Predicting the extension of equivalence classes from primary generalization gradients: the merger of equivalence classes and perceptual classes.

In Experiment 1, 6 college students were given generalization tests using 25 line lengths as samples with a long line, a short line, and a “neither” option as comparisons. The neither option was to be used if a sample did not go with the other comparisons. Then, four-member equivalence classes were formed. Class 1 included three nonsense words and the short line. Class 2 included three other nonsense words and the long line. After repeating the generalization test for line length, additional tests were conducted using members of the equivalence classes (i.e., nonsense words and lines) as comparisons and intermediate-length lines as samples. All Class 2 comparisons were selected in the presence of the test lines that also evoked the selection of the long line in the generalization test that had been given before equivalence class formation. Class 1 yielded complementary findings. Thus, the preclass primary generalization gradient predicted which test lines acted as members of each equivalence class. Regardless of using comparisons that were nonsense words or lines, the post-class-formation gradients overlapped, showing the substitutability of class members. Experiment 2 assessed the discriminability of the intermediate-length test lines from the Class 1 (shortest) and Class 2 (longest) lines. The test lines that functioned as members of an equivalence class were discriminable from the line that was a member of the same class by training. Thus, these test lines also acted as members of a dimensionally defined class of “long” or “short” lines. Extension of an equivalence class, then, involved its merger with a dimensionally defined class, which converted a close-ended class to an open-ended class. These data suggest a means of predicting class membership in naturally occurring categories.     

Robert Kilwardby's Disputational Logic

The article is concerned with the account of Aristotle's theory of disputation given by Robert Kilwardby (d. 1279) in his commentary, composed in Paris during the 1240s, on Aristotle's Prior Analytics. Specifically, I show that Kilwardby covers demonstrative as well as dialectical disputations, and gives an elementary account of the rules governing such disputations, in their adversarial forms as well as in an idealized form where the interlocutors engage in a cooperative activity. I describe the resemblances and the differences between disputations as theorized by Kilwardby and the game of obligationes as understood by some of his contemporaries.     

The structure of equivalence classes

THE STRUCTURE OF EQUIVALENCE CLASSES CAN BE COMPLETELY DESCRIBED BY FOUR PARAMETERS: class size, number of nodes, the distribution of "singles" among nodes, and directionality of training. Class size refers to the number of stimuli in a class. Nodes are stimuli linked by training to at least two other stimuli. Singles are stimuli linked by training to only one other stimulus. The distribution of singles refers to the number of singles linked by training to each node. Directionality of training refers to the use of stimuli as samples and as comparison stimuli in training. These four parameters define the different ways in which the stimuli in a class can be organized, and thus provide a basis for systematically characterizing the properties of stimuli in a given equivalence class. The four parameters can also be used to account for the development of individual differences that are commonly characterized in terms of "understanding" and connotative meaning.Methods are described for generating all possible combinations of parameter values, and a formula is introduced which specifies all of the parameter values for an equivalence class. Its utility for interrelating experimental procedures is demonstrated by analyzing a number of representative experiments that have addressed equivalence-class formation.     

Stimulus equivalence and transitive associations: A methodological analysis

When a number of two-stimulus relations are established through training within a set of stimuli, other two-stimulus relations often emerge in the same set without direct training. These, termed "transitive stimulus relations," have been demonstrated with a variety of visual and auditory stimuli. The phenomenon has served as a behavioral model for explaining the emergence of rudimentary comprehension and reading skills, and the development of generative syntactic repertoires. This article considers the range of relations that can arise between a given number of stimuli in a class, the number of directly established two-stimulus relations necessary for the emergence of transitive relations, the forms that training sets of stimuli can take, and the number of transitive two-stimulus relations that can be induced without direct training. The procedures needed to establish and assess transitive stimulus control, the possible interactions between the training and testing procedures, and the constrainst these interactions place upon the analysis of transitive stimulus control are also examined. The present analysis indicates that in a transitivity test, choice among such stimuli may be controlled by (1) the relation between the sample and the positive comparison stimulus (transitive stimulus control), (2) the relation between the sample and the negative comparison stimulus (S- rule control), and (3) possible discriminative properties that may inadvertently be established in the positive and negative comparison stimuli (valence control). Methods are described for distinguishing these three forms of stimulus control.