Mathematical concepts for analyzing ambiguous situations

Abstract

We examine some mathematical tools for dealing with ambiguous situations. The main tool is the use of non-standard logic with truth-values in what is called a locale. This approach is related to fuzzy set theory, which we briefly discuss. We also consider probabilistic concepts. We include specific examples and describe the way a researcher can set up a suitable locale to analyse a concrete situation.     

Behavioral Skills Training With Teachers: Booster Training for Improved Maintenance

Behavioral skills training (BST) has been employed within many different populations for the reduction of problem behaviors and acquisition of new skills; however, these changes have not always maintained over time. This study evaluated the effects of a booster training for re-establishing the classroom management proficiencies that teachers had acquired previously, but had declined over a 12-month period. The single-subject experimental results showed the booster training to be effective in re-establishing teacher performance of skills as measured by both analogue role-play and in situ postassessments within the classroom. These findings suggest that a booster training utilizing BST may be an important strategy for maintaining skill performance over time.     

Author biographies

This paper describes how an application of psychodynamic principles and techniques in a school setting can open up perspectives and insights into aspects of loss and mourning, which result in the personal and professional growth of individuals. It discusses applied psychoanalytic work in relation to its clinical base in the consulting room, in particular family work, and exemplifies this with a variety of cases.     

A General Setting for Dedekind's Axiomatization of the Positive Integers

A Dedekind algebra is an ordered pair (B, h), where B is a non-empty set and h is a similarity transformation on B. Among the Dedekind algebras is the sequence of the positive integers. From a contemporary perspective, Dedekind established that the second-order theory of the sequence of the positive integers is categorical and finitely axiomatizable. The purpose here is to show that this seemingly isolated result is a consequence of more general results in the model theory of second-order languages. Each Dedekind algebra can be decomposed into a family of disjoint, countable subalgebras called the configurations of the algebra. There are ?₀ isomorphism types of configurations. Each Dedekind algebra is associated with a cardinal-valued function on ω called its configuration signature. The configuration signature counts the number of configurations in each isomorphism type that occurs in the decomposition of the algebra. Two Dedekind algebras are isomorphic iff their configuration signatures are identical. The second-order theory of any countably infinite Dedekind algebra is categorical, and there are countably infinite Dedekind algebras whose second-order theories are not finitely axiomatizable. It is shown that there is a condition on configuration signatures necessary and sufficient for the second-order theory of a Dedekind algebra to be finitely axiomatizable. It follows that the second-order theory of the sequence of the positive integers is categorical and finitely axiomatizable.     

Future Contradictions

A common and much-explored thought is ?ukasiewicz's idea that the future is ‘indeterminate’—i.e., ‘gappy’ with respect to some claims—and that such indeterminacy bleeds back into the present in the form of gappy ‘future contingent’ claims. What is uncommon, and to my knowledge unexplored, is the dual idea of an overdeterminate future—one which is ‘glutty’ with respect to some claims. While the direct dual, with future gluts bleeding back into the present, is worth noting, my central aim is simply to sketch and briefly explore an alternative glutty-future view, one that is conservative—indeed, entirely classical—with respect to the present.

The structure of the paper runs as follows. §1 briefly sketches the target gap picture of an indeterminate future yielding gappy claims at the present. §2 presents the direct dual idea—a glut picture of an overdeterminate future yielding glutty claims at present. §3 sketches the central idea, a more interesting glut picture in which the future contains contradictory states but the present remains entirely classical. §4 contains a general defence of the idea, leaving it open as to whether the gappy-future view enjoys substantive virtues over the proposed glutty-future view of §3.     

On applying cognitive psychology

Recent attempts to assess the practical impact of scientific research prompted my own reflections on over 40 years worth of combining basic and applied cognitive psychology. Examples are drawn principally from the study of memory disorders, but also include applications to the assessment of attention, reading, and intelligence. The most striking conclusion concerns the many years it typically takes to go from an initial study, to the final practical outcome. Although the complexity and sheer timescale involved make external evaluation problematic, the combination of practical satisfaction and theoretical stimulation make the attempt to combine basic and applied research very rewarding.     

Probabilities on Sentences in an Expressive Logic

Automated reasoning about uncertain knowledge has many applications. One difficulty when developing such systems is the lack of a completely satisfactory integration of logic and probability. We address this problem directly. Expressive languages like higher-order logic are ideally suited for representing and reasoning about structured knowledge. Uncertain knowledge can be modeled by using graded probabilities rather than binary truth values. The main technical problem studied in this paper is the following: Given a set of sentences, each having some probability of being true, what probability should be ascribed to other (query) sentences? A natural wish-list, among others, is that the probability distribution (i) is consistent with the knowledge base, (ii) allows for a consistent inference procedure and in particular (iii) reduces to deductive logic in the limit of probabilities being 0 and 1, (iv) allows (Bayesian) inductive reasoning and (v) learning in the limit and in particular (vi) allows confirmation of universally quantified hypotheses/sentences. We translate this wish-list into technical requirements for a prior probability and show that probabilities satisfying all our criteria exist. We also give explicit constructions and several general characterizations of probabilities that satisfy some or all of the criteria and various (counter)examples. We also derive necessary and sufficient conditions for extending beliefs about finitely many sentences to suitable probabilities over all sentences, and in particular least dogmatic or least biased ones. We conclude with a brief outlook on how the developed theory might be used and approximated in autonomous reasoning agents. Our theory is a step towards a globally consistent and empirically satisfactory unification of probability and logic.     

A Contemporary View of Applied Relaxation for Generalized Anxiety Disorder

Applied relaxation (AR), originally developed by Lars-Göran Öst, is a long-standing, efficacious treatment for generalized anxiety disorder (GAD). While newer treatments are continuing to be developed, AR remains one of the most efficacious treatments for GAD. However, AR has received less in-depth attention more recently, particularly in terms of potential mechanisms of action. This article is written to honor the development and history of AR and to highlight the ways that it has continued to be adapted. In this article, AR treatment strategies are presented, which include: noticing early signs of anxiety, learning relaxation skills, and applying relaxation at the first sign of anxiety. Then, additional adaptations to AR are presented along with recommendations of how AR may be enhanced by understanding potential mechanisms of change. Finally, recommendations are made for the continued evolution of AR as a powerful and efficacious treatment for GAD.