On Categorical Equivalences of Commutative BCK-algebras

A commutative BCK-algebra with the relative cancellation property is a commutative BCK-algebra (X;*,0) which satisfies the condition: if a ≤ x, a ≤ y and x * a = y * a, then x = y. Such BCK-algebras form a variety, and the category of these BCK-algebras is categorically equivalent to the category of Abelian ℓ-groups whose objects are pairs (G, G ₀), where G is an Abelian ℓ-group, G ₀ is a subset of the positive cone generating G ⁺ such that if u, v ∈ G ₀, then 0 ∨ (u - v) ∈ G ₀, and morphisms are ℓ-group homomorphisms h: (G, G ₀) → (G′,G′₀) with f(G ₀) ⫅ G′₀. Our methods in particular cases give known categorical equivalences of Cornish for conical BCK-algebras and of Mundici for bounded commutative BCK-algebras (= MV-algebras). This revised version was published online in June 2006 with corrections to the Cover Date.     

The Classification of Propositional Calculi

We discuss Smirnovs problem of finding a common background for classifying implicational logics. We formulate and solve the problem of extending, in an appropriate way, an implicational fragment H of the intuitionistic propositional logic to an implicational fragment TV of the classical propositional logic. As a result we obtain logical constructions having the form of Boolean lattices whose elements are implicational logics. In this way, whole classes of new logics can be obtained. We also consider the transition from implicational logics to full logics. On the base of the lattices constructed, we formulate the main classification principles for propositional logics.     

Adding Involution to Residuated Structures

Two constructions for adding an involution operator to residuated ordered monoids are investigated. One preserves integrality and the mingle axiom x ²x but fails to preserve the contraction property xx ². The other has the opposite preservation properties. Both constructions preserve commutativity as well as existent nonempty meets and joins and self-dual order properties. Used in conjunction with either construction, a result of R.T. Brady can be seen to show that the equational theory of commutative distributive residuated lattices (without involution) is decidable, settling a question implicitly posed by P. Jipsen and C. Tsinakis. The corresponding logical result is the (theorem-) decidability of the negation-free axioms and rules of the logic RW, formulated with fusion and the Ackermann constant t. This completes a result of S. Giambrone whose proof relied on the absence of t.     

A non-finitely based quasi-variety of De Morgan algebras

In this paper we exhibit a non-finitely based, finitely generated quasi-variety of De Morgan algebras and determine the bottom of the lattices of sub-quasi-varieties of Kleene and De Morgan algebras.Supported by Vicerrectoría Académica de la Facultad de Ciencias and by División de Investigación, Sede Bogotá of the Universidad Nacional de Colombia.Special issue of Studia Logica: Algebraic Theory of Quasivarieties Presented by M. E. Adams, K. V. Adaricheva, W. Dziobiak, and A. V. Kravchenko     

Working with Families from the Indian Sub-Continent: An Indo-German Experience*

Whereas most Western European therapists and clients consider emotional distance and abstinence as desirable and conducive in the therapeutic process, this may not always be the case with clients from India. Cultural components such as the contrast between linear and cyclic world-views and the traditional view of the psychologist as an advisor in India present a challenge to professionals with a Western background. Some factors such as the situation of women in society and seemingly too close familiy ties can mean that a Western therapist fails to promote the changes an Indian client is desiring for his or her family. However, besides cultural awareness and caution, one of the most helpful tools in work across cultures is curiosity in its most positive sense and the readiness to be surprised by ones’ clients.I would like to thank all my Indian colleagues who supported and advised me in my work in India.     

Complex learning and information processing by pigeons: a critical analysis

Three models of conditional discrimination learning by pigeons are described: stimulus configuration learning, the multiple-rule model, and concept learning. A review of the literature reveals that true concept learning is not characteristic of the behavior of pigeons in matching-to-sample, oddity-from-sample, or symbolic matching studies. Instead, pigeons learn a set of sample-specific S^D rules. Transfer of the discrimination to novel stimuli, at least along the hue dimension, is predicted by a “coding hypothesis”, which holds that pigeons make a unique, but usually unobserved response, R₁, to each sample, and that the comparison stimulus chosen depends on which R₁ was emitted in the presence of the sample. Convincing evidence is found that pigeons do code sample hues, but there is little evidence that allows one to infer that the “coding event” must have behavioral properties. Parameters of the conditional discrimination paradigm are identified, and it is shown that by appropriate parametric manipulation, a variety of analogous tasks may be generated for both human and animal subjects. The tasks make possible the comparative study of complex learning, attention, memory, and information processing, with the added advantage that behavior processes may be compared systematically across tasks.     

A demonstration of the value of interchangeable observers

The remarkably high agreement between observers using the SRIC, the TSBC, and the CFRS makes the observers interchangeable. This agreement is a product of the intensive and extensive training and monitoring of full-time observers, the use of categories rather than continua, the low degree of interpretation required by the procedures, the use of lay language, the immediate recording after short observational periods, and the familiar and standard setting. Other important features include use of rates, emphasis on time, concern for positive behavior, and close linkages between concepts and categories so that concepts are explicitly specified. Taken together, these characteristics make these systems powerful instruments for the guidance of treatment and the prediction of outcomes. The key to their success seems to be the recording of simple actions and action sequences (the basic phenomena of psychology) as the basic data from which a variety of useful indexes can be readily formed.Preparation of this article was supported by Grant MH-30654 from the National Institute of Mental Health to S. Duncan and D. Fiske.Presented at the 87th Annual Meetings of the American Psychological Association, New York City, September 1979, as part of a symposium on New assessment systems for residential treatment, management, research, and evaluation.     

Transfer of oddity-from-sample performance in pigeons

Four pigeons were trained on a modified three-key oddity-from-sample task in which an observing response to the sample (center-key) stimulus lighted a single comparison (side-key) stimulus. If the comparison stimulus was different from the sample stimulus, a single peck to the lighted comparison was reinforced. If the comparison and sample stimuli were identical, the pigeons had to refrain from pecking the comparison for 4.6 seconds to terminate the matching comparison and to produce immediately a nonmatching comparison on the remaining side key. Each peck to the matching comparison reset the 4.6-second delay interval. Three hues were used during acquisition. During tests for transfer of the oddity performance, two novel hues were substituted either individually or together for one or two of the original training hues. For three birds, latencies to novel nonmatching hues were identical to baseline nonmatching latencies. Latencies to novel matching hues were shorter than baseline matching latencies but were consistently longer than novel nonmatching latencies. These transfer data demonstrate that the pigeons learned the oddity concept.     

儿童理解程度副词的特征

本实验用模糊数学方法研究4—9岁儿童对程度副词的理解特征。结果说明9岁儿童的理解达到了成熟水平;幼儿偏向于夸大理解程度副词的词义;7、8岁是发展的转折期。     

Psychotherapists' efforts to increase awareness of social privilege

Understanding the role of privilege in systemic oppression positions psychotherapists to engage in culturally oriented practice. Twenty-five practising psychotherapists were asked: ‘How can psychotherapists increase their awareness of social privilege?’ Participants sorted 68 unique responses into six concepts. The concepts included: make yourself uncomfortable; embrace humility; learn from the community; involvement in social action; seek critical knowledge; and find like-minded others in the profession. Recommendations for student training are offered.