A Relational Theory of the Visible

Axiomathes - On the basis of previous studies in relational biology and the phenomenological calculus, in my contribution I outline the mathematical foundations of biological perception generally,...     

Varieties of Linear Calculi

A uniform calculus for linear logic is presented. The calculus has the form of a natural deduction system in sequent calculus style with general introduction and elimination rules. General elimination rules are motivated through an inversion principle, the dual form of which gives the general introduction rules. By restricting all the rules to their single-succedent versions, a uniform calculus for intuitionistic linear logic is obtained. The calculus encompasses both natural deduction and sequent calculus that are obtained as special instances from the uniform calculus. Other instances give all the invertibilities and partial invertibilities for the sequent calculus rules of linear logic. The calculus is normalizing and satisfies the subformula property for normal derivations.     

Completeness Theorems via the Double Dual Functor

The aim of this paper is to apply properties of the double dual endofunctor on the category of bounded distributive lattices and some extensions thereof to obtain completeness of certain non-classical propositional logics in a unified way. In particular, we obtain completeness theorems for Moisil calculus, n-valued Łukasiewicz calculus and Nelson calculus. Furthermore we show some conservativeness results by these methods. This revised version was published online in June 2006 with corrections to the Cover Date.     

Hegel and Deleuze on the metaphysical interpretation of the calculus

The aim of this paper is to explore the uses made of the calculus by Gilles Deleuze and G. W. F. Hegel. I show how both Deleuze and Hegel see the calculus as providing a way of thinking outside of finite representation. For Hegel, this involves attempting to show that the foundations of the calculus cannot be thought by the finite understanding, and necessitate a move to the standpoint of infinite reason. I analyse Hegel’s justification for this introduction of dialectical reason by looking at his responses to Berkeley’s criticisms of the calculus. For Deleuze, instead, I show that the differential must be understood as escaping from both finite and infinite representation. By highlighting the sub-representational character of the differential in his system, I show how the differential is a key moment in Deleuze’s formulation of a transcendental empiricism. I conclude by dealing with some of the common misunderstandings that occur when Deleuze is read as endorsing a modern mathematical interpretation of the calculus.     

The finite model property for BCI and related systems

We prove the finite model property (fmp) for BCI and BCI with additive conjunction, which answers some open questions in Meyer and Ono [11]. We also obtain similar results for some restricted versions of these systems in the style of the Lambek calculus [10, 3]. The key tool is the method of barriers which was earlier introduced by the author to prove fmp for the product-free Lambek calculus [2] and the commutative product-free Lambek calculus [4].Presented by H. Ono     

The woman in the painting and the image in the penny: an investigation of phenomenological doubleness, seeing-in, and “reversed seeing-in”

The experience of looking at a tilted penny involves a “phenomenological doubleness” in that it simultaneously seems to be of something circular and of something elliptical. In this paper, I investigate the phenomenological doubleness of this experience by comparing it to another case of phenomenological doubleness––the phenomenological doubleness of seeing an object in a painting. I begin by pointing out some striking similarities between the phenomenological characters of these two experiences. I then argue that these phenomenological characters have a common explanation. More specifically, I argue that the psychological mechanism that explains the phenomenological doubleness of the experience of seeing an object in a painting can be extended to also explain the phenomenological doubleness of the experience of seeing a tilted penny.

Completeness of quantum logic

This paper is based on a semantic foundation of quantum logic which makes use of dialog-games. In the first part of the paper the dialogic method is introduced and under the conditions of quantum mechanical measurements the rules of a dialog-game about quantum mechanical propositions are established. In the second part of the paper the quantum mechanical dialog-game is replaced by a calculus of quantum logic. As the main part of the paper we show that the calculus of quantum logic is complete and consistent with respect to the dialogic semantics. Since the dialog-game does not involve the excluded middle the calculus represents a calculus of effective (intuitionistic) quantum logic.In a forthcoming paper it is shown that this calculus is equivalent to a calculus of sequents and more interestingly to a calculus of propositions. With the addition of the excluded middle the latter calculus is a model for the lattice of subspaces of a Hilbert space.On leave of absence from the Institut für Theoretische Physik der Universität zu Köln, W.-Germany.     

Pragmatism, Naturalism, and Phenomenology

Pragmatism’s naturalism is inconsistent with the phenomenological tradition’s anti-naturalism. This poses a problem for the methodological consistency of phenomenological work in the pragmatist tradition. Solutions such as phenomenologizing naturalism or naturalizing phenomenology have been proposed, but they fail. As a consequence, pragmatists and other naturalists must answer the phenomenological tradition’s criticisms of naturalism.     

Phenomenological methodology in the human sciences

The author suggests that phenomenological methodology differs from traditional methodologies both in purpose and procedure. The task of a phenomenological researcher is to see the logic or meaning of an experience, for any subject, rather than to discover causal connections or patterns of correlation. The nature of the task demands extensive study of a small sample, allowing the subjects to speak for themselves and to reveal the logic of their experience as lived. The author reviews verification procedures relevant to phenomenological studies and discusses the limitations inherent in phenomenological research.     

Compatible Functions in Algebras Associated to Extensions of Positive Logic

In [4], Caicedo and Cignoli study compatible functions on Heytingalgebras and the corresponding logical properties of connectivesdefined on intuitionistic propositional calculus. In this paperwe study some aspects of compatible functions on the algebrasassociated to positive propositional calculus and successiveextensions of it: intuitionistic calculus itself, the modalsymmetric propositional calculus of Moisil and n-valued ukasiewiczpropositional calculus.     

Human/nature: Toward a critical ecology

Abstract

Gadamer's notion of statements, distinguished from the otherwise speculative nature of language, is explored with regard to its implications for reflexivity in qualitative research. After reviewing the importance of ‘speculative’ language for phenomenological research, this paper discusses the problem that statements pose for phenomenological methods. The methodological challenges faced in trying to facilitate experiential accounts rather than statements are examined, with particular attention to the central role of the researcher‐participant conversation. The methodological implications of a conversational approach to phenomenological research are then presented, and this approach is discussed in terms of reflexivity.     

The theory of random propositions

The theory of random propositions is a theory of confirmation that contains the Bayesian and Shafer—Dempster theories as special cases, while extending both in ways that resolve many of their outstanding problems. The theory resolves the Bayesian problem of the priors and provides an extension of Dempster's rule of combination for partially dependent evidence. The standard probability calculus can be generated from the calculus of frequencies among infinite sequences of outcomes. The theory of random propositions is generated analogously from the calculus of frequencies among pairs of infinite sequences of suitably generalized outcomes and in a way that precludes the inclusion of contrived orad hoc elements. The theory is also formulated as an uninterpreted calculus.I am grateful to Frank Arntzenius, John Earman, Allan Franklin, Teddy Seidenfeld, Brian Skyrms, Sandy Zabell and an anonymous referee for helpful discussion.     

Undecidability of the Problem of Recognizing Axiomatizations of Superintuitionistic Propositional Calculi

We give a new proof of the following result (originally due to Linial and Post): it is undecidable whether a given calculus, that is a finite set of propositional formulas together with the rules of modus ponens and substitution, axiomatizes the classical logic. Moreover, we prove the same for every superintuitionistic calculus. As a corollary, it is undecidable whether a given calculus is consistent, whether it is superintuitionistic, whether two given calculi have the same theorems, whether a given formula is derivable in a given calculus. The proof is by reduction from the undecidable halting problem for the so-called tag systems introduced by Post. We also give a historical survey of related results.     

Quantum logical calculi and lattice structures

In a preceding paper [1] it was shown that quantum logic, given by the tableaux-calculus T _eff, is complete and consistent with respect to the dialogic foundation of logics. Since in formal dialogs the special property of the value-definiteness of propositions is not postulated, the calculus T _eff represents a calculus of effective (intuitionistic) quantum logic.Beginning with the tableaux-calculus the equivalence of T _eff to calculi which use more familiar figures such as sequents and implications can be investigated. In this paper we present a sequents-calculus of Gentzen-type and a propositional calculus of Brouwer-type which are shown to be equivalent to T _eff. The effective propositional calculus provides an interpretation for a lattice structure, called quasi-implicative lattice. If, in addition, the value-definiteness of quantum mechanical propositions is postulated, a propositional calculus is obtained which provides an interpretation for a quasi-modular orthocomplemented lattice which, as is well-known, has as a model the lattice of subspaces of a Hilbert space.     

Synchronized Linear-Time Temporal Logic

A new combined temporal logic called synchronized linear-time temporal logic (SLTL) is introduced as a Gentzen-type sequent calculus. SLTL can represent the n-Cartesian product of the set of natural numbers. The cut-elimination and completeness theorems for SLTL are proved. Moreover, a display sequent calculus ??SLTL is defined.     

BRIDGING PHYSIOLOGY AND PHENOMENOLOGY IN DYNAMIC PSYCHOLOGY

R asmussen , E. T ranekjær . Bridging physiology and phenomenology in dynamic psychology. Scand. J. Psychol ., 1961, 2, 161–166.—It is important to distinguish between the immediately experienced (the phenomenological) and the physiological-neurological basis for the phenomenological world, e.g. between beta needs (subjectively experienced states of needing or wanting) and alpha needs (objectively defined, physiological needs). The concept of psychon is introduced and defined as a necessary and sufficient condition in the physiological-neurological system for the existence of a subjective experience; psychon bridges the gap between the physiological and the phenomenological world.     

A note on the proof theory the λII-calculus

The II-calculus, a theory of first-order dependent function types in Curry-Howard-de Bruijn correspondence with a fragment of minimal first-order logic, is defined as a system of (linearized) natural deduction. In this paper, we present a Gentzen-style sequent calculus for the II-calculus and prove the cut-elimination theorem.The cut-elimination result builds upon the existence of normal forms for the natural deduction system and can be considered to be analogous to a proof provided by Prawitz for first-order logic. The type-theoretic setting considered here elegantly illustrates the distinction between the processes of normalization in a natural deduction system and cut-elimination in a Gentzen-style sequent calculus.We consider an application of the cut-free calculus, via the subformula property, to proof-search in the II-calculus. For this application, the normalization result for the natural deduction calculus alone is inadequate, a (cut-free) calculus with the subformula property being required.This paper was written whilst the author was affiliated to the University of Edinburgh, Scotland, U.K. and revised for publication whilst he was affiliated to the University of Birmingham, England, U.K.Presented byDaniele Mundici     

On the consistency of a three-valued logical calculus

[This résumé was published in English in Matematicheskii Sbornik along with the article.]The present paper contains an investigation of a three-valued logical calculus (the system) previously described by the author [Recueil Mathématique 4 (46), 2 (1938)].A constructive consistence proof is given for a part of this calculus rendering the results previously published concerning the Russell paradox. A method for a non-constructive completeness proof for the complete calculus is briefly indicated.     

Corporal punishment in the public schools

Abstract

A phenomenological approach to psychopathology is illustrated through an inquiry into the meanings of agoraphobia. A classical and a contemporary phenomenological comprehension are presented. Utilizing the thought of Erwin Straus, the classical approach articulates the problem in terms of lived body and spatiality. the contemporary version suggests the necessity of combining existential and psychoanalytic understandings to be faithful to the phenomenon as disclosed in clinical praxis. General implications for a contemporary phenomenological psychopathology are discussed.