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Controlling relations in baseline conditional discriminations as determinants of stimulus equivalence.

According to mathematical set theory, relations that possess the properties of symmetry, transitivity, and reflexivity are relations of equivalence. Sidman and Tailby (1982) applied this mathematical notion to relations between stimuli and argued that conditional relations between stimuli that are reflexive, symmetrical, and transitive document the formation of classes of equivalent stimuli. Sidman and Tailby (see also Sidman, 1986, 1994) also argued that equivalence is the defining property of symbolic relations. Therefore, according to Sidman and Tailby, to state that a stimulus is a symbol for another, it is necessary to demonstrate that these stimuli share membership in an equivalence class. Recent studies confirmed that stimuli in an equivalence class are semantically related. This has been shown through behavioral evaluations of meaning (e.g., Barnes-Holmes et al., 2005; Bortoloti & de Rose, 2009, 2011, 2012) or through electrophysiological measures (Barnes-Holmes et al., 2005; Haimson, Wilkinson, Rosenquist, Ouimet, & Mcllvane, 2009).

Laboratory studies ot equivalence class tormation typically comprise a training pnase, in which participants learn related conditional discriminations, such as AB and BC, and a phase that tests emergent conditional discriminations, not directly trained but derived from the trained ones, that attest the properties of symmetry (such as BA and CB), transitivity (such as AC), and combined symmetry and transitivity (such as CA). The property of reflexivity is tested by identity matching probes (such as AA, BB, and CC). However, these probes are ambiguous because choosing the identical stimulus may be attributed to generalized identity matching (see R. R. Saunders & Green, 1999). Because of this ambiguity, reflexivity tests are often omitted in studies of stimulus equivalence.

Stimulus equivalence is a robust phenomenon. Human participants that master baseline conditional discriminations often show the formation of equivalence classes. Many participants do so promptly, showing high scores in emergent conditional discriminations at the beginning of the testing phase. Often, however, participants show delayed emergence of equivalence classes: Performance in emergent conditional discriminations is defective at the beginning of testing, and scores eventually reach a criterion as tests are repeated, sometimes with baselines retrained (e.g., Lazar, Davis-Lang, & Sanchez, 1984; Sidman, Kirk, & Wilson-Morris, 1985; Spradlin, Cotter, & Baxley, 1973). Most participants show formation of equivalence classes, either immediate or delayed. However, a proportion of participants may master the baselines and do not show the formation of equivalence classes (with the amount of testing provided). Failures to exhibit equivalence indicate that mastery of baseline conditional discriminations is a necessary but not sufficient condition for stimulus equivalence. Researchers disagree about necessary conditions for equivalence other than baseline training. Some propose that other behavioral repertoires, such as language (Devany, Hayes, & Nelson, 1986) or naming (Horne & Lowe, 1996), are necessary. These authors attribute the failure to show equivalence in tests conducted after the baseline has been learned to deficiencies in the participants' repertoires. Other researchers have emphasized variables related to the training procedure, such as the number of nodes intervening between related stimuli (e.g., Moss-Lourenco & Fields, 2011) or the logical design of conditional discriminations trained (e.g., Arntzen, Grondahl, & Eilifsen, 2010; R. R. Saunders & Green, 1999).

Other authors (e.g., Carrigan & Sidman, 1992; de Rose, 1996; Mcllvane, 2012) have pointed out, however, that the description of conditional discrimination performance as a relation between the sample and the correct comparison may mask different forms of learning that may actually take place. Figure 1 shows two different stimulus control relations that may satisfy the contingencies of a conditional discrimination with two samples and two comparison stimuli. Although many other controlling relations might result in successful performance (and many more will result in unsuccessful performance), the present analysis will focus on a combination of the two controlling relations illustrated in Figure 1. The term controlling relations will be retained because it is more familiar, although the use of the term stimulus control topographies has been convincingly proposed by some (e.g., Mci'vane & Dube, 1992).

The left panel illustrates responding based on a relation between the sample and the correct comparison (S+). This sample-S+ relation (represented by an arrow pointing from the sample to the S+) controls selection of the S+. In selections based on only this relation, the participant may not even notice the features of the incorrect stimulus (S--). The right panel illustrates responding based on a sample-S-relation that controls responding away from the S-- Since only one alternative remains, the participant responds to the S+, but this responding is controlled by the S--, and the participant may not even notice the features of the S+. An arrow with an oval substituted for the tip, pointing from the sample to the S--, represents this sample-S-relation.

When participants learn a set of conditional discriminations, they may learn to select each comparison stimulus in the presence of the corresponding sample on the basis of either a sample-S+ relation or a sample-S--relation, or both. Figure 2 illustrates several hypothetical possibilities for a set of three conditional relations--AB, BC, and CD--between sets of two stimuli. From top to bottom, the first diagram represents a case in which each sample is related only to the corresponding S+. The second diagram represents a case in which each sample is related only to the corresponding S--. The third diagram represents both sample-S+ and sample-S--relations throughout. We may say that stimulus control relations are homogeneous in all of these three cases: The type of controlling relation is the same for all samples. The fourth diagram represents a heterogeneous case: Selections of some comparisons are based on sample-S+ relations; selections of other comparisons are based on sample-S--relations, and selections of yet other comparisons are based 9n both sample-S+ and sample-S--relations.

Sidman and colleagues showed that sample-S+ controlling relations are necessary tor equivalence formation, and the expected classes will not form when participants learned only sample-S--relations (Carrigan & Sidman, 1992; Johnson & Sidman, 1993). The influence of controlling relations on equivalence seems well established. It is not clear, however, how it interacts with other conditions presumed to be influential on equivalence outcomes.

Kato, de Rose, and Faleiros (2008) found that the topography of response in conditional discrimination considerably influenced equivalence outcomes. Most participants who used the computer's mouse to move a cursor to select stimuli showed class formation. However, class formation was significantly less probable among students who selected stimuli by pressing keys on the computer's keyboard. In another experiment, these authors substituted a black mask for one of the comparison stimuli (either the S+ or the S--) on probe trials. They found that seven out of eight participants who used the mouse continued to correctly select the S+ when the mask covered the S--but selected the mask when it covered the S+. This indicated that these participants had learned both sample-S+ and sample-S--relations for all samples. From eight participants who used the keyboard to select stimuli, only two learned both sample-S+ and sample-S--relations for all samples. There were, therefore, nine participants who learned both sample-S+ and sample-S--relations for all samples, and seven of these showed class formation. There were seven participants who did not learn sample-S+ and sample-S--relations for all samples, and only two of these showed class formation.

Research on variables that influence class formation has focused mostly on the linguistic repertoire of participants and on the logical design of trained conditional discriminations. However, the results of Kato et al. (2008) suggest that topographical variables related to the

stimulus display and required responding may influence baseline controlling relations and, consequently, equivalence outcomes. It is possible, therefore, that at least part of the variability in equivalence outcomes may be clue to hitherto unrecognized variables that influence the controlling relations in baseline conditional discriminations.

With the usual methods to train conditional discriminations, the controlling relations will be unknown to the researcher unless special probes are used. A study by Carrigan (1986), however, manipulated experimentally baseline controlling relations. He used a training procedure designed to prevent sample-S+ relations. This produced a homogeneous pattern of sample-S--relations and confirmed the absence of expected equivalence classes. Actually, Carrigan found different classes (that will not be described here, because their logical basis would introduce unnecessary complexity for the purposes of the present study; the reader is referred to Carrigan & Sidman, 1992, for further information). Johnson and Sidman (1993) used a similar procedure and confirmed the earlier findings.

The present study used a within-subject design in which children had conditional discrimination training followed by equivalence tests, with two or more different sets of stimuli. Two training procedures were compared. One induced a homogeneous pattern of both sample-S+ and sample-S--relations. The other attempted to induce a heterogeneous pattern of controlling relations, with exclusive sample-S--control in one of the conditional discriminations.



Four children, with ages ranging from 7 to 9 years, participated. Alice and Fabio attended, respectively, second and third grades in local public schools. They were also participating in a program to teach reading skills (de Rose, de Souza, & Hanna, 1996). Based on their everyday behavior and performance in lab tasks, researchers concluded that their school difficulties were due to inadequacy of school teaching and not to any significant developmental delay. Fred and Barb attended, respectively, first and third grades in local private schools.

Setting and Equipment

Participants' parents brought them to the lab 3 or 4 days per week. Two experimental rooms in the lab housed Macintosh computers (Performa 630 and PowerPC) with 14" color monitors. One of these rooms was a 3 in by 2 m cubicle that had no windows. This cubicle had air conditioner and a fan, and there were cabinets suspended on two of the walls and a desk with the computer on top. The other room was 6 in by 4 m, with the entrance door at the left side of one of the smaller walls. Wooden cabinets covered the rest of this wall. The opposite wall had windows that were covered by curtains, with the air conditioner above them. The longer wall to the left of the door had a one-way mirror, covered by a curtain. The opposite wall had suspended wooden cabinets. The room contained two desks on the rear corners and a round table near the front corner. One desk contained the computer used in the study, and the other contained a PC-compatible microcomputer, which was not used in this experiment. Participants had sessions in either room, depending on availability at arrival time. Fred, who was the son of one of the experimenters, did not come to the lab and had sessions three or four times per week at the study in his own home, with an Apple iMac microcomputer.

Session execution was controlled by the MTS v. 11.6.7 software (Dube & Hiris, 1999), which presented stimuli, recorded responses, and delivered programmed consequences. The software displayed stimuli on the monitor's screen, on five 3 cm by 3 cm white "windows" on a gray background, at the center, top left, top right, lower left, and lower right of the screen. In the experimental phases, stimuli were line drawings of abstract pictures (see Figure 3). Familiar pictures appeared only in the pretraining phase (see the next section). Participants responded to the stimuli by moving the mouse, positioning the cursor on the window, and clicking the mouse. Barb had additional conditions in which she responded by pressing keys on the numeric keypad on the right-hand side of the computer's keyboard, corresponding to the windows' locations. Active keys were 7, 9, 5, 1, and 3, corresponding, respectively, to windows located at the top left, top right, center, bottom left, and bottom right. Correct responses produced a visual display of moving colored stars, together with a sequence of tones. Incorrect responses produced a 3-s timeout, during which the screen remained dark. In some phases of the study, no differential consequences followed correct or incorrect responses (see the next section).

Participants received points corresponding to the percentage of correct responses in each session. Participants Fred, Alice, and Barb received periodic monetary payments based on the number of accumulated points. Alice and Barb could also choose a present from an assortment of pens, letter paper, stickers, and so forth. Fabio exchanged points for only presents; he received no monetary payments.


Experimental design. The study comprised a pretraining and two training conditions, with different stimuli appearing in each. Each condition taught a set of conditional discriminations, with two samples and two comparisons: AB, BC, and CD. Probes then verified emergent relations DA, CA, and DB, testing for transitivity and symmetry of the conditional relations. The single comparison method was used throughout training. In all conditional discriminations of Condition A, the mask covered the correct comparison in half of the trials and the incorrect one in the other half. This ensured that children would learn to perform the conditional discriminations under both sample-S+ and sample-S--controlling relations. The same procedure taught conditional discriminations AB and CD in Condition B. However, in training and all subsequent trials of conditional discrimination BC, the incorrect stimulus was always displayed, whereas the correct one was covered with the mask. The purpose of this teaching procedure was to establish conditional discrimination BC exclusively under sample-S--controlling relations, preventing the establishment of sample-S+ controlling relations. Probes after this training showed that children were indeed rejecting C2 in the presence of sample B1 and rejecting Cl in the presence of sample B2, and that they were not merely selecting the mask conditionally upon both samples B1 and B2.

Fred and Fabio ended the experiment after Conditions A and B. With Barb, Conditions A and B were replicated with new sets of stimuli and responses through the keyboard. For Alice, Condition B was replicated with a different set of stimuli.

Pretraining. The purpose of the pretraining was to give children ample experience with conditional discriminations with two samples and two comparisons, both in the two-comparison and the single-comparison formats. In this way, when children began the training with conditional discriminations AB, BC, and CD, they would already have a repertoire of generalized conditional discrimination (see Williams, Saunders, Saunders, & Spradlin, 1995).

Pretraining initially taught children to respond to the stimulus display, clicking on the window with a stimulus presented on it. They next learned to respond to the sample and then to select a comparison. A blocked trial procedure (K. J. Saunders & Spradlin, 1989) then taught children to match two familiar samples to two thematically related familiar comparisons (see Pilgrim, Jackson, & Galizio, 2000). Next, children learned to match two abstract samples to two abstract comparisons, with the blocked trial procedure. The next phase verified whether they would now learn another matching task, with novel abstract pictures, without the blocked trial procedure. A black mask was then faded in, covering one comparison on each trial. The next phase taught another matching task, between novel arbitrary pictures, now covering one of the comparisons on each training trial. After children learned a new matching performance with a single comparison presented on each trial, the pretraining ended.

In the initial session, participants sat facing the computer, with the picture of a moon presented in one of the outer windows. The experimenter instructed children to move the mouse, clicking on the window with the picture on it. The computer presented the tones and display of stars, and the experimenter told the child that this meant the choice had been correct. Responses on windows without a picture had no consequences. After a block of eight trials in which the picture of the moon appeared twice in each of the four outer locations in a randomized order, another block of eight trials followed, and all trials began with the picture of a sun in the center window. A response to the sample produced the presentation of the moon in any of the outer windows, and a response to the moon produced the display with stars.

The following trial blocks taught a conditional discrimination with two samples and two comparisons: When the sample was the sun, the correct comparison was the moon; when the sample was a picture of grapes, the correct comparison was a picture of a pear. An adaptation of the blocked trial procedure (K. J. Saunders & Spradlin, 1989) was used to teach this performance.

In the following block, which had eight trials, clicking the center window with the picture of the sun produced two pictures--the moon and a pear--presented in randomized positions across the outer windows. This block repeated until participants responded correctly on all trials. The same criterion of 100% correct applied to subsequent trial blocks unless explicitly mentioned. The next block had 16 trials; the first 8 presented a picture of grapes as the sample, and in the last 8, a picture of the sun was the sample. The next block, which had 12 trials, alternated the two samples every three trials. This phase of the pretraining ended with a block of 12 trials that presented the two samples in a randomized sequence.

The two subsequent phases of the pretraining used the blocked training procedure to teach two conditional discriminations between arbitrary pictures, with two samples and two comparisons each, designated as XY and WZ. After a review of conditional discriminations XY and WZ, the next phase presented conditional discrimination WZ and used a 16-step fading procedure to cover one comparison stimulus with a mask on every trial. A trial with no mask was the first step. The second step presented a small gray mask at the center of one comparison. The mask appeared to be behind the stimulus, which remained visible. The following six steps increased the size of this mask, until the mask was 180 by 180 pixels and larger than the stimulus, which did, however, remain visible. The eighth step presented this full-sized gray mask and a small black square at the center. The black square appeared to be in front of the stimuli and therefore obliterated the view of the part of the stimulus that it overlapped. The seven subsequent steps increased the size of the black square, until the whole mask of 180 by 180 pixels became black and completely covered the stimulus. During fading, the mask covered either the S+ or the S-, each in half of the trials.

The next block presented trials of conditional discrimination XY, with one of the comparisons always covered by the full-sized black mask.

The subsequent pretraining phase repeated the blocked training procedure to teach a new conditional discrimination, PQ. All trials in this training covered one of the comparisons with the mask. The S+ was displayed with the mask in half of the trials, and the S- was displayed with the mask in the other half.

The final phase of the pretraining taught conditional discrimination MN, without the blocked trial procedure and with a single comparison displayed (the second one was covered by the mask). A block of 12 trials presented samples M1 and M2 in a randomized sequence. In the six trials displaying M1 as the sample, the mask covered Ni (the S+) or N2 (the S-), each in three trials. The same occurred in the six trials displaying sample M2.

Experimental conditions.

Condition A. A block of 16 trials presented comparisons B1 and B2, randomized across positions, with samples Al and A2 presented in an irregular order. On trials with either sample Al or A2, the mask covered each comparison in half of the trials. This block repeated until children made correct selections in at least 15 trials, choosing the correct comparison on trials that displayed it or choosing the mask on trials that displayed the incorrect comparison. In the next trial block, the same procedure taught conditional discrimination BC. A block of 16 trials followed, mixing eight trials of conditional discrimination AB and eight trials of conditional discrimination BC in a randomized sequence. The block repeated until children responded correctly in 15 trials. The next block taught conditional discrimination CD, with the same procedure used for AB and BC. Then a block of 24 trials followed, mixing eight trials of each conditional discrimination taught, AB, BC, and CD, in a randomized sequence. This block repeated until children responded correctly in at least 22 trials. A message then appeared on the screen, saying that the computer would no longer tell whether responses were correct or not. The experimenter helped the child to read the message. The following block also had eight trials each of conditional discriminations AB, BC, and CD, in a randomized order, but selections of either the correct or incorrect alternative produced only the intertrial interval.

After this training, tests for emergent conditional relations DA, CA, and DB verified combined transitivity and symmetry of the trained conditional discriminations. The DA conditional discrimination could emerge only if all trained conditional discriminations were symmetrical and transitive and would be logically sufficient to document equivalence classes. However, probes for emergent conditional discriminations CA and DB were conducted to further validate inferences of equivalence, through the consistency in the results of different probes. Each probe block contained probes for only one of the emergent conditional discriminations. Probe blocks contained 16 probe trials mixed with 12 trials of the relevant baseline. Probe trials did not present the mask, and the two comparisons were visible on them. However, the mask was presented in the 12 baseline trials interspersed within each probe block. A block of trials of the relevant baseline anteceded each probe block. Conditional discriminations AB, BC, and CD composed the relevant baseline for the DA probes because mastery of those relations was logically required for the emergence of DA. Only AB and BC composed the baseline logically required for emergence of CA. Only BC and CD composed the baseline logically required for emergence of DB. Responses did not produce differential consequences in probe blocks and in the preceding baseline blocks. Baseline blocks preceding a probe block repeated if more than one incorrect response occurred. There was no such criterion for probe blocks. The first probe session contained DA and CA probes. This session was usually repeated and followed by one or two sessions with probes for DB.

Condition B. There were only three differences between Condition B and Condition A. First, a different set of stimuli was used. Second, all trials of conditional discrimination BC displayed the incorrect stimulus along with the mask. Thus, trials displaying sample B1 presented C2 and the mask as choices, and trials displaying sample B2 presented Cl and the mask. Therefore, in all trials displaying either sample B1 or sample B2, selections of the mask were considered correct. This training could establish sample-S--relations between B1 and C2 and between B2 and Cl. However, the training could also establish sample-S+ relations between B1 and the mask, as well as between B2 and the mask. For this reason, a third difference in Condition B was that, after the training, a probe block presented four trials of each trained conditional discrimination, in the absence of the mask. Correct responses in the BC trials of this block, in the absence of the mask, would demonstrate that the mask was not the controlling stimulus for responding. Since the BC samples had never been displayed together with the respective S+'s, it was believed that this would demonstrate that participants were indeed rejecting the S--'s. For conditional relations AB and CD, training was conducted as in Condition A; that is, the mask covered the S+ in 50% of the trials and the S--in the remaining 50% of the trials.

All participants were first exposed to Condition A, with Stimulus Set 1, and then to Condit ion B, with Stimulus Set 2 (see Figure 3).

Intrasubject Replications of Conditions A and B. Conditions A and B were replicated with Barb, with Stimulus Sets 3 and 4, respectively (see Figure 3), and responding through the keyboard. Condition B was replicated with Alice, with Stimulus Set 4.

Probes for controlling relations. Because there were unexpected results in Condition B for Alice and Fabio (see the Results section), probes with novel stimuli (e.g., Stromer & Osborne, 1982) were conducted in an attempt to confirm the controlling relations in Condition B. These probes were conducted with Alice after her two exposures to Condition B. Fabio left the study after the first exposure to Condition B and was not available for further testing. Details of these probes are presented with the results.


The graded difficulty of the pretest was successful to establish required performances with a negligible amount of errors. Very few errors occurred in the training phases as well. The results of interest are from the stimulus equivalence probes and additional probes.

Figure 4 shows children's performance in probes for emergent conditional discriminations. Each circle corresponds to a block of probe trials. Responses consistent with the probed conditional discrimination will be arbitrarily designated as correct. Black circles indicate emergence of the probed conditional discrimination, and the unfilled ones indicate non-emergence of the probed conditional discrimination, whereas gray circles are ambiguous cases. These may often be clarified by the overall pattern of probe results in the particular phase.

In Condition A, when training ensured that the students acquired both sample-S+ and sample-S--relations, all participants showed immediate emergence of conditional discriminations DA, CA, and DB, which allows one to make the inference that equivalence classes were immediately formed. In Condition B, with training designed to prevent sample-S+ in the BC conditional discrimination, Fred and Barb did not show emergent conditional discriminations, as predicted on the basis of the analysis of Carrigan and Sidman (1992). Surprisingly, however, Alice and Fabio showed indications of equivalence class formation. Alice performed above 90% correct responses in all these probes except one, in which she performed above 80% correct. Fabio performed above 90% in probe DA and above 80% in probes CA and DB.

After showing equivalence in Condition B and in the replication of this condition, Alice had a block of 18 probe trials to verify controlling relations in all trained conditional discriminations. There were three trials with each sample, each displaying two comparison stimuli and the mask. With sample Bl, for instance, one trial displayed Cl, C2, and the mask; another trial substituted a novel stimulus (N) for C2 and thus displayed Cl, N, and the mask; and the other trial substituted a novel stimulus for Cl and thus displayed C2, N, and the mask. There were four such blocks of probes (in four consecutive sessions) after the first implementation of Condition B and two such blocks (in one session) after the replication. Table 1 shows how many times each comparison was selected (number of selections by total number of probes of a particular type).

Table 1
Comparisons Selected by Alice in Probes for Controlling Relations,
in the BC Conditional Discrimination, After the Initial Exposure
to and the Replication of Condition B

          Trial type             Cl   C2    N   Mark

Initial exposure to Condition B

B1: Cl/C2/MASK                   3/4  0/4   --   1/4

B1: Cl/N/MASK                    1/4   --  1/4   2/4

B1: C2/N/MASK                     --  0/4  0/4   4/4

B2: C2/C1/MASK                   0/4  4/4   --   0/4

B2: C2/N/MASK                     --  4/4  0/4   0/4

B2: C1/N/MASK                    0/4   --  0/4    44

Replication of Condition B

B1: C1/C2/MASK                   2/2  0/2   --   0/2

B1: Cl/N/MASK                    2/2   --  0/2   0/2

B1: C2/N/MASK                     --  0/2  0/2   2/2

B2: C2/C1/MASK                   0/2  2/2   --   0/2

B2: C2/N/MASK                     --  2/2  0/2   0/2

B2: Cl/N/MASK                    0/2   --  0/2   2/2

Training trials never displayed Cl with B1 nor C2 with B2. After the first exposure to Condition B, Alice selected C2 in the four probe trials displaying sample B2 and comparisons Cl and C2 with the mask. She also selected C2 in the four probe trials displaying comparisons C2. N, and the mask. This indicates a sample-S+ relation. Results were not so clear when the sample was Bl. She selected Cl in three of the four trials displaying comparisons Cl, C2, and the mask. She selected the mask in one of these trials. She selected Cl in only one trial displaying comparisons Cl, N, and the mask. She selected N in one trial and the mask in two trials. It is not possible to conclude that there was a sample-5+ relation for sample Bl. On trials that displayed the S--with a novel stimulus and the mask, Alice always selected the mask. This confirms that sample-S--relations developed for both sample B1 and sample B2. Results were much clearer after the replication of Condition B. Alice always selected the comparisons compatible with both sample-S+ and sample-S--relations.


In Condition A, with training designed to establish both sample-S+ and sample-S--controlling relations in all baseline conditional discriminations, children always showed immediate emergence of the DA, CA, and DB conditional discriminations, documenting combined symmetry and transitivity of the trained conditional discriminations. This leads to the conclusion that children promptly formed equivalence classes when they learned the relation of each sample with both the corresponding S+ and S--.

Although the number of participants was relatively small, the results are striking because the "linear" training design used in this experiment is considered the least effective for producing formation of equivalence classes. For instance, Arntzen and colleagues (Arntzen et al., 2010; Arntzen & Ho1th, 1997) compared the efficacy of different training designs to promote formation of equivalence classes and showed the linear design to be the least effective. In the present study, by contrast, all participants showed immediate formation of equivalence classes with a linear design.

The literature shows considerable variability in the outcomes of equivalence tests. Among participants exposed to similar training, it is common to find that some show immediate formation of equivalence classes, whereas others show delayed emergence of equivalence and still others do not show class formation at all. In the present study, a training that guaranteed both sample-S+ and sample-S--controlling relations in the baseline produced highly consistent outcomes. All four children showed immediate formation of equivalence. According to the analysis of Carrigan and Sidman (1992), a necessary condition for equivalence is sample-S+ control in all conditional discriminations. Results of the present study are in principle compatible with this analysis, since participants fulfilled the necessary condition of sample-S+ control for every sample.

It is possible that the sample-S-- relations that children also learned were irrelevant to the equivalence outcomes, but this should be confirmed by additional research. The present data do not rule out, however, the importance of sample-S-- relations. It is not possible to rule out the hypothesis that sample-S-- relations, together with sample-S+ relations, increase the probability of equivalence class formation. To verify this, it would be necessary to compare Condition A of the present study with a training that established only sample-S+ relations. This training might be a reciprocal of Condition B of the present study, in which sample-S-- relations were prevented.

Kato et al. (2008) suggested that different response topographies for selections in conditional discrimination may affect the probability of controlling relations. They found that participants who responded with the mouse were more likely to show both sample-S+ and sample-S-- relations than were participants responding with the keyboard. Also, participants who showed both sample-S+ and sample-S-- controlling relations were more likely to show class formation.

Barb was the only participant who responded using the keyboard in the present study. She showed class formation in Condition A, responding with both the mouse and the keyboard, and did not show class formation in Condition B, with either the mouse or the keyboard. The present study is not, however, directly comparable to Kato et al. (2008) because Condition A of the present study ensured that both sample-S+ and sample-S-- controlling relations were acquired, regardless of response topography.

Condition B attempted to prevent sample-S+ control in one of the baseline conditional discriminations, BC. Analysis of Carrigan and Sidman (1992) predicted that stimulus equivalence would not emerge if performance in the BC relation were based only on sample-S-- relations. Results for Fred and Barb were consistent with this prediction. Alice and Fabio, on the other hand, showed indications of equivalence even with this training. However, probes indicated that Alice learned sample-S+ relations in the BC conditional discrimination. In the first exposure to Condition B, Alice learned at least the relation between sample B2 and comparison C2. In the replication of Condition B, she learned both the relations between sample B1 and comparison Cl, and between sample B2 and comparison C2. These results do not, therefore, conflict with the analysis by Carrigan and Sidman. A sample-S+ relation between B2 and C2 in the first exposure to Condition B was also sufficient to produce correct responding in all probe trials. Since sample-S+ relations were explicitly trained between A2 and B2 and between C2 and D2, and sample-S+ relations were also formed between B2 and C2, there were sample-S+ relations between all members of class 2. This would lead to selections of the comparison of class 2 (composed of A2, B2, C2, and D2) in all probe trials with a sample from class 2. The extensive history with conditional discriminations and equivalence probes in conditional discrimination format would be sufficient for the child to avoid choosing class 2 comparisons in the presence of samples of class 1, and the only alternative would be to choose the comparison from class 1. It is likely that Fabio also learned sample-S+ relations in Condition B, but this participant left the study after the first exposure to Condition B and was not available for further probing.

Why did Alice and Fabio learn sample-S+ relations even with a training procedure that never displayed the sample with the S+ in a conditional discrimination? The pretraining was designed to establish generalized conditional responding (Williams et al., 1995), in which participants would respond conditionally to novel stimuli presented in conditional discrimination format. Participants had a history with several "problems" of conditional discrimination, so that each problem presented new stimuli to be conditionally related. However, most of these problems were presented in the single comparison format, requiring children to learn both sample-S+ and sample-S-- relations. All these problems had two samples (e.g., B1 and B2) and two comparison stimuli (e.g., Cl and C2). Whenever a given comparison, such as C2, was incorrect in the presence of a given sample, such as Bl, it was always correct in the presence of the other sample, B2. Thus, a relation between the sample and its corresponding S+ may have been established when this stimulus was displayed, as S--, along with the other sample. Probes to verify conditional relations in Condition B may have also contributed to induce sample-S+ relations: To ensure that Condition B had actually generated sample-S--relations in the BC conditional discrimination, rather than sample-S+ relations between samples and the mask, probes were conducted with the sample and both comparisons. Four probes of the BC relation were conducted, mixed with probes of the other relations. Although responses in these probes were unreinforced, they provided opportunity for selections of the S+ in the presence of samples B1 and B2 and could contribute to establish sample-S+ relations, particularly with the extensive training with conditional responding that had been provided to participants.

This study did not attempt to balance the order of conditions and stimulus sets. It was expected that Condition A would produce equivalence class formation and that Condition B would not. However, if Condition B had been implemented first, negative results in Condition B followed by positive results in Condition A would be difficult to interpret. They might reflect merely increased familiarity with task demands. Positive results in Condition A followed by negative results in Condition B could not be explained on that basis. The successful replication of these results with two additional stimulus sets, with Barb, is additional evidence that training conditions, and not order or particular features of stimulus sets, are responsible for the results. However, replications of the present study that balance condition order and stimulus sets would strengthen the conclusions.

The pretraining may have also contributed to increase the efficacy of Condition A for equivalence class formation. In this study, participants mastered the conditional discrimination task before the experiment proper began. When Condition A started, participants were no longer learning what the conditional discrimination task required of them. They had to learn only the particular arbitrary relations with a given set of stimuli, after having learned other similar relations with different stimuli. It is possible that variability of outcome is increased in studies of equivalence when baseline relations are taught without such pretraining because, in this case, learning the specific baseline relations is confounded with learning the conditional discrimination task in general. Perhaps delayed emergence of stimulus equivalence may reflect, at least in part, increased familiarity with the task required in the baseline. This seems also to be an important topic for further investigation.

Stimulus equivalence, like so many conceptual tasks, requires participants to exhibit behavior different from the behavior directly trained. The variability of outcomes in these tasks is a concern for teachers and researchers. A proportion of students learn all exemplars taught in a program to teach concepts (i.e., the baseline) and fail tests with new exemplars (i.e., tests for emergent behavior). On the basis of the present results, one may raise the hypothesis that different controlling relations may underlie baseline performance. It is possible that apparently similar baseline performances mask different controlling relations. To the extent that the present results may be generalized to other forms of conceptual learning, variability may be reduced and positive outcomes may be increased when teachers and researchers identify the baseline controlling relations required by different forms of conceptual learning and subsequently develop procedures to guarantee those controlling relations. This may increase the effectiveness of programs to teach conceptual behavior, implemented either by teachers or by computers.


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Matheus Hidalgo is now at Universidade Federal de Sergipe, and Mariliz Vasconcellos is at Universidade Nove de Julho.

Data collection was supported by the State of Sao Paulo Foundation for Research Support (FAPESP, Grant 2003/09928-4). Preparation of the manuscript was supported by the Brazilian National Research Council (Grant 573972/2008-7) and FAPESP (Grant 08/57705-8), both for the National Institute of Research and Technology on Behavior, Cognition and Teaching (Chairperson: Deisy G. de Souza). The first author has a Research Productivity Grant from CNPq, and the second author had an undergraduate Science Initiation Grant from CNPq.

We are grateful to Deisy de Souza, Bill McIlvane, Bill Dube, Ana Arantes, and Priscila Grisante for their collaboration on this project.

Correspondence concerning this article should be addressed to Julio C. de Rose, Departamento de Psicologia, Universidade Federal de Sao Carlos, Caixa Postal 676, 13565-905 sao Carlos, SP, Brazil. E-mail:

Julio C. de Rose, Matheus Hidalgo, and Mariliz Vasconcellos

Universidade Federal de Sao Carlos
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Author:Rose, Julio C. de; Hidalgo, Matheus; Vasconcellos, Mariliz
Publication:The Psychological Record
Article Type:Report
Geographic Code:3BRAZ
Date:Jan 1, 2013
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