Matching Derived Functionally-Same Stimulus Relations ... · During the first test, referred to as the equivalence test, B stimuli served as samples and C stimuli as comparisons,

The Psychological Record, 2004, 54, 255-273

MATCHING DERIVED FUNCTIONALLY-SAME STIMULUS RELATIONS: EQUIVALENCE-EQUIVALENCE

AND CLASSICAL ANALOGIES

FRANCK CARPENTIER and PAUL M. SMEETS Leiden University

DERMOT BARNES-HOLMES National University of Ireland at Maynooth

IAN STEWART National University of Ireland at Galway

Previous studies have shown that, after being trained on A-B and A-C matching tasks, subjects match not only functionally-same Band C stimuli (stimulus equivalence), but also BC compounds with sameclass elements and BC compounds with different-class elements (equivalence-equivalence). Similar performances are required in classical analogies (a : b :: c : d). Therefore, some researchers have argued that equivalence-equivalence can serve as a behavior analytic model for analogical reasoning. Recent studies, however, have shown that compounds with same-class elements and different-class elements have different discriminative (S+, S-) properties. Hence, matching of same discriminative functions may have occurred. The present study aimed to design an equivalence-equivalence test in which the designated correct performances cannot be attributed to a process other than matching functionally-same relations. In Experiment 1 , subjects were trained to relate X and Y stimuli to colors and X and l stimuli to forms. After equivalence was assessed (Y-Xl) , the subjects received an equivalence-equivalence test in which only compounds with same-class elements were used: an XY or Xl compound as sample and an XY and Xl compound as comparisons (e.g., X1Y1-X2Y2IX2Z2). All subjects passed the equivalenceequivalence test. However, as reported by 1 subject, and was later demonstrated in Experiment 2, these equivalence-equivalence tasks could be solved by matching functionally-same stimuli (e.g., Y1-colorY2, hence Y1-Y2). Experiment 3 demonstrated that this problem also exists in classical analogy tasks. When given the analogy tasks used by Goswami and Brown (1990), all subjects selected the correct dterm option on the basis of the b-term alone (equivalence). In Experiment 4, the equivalence-equivalence test was further modified to permit differentiation of matching functionally-same relations from matching functionally-same stimuli. All 5 subjects readily matched functionally-same equivalence relations, thus evidenced equivalenceequivalence or analogical reasoning.

Correspondence should be sent to Paul M. Smeets, Department of Psychology, Leiden University, P. O. Box 9555, Leiden, The Netherlands. (E-mail : [email protected]) .

256 CARPENTIER ET AL.

Recent studies reported that, after being trained on multiple arbitrary conditional discrimination tasks, humans not only match functionallysame stimuli , but also functionally-same stimulus relations (Barnes, Hegarty, & Smeets, 1997; Stewart, Barnes-Holmes, Roche, & Smeets, 2001, 2002). For example, in Experiment 1 of the study by Barnes et al. (1997) , 5 adults and a 12-year-old boy were trained on conditional discrimination tasks with A stimuli as samples and B stimuli and C stimuli as comparisons (A 1-B 1, A2-B2, A3-B3, A4-B4; A 1-C 1, A2-C2, A3-C3, A4-C4). Then they received two tests. During the first test, referred to as the equivalence test, B stimuli served as samples and C stimuli as comparisons, and vice versa. This test assessed whether the subjects matched B with C stimuli that had been related to a same A stimulus (e.g., B1 -C1, B2-C2, C1-B1, C2-B2) . During the second test, referred to as the equivalence-equivalence test, BC compounds were used as samples and as comparisons (BC-BC). This test measured whether the subjects matched compounds with same functional relations between elements: equivalence (e.g., B1 C1-B3C3) or nonequivalence (e.g., B1 C2-B3C4). All subjects demonstrated equivalence-equivalence (and nonequivalencenonequivalence). Similar findings were reported when, in Experiment 2 of that study (5 adults and a 9-year-old boy), equivalence-equivalence was tested before equivalence.

Equivalence-equivalence may be an important phenomenon not only because it extends equivalence (Barnes, 1994; Saunders & Green, 1992; Sidman, 1994, 2000; Sidman & Tailby, 1982) but because of its potential to serve as a behavior analytic model for analyzing analogical reasoning , a faculty that prominent philosophers (Aristotle, see Pelegrino, 1985), psychologists (Inhelder & Piaget, 1985; James, 1890), and physicists (Oppenheimer, 1956) have long been regarded as one of the foundation pillars of human intellectual life and creativity. Oppenheimer (1956, p. 130), for example, emphasized the importance of analogy in science as follows, "Science is an immensely creative and enriching experience; and is full of novelty and exploration; and it is in order to get to these that analogy is an indispensable instrument. Even analysis, even the ability to plan experiments, even the ability to sort things out and pick them apart presupposes a good deal of structure, and that structure is characteristically an analogical one."

The equivalence-equivalence tests used in the aforementioned studies closely correspond, structurally and functionally, with classical analogies (a : b :: c : d). Consider the following type of classical analogy task, also mentioned in previous reports (e.g., Barnes et ai., 1997; Stewart et ai., 2001), apple : orange :: dog : sheep/book. The argument was that if apple and orange are equivalent in the context of fruit, and dog and sheep equivalent in the context of animals, the subjects should select sheep rather than book. If the b-term had been toy, the subject would probably select book. Thus, equivalence-equivalence and classical analogies both require subjects to match functionally-same relations. Also the developmental trends obtained with both types of tasks are

EQUIVALENCE-EQUIVALENCE 257

remarkably similar. Recent studies by Carpentier, Smeets, and BarnesHolmes (2002b, 2003b) have shown that equivalence-equivalence, like analogical reasoning, is a late rather than early childhood phenomenon (Goldman, Pellegrino, Parseghian, & Sallis, 1982; Levinson & Carpenter, 1974; Piaget, Montangero, & Billeter, 1977; Sternberg & Nigro, 1980; Sternberg & Rifkin, 1979; but see Alexander et aI., 1989; Goswami & Brown, 1989, 1990).

In spite of these commonalties, the validity of the existing equivalenceequivalence tests as a model for classical analogy tests can be challenged. The problem relates to the fact that the equivalence-equivalence tests required the subjects to choose between a compound with two-same-class elements and one with two different-class elements (e.g. , B1 C1-B3C3/B2C3). Although this feature is also present in the aforementioned analogy task in which the subjects have to choose between sheep-dog (both items of the class animals) and sheep-book (different-class items), in most well designed classical analogy tasks, all d-term options are in some way or another related to, hence "go with," the c-term (e.g., automobile: gas :: sailboat: travellwindlsailslrudder [Gallagher & Wright, 1977]; spider: web :: bee : hivelhoneylantif/y [Goswami & Brown, 1990], and shoes: feet :: hat: headlbucketiclothes/cap [Sternberg & Nigro, 1980]). Unlike the equivalenceequivalence tests, these analogy tasks require the subjects to identify which preestablished c-d relation (e.g. , bee-hive, bee-honey, bee-ant, bee-fly) is functionally the same as that between the a- and b-terms (spider-web). Furthermore, recent findings strongly suggest that the equivalenceequivalence (BC-BC) performances were not based on matching same functional stimulus relations, but on a much simpler process, that is, matching same discriminative (S+, S-) functions. Carpentier et al. (2003b) reported that, after being trained on A-B and A-C matching tasks, 5-year-old children related BC compounds with same-class elements (e.g ., B1C1) to a picture of a Happy Face (for most children an S+), and BC compounds with different-class elements (B1 C2) to the picture of a Sad Face (S-). Similar findings were obtained in a follow-up study with adults (Carpentier, Smeets, & Barnes-Holmes, 2004). After being trained on conditional A-B and A-C discrimination tasks and on a simple O-discrimination task (01+/02-), most subjects (a) selected BC compounds with same-class elements in a BC discrimination test (e.g., B1C1+/B1C2-), and (b) matched BC compounds with other BC compounds and with unitary 0 stimuli of same discriminative functions (e.g., B1C1-B3C3, B1C2-B2C3; 01-B3C3, 02-B2C3). Because (a) the BC-Face (Carpentier et aI., 2003b) and O-BC performances (Carpentier et aI., 2004) could have resulted only from matching same discriminative functions, and (b) previous studies have shown that adults and children readily relate stimuli to any other stimuli of same discriminative functions (Carpentier, Smeets, & Barnes-Holmes, 2002a, 2002c, 2003a; Perez-Gonzalez, 1994; Perez-Gonzalez & Serna, 2003), it should be assumed that the BC-BC performances were based on the same process.

The present study was designed to adapt the equivalenceequivalence tests more closely to the classical analogy tasks by using


only compounds with same-class elements. In Experiment 1, the subjects were trained to relate X and Y stimuli to a same color (Xi ~Redf-Y1, X2~Greenf-Y2 , X3~Bluef-Y3) and to relate the X and Z stimuli to a same form (Xi ~ Trianglef-Z1, X2~Circlef-Z2, X3~Squaref-Z3). After assessing equivalence (Y-X-Z) , equivalence-equivalence was tested. These tests involved three compounds, a sample with two color-related elements (e.g. , Xi Y1) or two form-related elements of one class (e.g., Xi Z1) and two comparisons with elements of another class, one with color-related elements (X3Y3) and one with form-related elements (X3Z3). Note that these tasks closely parallel analogy tasks in which all d-terms are associated or "go with" the c-term, hence do not permit matching same discriminative functions. Would the subjects relate colorbased equivalence relations to other color-based equivalence relations (e.g., match Xi Y1 with X3Y3, not X3Z3) and relate form-based equivalence relations to other form-based equivalence relations (match X1Z1 with X3Z3, not X3Y3)? The following three experiments (a) assessed whether these equivalence-equivalence performances and corresponding classical analogies could be based on equivalence responding (Experiments 2 and 3), or (b) were designed to further improve the equivalence-equivalence test (Experiment 4) .

Experiment 1

This experiment evaluated an equivalence-equivalence test with two compounds as comparisons, one in which the equivalence relation between elements was based on the same dimension as that between the sample elements (e.g., color), and one in which the equivalence relation between elements was based on a different dimension (form).

Method Subjects. PartiCipants were 4 students from Leiden University, 3

female and 1 male. Their ages ranged from 20 to 26 years. The subjects were recruited

through notice board advertisements, had no prior experience with stimulus equivalence research , and were paid for their participation. The sessions were arranged such that the subjects did not meet with one another in the vicinity of the laboratory.

Apparatus and materials. All subjects were trained and tested individually in a quiet room containing a microcomputer which displayed stimuli (4 x 4 cm) on a white background (see Figure 1). The stimuli consisted of three dollar signs of different colors (red, green, or blue), three black geometric forms (triangle, circle, square) , and nine black symbols. For ease of exposition, the symbols will be referred to by alphanumeric codes (e.g., Xi , Y2, Z3) . The subjects never saw these codes.

Sessions, tasks, and feedback. Sessions lasted from 126 to 163 min, sometimes interrupted by a 5- to 10-min break. The need for and the timing of the break were determined by the subjects. All subjects completed the experiment in 1 session.


Red Green Blue

$ $ $

1 2 3

x

y\}J

z 11 * Figure 1: Experimental stimuli.

Match-to-sample tasks were used for training and testing. On each trial, three or four stimuli appeared on the screen, a sample stimulus at the center of the screen and two or three comparison stimuli near the bottom of the screen (left and right, or left, middle, and right). The locations of the comparisons were counterbalanced across trials.

Table 1

Test and Training Phases in Experiments 1, 2, and 4 .- - -- --- - T rai n -- -- -- - --- - - -- - -- -- - - -Ir ------------ -- - train -- - -- - - - ---- - -- - -- - - ¥ --- t -raln -- - ---------------- ---- r -Phases Test Tasks Trials Phases Test Tasks Trials Phases Test Tasks

1 Train Color-X 18 9A Test Equivalence 1 27 10B Test Equivalence-Equiv 2 24 Rd-X1/X2/X3 V-X, X-Z, Z-Y XV-XV, XZ-XZ Gn-X2/X3/X1 Y1-X1/X2/X3 X1 Y1-X2Y2/X2Y3/X2Z2/X2Z3 BI-X3/X1/X2 Y2-X2/X3/X1 X1 Z1-X2Z2IX2Y2IX2Y3/X2Z3

Y3-X3/X1/X2 2 Train Color-Y 18 X2Y2-X1 Y1/X1Z3/X1Z1/X1 Y3

Rd-Y1!Y2!y3 X1-Z1/Z2IZ3 X2Z2-X1Z1/X1 Y1/X1Z3/X1 Y3 Gn-Y2!y3!Y1 X2-Z2/Z3/Z1 BI-Y3!Y1!Y2 X3-Z3/Z1/Z2 X1 Y1-X3Y3/X3Y2/X3Z3/X3Z2

X1 Z1-X3Z3/X3Y3/X3Y2/X3Z2 3 Train Color-X& Y mixed 36 Z1-Y1!y2/Y3

Z2-Y2!y3!Y1 X3Y3-X1 Y1/X1 Y2IX1Z1/X1Z2 4 Train Form-X 18 Z3-Y3!Y1!Y2 X3Z3-X1 Z1/X1 Y1 /X1 Y2/X1 Z2

Tr-X1/X2IX3 Ci-X2IX3/X1 9B Test Equivalence 2 27 X2Y2-X3Y3/X3Y1/X3Z3/X3Z1 Sq-X3/X1/X2 X-V, Z-X, Y-Z X2Z2-X3Z3/X3Y3/X3Y1/X3Z1

5 Train Form-Z 18 10A Test Equivalence-Equiv 1 24 X3Y3-X2Y2IX2Y 1 /X2Z2IX2Z1 Tr-Z1/Z2/Z3 XV-XV, XZ-XZ X3Z3-X2Z2IX2Y2IX2Y1/X2Z1 Ci-Z2/Z3/Z1 X1 Y1-X2Y2/X2Z2 Sq-Z3/Z1/Z2 X1Z1-X2Z2IX2Y2 11 Test Equivalence 2: 12

Y-Y,Z-Z 6 Train Forms-X&Z mixed 36 X2Y2-X3Y3/X3Z3 Y1-Y2/Z2

X2Z2-X3Z3/X3Y3 Z1-Z2!Y2 7 Train Color-X& Y and 36

Form-X&Z mixed X3Y3-X1Y11X1Z1 Y2-Y3/Z3 X3Z3-X1Z1/X1Y1 Z2-Z3!Y3

8 Test Color-X& Y and 36 Form-X&Z mixed

Note: RD = Red, GN = Green, BI = Blue, Tr = Triangle, Ci = Circle, Sq = Square. Sample and correct comparison are shown immediately left and right of the horizontal bar (-) , Respectively.

I\.l (j) a

0 » ::D "'0 m Z :::! m ::D m -J » !


All training trials were followed by visual feedback (1 sec), "Right" or "Wrong," and a 1-sec intertrial interval. Test trials were followed by a 2-sec interval (no feedback).

Training and testing program. The program consisted of 10 phases (see Phases 1 to 10A in Table 1). Phases 1 to 8 were directed at establishing three 5-member equivalence classes: Y1-Red-X1-TriangleZ1, Y2-Green-X2-Circle-Z2, and Y3-Blue-X3-Square-Z3. After demonstrating equivalence in Phase 9 (X-Y-Z), the subjects were given the opportunity to demonstrate equivalence-equivalence in Phase 10A. This test assessed whether the subjects related color-based equivalence relations to other color-based equivalence relations (e.g., X1 Y1-X2Y2/X2Z2), and form-based equivalence relations to other form-based equivalence relations (X1 Z1-X2Z2/X2Y2).

Phases 1-8: Baseline training and testing (Color-X, Color-Y, Form-X, Form-Z). The Color-X relations were trained first (Phase 1). Blocks of 18 trials were used, with a red, green, or blue dollar sign as sample, and X1 , X2, and X3 as comparisons. Immediately before each block, the following instruction appeared on the screen, "You are about to see four objects on the screen, one at the center, and three below. Look at the object at the center before clicking one of the objects below. Start the program by pressing anyone key." Subjects received positive feedback when selecting X1 when given red, X2 when given green, and X3 when given blue (6 trials each). Training continued until 17 or more trials of a block were correct, at which point they proceeded to the next phase. After receiving the same training with the Y stimuli as comparisons (Red-Y1, Green-Y2, Blue-Y3) in Phase 2, the subjects received mixed training (Color-X& Y) in Phase 3. The procedures were the same as in both previous phases except that blocks of 36 trials were used (6 trials on each task). Subjects who responded correctly on at least 35 trials proceeded to the next step. Phases 4, 5, and 6 were the same as Phases 1, 2, and 3, respectively, except that Form-X relations were trained in Phase 4 (Triangle-X1, Circle-X2, Square-X3), Form-Z relations in Phase 5 (Triangle-Z1, Circle-Z2, Square-Z3), and Form-X&Z relations (mixed training) in Phase 6. After receiving mixed training on all baseline tasks in Phase 7, the subjects proceeded to Phase 8 (baseline test). This test assessed if the trained performances remained intact without feedback. The procedures were the same as in Phase 7 but without feedback. Each block started with the instruction, "Now, you will no longer see whether your selections are right or wrong. Do your best. Start the program by pressing anyone key." Criterion was set on at least 34/36 responses (94%) correct. Subjects who passed the test proceeded to Phase 9. Those who failed the test returned to Phase 7 (mixed training), after which they received Phase 8 again.

Phase 9: Equivalence Test 1 (Y-X-Z). This test assessed whether the subjects matched same-class X, Y, and Z stimuli. Again, all trials involved three comparisons. Two versions were used, 9A (Subjects 1 and 3) and 9B (Subjects 2 and 4). V-X, X-Z, and Z-Y were tested in Phase 9A, and


X-V, Z-X, and Y-Z in Phase 98 (see Table 1). The test consisted of 27 trials, 3 on each task. Equivalence was assumed when subjects responded correctly on at least 25/27 trials. The test was presented three times, each time followed by a return to Phase 7 (mixed baseline training) . At that point, the subjects proceeded to Phase 10A.

Phase 10A: Equivalence-Equivalence Test 1 (XY-XY, XZ-XZ). This test assessed whether the subjects matched equivalence relations that were based on color (e.g., X1Y1-X2Y2/X2Z2) or form (X1Z1-X2Z2/X2Y2). Six matching tasks were used (see Table 1). The test consisted of 24 trials, 4 trials per task. Equivalence-equivalence was assumed when a subject responded correctly on at least 22/24 trials. The test was presented three times, each time followed by a return to Phase 7 (mixed baseline training).

Results and Discussion Table 2 shows the required number of baseline trials (Phases 1-8), and

the percentages of correct responses during each presentation of the equivalence and equivalence-equivalence test. The retraining data of Phase 7 (after each equivalence and equivalence-equivalence test) are not presented because these performances were always (near) perfect. All 4 subjects learned the baseline tasks (270-414 trials) and passed the equivalence and equivalence-equivalence tests, most of them immediately.

Although these data suggested that the subjects matched functionally-same stimulus-stimulus relations, a post hoc conversation with the last subject (4) revealed that this conclusion was not justified. When given some extra equivalence-equivalence trials, she indicated that she had ignored the X elements and simply matched elements that had been related to a same dimension. For example, when given trial X1 Y1-X2Y2/X2Z2 , she selected X2Y2 because, Y1 and Y2 "go with" color. Likewise, when given trial X1 Z1-X2Z2/X2Y2, she selected X2Z2 because Z1 and Z2 were related to form. Thus, instead of matching functionally-same relations (equivalence-equivalence), this subject had been matching functionally-same stimuli (equivalence). The same, of course, might have occurred with the other 3 subjects. This issue was addressed in Experiment 2.

Table 2

Required Number of Baseline Trials and Percentages of Test Trials Correct in Experiment 1

Train Subjects

Phases Tasks Test 2 3 4

1-8 Baseline Train+Test 270 288 321 414

9 Equivalence 1 Test 1 100 100 93 100 Test 2 96 100 100 96 Test 3 100 96 100 100

10A Equiv-Equiv 1 Test 1 96 92 88 96 Test 2 100 96 96 92 Test 3 100 96 100 96


Experiment 2

This experiment examined whether, in Experiment 1, most subjects could have solved the equivalence-equivalence tasks on the basis of equivalence (Y-Y, Z-Z).

Method Participants were 4 new subjects, 2 male and 2 female students

between 19 and 23 years. The recruitment and experimental procedures were the same as in Experiment 1, except that the equivalence-equivalence test (Phase 10A) was replaced by a second equivalence test (Equivalence Test 2, Phase 11) in which the subjects were given the opportunity to match color-related and form-related stimuli. Each trial involved a Y or Z stimulus of one class as sample, and a Y and Z stimulus of another class as comparisons (e.g., Y1-Y2/Z2, Z1-Z2IY2) (see Table 1). The test consisted of 12 trials, 3 trials per task. The test was presented three times, each time followed by a return to Phase 7 (mixed baseline training).

Results The results are shown in Table 3. All 4 subjects learned the baseline

tasks (Phases 1 - 8) in 270 to 414 trials, and demonstrated criterion performance in both equivalence tests, most of them immediately. These findings documented the formation of overlapping stimulus classes and made it plausible that the equivalence-equivalence performances in Experiment 1 were based on equivalence rather than on equivalence-equivalence.

Discussion Experiment 2 showed that, even though the equivalence-equivalence

tasks were structured after analogy tasks in which all d-terms were related to the c-term, these tasks fell short of measuring the matching of functionally-same relations. Instead, they simply measured previously acquired stimulus relations (equivalence) or "associations" (e.g., Goswami & Brown, 1990). Did this problem result from the way we had constructed our equivalence-equivalence tasks, or did we touch upon a problem that also exists in the analogy tasks we used as a model?

Table 3


Train Subjects

Phases Tasks Test 1 2 3 4

1-8 Baseline Train+Test 414 288 270 270




Researchers in cognitive psychology recognized this problem of restricted stimulus control a long time ago. Rosenblum (1931), Zirkle (1941), and Willner (1964) demonstrated that many (in Willner's study, up to 85%) of the classical analogy tasks used in IQ and other aptitude tests can be solved on the basis of the c-term alone. For example, subjects not only respond to foot when given the full analogy task, hat: head :: shoe: foot/arm/table/lamp, but also when given the task, shoe : footlarm/table/lamp. Although this problem would be expected in analogy tasks in which only one d-term option (foot) is related to the c-term (shoe), Experiment 2 showed that similar problems may occur when all d-term options are functionally related to the c-term because that still permits the subjects to base their selections on another term, in this case the b-term (i.e., when given X1 : Y1 :: X3 : Y3/Z3, the subjects match Y1-Y3). This inadvertent control by the b-term might also exist in classical analogy tasks used in prominent studies on cognitive development. Consider the following analogy task used by Goswami and Brown (1990), spider: web :: bee : hive/honey/antlfly. If this task had been presented as web : hive/honey/antlfly or as X : web :: Y : hive/honey/antlfly, most people would be expected to select hive. The same applies to the geometric analogy tasks (e.g., big red circle: small blue circle :: big red square: small blue square/small red square/small red triangle/big yellow circle) used by Alexander et al. (1989). Analysis of the examples of the analogy tasks given in that study revealed that, without exception, the correct dterm options can be identified on the basis of the color of the b-term alone. If this had occurred, these analogy tasks simply measured color matching (e.g., blue-blue). Thus, it should not come as a surprise that young children can solve these "analogy" tasks. This issue was addressed in Experiment 3.

Experiment 3

This experiment assessed whether classical analogy tasks used in studies on cognitive development could have been solved on the basis of the b-term alone. For this purpose, we selected the analogy test used by Goswami and Brown (1990) with children because it is one of the few studies in which the analogy tasks and procedures are sufficiently specified. The test consisted of 10 analogy tasks (see Table 4). Each task consisted of seven picture cards and a blank card showing a question mark. The cards were divided into two rows, a top row of three pictures representing the a-, b-, and c-terms (e.g., spider, web, and bee) and the blank card, and a lower row of four pictures representing the d-term options (hive, honey, ant, fly). The children were asked to complete the top row by selecting a card from the bottom row. Experiment 3 examined if these analogy tasks could be solved by replacing the items representing the a- and c-terms by the letters X and Y (X : web :: Y : hive/honey/antlfly). This procedure was selected rather than presenting the b-term alone (web: hive/honey/antlfly) because it left the format of the 4-term analogy intact.


Table 4

Analogy Tasks Used by Goswami and Brown (1990)

Terms Tasks a : b :: c d1 d2 d3 d4

1 spider : web :: bee: hive honey ant fly 2 pencil : pencil eraser :: chalk: chalk board chalk board paints crayons

eraser 3 dog : dog basket :: baby : crib bottle kitten doll 4 tram : tracks :: boat : water sailor car canoe 5 dress : hanger :: coat: hook hat sweater coat 6 bird: nest :: dog: kennel bone cat other dog 7 gloves: hands :: shoes : feet socks boots shoes 8 cow: milk :: hen : egg feather duck rooster 9 king : crown :: policeman: police hat police car fireman policewoman 10 bird : plane :: fish: boat net crab fish

Method Participants were 12 adults, 4 male and 8 female, all Leiden

University students and staff. Their ages ranged from 21 to 59 years. The subjects completed the test in a quiet room or in their own office, and they were asked not to consult anybody or inform others about the tasks. All subjects completed the test in 15 min.

Table 5 shows a translated version of the test form. The test items are the same as those used by Goswami and Brown (1990) except that (a) the pictures are replaced by words (same wording as in Table 1 of the study by Goswami and Brown, 1990), and (b) in the 6th task, the d-term option "other dog" was replaced by "leash."

Table 5

Test Used in Experiment 3

Below, you will find 10 comparison tasks, each with two undefined terms, X and Y. Each task involves 4 alternatives (printed in bold). Encircle the correct alternative for each task. Before you start, please fill out your sex and age. Thank you for your cooperation .

Sex: Age:

1. X is to web as Y is to hive, honey, fly, or ant.

2. X is to pencil eraser as Y is to paints, blackboard eraser, blackboard, or crayons.

3. X is to dog basket as Y is to kitten , bottle, doll , or crib.

4. X is to tracks as Y is to sailor, water, car, or canoe.

5. X is to hanger, as Y is to sweater, hat, hook, or coat.

6. X is to nest, as Y is to kennel, bone, cat, or leash.

7. X is to hands, as Y is to shoes, boots, socks, or feet.

8. X is to milk, as Y is to duck, egg, rooster, or feather.

9. X is to crown, as Y is to police hat, policewoman, police-car, or fireman.

10. X is to plane as Y is to crab, net, fish , or boat.


Results and Discussion The overall rate of correct responses was 97%. Ten subjects

responded correctly on all 10 items, 1 on 9 items (she encircled two options, including the correct one), and 1 on 7 items (all three incorrect selections were based on the first d-term). These findings clearly demonstrated that, in contrast to what had been assumed (Goswami & Brown, 1990), these analogy tasks, and possibly also those in other studies on analogical reasoning, may have been solved on the basis of equivalence (e.g., web-living quarter, hive-living quarter; hence: webhive) or "associations."

Of course, this conclusion could be qualified as an "easy shot" because several of the analogies could be considered as "give aways," in that the b- and the d-terms made the a- and c-terms redundant. For example, in the task spider: web :: bee: hive/honey/antlfly, the b-term web is uniquely related to spider and the d-term hive uniquely related to bee. Hence, web and bee were redundant. Moreover, one might argue that, because these analogies were designed for young children, most adults should find them easy to solve even though some terms were missing. This argument, however, negates the implicit notion that in 4-term analogies, no matter how easy they are, finding the 4th term should be impossible unless all other three terms are specified.

Even so, one might argue that there are no reasons to assume why, when all three terms (a, b, c) are given, most subjects would base their responses exclusively on the b-term. This argument may not hold either. Given that all d-term options are related to the c-term (e.g., bee-hive, beehoney, bee-ant, bee-fly), subjects could easily shift their attention to the next term (web) to find the appropriate cue. More informally, if the c-term provides no cue, then perhaps the b-term does. In fact, if this systematic scanning for cues correlates with age, it should not come as a surprise that older children perform better on analogy tasks than young children, not because they have more analogical competence, but because they are more competent in identifying cues for making fitting "associations."

Experiment 4

This experiment evaluated a modified version of the equivalenceequivalence test that permitted equivalence-equivalence based responses to be differentiated from the color- and form-based equivalence responses described in Experiment 2. This was achieved by adding two more compounds as comparisons, one XY and one XZ, but with different-class elements (e.g., X2Y3 and X2Z3). For example, when given sample X1 Y1, the subjects were to choose between X2Y2, X2Z2, X2Y3, and X2Z3. 1 If the subjects' responses were based on matching

1Although three comparisons would have been sufficient (e .g., X1 Y1 -X2Y2/X2Z21X2Y3) , the inclusion of the fourth comparison (e .g. , X2Z3) served to keep the format of the equivalence-equivalence tasks the same as that in most 4-term analogy tasks.


functionally-same stimuli, they would be expected to select X2Y2 or X2Y3 because each of these two compounds includes, like the sample, a colorrelated Y element (X1Y1-X2Y2 because Y1-Y2; X1Y1-X2Y3 because Y1-Y3). If their responses were based on matching functionally-same relations, they should respond only to X2Y2 because this is the only compound in which, like in sample X1Y1, both elements are related to a same color (X1 and Y1: both red; X2 and Y2: both green).

Method Participants were 5 students (20 - 24 yrs) from Leiden University, 2

male and 3 female. The recruitment procedures were the same as before. The experimental procedures were the same as in Experiment 1 except that, after the completion of Phase 9 (Equivalence Test 1: Y-X-Z), the subjects proceeded to Phase 10B in which equivalence-equivalence was measured with Equivalence-Equivalence Test 2 (see Table 1). The test consisted of 12 tasks, 6 measuring XV-XV relations and 6 measuring XZXZ relations. Each task was presented twice (i.e., a total of 24 trials). The test was presented three times. Each presentation started with the following instruction on the screen: "Now you will see five objects, one at the center, and four objects to choose from, two at the top and two at the bottom. Click one of these four objects. You will not get any right/wrong messages. Start the program by pressing anyone key. Do your best." Each presentation was followed by a return to Phase 7 (mixed baseline training). Finally, the subjects received Equivalence Test 2 (Phase 11), measuring Y-Y and Z-Z (same as in Experiment 2) . Table 6 shows an overview of the testing and training phases in Experiments 1, 2, and 4.

Results and Discussion The results are shown in Table 7. All 5 subjects learned the baseline

tasks (Phases 1-8) in 288 to 342 trials and passed both equivalence tests and the equivalence-equivalence test, 3 (1, 3, 5) immediately, and 2 (2, 4) after repeated testing (see also Barnes et aI., 1997). This is the first demonstration of equivalence-equivalence that unequivocally demonstrates matching functionally-same stimulus relations.

Table 6

Training and Testing Phases in Experiments 1, 2, and 4

Experiments

Phases 2 4

Baseline 1-8 x x x Equivalence 1 9 x x x Equiv-Equiv 1 10A x Equiv-Equiv 2 10B x Equivalence 2 11 x x


Table 7


Train Subjects

Phases Tasks Test 2 3 4 5

1-8 Baseline Train+Test 288 288 342 306 288

9 Equivalence 1 Test 1 96 100 85 89 100 Test 2 100 100 96 100 100 Test 3 100 100 100 100 100

10B Equiv-Equiv 2 Test 1 92 21 10 0 92 Test 2 100 100 100 79 100 Test 3 100 100 100 100 100

11 Equivalence 2 Test 1 100 100 100 92 100 Test 2 100 100 100 100 100 Test 3 92 100 100 100 100

General Discussion

The present study was directed at designing an equivalenceequivalence test in which the designated correct performances cannot be attributed to a process other than matching functionally-same relations. In Experiment 1, the subjects were trained to relate X and Y stimuli to colors and X and Z stimuli to forms. Then, equivalence was assessed (Y-X-Z). Finally, the subjects received an equivalence-equivalence test involving XY and XZ compounds with same-class elements. These configurations were designed to prevent the subjects from matching same discriminative functions. Each equivalence-equivalence trial involved three compounds, a sample with two elements relating to a specific color (e.g. , X1 Y1) or form (X1Z1), and two comparisons, one with two elements relating to another color (X2Y2), and one with two elements relating to another form (X2Z2). All subjects passed the equivalence-equivalence test. However, as reported by 1 subject and was later demonstrated in Experiment 2, these performances were not based on matching dimension-related equivalence relations (e.g., X1 Y1-color, X2Y2-color, hence X1 Y1-X2Y2), but on dimension-related stimuli (Y1-color, Y2-color, hence Y1-Y2). In terms of the 4-term analogy task, the subjects in Experiment 1 did not match b-d with a-c, but simply b with d. Experiment 3 demonstrated that this problem is also immanent in 4-term analogy tasks used in core studies on cognitive development. With very few exceptions, all subjects selected the correct d-term options of the analogy tasks used in a study by Goswami and Brown (1990) when presented without the required aand c-terms. Thus, unlike what had been assumed, these analogy tasks can be "associatively" solved on the basis of the b-term alone (b-d equivalence). In Experiment 4, therefore , a modified test was used that permitted differentiation of equivalence-equivalence from simple equivalence responding . This was achieved by adding two more compounds with different-class elements as comparisons , one permitting


equivalence responding and one that did not (e.g., X1 Y1-X2Y2/X2Z2/X2Y3/X2Z3). All 5 subjects readily matched same-dimension equivalence relations (e.g., X1 Y1-X2Y2, X1 Z1-X2Z2).

The present findings, notably those obtained in Experiment 4, strongly suggest that equivalence-equivalence can serve as a behavior analytic model for analyzing analogical reasoning (Barnes et ai., 1997). In fact, the structure of the equivalence-equivalence tasks used in that experiment may serve as a model to construct 4-term analogy tasks that cannot be solved "associatively." Consider, for example Goswami and Brown's analogy task (1990), spider: web :: bee: hive/honey/ant!f/y. Had this task been constructed in accordance with the equivalenceequivalence tasks in Experiment 4, that is, spider : web :: bee : hive/honey/(bird's) nest! (bird) seed (X1 Y1-X2Y2/X2Z2/X2Y3/X2Z3), the outcome of this task probably could not be solved without the a- and cterms (X: web:: Y: hive/honey/nest!seed). From this, it would follow that the conceptually phrased definitions of classical analogies should be supplemented by stringent criteria for measuring adequate relational control. One such criterion would be that 4-term analogy tasks require that the selection of the d-term should only be possible on the basis of aI/ three other terms. Unfortunately, most studies on analogical reasoning do not carry out such analyses. As formulated by Barnes and Whitely (1981, p. 412), "Current models of analogical reasoning either assume that analogies are well-structured or ignore the issue."

Although the procedures are quite laborious, equivalence-equivalence may serve as an alternative to classical analogies when subjects, especially young children, may be at risk of failing these tasks due to lack of knowledge or understanding. For example, when given the analogy task bicycle : handlebars :: ship: ?, Piaget et al. (1977) found that children frequently responded with bird (often seen around ships) or vacuum cleaner (looks like a ship), rather than with rudder. These responses frequently have been taken as evidence that young children are not capable of matching same functional stimulus relations. Although this conclusion may be correct, these youngsters may not know that rudders steer ships like handlebars steer bicycles. If so, they should not be able to equate the a-b and cod relations (Goswami & Brown, 1989). The equivalence-equivalence procedure prevents this problem by training novel prerequisite (baseline) relations and assessing equivalence first.

Present findings are also consistent with Sidman's views on the events that make up equivalence relations. According to Sidman (1994, p. 384), "An equivalence relation is made up of pairs of events, with no restriction on the nature of the events that make up the pairs." These pairs may consist of stimulus-stimulus (e.g., Saunders, Drake, & Spradlin, 1999; Sidman & Tailby, 1982), stimulus-response (e.g., Smeets, BarnesHolmes, & Roche, 2001; Vaughan, 1988), and stimulus-feedback (Dube, Mcllvane, Maguire, Mackay, & Stoddard, 1989; Schenk, 1994). Apparently, this also applies to pairs of functionally-same stimulus relations. In effect, similar findings would be expected when testing


opposite-opposite relations (e.g., black: white :: fire: water/wood). Thus, equivalence-equivalence can be seen as simply another demonstration of equivalence. In stimulus equivalence tests, subjects match functionally-same stimuli, in equivalence-equivalence tests, subjects match functionally-same stimulus relations.

Finally, the question remains of how the matching of functionally-same relations (analogical reasoning, equivalence-equivalence) and stimulus class formation (stimulus equivalence) are related to one another. Carpentier et al. (2002a, 2003b) reported that 5-year-old children readily pass equivalence tests but, in contrast to older children and adults, seldom pass equivalence-equivalence tests. These findings are consistent with many studies in which analogical performance increases with age and indicate that stimulus equivalence precedes equivalence-equivalence (Stewart & Barnes-Holmes, 2001). From this perspective, equivalence and relating equivalence-relations are learned phenomena or generalized operant response classes, that are established across multiple-exemplars (Barnes-Holmes & Barnes-Holmes, 2000; Barnes-Holmes, BarnesHolmes, Roche, & Smeets, 2001; Cullinan, Barnes-Holmes, & Smeets, 2001; Schusterman & Kastak, 1993). The exemplar training that is needed to establish the operant of equivalence responding, such as name training, likely precedes the early educational training that is involved in establishing equivalence-equivalence responding. Thus, the finding that young children readily demonstrate equivalence but not equivalenceequivalence sits well with the exemplar-based, generalized operant interpretation of these performances.

This behavioral interpretation of analogical reasoning, in terms of relating derived relations, contrasts with the ideas expressed in the cognitive literature on analogical reasoning. In particular, some cognitive theorists have argued that basic category formation itself may be based on reasoning by analogy, or "analogical transfer" (Robertson, 2001). Consider, for example, the findings of Schusterman and Kastak (1993). They reported that after a California sea lion had been trained on multiple sets of baseline relations (A-B, B-C), and had been tested and trained on symmetry (A-C, C-B) and equivalence tasks (A-C, C-A), it eventually evidenced symmetry and equivalence. Although it remains to be seen whether the animal would also be capable of demonstrating equivalenceequivalence, from a cognitive perspective, the testing conditions are not fundamentally different. In equivalence-equivalence tests, subjects are required to match two sets of simultaneously presented derived stimulus relations. During the probes used in the multiple exemplar training procedure, the subjects are required, according to the "analogical transfer" interpretation, to match an untrained stimulus relation with a previously trained relation that is no longer present. Thus, stimulus class formation is the product of analogical reasoning (e.g., Gentner & Holyoak, 1997; Goswami, 1992). In the words of Gentner and Holyoak (1997, p. 32): "how do categories first get formed? One basic mechanism is analogy-the process of understanding a novel situation in terms of one


that is already familiar. In the course of reasoning by analogy, the novel target comes to be seen as another example of 'the same kind of thing' as the familiar analog. And the analogy between the two specific situations may provide the 'seed' for learning a more general category or schema that encompasses both."

Although much more work needs to be done, the current research and the previous studies indicate that equivalence-equivalence responding (or analogical reasoning) is the product of more basic relational learning, and not the basis for that learning as cognitive researchers have suggested.

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Matching Derived Functionally-Same Stimulus Relations ... · During the first test, referred to as the equivalence test, B stimuli served as samples and C stimuli as comparisons,