A traditional account of stimulus equivalence.

In matching-to-sample tasks, two or more stimuli, the samples, are presented successively. In the presence of each sample, two or more comparison stimuli are arranged. Responding to one comparison is reinforced; this depends on the sample. For example, a matching-to-sample task may present sample A1 or A2 with comparison stimuli B1 and B2. Choosing B1 may be reinforced in the presence of A1, and choosing B2 in the presence of A2.

Suppose that a particular sample is repeatedly presented together with comparisons. It may be found on those occasions that the choice of a particular comparison predominates. One can then speak of a conditional relation between the sample and the comparison (Sidman & Tailby, 1982). If choosing comparison B1 is reinforced in the presence of sample A1, and choosing comparison B2 in the presence of sample A2, then conditional relations between A1 and B1, and between A2 and B2, may develop. These conditional relations may be designated A1 B1 and A2B2.

Tests conducted with humans frequently reveal that matching-to-sample tasks bring about novel conditional relations, in addition to the conditional relations that were taught explicitly (see Hayes & Hayes, 1992, and Sidman, 1990, for reviews). Consider the following pair of matching-to-sample tasks as an example. In the first task, choosing comparison B1 is reinforced in the presence of sample A1, and choosing comparison B2 in the presence of sample A2. In the second task, choosing comparison C1 is reinforced in the presence of sample B1, and choosing comparison C2 in the presence of sample B2. Conditional relations A1B1, A2B2, B1C1, and B2C2 thus are taught explicitly. After these conditional relations have been established, a test may be arranged with samples C1 and C2 and comparisons A1 and A2; reinforcement is not presented. Novel conditional relations C1A1 and C2A2 may be found on this test.

Sidman and Tailby (1982) have proposed a specification of the conditional relations produced by matching-to-sample tasks. They suggest that the conditional relation can be an equivalence relation: a relation that is reflexive, symmetric, and transitive. When the conditional relation is an equivalence relation, stimuli form classes, such that conditional relations are shown between stimuli in the same class. There are two classes in the example given above: [A1, B1, C1] and [A2, B2, C2]. Stimuli in the same class are called equivalent, and the actual existence of classes of equivalent stimuli is called stimulus equivalence.

Stimulus equivalence has been found in both children and adults. It has not been found in young mentally retarded children who lack expressive language skills (Devany, Hayes, & Nelson, 1986). Also, it has not been found in animals (D'Amato, Salmon, Loukas, & Tomie, 1985; Dugdale & Lowe, 1990; Hogan & Zentall, 1977; Kendall, 1983; Lipkens, Kop, & Matthijs, 1988; Rodewald, 1974; Sidman, Rauzin, Lazar, Cunningham, Tailby, & Carrigan, 1982).

Equivalent stimuli often function in similar ways even when they occur outside the matching-to-sample context. This has been found both for discriminative stimulus functions (e.g., Barnes & Keenan, 1993; de Rose, Mcllvane, Dube, Galpin, & Stoddard, 1988) and for functions of consequences (reinforcing or punishing effects; Hayes, Devany, Kohlenberg, Brownstein, & Shelby, 1987; Hayes, Kohlenberg, & Hayes, 1991). The transfer of discriminative functions within equivalence classes has been found in both children and adults. The transfer of consequential functions has until now been investigated in adults only.

The purpose of the present paper is to show that three-term contingencies of reinforcement can explain the phenomena of stimulus equivalence (the novel conditional relations and the transfer of functions). The basic idea has been put forward by Rodewald (1974). Rodewald had studied matching to sample in pigeons. After he had established conditional relations that may be designated A1 B1 and A2B2 he did not find novel conditional relations B1A1 and B2A2. Rodewald then proposed a training history that could lead to novel conditional relations. This history can be described as follows. First, A1B1 and A2B2 are taught. Next, B1A1 and B2A2 are also taught. In other words, a pair of tasks is arranged in which symmetric conditional relations are taught. Later, A3B3 and A4B4 are taught, followed by B3A3 and B4A4; A5B5 and A6B6 are taught, followed by B5A5 and B6A6, and so forth. This history may lead to a generalized performance such that a novel BA relation is found after the training of an AB relation, whatever A and B.

The phenomena of stimulus equivalence in humans can be explained with the assumption that training histories like the one proposed by Rodewald take place before the experiments in which the phenomena are observed. This is the key assumption of an account proposed by Hayes and Hayes (1989; see also Dugdale & Lowe, 1990; Hayes, 1991; Hayes & Hayes, 1992; Sidman et al., 1982). Suppose the behavior of a person is studied in an experiment. After conditional relations A1B1 and A2B2 have been established, novel conditional relations B1A1 and B2A2 are found. According to Hayes and Hayes (1989), this result may be caused by the training of symmetric conditional relations in preexperimental pairs of tasks. Other stimulus equivalence phenomena can be explained in the same way.

The following is a three-term contingency formulation of the idea that the phenomena of stimulus equivalence depend on preexperimental training histories. It will first be shown how three-term contingencies can account for the novel conditional relations. Next, the transfer of functions through the equivalence class will be considered. Tactics for testing the account will then be presented, and the account will be compared with the account proposed by Hayes and Hayes (1989).

Novel Conditional Relations

According to the present formulation, the novel conditional relations found in studies of stimulus equivalence are instances of generalized performances. The generalized performances are generalized in the sense that they are shown for a variety of sample and comparison stimuli. It is assumed that the generalized performances are produced by reinforcement histories in the usual (extra-experimental) environments of humans. The generalized performances will be defined first. Reinforcement histories that explain the generalized performances will then be presented.

Generalized Performances

The generalized performances will be divided into two categories. One category consists of three generalized performances, which will be called generalized identity matching, generalized symmetric responding, and generalized transitive responding. The second category consists of an unlimited number of other generalized performances, which will be called generalized equivalences. Expressions of the form x [right arrow] y will be used in the definitions of all generalized performances, except generalized identity matching. The expression x [right arrow] y stands for a choice of comparison y in the presence of sample x.

In generalized identity matching, the presentation of sample x with comparison x is usually followed by the choice of comparison x. This is the case for a variety of stimuli x. In generalized symmetric responding, x [right arrow] y is usually followed by y [right arrow] x. This is the case for a variety of stimuli x and y. In generalized transitive responding, x [right arrow] y and y [right arrow] z are usually followed by x [right arrow] z. This is the case for a variety of stimuli x, y, and z.

The second category of generalized performances, the generalized equivalences, will be introduced with the help of examples. The examples are shown in Figure 1. Arrows point from samples to comparisons. Solid arrows indicate choices that occur initially; broken arrows indicate choices that occur finally. In one generalized equivalence (left), x [right arrow] y and y [right arrow] z are usually followed by z [right arrow] x. This is the case for a variety of stimuli x, y, and z. In another generalized equivalence (middle), x [right arrow] y and x [right arrow] z are usually followed by y [right arrow] z. Again, this is the case for a variety of stimuli x, y, and z. In a third generalized equivalence (right), w [right arrow] x, x [right arrow] y, and y [right arrow] z are usually followed by w [right arrow] z. This is the case for a variety of stimuli w, x, y, and z. In each of these examples, the following two conditions are met: (1) The final choice is predicted, if the conditional relation is an equivalence relation, on the basis of the initial choices; (2) the final choice does not follow immediately, by a single application of symmetry or transitivity, from the initial choices. These are the defining properties of generalized equivalences. The first property is shared with generalized symmetric responding and generalized transitive responding. The second property distinguishes the generalized equivalences from generalized symmetric responding and generalized transitive responding.

The initial choices in generalized symmetric responding, generalized transitive responding and the generalized equivalences may be reinforced, but that is not necessary. They might also be test trial responses. For example, generalized symmetric responding is present when x [right arrow] y choices tend to be followed by y [right arrow] x choices, whether the x [right arrow] y choices are reinforced or not.

It might be thought that the generalized equivalences can be reduced to generalized symmetric responding and generalized transitive responding, so that it would not be strictly necessary to introduce the generalized equivalences. This cannot be done, as the following example illustrates. Suppose that conditional relations A1B1 and B1C1 are taught explicitly. After these conditional relations have been established, a test of conditional relation C1A1 is arranged; no other conditional relations are tested. Suppose conditional relation C1A1 is indeed found. This conditional relation can be interpreted as an instance of a generalized equivalence. This is the case because A1 [right arrow] B1 and B1 [right arrow] C1 are followed by C1 [right arrow] A1. There are no other interpretations available. Conditional relation C1A1 could have been interpreted as an instance of generalized symmetric responding, had conditional relation A1C1 been found before conditional relation C1A1 (A1 [right arrow] C1 would have been followed by C1 [right arrow] A1). Because conditional relation A1C1 has not been tested, no choices of the type A1 [right arrow] C1 are present in the performances. Therefore, an interpretation that refers to generalized symmetric responding is not available. Similarly, conditional relation C1A1 could have been interpreted as an instance of generalized transitive responding, had conditional relations B1A1 and C1B1 been found before conditional relation C1A1 (B1 [right arrow] A1 and C1 [right arrow] B1 would have been followed by C1 [right arrow] A1). Because conditional relations B1A1 and C1B1 have not been tested, no choices of the types B1 [right arrow] A1 and C1 [right arrow] B1 are present in the performances. Therefore, an interpretation that refers to generalized transitive responding is also not available. Thus, the novel performance of this example cannot ultimately be explained as generalized symmetric responding or generalized transitive responding. (Novel performances of this kind have often been reported; see Devany et al., 1986; Hayes, Tilley, & Hayes, 1988; Sidman & Tailby, 1982; Sidman, Willson-Morris, & Kirk, 1986; Spradlin, Cotter, & Baxley, 1973; Wetherby, Karlan, & Spradlin, 1983.)

Histories That Explain the Generalized Performances

Table 1 presents schematic descriptions of reinforcement histories that explain generalized identity matching, generalized symmetric responding, generalized transitive responding, and one generalized equivalence. Each row shows a contingency: an antecedent stimulus, a response, and a consequence. The antecedent stimuli are sequences of events. The last event of a sequence is the presentation of comparisons in the presence of a sample. An expression of the form x(y) stands for TABULAR DATA OMITTED the presentation of comparison y, together with other comparisons, in the presence of sample x. For example, B1(A1) stands for the presentation of comparison A1, together with other comparisons, in the presence of sample B1. The responses are choices of comparisons; the chosen comparison is included in the specification of the response. That is, the responses are specified as "Choose A1," "Choose A2," and so forth. The responses are thus specified in terms of their antecedent stimuli. The consequences are presentations of reinforcers. (The units of the analysis are discriminated operants: classes of responses defined in terms of shared environmental effects as well as shared antecedent stimuli; see Catania, 1968, 1973. Each of the contingencies shown in Table 1 defines a discriminated operant. The generalized performances are also discriminated operants. In the generalized performances, responding is occasioned by a property of antecedent stimuli.)

The first history of Table 1 explains generalized identity matching. In the first contingency of this history, sample A1 is presented and a choice between comparison A1 and other comparisons is offered. Choosing comparison A1 is reinforced. Next, in the presence of sample A2, choosing comparison A2 is reinforced, and so forth.

The second history explains generalized symmetric responding. Each antecedent stimulus consists of different parts, which occur at different times. The antecedent stimulus of the first contingency begins with a choice of comparison B1 in the presence of sample A1. Later, sample B1 is presented together with comparisons; one comparison is A1. Choosing A1 is reinforced. The antecedent stimulus of the second contingency begins with a choice of comparison B2 in the presence of sample A2. It ends with the presentation of sample B2 and comparison A2. Choosing A2 is reinforced, and so forth.

The third history explains generalized transitive responding. Here, each antecedent stimulus begins with two choices. The antecedent stimulus of the first contingency begins with a choice of comparison B1 in the presence of sample A1, and a choice of comparison C1 in the presence of sample B1. Later, sample A1 is presented with C1 and other comparisons; choosing C1 is reinforced. The antecedent stimulus of the second contingency begins with a choice of comparison B2 in the presence of sample A2 and a choice of comparison C2 in the presence of sample B2. Later, sample A2 is presented with comparison C2 and choosing comparison C2 is reinforced, and so forth.

The last history explains one generalized equivalence. The generalized equivalence is shown in the left-hand panel of Figure 1. The contingencies are the same as in the history that explains generalized transitive responding, except for the final choices: Choosing comparison A1 is reinforced in the presence of sample C1, choosing A2 in the presence of C2, and so forth. Similar histories explain other generalized equivalences.

A Traditional Account of the Novel Conditional Relations

It is assumed here that histories like those shown in Table 1 take place in the usual environments of humans. This is assumed for generalized identity matching, generalized symmetric responding, generalized transitive responding, and a limited number of generalized equivalences. Further, it is assumed, for all histories, that a limited number of contingencies is sufficient for generalization to a variety of novel samples and comparisons, including those arranged in stimulus equivalence experiments. These assumptions lead to a traditional account of the novel conditional relations: The conditional relations are shown because similar performances have been reinforced in the past.

The generalized performances are similar to abstract tacts, such as the verbal response "red" under the control of redness (Skinner, 1953, 1957). In both cases, responding is under discriminative control of a stimulus property. The properties that control responding in the generalized performances are patterns shared by antecedent stimuli. For example, in generalized symmetric responding, the property is the presence of a sample y with a comparison x after comparison y has been chosen in the presence of sample x. This pattern may be designated x [right arrow] y, y(x).

The shaping of a generalized performance may be facilitated by nonreinforcement of responses that do not belong to the performance. Nonreinforcement occurs when the comparisons chosen differ from those given in the response specifications of Table 1. For example, choosing a comparison other than A1 goes unreinforced after the antecedent stimulus A1 [right arrow] B1; B1 (A1). The absence of choices of other comparisons may be facilitated by nonreinforcement of those choices.

A generalized performance may be stronger in some contexts than in others. For example, it might be found that a child's generalized symmetric responding is strong when mother and child are looking through picture books together, and weaker in other situations. When the strength of a generalized performance varies between contexts, a three-term contingency interpretation can be that responding is under discriminative control of the context together with the pattern shared by the antecedent stimuli. For example, the picture book context and the pattern x [right arrow] y; y(x) may jointly control choosing comparison x. This interpretation is plausible if generalized symmetric responding has been reinforced in the picture book context, and not, or to a lesser extent, in other contexts. Although contextual control is possible, some degree of generalization between contexts must be assumed to explain why performances reinforced in usual environments are also shown in experiments.

The environments in which the effective contingencies occur remain to be specified. Only a few possibilities will be suggested here. One possibility is that the contingencies occur in situations where one object is placed near another object. Here, the object that is moved corresponds to a comparison in a matching-to-sample procedure; the other object corresponds to the sample. When the comparison-object is placed near the sample-object, the subject touches one object (the comparison) in the neighborhood of another object (the sample), just as in a matching-to-sample procedure. Reinforcement of this behavior often takes place with different pairs of objects within a set of more or less similar objects. Consider a two-year-old boy playing with toy cars. Occasionally, he puts one car near another in a garage, which is followed by automatic or social reinforcement. In cases like these, many different placements of objects are taught with objects in the same set. Suppose all possible relations between different pairs of objects are taught with objects A1, B1, and C1. Thus, placing B1 near A1 is reinforced; placing A1 near B1 also. This contributes to generalized symmetric responding. Placing B1 near A1, placing C1 near B1, and C1 near A1 are reinforced. This contributes to generalized transitive responding. Placing B1 near A1, C1 near B1, and A1 near C1 are reinforced. This contributes to one generalized equivalence, and so forth. Further, generalized identity matching will be strengthened when the objects placed together resemble one another in certain ways (e.g., they are all toy cars). These contingencies occur in play, in tidying up, and in other activities.

Another possibility is that the effective contingencies occur in situations where names of objects are spoken. This possibility will be discussed below.

Other Contingencies That Explain the Novel Conditional Relations

The present account assumes that the novel conditional relations depend on stimulus generalization in more than one way. Two of these have already been mentioned. First, responding is assumed to generalize from the sample and comparison stimuli in the reinforcement histories of Table 1 to the unfamiliar sample and comparison stimuli arranged in studies of stimulus equivalence. Second, responding is assumed to generalize from preexperimental to experimental contexts.

A third kind of generalization applies to the choices in the antecedent stimuli of generalized symmetric responding, generalized transitive responding, and generalized equivalences. Consider the history shown for generalized symmetric responding in Table 1. The antecedent stimuli begin with choices A1 [right arrow] B1, A2 [right arrow] B2, and so forth. It is assumed here that generalized symmetric responding can also be produced if the contiguous occurrences of A1 and B1, of A2 and B2, and so forth, take the place of the choices. In other words, antecedent stimuli that contain choices can occasion symmetric responding after a reinforcement history in which antecedent stimuli contained contiguous occurrences of stimuli. This seems a reasonable assumption because the two kinds of antecedent stimulus have a structure in common. Similar assumptions are made for generalized transitive responding and for generalized equivalences.

The contiguous occurrence of stimuli may involve spoken names. A spoken name occurs in the following contingency, which might contribute to generalized symmetric responding. Stimulus A1 is presented, and its name (B1) is spoken. Later, name B1 is spoken again and choosing A1, which is now a comparison, is reinforced. Here, name B1 may be spoken by the subject whose responding is finally reinforced, or by another person. This possibility exists for both the first and the second occurrence of the name in the antecedent stimulus. Therefore, four contingencies may be distinguished. A contingency that teaches auditory comprehension is obtained if another person speaks both names. For example, a mother shows her two-year-old child a picture of a house and says: "It's a house." Later, she asks: "Where's the house?" and after the child has pointed to the house she praises the child. A contingency that teaches transfer from expressive to receptive use of names is obtained if the first name is spoken by the subject and the second by another person. For example, a mother shows her child a picture, which the child names. Later, the mother names the picture and she reinforces after the child has chosen appropriately. The other two contingencies, in which the subject speaks the second name, may also contribute to the development of generalized symmetric responding.

Contingencies that contain spoken names may also contribute to generalized transitive responding and generalized equivalences. In the following contingency, which might contribute to generalized transitive responding, B1 is the name of both A1 and C1. Stimulus A1 is presented, and name B1 is spoken. Later, name B1 is spoken again, and C1 is chosen. Finally, choosing C1 is reinforced in the presence of A1. For example, a mother shows her child a picture of a house (A1) and says: "a house" (B1). On another occasion, she asks: "Where's the house?" (B1), and the child points to another house (C1; A1 and C1 have different shapes). Finally, the mother points to the first house (A1), asks "Where's another one?" and praises her child after he or she has pointed to the second house (C1). Similar contingencies contribute to generalized equivalences.

Transfer of Functions

The account of the transfer of functions in equivalence classes is similar to the account of the novel conditional relations. It is assumed that the transfer of functions depends on preexperimental contingencies of reinforcement. In these contingencies, the transfer of functions is reinforced.

Table 2 shows two reinforcement histories that explain transfer of discriminative functions. The first history explains the transfer of a discriminative function from a sample to a comparison chosen in the presence of the sample. The antecedent stimulus of the first contingency begins with a choice of comparison B1 in the presence of sample A1, and a response r1 to stimulus A1. It ends with a presentation of B1. Response r1 is reinforced. The antecedent stimulus of the second contingency begins with a choice of comparison B2 in the presence of sample A2, and a response r2 to stimulus A2. It ends with a presentation of B2. Response r2 is reinforced, and so forth. As a result of these contingencies, a generalized performance is strengthened in which an antecedent stimulus that has the form x [right arrow] y, x [right arrow] r; y occasions response r.

The second history explains a more complex case. In this case, there is a transfer between stimuli that do not occur together in a single choice. The antecedent stimulus of the first contingency begins with a TABULAR DATA OMITTED choice of comparison B1 in the presence of sample A1, a choice of comparison C1 in the presence of sample B1, and a response r1 to stimulus A1. It ends with a presentation of C1. Response r1 is reinforced. The antecedent stimulus of the second contingency begins with a choice of comparison B2 in the presence of sample A2, a choice of comparison C2 in the presence of sample B2, and a response r2 to stimulus A2. It ends with a presentation of C2. Response r2 is reinforced, and so forth. As a result, a generalized performance is strengthened in which an antecedent stimulus that has the form x [right arrow] y, y [right arrow] z, x [right arrow] c, z occasions response r. These generalized performances are only two possibilities. An unlimited number of generalized performances is possible.

There is an overlap between the contingencies that produce transfer of discriminative functions and the contingencies that produce novel conditional relations. The overlap exists because the response occasioned by a stimulus can be the choice of a comparison. As an example of this, consider generalized transitive responding. In this performance, an antecedent stimulus that has the form x [right arrow] y, y [right arrow] z; x(z) occasions the choice of z. This can be viewed as a kind of transfer of function: the choice of comparison z transfers from stimulus y to stimulus x. The contingencies that produce generalized transitive responding thus teach the transfer of a discriminative function from a comparison (y) to the sample that went with the comparison (x). The overlap between the two kinds of contingencies is not complete: Some of the contingencies that lead to novel conditional relations do not lead to transfer of discriminative functions. Examples are the contingencies of generalized identity matching and generalized symmetric responding. In these cases, there is no discriminative function that transfers from one stimulus to another.

The kind of discriminative function that is strengthened in a contingency is probably important. As an example of this, consider two discriminative functions: (a) the choice of a comparison in the presence of a sample, and (b) a vocalization in response to the presentation of an object. Contingencies in which the reinforced response is the choice of a comparison will contribute primarily to generalized performances in which the response is also the choice of a comparison. Contingencies in which the reinforced response is a vocalization in response to an object will contribute primarily to generalized performances in which the response is also a vocalization. The contingencies will also contribute to the other kind of generalized performance, but this contribution will be smaller.

The contingencies of reinforcement thus far assumed account for the transfer of discriminative functions. The transfer of consequential functions remains to be discussed. Table 2 shows a reinforcement history that explains the transfer of a reinforcing function from a sample to a comparison. The antecedent stimulus of the first contingency begins with three events: (a) the choice of comparison B1 in the presence of sample A1, (b) the contiguous occurrence of A1 and the spoken word "Good," and (c) a sequence in which stimulus C1 is followed by response r1, which is followed by B1. The antecedent stimulus ends with C1. This antecedent stimulus may be described as: A1 [right arrow] B1, A1 "Good," C1 [right arrow] r1 [right arrow] B1; C1. It is followed by r1, which is reinforced. The antecedent stimulus of the second contingency may be described as: A2 [right arrow] B2, A2 "Good," C2 [right arrow] r2 [right arrow] B2; C2. Response r2 is reinforced, and so forth. In these contingencies, the transfer is from stimulus A to stimulus B. Stimulus A (A1, A2, etc.) occurs together with the word "Good," and may therefore be a conditioned reinforcer. Stimulus B occurs together with stimulus A in the choice A [right arrow] B, and it follows response r in the antecedent stimulus. Stimulus C is an occasion upon which response r may occur. Ultimately, stimulus C will control a response (r) which has had a certain consequence (B) in the past. Similar reinforcement histories explain more complex kinds of transfer, such as the transfer of a reinforcing function from stimulus y to stimulus z, after choices x [right arrow] y and x [right arrow] z have been taught (see Hayes et al., 1987; 1991).

Tactics for Testing the Account

The following three tactics for testing the present account are based on suggestions made by Baer (1973), who considered ways to test behavior-analytic accounts of developmental phenomena. The first tactic is to arrange the contingencies of Tables 1 and 2, in an attempt to strengthen the generalized performances. The subjects might be young normally capable children, or older mentally retarded children (see Devany et al., 1986). Evidence that generalized identity matching can be strengthened in normally capable children was obtained by Sherman, Saunders, and Brigham (1970). In this study, a novel instance of identity matching was established by reinforcing other instances. The tactic remains to be applied to the other generalized performances.

The second tactic is to arrange operant procedures for weakening the generalized performances. The procedures are: letting the performances go unreinforced, punishing them, or reinforcing incompatible performances. (The incompatible performances are oddity from sample, the opposite of generalized symmetric responding, the opposite of generalized transitive responding, and so forth. For example, the opposite of generalized symmetric responding means that after an antecedent stimulus with the pattern x [right arrow] y; y(x) a comparison other than x is chosen.) Here, the reasoning would be that if the performances can be produced with operant procedures it should also be possible to eliminate them with operant procedures. Evidence that generalized identity matching can be weakened was obtained by Sherman et al. (1970). Like the first tactic, the second tactic remains to be applied to the other generalized performances. Of course, evidence that operant procedures can be used to weaken the performances does not prove that the performances were originally established with operant procedures. Therefore, the first tactic is better than the second one.

The third tactic is a two-step approach. It begins with a search, in the usual environments of humans, for contingencies that might cause the phenomena of stimulus equivalence. Once such contingencies have been found, their functionality in causing the phenomena is assessed. This can be done by arranging the contingencies for subjects who do not yet show the phenomena. A systematic search remains to be done, but it is plausible that appropriate contingencies occur when objects are placed together or names of objects are spoken.

Comparison with the Account Proposed by Hayes and Hayes

The present account of stimulus equivalence may be compared with an account proposed by Hayes and Hayes (1989; see also Hayes, 1991; Hayes & Hayes, 1992). There are similarities and differences between the two accounts. Both accounts assume that the phenomena of stimulus equivalence depend on preexperimental training histories. Hayes and Hayes (1989, p. 174) suggest that "a history of arbitrary matching-to-sample that reinforced symmetry, reflexivity, and transitivity" might lead to stimulus equivalence. If symmetry, reflexivity, and transitivity here refer to generalized symmetric responding, generalized identity matching, and generalized transitive responding, then it can be concluded that the present account makes a similar suggestion. In both Hayes (1991, p. 25) and the present account, it is assumed that the transfer of functions within equivalence classes is learned. Further, Hayes and Hayes (1989, p. 167) suggest that generalized symmetric responding might result from a training history with names, as follows. First, a child is taught to speak Name 1 when Object 1 is shown, and to choose Object 1 when Name 1 is spoken. Thereafter, the child is taught to speak Name 2 when Object 2 is shown, and to choose Object 2 when Name 2 is spoken, and so forth. This suggestion is also present in the present account of stimulus equivalence.

One difference between the two accounts concerns the nature of the responding brought about by the training histories. Hayes and Hayes assume that stimulus equivalence is a kind of higher order responding, called a "relational frame of coordination" (see Hayes & Hayes, 1989, p. 172; Hayes, 1991, p. 29). This kind of responding includes generalized identity matching, generalized symmetric responding, generalized transitive responding, and an unlimited number of generalized equivalences. It is suggested that the training of reflexivity, symmetry, and transitivity allows the child to "derive" the frame of coordination (Hayes & Hayes, 1989, p. 174). According to the present account, no such higher order kind of responding is brought about.

The following example illustrates this difference between the two accounts. Suppose a child has been exposed to histories of reinforcement in which generalized identity matching, generalized symmetric responding, and generalized transitive responding have been established. No instances of generalized equivalences have been reinforced. The child then receives a training in which conditional relations between unfamiliar stimuli are taught. The conditional relations are A1B1 and B1C1. After these conditional relations have been established, a test of conditional relation C1A1 is arranged. The account proposed by Hayes and Hayes (1989) presumably leads to the prediction that C1A1 will be found, because generalized identity matching, generalized symmetric responding, and generalized transitive responding had been established, which allows the child to derive the frame of coordination. The present account predicts that C1A1 will not be found, because no instances of generalized equivalences had been reinforced.

Another difference concerns the role of context. Hayes (1991, p. 37) states that "the combination of the contextual stimuli controlling relational responding and the stimuli to be related produce the results seen." In other words, responding is controlled by the contextual stimuli together with the particular sample and comparison stimuli. At some points, Hayes and Hayes suggest that contextual control is a logical necessity. For example, Hayes and Hayes (1989, p. 168) state that "a relational response must, by definition, be brought to bear on the situation by stimuli other than the relatae themselves. Thus, it must be contextually controlled."

According to the present account, these "stimuli other than the relatae themselves" are patterns shared by antecedent stimuli, such as the pattern x [right arrow] y; y(x) in generalized symmetric responding. It will be necessary to assume contextual control over the generalized performances only if the same pattern is followed by different responding in different contexts. For example, in one context, the pattern x [right arrow] y; y(x) might be followed by choosing x, and in another by choosing a comparison other than x. Here, a three-term contingency interpretation could be that responding is under discriminative control of the context together with the pattern.

Concluding Remarks

According to the present formulation, the performances of stimulus equivalence are instances of generalized performances that have been shaped by their consequences in the usual environments of humans. In all generalized performances, except generalized identity matching, behavior is under discriminative control of earlier responding by the same subject.

The assumption that behavior can be brought under discriminative control of earlier responding by the same subject has also been made in accounts of other behavioral phenomena. The phenomena include characteristics of responding under schedules of reinforcement (Zeiler, 1977), discrimination learning sets (Mackintosh, 1983), and various phenomena of verbal behavior (Branch, 1977; Hineline & Wanchisen, 1989). An example is the phenomenon of do-say correspondence: a correspondence between a person's earlier behavior ("doing") and his or her later reporting about it ("saying"). For example, do-say correspondence exists when somebody accurately reports about his or her own earlier nonverbal behavior. According to a three-term contingency interpretation, accurately answering a question like "What have you been doing?" is verbal behavior under discriminative control of the earlier behavior (Hineline & Wanchisen, 1989).

Evidence for a three-term contingency interpretation of do-say correspondence has been provided by Risley and Hart (1968), who allowed four- to five-year-old children to play with various materials. Approximately one and a half hours after a play session, each child was asked: "What did you do that was good today?" Risley and Hart found that they could bring the answer to the question under the control of the earlier behavior by reinforcing only accurate reports. This finding suggests that the kind of stimulus control assumed by the present account of stimulus equivalence is feasible.

Until now, animals have not shown stimulus equivalence. The reason for this may be that the animals studied so far had not been exposed to the contingencies assumed by the present account. On the basis of other findings, however, it should be expected that these contingencies will not produce stimulus equivalence in animals. There is evidence that temporally extended stimuli, in which previous choices occur, can acquire discriminative control over animal behavior only with difficulty, if at all. This evidence has been obtained in studies of double-alternation in mazes (Woodworth & Schlosberg, 1954, pp. 627, 628) and in studies of discrimination learning sets (Mackintosh, 1983, p. 273). The important difference between animals and humans, as far as stimulus equivalence is concerned, may be the kind of discriminative control that can be established, rather than the possession of language or some other behavioral repertoire (see Dugdale & Lowe, 1990, and Hayes & Hayes, 1989, for discussion).

In the present account of stimulus equivalence, matching to sample is viewed as responding to stimulus compounds. This can be seen in Table 1, in the contingencies that lead to generalized identity matching. The stimulus compound is a sample together with comparisons available for choice; the response is choosing a comparison. Sidman (1986) has pointed out that stimulus compound formulations may not deal adequately with the phenomena of stimulus equivalence. It will here be shown that this criticism does not apply to the present account. Sidman's reasoning is the following:

In tests for reflexivity, symmetry, and transitivity, and in the global test for equivalence, each stimulus serves its new function effectively in the derived units without ever before having been paired with the other stimulus component of the new unit. This observation documents the earlier assertion that it can be incorrect to treat sample and comparison stimuli as a unitary compound. (p. 236)

There are two important differences between the present account of stimulus equivalence and the view criticized by Sidman. First, the stimulus compounds of the present account are not unitary compounds, in the sense of totally new stimuli. According to the present account, compounds A1(A1) and A2(A2) are not simply two different stimuli. They share a property, which may acquire control over responding. Second, the present account does not consist only of the idea that sample and comparison stimuli may be treated as compounds. It combines this idea with the assumption that the performances of stimulus equivalence are produced by reinforcement histories in the usual environments of humans.

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Appendix 1

Let S be a set of objects. We can speak of relation R on S if we can say for each pair of objects x and y in S whether x stands in the relation R to y, or does not stand in the relation R to y. The expression xRy can be used to indicate that x stands in the relation R to y. A relation R on a set S is an equivalence relation when R is reflexive (that is, xRx for all x in S), symmetric (if xRy then yRx, for all x and y in S), and transitive (if xRy and yRz then xRz, for all x, y, and z in S). These definitions can be found in texts on algebra (e.g., Birkhoff & MacLane, 1965).

The concept of the equivalence relation can be applied to conditional relations in the following way. Define xRy to mean that conditional relation xy is shown. In other words, xRy means that the choice of comparison y predominates when x is the sample and y one of the comparisons. Further, let S be the set of all stimuli that can be arranged both as samples and as comparisons in matching-to-sample tasks. It might then be proposed that the relation R on S is an equivalence relation. This would mean: (1) conditional relation xx is shown, for all x in S; (2) if conditional relation xy is shown then conditional relation yx is shown, for all x and y in S; and (3) if conditional relations xy and yz are shown then conditional relation xz is shown, for all x, y, and z in S. These three statements give the usual meanings of the terms reflexivity, symmetry, and transitivity in the literature on stimulus equivalence.

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