On the algebraization of a Feferman's predicate

Summary This paper is devoted to the algebraization of an arithmetical predicate introduced by S. Feferman. To this purpose we investigate the equational class of Boolean algebras enriched with an operation , which translates such predicate, and an operation , which translates the usual predicate Theor. We deduce from the identities of this equational class some properties of and some ties between and ; among these properties, let us point out a fixed-point theorem for a sufficiently large class of - polynomials. The last part of this paper concerns the duality theory for - algebras.Allatum est die 24 Maii 1976     

Free Modal Lattices via Priestley Duality

A Priestley duality is developed for the variety j of all modal lattices. This is achieved by restricting to j a known Priestley duality for the variety of all bounded distributive lattices with a meet-homomorphism. The variety j was first studied by R. Beazer in 1986.The dual spaces of free modal lattices are constructed, paralleling P.R. Halmos' construction of the dual spaces of free monadic Boolean algebras and its generalization, by R. Cignoli, to distributive lattices with a quantifier.     

Birkhoff-like sheaf representation for varieties of lattice expansions

Given a variety we study the existence of a class such that S1 every A can be represented as a global subdirect product with factors in and S2 every non-trivial A is globally indecomposable. We show that the following varieties (and its subvarieties) have a class satisfying properties S1 and S2: p-algebras, distributive double p-algebras of a finite range, semisimple varieties of lattice expansions such that the simple members form a universal class (bounded distributive lattices, De Morgan algebras, etc) and arithmetical varieties in which the finitely subdirectly irreducible algebras form a universal class (f-rings, vector groups, Wajsberg algebras, discriminator varieties, Heyting algebras, etc). As an application we obtain results analogous to that of Nachbin saying that if every chain of prime filters of a bounded distributive lattice has at most length 1, then the lattice is Boolean.We wish to thank Lic. Alfredo Guerin and Dr. Daniel Penazzi for helping us with linguistics aspects. We are indebted to the referee for several helpful suggestions. We also wish to thank Professor Mick Adams for providing us with several reprints and useful e-mail information on the subject.Suported by CONICOR and SECyT (UNC).     

A partially ordered extention of the integers

This paper presents a monotonic system of Post algebras of order +* whose chain of Post constans is isomorphic with 012 ... -3-2-1. Besides monotonic operations, other unary operations are considered; namely, disjoint operations, the quasi-complement, succesor, and predecessor operations. The successor and predecessor operations are basic for number theory.The editing of this unified version of two previous papers [Epstein, Rasiowa 1990, 1991] by the authors was completed, except for footnotes, while the first author visited at Warsaw University during June–July, 1994. The footnotes were added by the first author in September, 1994.Presented byRyszard Wójcicki     

Empathy and justice motivation

Empathic distress is defined as an aversive feeling contingent on another's physical, emotional, or economic distress. The paper (1) summarizes a developmental scheme consisting of four stages of empathic distress; (2) suggests that causal attributions may partly transform empathic distress into sympathy, empathic anger, feeling of injustice, and guilt feeling; (3) notes the evidence that these empathic affects often serve to motivate moral behavior, and therefore that they qualify as moral motives; (4) points up limitations of these affects/motives and the need to embed them in justice principles; (5) discusses links between empathic affects/motives and principles of distributive justice/ (6) argues that Rawls' theory of justice, which excludes empathy, may nevertheless require it for the difference principle to influence behavior in real life; (7) hypothesizes a functional equivalence between empathy and the veil of ignorance; and (8) speculates that the conjunction of empathic affect and justice-principle thinking — in life and in abstract didactic contexts like Rawls' original position — may produce a principle having the motivational and stabilizing properties of a hot cognition.     

“Soul-making” in a schizophrenic saint

Soul-making (cf. Jung and Hillman) is the process of integrating spiritual and bodily imagery into an intimated wholeness transcending conscious comprehension. Pierre Janet's case of the psychotic mystic, Madeleine, reveals that the patient had been making her own-soul even though his theory had no provision for soul. Janet's soul-stripping theory is contrasted with a soul-making approach, primarily in their respective interpretations of Madeleine's altered states of consciousness. Religious ecstasy is a stretching of soul, an expansion into the realm of spirit, which requires a subsequent descent into and reconciliation with tradition, society, outer world, and body.     

Some modifications of Scott's theorem on injective spaces

D. Scott in his paper [5] on the mathematical models for the Church-Curry -calculus proved the following theorem.A topological space X. is an absolute extensor for the category of all topological spaces iff a contraction of X. is a topological space of Scott's open sets in a continuous lattice.In this paper we prove a generalization of this theorem for the category of , -closure spaces. The main theorem says that, for some cardinal numbers , , absolute extensors for the category of , -closure spaces are exactly , -closure spaces of , -filters in , >-semidistributive lattices (Theorem 3.5).If = and = we obtain Scott's Theorem (Corollary 2.1). If = 0 and = we obtain a characterization of closure spaces of filters in a complete Heyting lattice (Corollary 3.4). If = 0 and = we obtain a characterization of closure space of all principial filters in a completely distributive complete lattice (Corollary 3.3).     

From <Emphasis Type="Italic">Pintele Yid to Racenjude</Emphasis>: Chaim Zhitlovsky and racial conceptions of Jewishness

The Russian Jewish intellectual, Chaim Zhitlovsky (1865–1943), a leading architect of secular Jewish culture and thought, was a central figure in the progressive Jewish intelligentsia of the late nineteenth and early twentieth century. In an essay written in 1927, Yidn un Yiddishkayt (Jews and Jewishness), he sought to define the secular essence of what he calls Yiddishkayt. This essay is not the first in Zhitlovskys long publicistic career in which he searches for new, secular definitions of Jewish identity and culture. But this essay differs, since it is marked by Zhitlovskys use of contemporary social scientific notions of race and racial traits to conceptualize what he believes constitutes Jewishness in a non-religious context, along with his adoption of the mystical Jewish concept of the pintele yid, the theory of an innate Jewishness embodied by a Jewish spark. Zhitlovskys desire to craft a truly secular theory of Jewish identity led him ironically to accept models of Jewish identity at odds with his stated larger vision. In turning to contemporary racial theory, as well as long nurtured mystical models of Yiddishkayt, Zhitlovsky reveals the wide range of ideological discourses that led him to innovative and controversial notions of modern Jewish identity.     

The Self Psychology of Heinz Kohut: What's It All About Theologically?

As a psychoanalytic thinker who offered by means of his self psychology a new paradigm of psychological development and functioning, Heinz Kohut was also a theologian manqué. With the help of the method of interpretation devised by Paul Tillich and David Tracy, Kohut's limit-concepts of tragic man, the self-object, and empathy, all set within his theory of narcissism, are elucidated as theological constructs. These are critiqued for adequacy from a Christian perspective. The conclusion is that Kohut's understanding of the human dilemma and of the way of salvation correlates well with Christianity, while his view of empathy as the means of salvation has created some confusion. Kohut has thus left an unfinished, profoundly important, agenda for theologians and clinicians.     

Modal logic with names

We investigate an enrichment of the propositional modal language with a universal modality having semanticsx iff y(y ), and a countable set of names — a special kind of propositional variables ranging over singleton sets of worlds. The obtained language _c proves to have a great expressive power. It is equivalent with respect to modal definability to another enrichment () of, where is an additional modality with the semanticsx iff y(y x y ). Model-theoretic characterizations of modal definability in these languages are obtained. Further we consider deductive systems in _c. Strong completeness of the normal _c-logics is proved with respect to models in which all worlds are named. Every _c-logic axiomatized by formulae containing only names (but not propositional variables) is proved to be strongly frame-complete. Problems concerning transfer of properties ([in]completeness, filtration, finite model property etc.) from to _c are discussed. Finally, further perspectives for names in multimodal environment are briefly sketched.     

Review article

Conclusions Using the definition of education as a practice, the only coherent interpretation of Heslep's central thesis of Education's moral role was found to need the notion of a social institution of Education. This in turn made sense only by positing a notion such as a Political Proficiency Certificate, with its concomitant drastic government intervention. Heslep himself did not suggest this, presumably because he saw the tension that would appear between such an institution and the voluntary actions of citizens. This tension was then shown to exist even in his notion of the public interest, which brought into question his argument of why Education needs to be standardized. This was reinforced by the realisation that, if the public interest required votes to be aggregated into a social choice, then it is unlikely to be realised by any current voting system. This suggested it was an unsuitable foundation on which to rest either a theory of democracy or a justification for the need for standardized Education. With that justification brought into question, the need for a social institution of Education disappears; without that, as we have said, Heslep's central thesis is without a coherent interpretation.     

The family as portrayed on prime-time television, 1947–1990: Structure and characteristics

Successful family series across four decades of American prime-time television were examined. Family portrayals were defined as either conventional or nonconventional. Conventional families were categorized as couples without children and couples with children. Nonconventional families were categorized as single parent or contrived. Additional family characteristics were also recorded including sex of single parent, reason for singleness, social class status, females employed outside the home, live-ins, race, and whether the presentation was dramatic or comedic. The data show a trend toward more equal presentation of conventional and nonconventional families, few divorced or female single parents, and few minority families. Implications of findings are discussed and future research questions suggested.     

Oral production of linguistically complex sentences with meaning relationships of time

The purpose of this investigation was to determine the abilities of children to use the adjoining mechanism in combining two constituent sentences with the temporal adjoiners: after, before, until, when, and while. To elicit responses, a sentence repetition task was devised that included these five temporal adjoiners in four different syntactic environments: transitive sentences with the adjoiner and the subordinate clause following the main clause, transitive sentences with the adjoiner and the subordinate clause preceding the main clause, intransitive sentences with the adjoiner and the subordinate clause following the main clause, and intransitive sentences with the adjoiner and the subordinate clause preceding the main clause. The 30 were between the ages of 4O and 66 years. They were average children who were free from any known emotional disturbance, who were acquiring Standard American English as a native language, who had normal speech and hearing, and whose parents had neither very high nor very low socioeconomic status. To the extent that the children in this study were representative of normal-speaking children of their ages, certain general conclusions were drawn. Children begin to use the temporal adjoining mechanism early, but they do not master it by the age of 66 years. The ability to use the adjoiners, nor is it equal for different syntactic structures nor for all degrees of semantic complexity. After, before, and when appear earlier than while and until. A rapid period of growth in learning to use the temporal adjoining mechanism occurs between the ages of 4 and 5 years. However, a plateau of learning appears to be reached between the ages of 5 and 6 years. In general, children first learn to use the temporal adjoining mechanism in intransitive sentences with the adjoining link in the middle or at the beginning of the utterance. Next, they learn to use it in transitive sentences with the adjoining link at the beginning of the utterance. Finally, they learn to use it in transitive sentences with the adjoining link in the middle of the utterance. In transitive sentences, children appear to learn the rule for placing the subordinate clause at the beginning of the utterance when temporally adjoining two constituent sentences before they learn the base structure rule. In intransitive sentences, they appear to learn the rule for placing the subordinate clause at the beginning of the utterance when temporally adjoining two constituent sentences at the same time that they learn the base structure rule. The underlying semantic relationships that are expressed by specific temporal adjoiners are important determinants of children's abilities to use these adjoiners. In linguistic evaluations, one should consider the syntactic environment in which the temporal adjoiner occurs and assume that after, before, and when are developmentally earlier than while and until.     

Applications of object relations theory and self-psychology in current group therapy

In an effort to explore how group therapists can apply psychoanalytic theory in the group modality to advantage, two major developments in analytic theory are examined. 1) Kernberg stresses the neutral interpretive stance and utilizes concepts of protective identification and splitting in dealing with borderline pathology. This paper illustrates concretely how such theory can be used by the group therapist to encourage working through in the group situation. 2) Kohut's work stresses the empathic, nurturant analytic stance and focuses upon idealization, the grandiose self and the self-object. An understanding of these concepts helps the group therapist avoid obstacles in the treatment of narcissistic patients and can enrich the therapeutic work. The paper concludes that treatment of the difficult patient may be enhanced by the group situation in combination with specific psychoanalytic concepts.     

On Neat Reducts of Algebras of Logic

SC , CA , QA and QEA stand for the classes of Pinter's substitution algebras, Tarski's cylindric algebras, Halmos' quasipolyadic algebras, and quasipolyadic equality algebras of dimension , respectively. Generalizing a result of Németi on cylindric algebras, we show that for K {SC, CA, QA, QEA} and ordinals < , the class Nr K of -dimensional neat reducts of -dimensional K algebras, though closed under taking homomorphic images and products, is not closed under forming subalgebras (i.e. is not a variety) if and only if > 1.From this it easily follows that for 1 < < , the operation of forming -neat reducts of algebras in K does not commute with forming subalgebras, a notion to be made precise.We give a contrasting result concerning Halmos' polyadic algebras (with and without equality). For such algebras, we show that the class of infinite dimensional neat reducts forms a variety.We comment on the status of the property of neat reducts commuting with forming subalgebras for various reducts of polyadic algebras that are also expansions of cylindric-like algebras. We try to draw a borderline between reducts that have this property and reducts that do not.Following research initiated by Pigozzi, we also emphasize the strong tie that links the (apparently non-related) property of neat reducts commuting with forming subalgebras with proving amalgamation results in cylindric-like algebras of relations. We show that, like amalgamation, neat reducts commuting with forming subalgebras is another algebraic expression of definability and, accordingly, is also strongly related to the well-known metalogical properties of Craig, Beth and Robinson in the corresponding logics.     

A calculus for the common rules of ∧ and ∨

We provide a finite axiomatization of the consequence , i.e. of the set of common sequential rules for and . Moreover, we show that has no proper non-trivial strengthenings other than and . A similar result is true for , but not, e.g., for ⁺.To the memory of Jerzy Supecki     

Three prepositional calculi of probability

Attempts are made to transform the basis of elementary probability theory into the logical calculus.We obtain the propositional calculus NP by a naive approach. As rules of transformation, NP has rules of the classical propositional logic (for events), rules of the ukasiewicz logic ₀ (for probabilities) and axioms of probability theory, in the form of rules of inference. We prove equivalence of NP with a fragmentary probability theory, in which one may only add and subtract probabilities.The second calculus MP is a usual modal propositional calculus. It has the modal rules x x, x y x y, x x, x y (y x), (y x), in addition to the rules of classical propositional logic. One may read x as x is probable. Imbeddings of NP and of ₀ into MP are given.The third calculus P is a modal extension of ₀. It may be obtained by adding the rule ((xy)y) xy to the modal logic of quantum mechanics Q [5]. One may read x in P as x is observed. An imbedding of NP into P is given.     

In and behind a Lenneberg paradigm

Lenneberg suggested that a chimpanzee's linguistic ability could be tested by presenting sentences containing two prepositional phrases joined by a conjunction. This would involve joining two semantic propositions, and thus represents a more complex test of chimpanzee syntactic competence than previously attempted. Jane, a five year-old language trained chimpanzee, was tested on her ability to both produce and comprehend sentences involving a preposition (in or behind) and a conjunction (and). The results from production and comprehension were substantially the same. Jane showed the ability to appropriately deploy and, behind and in, but displayed very little flexibility in their use. It is suggested that a chimpanzee may be able to learn some rules of syntax but is not able to be creative with that syntax.     

Development of Gender Differences in Children's Responses to Animated Entertainment

This study examined gender differences in children's responses to animated scenes from an action adventure versus a sad film, and to animated previews of a prototypical male versus female movie. Females were more likely than males to express sadness in response to the sad scene, and gender differences in intensities of sadness increased with age. Children were much more likely to stereotype the male preview as most liked by other boys, whereas the majority of children perceived the female preview as liked by either gender equally. In terms of enjoyment of the male and female previews, gender differences in enjoyment of the male preview were apparent only among children who perceived the film as more appealing to boys, and gender differences in enjoyment of the female preview were apparent only among children who perceived the film as more appealing to girls. Implications for children's programming are discussed.     

Rational choices in wason's selection task

Sixty-eight subjects were asked to solve a series of six Wason selection problems by physically turning relevant cards over until they could say whether the rule was correct or false. In the first experiment most of the subjects misunderstood the rule they were asked to verify, different subjects misunderstanding in different ways, but commonly reading (a) top/underneath for one side/other side and (b) a dual conditional for if..., then.... However, a majority of subjects thereafter responded in complete logical consistency with the rule as they understood it to be. When the misunderstandings revealed in the first experiment were themselves presented as rules to be verified in a second experiment, performance was much more logical than is usually reported. It is concluded that the illogicality commonly associated with Wason's selection task is only apparent.