## Introduction to Stimulus Equivalence

Stimulus equivalence, as studied by behavior analysts, is a relatively new research area that has been rapidly expanding. Many behavior analysts are excited about the topic, and its possible far reaching implications have not gone unnoticed (Sidman, 1994). The research itself focuses on relations among stimuli. It is based on a mathematical definition, whereby "a relation R is an equivalence relation for set S if the properties of reflexivity, symmetry, and transitivity are satisfied for all members of S" (Hall & Chase, 1991, p. 108).

Reflexive properties involve the relation between a stimulus and itself (e.g., A = A). Symmetrical properties involve the relation between a pair of ordered stimuli, such that their order is reversible (e.g., if A = B, then B = A). Transitive properties involve two ordered pairs of stimuli, such that the second element of the first pair is the first element of the second pair, which in turn determines a third ordered pair (e.g., if A = B and B = C, then A = C). When stimuli have these relations to each other, they are said to form an equivalence class.

How do behavior analysts study stimulus equivalence, and why do they believe it to be so important?

In the standard stimulus equivalence paradigm, relations among stimuli are determined by the choices of a learner. Specifically, the learner is shown a sample stimulus together with some other stimuli that are to be compared to the sample. The learner's task is to select the comparison stimulus that matches the sample stimulus in accordance with a particular relation between the two. Not surprisingly, this paradigm is called match-to-sample.

Imagine we set up the following task for a child named Laura. We have three sets of stimuli: the English words cat, dog, horse (Set A); the French words chat, chien, cheval (Set B); and the Spanish words gato, perro, and caballo (Set C). Each word is written on a separate card. We show Laura the cat card from Set A, and beneath it, in random order, are the three French cards from Set B. We instruct Laura to choose the card from the bottom row (comparison stimulus) that goes with the card in the top row (sample stimulus). If she selects the chat card, she is praised; other choices are not praised. Over time, given the cat card in the top row, Laura reliably chooses the chat card from the bottom row. We can summarize her behavior by the rule "If you see cat, then select chat," or more generally, If Stimulus A, then select Stimulus B. Her performance implies that she has learned a relation between the two stimuli, typically abbreviated as AB, with A representing the sample stimulus and B representing the comparison stimulus whose selection is reinforced. The AB relation is illustrated in the top panel of Figure 1.

Now we teach Laura a second task. We present chat card from Set B as the sample stimulus. The comparison stimuli are the Spanish words from Set C. In this new task, Laura is reinforced for selecting the gato card from the bottom row, which she eventually learns to do. Consistent with the terminology above, we abbreviate the relation as BC, and it is illustrated in the bottom panel of Figure 1.

In the third part of our procedure, we arrange some test trials. In one type of test trial, sometimes we present cat card as the sample stimulus, along with the English words from Set A as the comparisons. Sometimes we present chat card as the sample stimulus, along with the French words from Set B as the comparisons. And sometimes we present gato card as the sample stimulus, along with the Spanish words from Set C as the comparisons. The typical finding is that in test trials like these Laura will select the comparison that is identical to the sample (e.g., If you see cat, select cat), even though she has not been reinforced for doing so. In this untrained performance, called reflexivity, a learner responds to a training stimulus on the basis of its sameness. Reflexivity test trials are illustrated in Figure 2.

In another type of test trial, we show Laura the chat card from Set B as the sample stimulus, along with the English words from Set A as the comparisons. Research suggests that Laura will select the cat card, even though she has not been reinforced for doing so. This untrained performance is called symmetry. Because the AB relation was trained, and behavior consistent with a BA relation emerged, we summarize this symmetrical performance by the rule If AB, then BA. We might also present the gato card from Set C as the sample, along with the French words from Set B as the comparisons. Similar to above, Laura will likely select the chat card, and this symmetrical performance is summarized by the rule If BC, then CB. Symmetry test trials are illustrated in Figure 3.

On still other test trials, we present the cat card from Set A as the sample stimulus along with the Spanish words from Set C as the comparisons. We are likely to discover that Laura will select the gato card, even though she has not been reinforced for doing so. This untrained performance called, transitivity, is summarized by the rule If AB and BC, then AC. The transitivity test trial is illustrated in the top panel of Figure 4.

The untrained relations indicative of reflexivity, symmetry and transitivity are called emergent relations; they are derivatives of other relations that have been trained. They are the defining features of an equivalence class. For Laura, the words cat, chat, and gato form an equivalence class because these stimuli were deemed to be reflexive, symmetrical, and transitive. As Sidman (1986) notes, "A consequence of such class formation is that a subject, when tested, will match any member of a class to any other, even without having encountered the tested relation before" (p. 233).

There is an abbreviated way that researchers sometimes use to assess stimulus equivalence. In our example, we would present the gato card as the sample, along with the English words from Set A as the comparisons. No other test trial type would be presented. If Laura is a typical subject, she will likely select the cat card, even though she has not been reinforced for doing so. Her untrained performance can be summarized by the rule If AB and BC, then CA. This abbreviated assessment is called the equivalence test (Sidman, 1994, p. 224). It is considered to be a sufficient test for stimulus equivalence because it simultaneously evaluates both symmetry and transitivity (reflexivity is assumed). Specifically, the combination of the AB and BC relations yields the AC relation via transitivity, and its symmetrical counterpart is the CA relation (but see Pilgram & Galizio, 1995). The equivalence test trial is illustrated in the bottom panel of Figure 4.

All trained and tested relations in a standard stimulus equivalence paradigm are summarized in Figure 5.

Why are behavior analysts excited about the stimulus equivalence? First, it seems clear that people know more than they have been directly taught. Stimulus equivalence research is discovering specific learning histories that can account for this fact (see also Alessi, 1987). The research is showing that a little bit of training goes a long way. In our example, Laura was only taught only two relations (AB and AC); consequently, she also proved capable of performing in accordance with seven others (AA, BB, CC, BA, CB, AC, CA). And consider this. Suppose we teach Laura only one more relation. Perhaps we show her a picture of a cat, along with the Spanish words from Set C as comparisons, and reinforce her for selecting the gato card (DC). She will then likely prove capable of six more untrained performances: DD, AD, DA, BD, DB, CD (e.g., Sidman & Tailby, 1982).

But this is only the beginning. When one member of an equivalence class acquires a new function via a standard behavioral process, other members in that class often take on that same function. This effect has been shown for SDs (Catania, Horne, & Lowe, 1989), reinforcers and punishers (Hayes, Kohlenberg & Hayes, 1991), and elicitors (Dougher, Augustson, Markham, Greenway, & Eldelgard, 1994). In our example, we might reinforce Laura for saying "meow" only in the presence of the cat card, a discrimination training procedure. Given that the printed word cat participates in an equivalence class with the printed words chien and gato, Laura may now be more likely to say "meow" when we show her either chien card or the gato card, even though reinforcement for such a response has never occurred in the presence of either of these stimuli. In other words, the chien and gato cards now act like discriminative stimuli in occasioning behavior without ever having been part of a discrimination training procedure themselves.

A second reason why stimulus equivalence has generated excitement is its apparent close link to our everyday terms language and meaning. Consider Laura's CA test trial performance. Laura had never before selected cat given gato; in fact, she had never even seen the two words together. However, she had related the words cat and chat together (AB training trials) and the words chat and gato together (BC training trials). On the CA test trial we might say that Laura responds to Spanish word gato as if it was the French word chat; or, she selects English word cat as if it was the French word chat. The three words appear to be interchangeable.

Sidman (1986) summarizes what may be the key contribution of stimulus equivalence research:

"By reacting to a word as an equivalent stimulus - the meaning of a word - a person can behave adaptively in an environment without having previously been exposed to it. The emergence of equivalence from conditionality permits Behavior Analysis to account for the establishment of at least simple semantic correspondences without having to postulate a direct reinforcement history for every instance. Instead of appealing to cognitions, representations, and stored correspondences to explain the initial occurrence of appropriate new behavior, one can find a complete explanation in the...units that are the prerequisites for the emergent behavior." (p. 236)

Stimulus equivalence is a complicated topic. Sidman (1994) has updated and refined his analysis in considerable detail. Alternative frameworks have also been formulated in recent years by two other research teams to account for the same phenomenon (Hayes & Hayes, 1992; Horne & Lowe, 1996).