Duality for algebras of relevant logics

This paper defines a category of bounded distributive lattice-ordered grupoids with a left-residual operation that corresponds to a weak system in the family of relevant logics. Algebras corresponding to stronger systems are obtained by adding further postulates. A duality theoey piggy-backed on the Priestley duality theory for distributive lattices is developed for these algebras. The duality theory is then applied in providing characterizations of the dual spaces corresponding to stronger relevant logics.The author gratefully acknowledges the support of the National Sciences and Engineering Research Council of Canada.     

WORD FREQUENCY AND WORD DIFFICULTY:

Abstract— This article compares word counts made using four different collections of text, including one based on collections of electronic text For each of the collections, standard word frequency indices were computed and compared with a carefully developed list of words ranked in order of difficulty as determined by vocabulary tests Correlations between the word frequency indices and word difficulty ranks show that word frequencies for all four corpora are highly correlated with word difficulty Despite these high correlations, the results show also that the difficulty of some words is not estimated accurately by word frequency The reasons for disparities between word frequency and word difficulty are not clear The high correlations obtained for the corpus based on electronic text suggest that this method of text sampling has potential but that caution is advisable in conducting such collections.     

A repertory grid study of therapist/couple interaction

A case of marital therapy is described in which the presenting problem, one of non-consummation, is treated by the standard Masters and Johnson's technique. Using repertory grids at the start and finish of treatment the changing perceptions of both partners and the therapist were determined. In spite of an ostensibly behavioural treatment, substantial changes occurred in the couple's understanding of the nature of their problem. The changes undergone in the couple's perceptions, and their differences from the therapist's assessments of change give rise to speculations about the nature of the therapeutic interaction and suggest further lines of exploration.     

The treatment of tension headache—I. Muscular abnormality and biofeedback

The specificity and efficacy of a short EMG biofeedback treatment were assessed in a selected group of chronic tension-headache cases—those having significantly elevated levels of muscle tension. The effects of training patients to raise/maintain EMG levels were compared to those obtained from patients who were trained to lower tension levels. The results showed little evidence of increased self-control of the muscle despite successful EMG control during six bio-feedback sessions. A progressive and significant reduction in resting level was found only in the groups trained to reduce EMG levels. The treatment proved ineffectual in reducing headache. The implications of the dissociation of muscle tension and headache are discussed both with respect to current views of tension headaches and the role of biofeedback in their treatment.     

The polar unities as a guide to psychotherapy

The author recognizes the need for a psychotherapy that aims to be either spiritual or wholistic to ground itself in appropriate philosophical and theological constructs. As one small step in this direction, the ontological elements as discussed by Paul Tillich are applied to some specific therapeutic issues. Striking parallels are found between Tillich's first two ontological elements (individualization-participation, and dynamics-form) and the two dimensions (cohesion and adaptability) of the circumplex model of family therapy. It is suggested that Tillich's third ontological element (freedom-destiny) throws useful light on the educational and psychotherapeutic issue of permissiveness versus authoritarianism.     

Concurrent schedules: undermatching and control by previous experimental conditions.

Five pigeons were trained on concurrent variable-interval schedules. A series of conditions in which the ratio of reinforcement rates on two keys was progressively increased and then decreased was arranged twice. The birds were then exposed to an irregular sequence of conditions. Each condition in which reinforcement was available on both keys lasted six sessions. Performance in the first, third, and sixth sessions after a condition change was analyzed. Following a condition change, preference was biased toward the preference in the last condition, but this effect largely disappeared before the sixth session of training. The birds' preferences also appeared less sensitive to reinforcement rates in early sessions after a transition. Preference in a session was a function of both the reinforcements in that session and the reinforcements obtained in as many as four or five previous sessions. The effects of reinforcements in previous sessions could be summarized by the performance in the immediately preceding session, giving a relatively simple relation between present performance and a combination of present reinforcement and prior session performance. While such hysteresis could cause undermatching when only a small number of sessions are arranged in a condition, undermatching in a stable-state performance probably arises elsewhere.