Linearization of the BCK-logic

In the paper we obtain a new characterization of the BCK-algebras which are subdirect product of BCK-chains. We give an axiomatic algebraizable extension of the BCK-calculus, by means of a recursively enumerable set of axioms, such that its equivalent algebraic semantics is definitionally equivalent to the quasivariety of BCK-algebras generated by the BCK-chains. We propose the concept of "linearization of a system" and we give some examples.     

First-Order Theories of Orthogonality Structures

We investigate certain aspects of the first-order theory oforthogonality structures - structures consisting of a domainof lines subject to a binary orthogonality relation. In particular,we establish definitions of various geometric and algebraicnotions in terms of orthogonality, describe the constructionof extremal subspaces using orthogonality, and show that thefirst-order theory of line orthogonality in the Euclidean n-spaceis not ₀-categorical for n 3.     

Projective Beth Property in Extensions of Grzegorczyk Logic

All extensions of the modal Grzegorczyk logic Grz possessing projective Beth's property PB2 are described. It is proved that there are exactly 13 logics over Grz with PB2. All of them are finitely axiomatizable and have the finite model property. It is shown that PB2 is strongly decidable over Grz, i.e. there is an algorithm which, for any finite system Rul of additional axiom schemes and rules of inference, decides if the calculus Grz+Rul has the projective Beth property. Dedicated to the memory of Willem Johannes Blok     

The Phenomenology of Early Psychosis Elicited in an Occupational Therapy Expressive Evaluation

Expressive assessments facilitate sharing, and the therapeutic understanding of youth. The experiences of nine young adults seeking help for psychosis were explored with an arts-based, occupational therapy expressive evaluation. The study used participant-observation methods. Data analysis drew on hermeneutic phenomenology. The findings illustrate the challenges these participants have in feeling connected to a shared world of doing, and in locating themselves within the temporal flow of time. Of significance was the potential to read the unfolding of creation in clay media as an acted narrative to tap into “constellations of meaning.” Implications for clinical practice and professional reasoning are highlighted.     

Russell's Early Theory of Denoting

The ancient Greek method of analysis has a rational reconstruction in the form of the tableau method of logical proof. This reconstruction shows that the format of analysis was largely determined by the requirement that proofs could be formulated by reference to geometrical figures. In problematic analysis, it has to be assumed not only that the theorem to be proved is true, but also that it is known. This means using epistemic logic, where instantiations of variables are typically allowed only with respect to known objects. This requirement explains the preoccupation of Greek geometers with questions as to which geometrical objects are ‘given’, that is, known or ‘data’, as in the title of Euclid's eponymous book. In problematic analysis, constructions had to rely on objects that are known only hypothetically. This seems strange unless one relies on a robust idea of ‘unknown’ objects in the same sense as the unknowns of algebra. The Greeks did not have such a concept, which made their grasp of the analytic method shaky.     

Point-Free Geometry and Verisimilitude of Theories

A metric approach to Popper’s verisimilitude question is proposed which is related to point-free geometry. Indeed, we define the theory of approximate metric spaces whose primitive notions are regions, inclusion relation, minimum distance, and maximum distance between regions. Then, we show that the class of possible scientific theories has the structure of an approximate metric space. So, we can define the verisimilitude of a theory as a function of its (approximate) distance from the truth. This avoids some of the difficulties arising from the known definitions of verisimilitude.     

The Undecidability of Grisin's Set Theory

We investigate a contractionless naive set theory, due to Grisin [11]. We prove that the theory is undecidable.     

子图问题学习对几何解题迁移的影响

将一个图形（称为母图）的若干条线段删除后得到的图形叫做母图的子图。运用实验方法探讨了子图结构学习和子图变式学习对解决问题的迁移效果。结果表明：子图结构学习仅对解决“图形结构相同、逻辑结构相同”问题有明显迁移作用;子图变式学习对解决“图形结构相同、逻辑结构相同”没有明显迁移作用,对“图形结构相似、逻辑结构相同”、“图形结构相同、逻辑结构相似”以及“图形结构相似、逻辑结构相似”四类问题均有明显迁移作用。     

The road to two theorems of logic

Work on how to axiomatize the subtheories of a first-order theory in which only a proper subset of their extra-logical vocabulary is being used led to a theorem on recursive axiomatizability and to an interpolation theorem for first-order logic. There were some fortuitous events and several logicians played a helpful role.     

Loosely Guarded Fragment of First-Order Logic has the Finite Model Property

We show that the loosely guarded and packed fragments of first-order logic have the finite model property. We use a construction of Herwig and Hrushovski. We point out some consequences in temporal predicate logic and algebraic logic.     

A Predicate Logical Extension of a Subintuitionistic Propositional Logic

We develop a predicate logical extension of a subintuitionistic propositional logic. Therefore a Hilbert type calculus and a Kripke type model are given. The propositional logic is formulated to axiomatize the idea of strategic weakening of Kripke's semantic for intuitionistic logic: dropping the semantical condition of heredity or persistence leads to a nonmonotonic model. On the syntactic side this leads to a certain restriction imposed on the deduction theorem. By means of a Henkin argument strong completeness is proved making use of predicate logical principles, which are only classically acceptable.     

The Impact of Early Loss on Depression

Abstract

This article reviews psychodynamic theoretical claims linking childhood losses to depression later in life and explores contemporary empirical studies that are congruent with those claims. Biologically mediated responses to grief, empirically supported pathways that link loss to depression, and factors that distinguish normal from pathological grief are identified. Trajectories that may lead to the development of psychotic-like, instead of neurotic-like, symptoms of depression are also explored, and empirically supported internal and external vulnerability factors that may lead to depression are reported. Several clinical vignettes are included to exemplify theoretical claims. Implications for assessment and treatment are discussed in light of these findings.     

Identical Twins,Deduction Theorems,and Pattern Functions: Exploring the Implicative BCSK Fragment of S5

We recapitulate (Section 1) some basic details of the system of implicative BCSK logic, which has two primitive binary implicational connectives, and which can be viewed as a certain fragment of the modal logic S5. From this modal perspective we review (Section 2) some results according to which the pure sublogic in either of these connectives (i.e., each considered without the other) is an exact replica of the material implication fragment of classical propositional logic. In Sections 3 and 5 we show that for the pure logic of one of these implicational connectives two – in general distinct – consequence relations (global and local) definable in the Kripke semantics for modal logic turn out to coincide, though this is not so for the pure logic of the other connective, and that there is an intimate relation between formulas constructed by means of the former connective and the local consequence relation. (Corollary 5.8. This, as we show in an Appendix, is connected to the fact that the ‘propositional operations’ associated with both of our implicational connectives are close to being what R. Quackenbush has called pattern functions.) Between these discussions Section 4 examines some of the replacement-of-equivalents properties of the two connectives, relative to these consequence relations, and Section 6 closes with some observations about the metaphor of identical twins as applied to such pairs of connectives.     

Commentary on “Transformations in hallucinosis and the receptivity of the analyst” by Civitarese

After summarizing Civitareses complex and stimulating thoughts, merits of his conceptualization are emphazised firstly before coming to drawbacks of it. As a supplementary perspective on the conceptualisation (TH) of Civitares the commentary is focused on a slightly different approach. Transformation into hallucinosis is linked to excessive forms of projective identification., which can have massive influence on the analyst, thus pressing him into (behavioral) patterns similar to what was phantazised, or hallucinated or deluded by the patient about the relationsship. If the analyst is after a shorter or longer period able to recognize, how near he is or was to the Patient's deluded or hallucinated Constructions of his world, this will give him a further source to sense and to examine the patients experience.     

Mathematical Pluralism: The Case of Smooth Infinitesimal Analysis

A remarkable development in twentieth-century mathematics is smooth infinitesimal analysis (‘SIA’), introducing nilsquare and nilpotent infinitesimals, recovering the bulk of scientifically applicable classical analysis (‘CA’) without resort to the method of limits. Formally, however, unlike Robinsonian ‘nonstandard analysis’, SIA conflicts with CA, deriving, e.g., ‘not every quantity is either = 0 or not = 0.’ Internally, consistency is maintained by using intuitionistic logic (without the law of excluded middle). This paper examines problems of interpretation resulting from this ‘change of logic’, arguing that standard arguments based on ‘smoothness’ requirements are question-begging. Instead, it is suggested that recent philosophical work on the logic of vagueness is relevant, especially in the context of a Hilbertian structuralist view of mathematical axioms (as implicitly defining structures of interest). The relevance of both topos models for SIA and modal-structuralism as appled to this theory is clarified, sustaining this remarkable instance of mathematical pluralism.     

Clinical applications of Matte Blanco's thinking

Ignacio Matte Blanco (1908–1995) left very few specific indications about the applications of his theoretical notions to his interpretative style. The author shows how he uses Matte Blanco to formulate some of his own interpretations. The first part of the paper uses clinical vignettes to illustrate some of Matte‐Blanco's concepts. Their theoretical vocabulary is thus made explicit. Then two psychoanalytic sessions are discussed at greater length, together with one from a therapy, so that the use of Matte‐Blanco's notions can be seen clearly, allowing for a fresh perspective on areas of psychoanalytic theory, particularly dreams, psychopathology viewed according to the proportions of asymmetrical and symmetrical functioning in the patient's bi‐logical mental system, the multidimensionality of the unconscious, the structural unconscious, the emotion‐thought relationship, projective identification, resistance, and negative therapeutic reaction. The practical consequences of all this are elaborated, particularly the ensuing possibility of ‘thinking with the patient' in the session. This enables the patient to introject a form of mental functioning in which the asymmetrical mode is not invaded by the symmetric mode (a parallel can be seen here with the Bionian concept of dialogue between the psychotic and non‐psychotic parts of the mind).