Fractal Images of Formal Systems

Formal systems are standardly envisaged in terms of a grammar specifying well-formed formulae together with a set of axioms and rules. Derivations are ordered lists of formulae each of which is either an axiom or is generated from earlier items on the list by means of the rules of the system; the theorems of a formal system are simply those formulae for which there are derivations. Here we outline a set of alternative and explicitly visual ways of envisaging and analyzing at least simple formal systems using fractal patterns of infinite depth. Progressively deeper dimensions of such a fractal can be used to map increasingly complex wffs or increasingly complex value spaces, with tautologies, contradictions, and various forms of contingency coded in terms of color. This and related approaches, it turns out, offer not only visually immediate and geometrically intriguing representations of formal systems as a whole but also promising formal links (1) between standard systems and classical patterns in fractal geometry, (2) between quite different kinds of value spaces in classical and infinite-valued logics, and (3) between cellular automata and logic. It is hoped that pattern analysis of this kind may open possibilities for a geometrical approach to further questions within logic and metalogic.\looseness=-1     

2. Models Presented

ABSTRACT This chapter introduces the two theoretical models to be applied to qualitative and quantitative data in the case study of Dodge Morgan. We present the five-factor model (FFM) and the life-story model (LSM) separately, with attention to their differing origins, assumptions, and utilities. We briefly trace the development and resurgence of the factor-analytic FFM and delineate the extent of its current influence in the field of personality. We discuss individual factors and consider the ways in which the model can be expected to be useful as well as limited in application to the individual. We next present the genealogy of the narrative, constructivist LSM and describe its components; we also discuss its contrasting strengths and limitations.     

6. The Life Voyage

ABSTRACT Chapter 6 applies the five-factor model and the life-story model as complementary perspectives to the multiple data sources collected for the case study of Dodge Morgan. We integrate the two models to create a new level of analysis that captures differing aspects of experience. We develop a holistic personality analysis of Morgan, interpreting his childhood, young manhood, adventures in Alaska and on the boat Coaster, his return to the mainland and embrace of business and entrepreneurship, and his solo circumnavigation. We address questions of voyage outcomes with reference to the quantitative analyses of his voyage log; we consider not only the external journey but also his internal psychological issues and developmental tasks.     

Ethnic differences in psychiatric disorders among adolescent substance abusers in treatment

The present study describes the diagnostic profile of over 800 adolescents admitted to residential therapeutic community programs for treatment of drug abuse and related problems. The relationship between race/ethnicity and psychiatric disturbance is examined, and the association among these factors to outcome is investigated. Over 90% of the sample had a DSM-III-R nonsubstance diagnosis. There were significant differences among the race/ethnic groups on attention deficit-hyperactivity and the disruptive behavior and affective disorders. In general, African-American adolescents yielded the lowest rates of disturbance across all categories of disorders. There was no relationship among race/ethnicity, psychiatric disturbance, and 1-year posttreatment outcomes.     

Positive and Negative Hypothesis Testing by Cooperative Groups

In a rule induction problem positive hypothesis tests select evidence that the tester expects to be an example of the correct rule if the hypothesis is correct, whereas negative hypothesis tests select evidence that the tester expects to be a nonexample if the hypothesis is correct. Previous research indicates the general effectiveness of a positive test strategy for individuals, but there has been very little research with cooperative groups. We extend the analysis of Klayman and Ha (Psychological Review, 1987) of ambiguous verification or conclusive falsification of five possible types of hypotheses by positive and negative tests by emphasizing the importance of further examples following hypothesis tests. In two experiments four-person cooperative groups solved rule induction problems by proposing a hypothesis and selecting evidence to test the hypothesis on each of four arrays on each trial. In different conditions the groups were instructed to use different combinations of positive and negative tests on the four arrays. Positive tests were more likely to lead to further examples than negative tests, and the proportion of correct hypotheses corresponded to the proportion of positive tests, in both experiments. We suggest that positive tests are more effective than negative hypothesis tests in generating further evidence, and thus in inducing the correct rule, in experimental rule induction tasks with a criterion of certainty imposed by the researcher.     

SEEING TWO AS ONE:

Abstract -Object tokens are episodic visual representations that mediate the ability to track visual events as they move about and change over time. Multiple tokens also allow the viewer to individuate multiple instances of a single type of object. In the present study, we established a functional link for object tokens in two seemingly disparate visual phenomena: apparent motion and repetition blindness [RB]. In RB, repeated items are more difficult to perceive than unre-peated items. Using displays in which two sets of alphanumeric characters streamed in opposite directions across a computer screen in apparent motion, we found increased RB for targets appearing within a single apparent motion stream, relative to targets in different apparent motion streams. The results are inconsistent with refractory period or memory retrieval accounts of RB and support the role of object tokens in both apparent motion and RB.     

BALANCING SOCIAL ACCEPTABILITY WITH TREATMENT EFFECTIVENESS OF AN INTRUSIVE PROCEDURE: A CASE REPORT

The debate over social acceptability of intrusive procedures has led some school districts to adopt policies allowing punishment for only the most extreme forms of destructive behavior. We investigated the effectiveness of selectively implementing punishment for only the most severe topographies of aggression and property destruction, while less extreme behaviors were ignored. Results indicated that severe behaviors were reduced to near-zero levels only when both severe and less severe behaviors were similarly punished.     

ACHIEVING AND MAINTAINING COMPLIANCE WITH THE KETOGENIC DIET

The ketogenic diet, a treatment for intractable epilepsy, is rarely initiated because it requires strict compliance with a diet that is perceived to be unpalatable. In a case study of a 15-year-old girl with uncontrolled epilepsy, we used a stimulus-choice procedure to assess relative preferences of 33 foods from the diet and to develop two treatments based on Premack's principle. The results of a multielement analysis showed that both treatments increased dietary compliance. Compliance was maintained with generalization of treatment across settings, and was associated with a 40% reduction in seizures.