Gleason's Theorem Has a Constructive Proof

Gleason's theorem for R ³ says that if f is a nonnegative function on the unit sphere with the property that f(x) + f(y) + f(z) is a fixed constant for each triple x,y,z of mutually orthogonal unit vectors, then f is a quadratic form. We examine the issues raised by discussions in this journal regarding the possibility of a constructive proof of Gleason"s theorem in light of the recent publication of such a proof.     

A First Order Nonmonotonic Extension of Constructive Logic

Certain extensions of Nelson's constructive logic N with strong negation have recently become important in arti.cial intelligence and nonmonotonic reasoning, since they yield a logical foundation for answer set programming (ASP). In this paper we look at some extensions of Nelson's .rst-order logic as a basis for de.ning nonmonotonic inference relations that underlie the answer set programming semantics. The extensions we consider are those based on 2-element, here-and-there Kripke frames. In particular, we prove completeness for .rst-order here-and-there logics, and their minimal strong negation extensions, for both constant and varying domains. We choose the constant domain version, which we denote by QN^c₅, as a basis for de.ning a .rst-order nonmonotonic extension called equilibrium logic. We establish several metatheoretic properties of QN^c₅, including Skolem forms and Herbrand theorems and Interpolation, and show that the .rst-oder version of equilibrium logic can be used as a foundation for answer set inference.     

A Note on Goodman's Theorem

Goodman's theorem states that intuitionistic arithmetic in all finite types plus full choice, HA + AC, is conservative over first-order intuitionistic arithmetic HA. We show that this result does not extend to various subsystems of HA, HA with restricted induction.     

Constructive Dialogical Pluralism: A Context of Interreligious Relations

This article presents current philosophical reflections on religious diversity and concomitant attitudes towards the interreligious situation. The motive behind this presentation is to show that in order to deal more efficiently with the phenomenon of religious plurality, there is a need for a development of the philosophy of religion, where new perspectives are opened up and explored. The very concept of religion as a belief system is put into question, since it has caused philosophical reflections on religious diversity to be confined to certain metaphysical and epistemological concerns. Instead of focusing on the noun ‘religion’, the article suggests a way to understand the adjective ‘religious’ and view religious plurality as a plurality of ways of being religious. This opens up a certain context of interreligious relations and interreligious dialogue, where this very dialogue itself can contribute to the development of philosophical tools, concepts and categories for dealing with the fact of plurality. I call this context constructive dialogical pluralism.     

Constructive Empiricism Now

Philosophical Studies -     

A Transtheoretical Approach to Case Formulation

In answer to the need for a comprehensive, systematic, yet flexible case formulation method, the present article describes the Causal Analysis and Synthesis of Events (CASE) system, a transtheoretical case formulation method. This method of functional case formulation is useful in terms of explicating the problem and identifying potential areas for intervention when dealing with clinical problems of both an intrapersonal and interpersonal nature. Further, this formulation method may improve communication between the therapist and the client during case conceptualization and facilitate dialogue between therapists of different theoretical orientations. This paper describes the case formulation method and includes an example incorporating this case formulation technique for a young man with social phobia.     

Hebrew and Buddhist Selves: A Constructive Postmodern Study

Our task will be to demonstrate that there are instructive parallels between Hebrew and Buddhist concepts of self. There are at least five main constituents (skandhas in Sanskrit) of the Hebrew self: (1) nepe? as living being; (2) rūah as indwelling spirit; (3) lēb as heart-mind; (4) bā?ār as flesh; and (5) dām as blood. We will compare these with the five Buddhist skandhas: disposition (samskāra), consciousness (vijñāna), feeling (vedanā), perception (samjñā), and body (rūpa). Generally, what we will discover is that both Buddhists and Hebrews have a ‘bundle’ theory of the self; both see the body as an essential part of personal identity; both overcome the modernist distinction of the inner and the outer; and both avoid language about the will as a distinct faculty. In sum, both present us with a fully somatic and nondualistic view of being human.     

Functioning of Epigenetically Evolved Emotion Systems: A Constructive Analysis

Abstract

The theory of emotions outlined in this paper follows a general theory that psychological processes maintain an adaptive equilibrium between one's construction system and sensory input from one's world. Persons build constructions to match any class of sensory data, and also build standards for (construe) varied levels of input which accompany the arousal-related activity associated with standard/input mismatch. Conscious identification of different emotional states reflects the use of constructions assigned to different standard/input relations. A constructivist may regard different emotional states or experiences as idealized configurations of attributes; and these configurations can be treated as prototypes, applying the same analyses which have been developed by cognitive scientists who have offered models for discussions of categorization processes. A model for discussing emotional development is provided, and directions of future constructivist investigations are suggested     

James Huchingson's Constructive Theology

James Huchingson's book, Pandemonium Tremendum: Chaos and Mystery in the Life of God , is an artistic improvisation on recurrent themes in the dialogue between religion and science. Around the cantus firmus of the Pandemonium Tremendum Huchingson composes a grand metaphysical composition that is glorious in its detail, magnificent in its overarching themes, and careful in its attention to context. Much like a suspended chord between two different harmonies, Huchingson's theological composition dangles the reader in the tensions of religion and science, modernity and postmodernity, particulars and universals, God and the world. Although this book is surely a cutting-edge development in the ongoing corpus of religion and science, I am most excited about its constructive theological provocations. This is a work in progress, a composition in the making.     

The Contemporary Constructive Task

Lutheran theology is noted for its excellent contributions to historical theology. Contemporary times, however, require that Lutheran theologians become attuned to the urgent demands of the present and take up the discipline of constructive theology to address these demands. Taking the lead from contemporary feminist Lutheran theologians, constructive approaches utilizing Lutheran theological resources must articulate vigorous critique of a corrosive neo‐liberal culture by articulating divine judgment on human greed as well as witnessing to divine grace that always favors truthful living.     

Intentionality Versus Constructive Empiricism

By focussing on the intentional character of observation in science, we argue that Constructive Empiricism—B.C. van Fraassen’s much debated and explored view of science—is inconsistent. We then argue there are at least two ways out of our Inconsistency Argument, one of which is more easily to square with Constructive Empiricism than the other.     

Notes on Constructive Negation

We put together several observations on constructive negation. First, Russell anticipated intuitionistic logic by clearly distinguishing propositional principles implying the law of the excluded middle from remaining valid principles. He stated what was later called Peirce’s law. This is important in connection with the method used later by Heyting for developing his axiomatization of intuitionistic logic. Second, a work by Dragalin and his students provides easy embeddings of classical arithmetic and analysis into intuitionistic negationless systems. In the last section, we present in some detail a stepwise construction of negation which essentially concluded the formation of the logical base of the Russian constructivist school. Markov’s own proof of Markov’s principle (different from later proofs by Friedman and Dragalin) is described.