to presentationsconcerned with advancing some particulartheory. I believe that such personal exposurewould also generate interest not just inequivalence but, more generally, in behavioranalysis. Much remains to be done both inequivalence research and its applications thatis independent of any particular theory. Anoveremphasis on theory has caused our fieldto overlook a number of interesting pub-lished extensions of equivalence relationsbecause the publications require readers towade through a complex theoretical back-ground in order to find out how the data wererelated to what was done rather than to theirtheoretical rationale. For the most part, then,I will just share a number of conjecturesabout the general significance of the phe-nomenon that our first experiments revealed.

First, however, a few things need sayingabout methodology. My concern with meth-odology arises from observations that themethods we use to gather and present evi-dence may not only influence scientific con-clusions and judgments but may also deter-mine what we do or fail to do next. Take, forexample, the equivalence triangle that weoften use to summarize a basic equivalenceprocedure and its findings (Figure 6). Here,A designates three of the dictated wordsamples we used in the first experiments; Bdesignates three of the pictures, and C

10 MURRAY SIDMAN

designates three of the printed words. Thetwo solid arrows indicate the auditory-visualword-to-picture (AB) and word-to-word(AC) matching that were explicitly taughtto the subjects. The two dashed arrows indi-cate the visual-visual word-to-picture (CB)and picture-to-word (BC) matching that em-erged without having been explicitly taught.

Unfortunately, the use of arrows in suchdiagrams suggests to many that equivalencerelations represent sequential processes.What is intended, however, is to indicatecontingencies, events that are true only undercertain conditions: If this, then that; if notthis, then not that. For example, if thedefined sample is dog and not any of theother possibilities, and if the picture of a dogand not any of the other pictures controls thedefined response (touching), then and onlythen will the defined reinforcer be forthcom-ing. If car and not any other possibility is thesample, then reinforcement becomes subject

to a different set of conditions. Summarydiagrams like Figure 6 do not show theactual contingencies, and the arrows mayeasily lead one to ignore the proceduralcomplexities and talk about temporal se-quences and associations rather than aboutsimultaneous options. One will then be lesslikely to ask questions about the contextualcontrol of equivalence relations, about therole of unintended instructional control of theemergence of new conditional discrimina-tions, about the number of possible classes asa determiner of how quickly new conditionaldiscriminations emerge and even as a criticalfactor in the generation of new conditionaldiscriminations without reinforcement, andabout a number of other likely extensions ofthe basic phenomenon outside the laboratory.In addition, as careful reading of many theo-retical discussions will reveal, oversimplifi-cation of methodological and procedural de-scriptions will produce oversimplification oftheoretical formulations also.

Methodological considerations not onlyprove relevant to the evaluation of data inequivalence experiments but sometimesprove interesting in their own right, withextensions also to other research areas. Forexample, studies of equivalence relations donot always require such complicated stimulias we have been looking at, or so manystimuli. Such technical simplification can,however, introduce complexities of datainterpretation. At one stage of our work, wetried to make matching to sample easier forour subjects by using only two comparisonstimuli per trial. For example, Figure 7 illus-trates the four possible trial displays from anattempt to teach subjects to match verticallines to vertical, and horizontal to horizontal,

Figure 6. The equivalence triangle. The solidarrows (AB and AC) designate conditional dis-criminations that were explicitly taught to thesubjects: auditory-visual word to picture (AB) andauditory-visual word to word (AC). The dashedarrows indicate conditional discriminations thatemerged without having been actually taught:visual-visual word to picture (CB) and visual-visual picture to word (BC).

Figure 7. The four possible stimulus displays inthe process of teaching horizontalvertical identitymatching.

EQUIVALENCE RELATIONS: TUTORIAL 11

a seemingly simple identity-matching task.The circles represent keys, and a vertical orhorizontal line appears on each key. In eachtrial display, the horizontal and verticalcomparison lines are at the top, above thesample; the left and right positions of thecomparisons vary from trial to trial. Subjectsproduce a reinforcer if they touch a compar-ison key that contains a line with the sameorientation as the sample key. What kind of aperformance would allow us to say that oursubject knows how to match vertical tovertical and horizontal to horizontal?

Suppose he or she achieves 75% correctover enough trials to make that statistic sig-nificantly different from chance. We wouldlike to think this average signifies that thesubject has matched correctly on 75% of alltrials, regardless of which line was thesample. With only two comparisons, howev-er, a score of 75% correct could have beenachieved in another way. Suppose that whenthe sample is vertical (as it is in the twodisplays on the left side of Figure 7), thesubject always selects the vertical compari-son; that would yield a score of 100% onvertical-sample trials. But suppose that whenthe sample is horizontal (as it is in the twodisplays on the right side of Figure 7), heor she always selects the left comparison,regardless of which stimulus is in that posi-tion. Because each comparison appears at theleft on half the trials, that would give a scoreof 50% correct when the sample is horizon-tal, even though really the subject neverselected the horizontal comparison at all; heor she just picked the left comparison key, nomatter which line was on that key. With100% correct on vertical trials and 50%recorded as correct on horizontal trials, theaverage score would be 75%, even thoughthe subject had never, on any trial, paid anyattention to the horizontal comparison. The75% score, even if statistically significant, isbehaviorally meaningless. Whenever I seean author claiming that an accuracy of 75%(even 80%) indicates that a subject haslearned a two-sample two-comparison con-ditional discrimination, I stop reading thatpaper (for a more detailed discussion, seeSidman, 1980).

With only two comparisons, a relatedmisconception can arise if a subject werealways to select the same comparison

stimulus. Suppose our student chooses verti-cal on every trial, regardless of which line isthe sample, giving a score of 100% on trialswith vertical samples and 0% with horizontalsamples. Although this yields an averagescore of 50%, statistically insignificant, sucha performance has been known to temptinvestigators into concluding that althoughthe student had completely failed to learn therelation between the horizontal stimuli, he orshe had learned to match vertical samplesperfectly. A student who always selects thesame comparison stimulus, however, cannotbe said to be matching either of the samples,horizontal or vertical. Such a performancecould indicate simply that the student paid noattention at all to the samples, that as far ashe or she was concerned, the samples did noteven exist. After all, picking the verticalcomparison all the time produced a reinforceron every other trial on the average; not a badpayoff for so little work.

What if our subject never makes a mistake,always picking the comparison line thatmatches the sample? Still, with only twocomparisons, even a seemingly perfect per-formance does not permit us to say for surethat he or she has matched both line orienta-tions. Let me describe how I might get areinforcer on every trial without matchingeach comparison line to its identical sample.Suppose that on every trial, I look for thevertical comparison. When I find it, I touch itif the sample is also vertical. But if thesample is not vertical, then, whatever else thesample may be, I touch the other comparison,whatever it may be, as long as it is notvertical. Note that all of my selections hereare controlled by just one of the stimuli, thevertical line; I either select the verticalcomparison or reject it, depending on wheth-er the sample is vertical. If you were then totest me by substituting even unfamiliarstimuli for the horizontal line, I would stillbe correct on every trial because the onlything that mattered to me was whether or notI was looking at vertical.

The recorded measure, accuracy, does notdistinguish between the two types of stimuluscontrol on correct trials: selection of onecomparison or rejection of the other. Theresponse that we record, touching the correctcomparison, does not tell us whether thatcomparison or the other one controlled our

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choice. If you assumed that my perfect scoremeant I had learned to match vertical andhorizontal comparisons to their identicalsamples, you would be mistaken. If you thentried to build on those stimulus controltopographies to teach me something else,you would run into serious problems. Forexample, the use of only two comparisonsper trial might then cause a failure to demon-strate equivalence relations, simply becausethe actual controlling stimuli are not the onesyou are testing for (Carrigan & Sidman,1992; Johnson & Sidman, 1993). The ques-tion, What are the actual controlling sti-muli? remains relevant not just in equiva-lence research but in every experiment thatinvolves stimulus control (McIlvane & Dube,2003; Ray & Sidman, 1970). Identifying theactual controlling stimuli is also critical inapplied situations, particularly when one istrying to remediate seeming failures to learn.

These are instances in which an aspect ofthe research methodology (e.g., the use ofonly two comparison stimuli) can greatlyincrease the ambiguity of ones conclusions.By presenting three comparisons with eachof three samples, one can reduce the likeli-hood of such a problem. Rather than learningto reject one of two comparisons, the subjectmust then learn to reject two of the threecomparisons on each trial; rejection becomesmore difficult than selection. I have, howev-er, seen subjects matching two of the com-parisons appropriately to their respectivesamples, by selection, and then rejectingboth of those comparisons on trials with thethird sample, thereby learning nothing aboutthe relation between the third sample andcomparison. In my own work, therefore, Ihave gone back to using at least four samplesand four comparisons per trial, and I nowrealize how lucky I was to have started withdisplays of eight comparisons.

Back to some additional but still elemen-tary matters of significance. The proceduresI have outlined have great generality. Anextension that to me seems obvious is theteaching of simple vocabularies, an extensionwhose utility has gone unnoticed within theeducation establishment. Indeed, very fewapplied behavior analysts seem to haverecognized this particular application. Thebasic equivalence paradigm provides a mostefficient way to add nouns to someones

reading vocabulary and probably to his orher speaking vocabulary also. Teach them tomatch spoken words both to their corre-sponding printed words and pictures, andwithout any more instruction, they are able tounderstand the printed words. The samecould easily be done with adjectives, ad-verbs, and by using the capabilities ofmodern computers, even verbs. Equivalenceclasses have been generated with manydifferent stimulus materials, not just thestrange stimuli that experimenters often useto reduce the likelihood of preexistingequivalence relations. For example, we havetaught children with retardation to matchboth a color and its printed name to the samedictated color name, and have then seen thechildren able to match colors and printednames to each other, in other words, tounderstand printed color names.

By teaching children to match dictatedletter names to both upper and lower caseletters (Figure 8, top), we have then seen

Figure 8. Like Figure 5 but with dictated letternames, upper case letters, and lower case letters asstimuli. After being taught to match dictatedletter-name samples to both upper case and lowercase comparisons (the two upper segments),posttests revealed the subjects emergent abilityto match the upper and lower case letters to eachother (the two lower segments).

EQUIVALENCE RELATIONS: TUTORIAL 13

them able, without further instruction, tomatch those visual stimuli to each other, as inthe lower segment of Figure 8, in otherwords, to understand the equivalence ofupper and lower case letters.

The application of the basic conditionaldiscrimination equivalence procedure tonumbers allows us to take a significant stepforward. Suppose we first teach students tomatch dictated number names both to digitsand to printed number names, as illustrated inthe leftmost and center segments of the uppersection of Figure 9. After such teaching,children are able to match both the digits andprinted words to each other, as shown in thetwo leftmost test segments.

Then, as shown in the rightmost teachingsegment, suppose we teach them to match thedictated number names also to quantities,represented here by quantities of dots. Afterthis additional teaching step, we find thatthe classes have enlarged. As indicated inthe four right segments of the test section(Figure 9), the children now can also match

both the printed digits and number namesto the dot quantities, and vice versa, eventhough they had not been directly taught todo so and had never before even seen thosestimuli (visual words, digits, and quantities)together. By learning to match a new stim-ulus to one member of a class, the childrenautomatically become able to match the newstimulus to all other members of the class. Ifwe were then to go on and teach our pupils tomatch dictated number names in anotherlanguage (say, French) to the digits, theywould automatically become able also tomatch the spoken French number names tothe printed English number names and to thequantities. This productivity is one of themost significant aspects of equivalence rela-tions. It is not a theory; it is a datum, anexciting datum. As a class enlarges, the di-rect addition of just one new member to theclass produces an enormous increase in thenumber of indirectly established new rela-tions. A small amount of teaching can yield atremendous amount of learning.

As far as I am aware, this explosive featureof equivalence relations has not been pur-posefully exploited in the teaching of second-language vocabularies. Nor has it been used,except in research, for the teaching of moneyskills like coin equivalences (McDonagh,McIlvane, & Stoddard, 1984; Stoddard,Bradley, & McIlvane, 1987). For example,by directly teaching students to match thedictated twenty-five cents with one quar-ter, five nickels, two dimes and a nickel, twodimes and five pennies, three nickels and 10pennies, and so on, we can then expect thatthe students, without additional training, willbe able to match each of those coin com-binations to all the others. To devise such aprogram would take a lot of planning andpreparation, for sure, but what a payoff!Anyone could do it, even if they had neverattended any of those confusing symposia onequivalence theory.

The design of our first experiments ledsome to assume that one must learn auditory-visual relations before showing visual-visualrelations like those involved in reading com-prehension. It is now clear, however, thatnew arbitrary visual-visual relations canemerge not just from auditory-visual rela-tions but also from relations that involve onlyvisual or several other stimulus modalities.

Figure 9. After learning to match dictatednumber-name samples to comparison digits,printed number names, and quantities (the upperthree segments), subjects show in subsequenttests their emergent ability to match each of theformer comparisons to each other (the lowersix segments).

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The auditory modality is not required. Classunion comes about when two sets of con-ditional discriminations have an element incommon. The process is general, regardlessof the stimulus modalities. Nevertheless, Iventure to suggest that the use of auditionand olfaction with nonhumans might proveespecially productive. For example, auditory-visual matching in studies with dolphinscould provide a key to our understanding oftheir communication, including both com-prehension and production of speech (that isto say, dolphin speech). With the sophisti-cated techniques that are now available forrecording and reproducing sounds, it shouldbe possible to use natural vocalizations asstimuli, relating them either to arbitrary stim-uli and responses or to referents that wesuspect are the actual ones for those animals.

Species generality of equivalence relationsstill requires investigation. The original andmost subsequent work on equivalence rela-tions involved humans, with emphasis onquestions about reading; about languagedevelopment; about the origin of symbols,classification, and concept formation; aboutthe sources of some seemingly untaught be-havior; and about implications for efficiencyin teaching. From the beginning, however, aprimary concern has been the possibility offinding a role for equivalence phenomena inthe behavior of other species. In spite of thefailure of the earliest attempts to demonstratespecies generality (e.g., Dugdale & Lowe,2000; Sidman et al., 1982), confirmation ofequivalence relations in individual nonhu-man subjects has come from Vaughans(1988) pigeon experiment, in which hepioneered a simple discrimination ratherthan a conditional discrimination techniquefor generating equivalence classes. Sidman,Wynne, Maguire, and Barnes (1989) reporteda systematic replication of that techniquewith human subjects. More recently, equiv-alence relations have been convincinglydemonstrated in several sea lion studies inSchustermans laboratory (e.g., Kastak,Schusterman, & Kastak, 2001). It has beensuggested (Sidman, 2008) that the successwith sea lions came about at least in partbecause the subjects, unlike those in otherexperiments, had previously been taught todo identity matching. Still to be ascertained,however, is even an approximate range of the

species that are capable of developingequivalence classes.

Most, if not all, stimuli are members ofmore than one class. For example, in someinstances green is in the class of color, inother instances it is in the class of beginner,and more recently, it is often in the class ofenvironmentally conscious. Stimulus mem-bership in more than one class does not, ofcourse, cause all of those classes to combineinto one. Contextual circumstancesthesubject of the particular book one is reading,the current topic of conversation, the partic-ular person with whom one is conversingdetermine which of several possible classes astimulus is in at any particular time. In somecontexts, classes that contain a member incommon will merge into one; class unionwill take place. In other contexts, classes willsimply intersect, remaining separate in spiteof a member common to both. In an exampleprovided by Bush, Sidman, and de Rose(1989), If we are discussing disciplines,Renoir, Constable, and Pollock go togetheras artists; Twain, Voltaire, and Byron aswriters; and Churchill, Kennedy, and DeGaulle as heads of state. If we are discussingnationality, Renoir, Voltaire, and De Gaullego together as French; Twain, Kennedy, andPollock as American; and Churchill, Consta-ble, and Byron as British (p. 31).

Just as the two-term operant reinforcementcontingency rarely, if ever, exists in real life(the discriminated operant is therefore thebasic unit of behavior analysis) so equiva-lence relations probably always come undercontextual control. Here is another fertilearea for investigation, particularly in extend-ing the significance of equivalence classesbeyond the laboratory.

When the topic of contextual control ofequivalence relations comes up, I often recallthe time I emphasized in a talk that an objectand its name are not always equivalent, thatcontext determines class membership. As anexample, I pointed out that although we swatflies, we do not swat the word fly. SteveHayes then proposed jokingly that he coulddisprove my theory of equivalence; heprinted the word fly on a piece of paper andthen batted the word with a fly swatter.Nevertheless, the equivalence of particularwords to what, in everyday speech, arereferred to as their referents is a sufficiently

EQUIVALENCE RELATIONS: TUTORIAL 15

everyday observation to provide a topiceven for popular cartoonists. For example, Iremember a comic strip in which onecharacter printed the word banana in thesand and another character then slipped onthe word and fell.

I think it is worth following up such bitsof folk wisdom. People do indeed behavetoward words and other symbols just as theybehave toward their referents. Much of whatwe do is determined by things and events thatwe cannot possibly have experienced direct-ly. For example, we can know historicalevents only through words; we can knowmost people only by what has been writtenabout them or by their photographs and otherrepresentations, like statues; words andpictures make it possible for us to knowabout events that take place too far away forus to observe them directly; we deal effec-tively with quantities that are representedonly by numbers on paper; we follow travelroutes shown as lines on maps; and so on.The opportunity to apply a scientific analysisto such phenomena is, to me, more reason todevelop and continue an interest in equiva-lence relations than is any possible personaltheoretical triumph.

Each to his or her own, however. Thosewith a stronger theoretical bent will also findmuch of interest in the data on equivalencerelations. (For different theoretical approach-es, see, e.g., Hayes, Barnes-Holmes, &Roche, 2001; Horne & Lowe, 1996; Sidman,1994, 2000.) But take care. One must judgetheories on grounds of parsimony (precision,neatness, and simplicity), coherence, andconsistency in their explanations of existingdata, and productivity in their predictions ofphenomena that have not yet been observed(see Sidman, 1997, particularly pp. 138143). Finally, a theory must be capable ofdisproof. The methodology of theory con-struction is just as demanding as is themethodology of experimentation. Further-more, do not go into theory constructionunder the illusion that you can escape fromthe technological constraints of rigorousexperimentation. Failure to attend to thesubtleties of experimental methodology willmake one unable to evaluate rigorously thedata that must inevitably be produced to testany theory.

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McDonagh, E. C., McIlvane, W. J., &Stoddard, L. T. (1984). Teaching coinequivalences via matching to sample.Applied Research in Mental Retardation,5, 177197.

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