Interpolation properties of superintuitionistic logics

A family of prepositional logics is considered to be intermediate between the intuitionistic and classical ones. The generalized interpolation property is defined and proved is the following. Theorem on interpolation. For every intermediate logic L the following statements are equivalent:

1. Craig's interpolation theorem holds in L,
2. L possesses the generalized interpolation property,
3. Robinson's consistency statement is true in L.

There are just 7 intermediate logics in which Craig's theorem holds. Besides, Craig's interpolation theorem holds in L iff all the modal companions of L possess Craig's interpolation property restricted to those formulas in which every variable is proceeded by necessity symbol.     

A conjunction in closure spaces

This paper is closely related to investigations of abstract properties of basic logical notions expressible in terms of closure spaces as they were begun by A. Tarski (see [6]). We shall prove many properties of ω-conjunctive closure spaces (X is ω-conjunctive provided that for every two elements of X their conjunction in X exists). For example we prove the following theorems:

1. For every closed and proper subset of an ω-conjunctive closure space its interior is empty (i.e. it is a boundary set).
2. If X is an ω-conjunctive closure space which satisfies the ω-compactness theorem and \(\hat P\) [X] is a meet-distributive semilattice (see [3]), then the lattice of all closed subsets in X is a Heyting lattice.
3. A closure space is linear iff it is an ω-conjunctive and topological space.
4. Every continuous function preserves all conjunctions.

     

Peirce's evolutionary pragmatic idealism

In this paper I synthesize a unified system out of Peirce's life work, and name it “Peirce's Evolutionary Pragmatic Idealism”. Peirce developed this philosophy in four stages:

1. His 1868–69 theory that cognition is a continuous and infinite social semiotic process, in which Man is a sign.
2. His Popular Science Monthly pragmatism and frequency theory of probabilistic induction.
3. His 1891–93 cosmic evolutionism of Tychism, Synechism, and Agapism.
4. Pragmaticism: The doctrine of real potentialities (“would-be's”), and Peirce's pragmatic program for developing concrete reasonableness.

Peirce's evolutionary conception of the cosmos is pantheistic, and he constructed it to reconcile religion with Darwinian evolution.     

Disjunctions in closure spaces

The main result of this paper is the following theorem: a closure space X has an 〈α, δ, Q〉-regular base of the power \(\mathfrak{n}\) iff X is Q-embeddable in \(B_{\alpha ,\delta }^\mathfrak{n} \) It is a generalization of the following theorems:

1. Stone representation theorem for distributive lattices (α = 0, δ = ω, Q = ω),
2. universality of the Alexandroff's cube for T ₀-topological spaces (α = ω, δ = ∞, Q = 0),
3. universality of the closure space of filters in the lattice of all subsets for 〈α, δ〉-closure spaces (Q = 0).

By this theorem we obtain some characterizations of the closure space \(F_\mathfrak{m} \) given by the consequence operator for the classical propositional calculus over a formalized language of the zero order with the set of propositional variables of the power \(\mathfrak{m}\) . In particular we prove that a countable closure space X is embeddable with finite disjunctions preserved into F _ω iff X is a consistent closure space satisfying the compactness theorem and X contains a 〈0, ω〉-base consisting of ω-prime sets. This paper is a continuation of [7], [2] and [3].     

Sentimentalism: Its Scope and Limits

The subject of this paper is sentimentalism. In broad terms this is the view that value concepts, moral concepts, practical reasons—some or all of these—can be analysed in terms of feeling, sentiment or emotion. More specifically, the paper discusses the following theses:

1. there are reasons to feel (‘evaluative’ reasons) that are not reducible to practical or epistemic reasons
2. value is analysable in terms of these reasons to feel.
3. all practical reasons are in one way or another grounded in evaluative reasons.

(i) and (ii) are accepted while (iii) is rejected.     

A free IPC is a natural logic: Strong completeness for some intuitionistic free logics

IPC, the intuitionistic predicate calculus, has the property

1. Vc(Γ?A ^c /x) ? Γ??xA.Furthermore, for certain important Γ, IPC has the converse property
2. Γ??xA ? Vc(Γ?A ^c /x).
3. may be given up in various ways, corresponding to different philosophic intuitions and yielding different systems of intuitionistic free logic. The present paper proves the strong completeness of several of these with respect to Kripke style semantics. It also shows that giving up (i) need not force us to abandon the analogue of (ii).

     

Three stages of medical dialogue

The negative consequences of physicians' failure to establish and maintain personal relationships with patients are at the heart of the “humanistic crisis” in medicine. To resolve this crisis, a new model of doctor-patient interaction is proposed, based on the ideas of Martin Buber's philosophy of dialogue. This model shows how the physican may successfully combine the personal (I-Thou) and impersonal (I-It) aspects of medicine in three stages. These “Three Stages of Medical Dialogue” include:

1. An Initial Personal Meeting stage, which initiates the doctor-patient relationship and involves mutual confirmation;
2. An Examination stage, which requires a shift from a personal to an impersonal style of interaction;
3. An Integration Through Dialogue or “Healing Through Meeting’ Stage, which involves the integration of the impersonal medical data into the ongoing dialogue between doctor and patient, as a basis for shared decision-making.

The use of the model, as well as common failures of doctor-patient dialogue are discussed.     

On the extension of intuitionistic propositional logic with Kreisel-Putnam's and Scott's schemes

LetSKP be the intermediate prepositional logic obtained by adding toI (intuitionistic p.l.) the axiom schemes:S = ((? ?α→α)→α∨ ?α)→ ?α∨ ??α (Scott), andKP = (?α→β∨γ)→(?α→β)∨(?α→γ) (Kreisel-Putnam). Using Kripke's semantics, we prove:

1. SKP has the finite model property;
2. SKP has the disjunction property.

In the last section of the paper we give some results about Scott's logic S = I+S.     

Framework for a new dialogue between psychoanalysis and neurosciences: is the combined neuro-psychoanalytic approach the missing link?

Freud's legacy deriving from his work The project for a scientific psychology (1895) could give a new impetus to the dialogue between psychoanalysis and neurosciences. A rapproachment phase is warrented. Based on the work of psychoanalysts who are themselves neuroscientists (such as Mauro Mancia, Martha Koukkou and Harold Shevrin) or have a long term dialogue with neuroscientists (Arnold Modell), three points of epistemological congruence are described:

1. dualism is no longer a satisfactory solution
2. cautions for the centrality of interpretation (hermeneutics)
3. the self-criticism of neuroscientists

     

Compatible Operations on Residuated Lattices

This work extend to residuated lattices the results of [7]. It also provides a possible generalization to this context of frontal operators in the sense of [9]. Let L be a residuated lattice, and f : L ^k ?? L a function. We give a necessary and sufficient condition for f to be compatible with respect to every congruence on L. We use this characterization of compatible functions in order to prove that the variety of residuated lattices is locally affine complete. We study some compatible functions on residuated lattices which are a generalization of frontal operators. We also give conditions for two operations P(x, y) and Q(x, y) on a residuated lattice L which imply that the function ${x \mapsto min\{y \in L : P(x, y) \leq Q(x, y)\}}$ when defined, is equational and compatible. Finally we discuss the affine completeness of residuated lattices equipped with some additional operators.     

Evidenzbasierte Psychotherapie der Panikst?rung mit und ohne Agoraphobie

From the evidence reported in the recent guidelines [Heinrichs et al. (2009) Evidenzbasierte Leitlinie zur Psychotherapie der Panikst?rung mit und ohne Agoraphobie. Hogrefe, G?ttingen] the following conclusions can be drawn with respect to the treatment of panic disorder with or without agoraphobia, as well as for agoraphobia without panic disorder:

1. Every psychotherapy should be preceded by suitable diagnostics and a case formulation.
2. The effectiveness of every psychotherapy should be evaluated with well established measures.
3. For the treatment of panic disorder without agoraphobia, cognitive behavioral treatment (CBT) and applied relaxation have been shown to be effective (evidence grade 1).
4. For panic disorder with agoraphobia, CBT, combination treatment (CBT plus medication), as well as panic-focused psychodynamic psychotherapy in the outpatient setting have shown short-term effectiveness (evidence grade 1). Longer lasting effects have been documented for CBT (evidence grade 1).
5. For agoraphobia without panic disorder, CBT with a focus on situational exposure can be recommended (evidence grade 1).

     

Axiomatic extensions of the constructive logic with strong negation and the disjunction property

We study axiomatic extensions of the propositional constructive logic with strong negation having the disjunction property in terms of corresponding to them varieties of Nelson algebras. Any such varietyV is characterized by the property: (PQWC) ifA,B εV, thenA×B is a homomorphic image of some well-connected algebra ofV. We prove:

• each varietyV of Nelson algebras with PQWC lies in the fibre σ^?1(W) for some varietyW of Heyting algebras having PQWC,
• for any varietyW of Heyting algebras with PQWC the least and the greatest varieties in σ^?1(W) have PQWC,
• there exist varietiesW of Heyting algebras having PQWC such that σ^?1(W) contains infinitely many varieties (of Nelson algebras) with PQWC.

     

Can Teleosemantics Deflect the EAAN?

Alvin Plantinga’s Evolutionary Argument Against Naturalism aims to show that the conjunction of contemporary evolutionary theory (E) with the claim that there is no God (N) cannot be rationally accepted. Where R is the claim that our cognitive faculties are reliable, the argument is:

1. The probability of R given N and E is low or inscrutable.
2. Anyone who sees (1) and accepts (N&E) has a defeater for R, and this defeater cannot be defeated or deflected.
3. Anyone who has an undefeated, undeflected defeater for R has an undefeated, undeflected defeater for everything she believes.
4. Therefore she has an undefeated, undeflected defeater for (N&E).

Plantinga (2011) defends the second premise. It examines and rejects several candidate defeater defeaters and defeater deflectors. One candidate is Millikan’s teleosemantics. I show that Plantinga’s motives for rejecting teleosemantics as a defeater deflector are inadequate. I then show that teleosemantics is not on its own an adequate defeater deflector. Then I offer an additional premise that constitutes a defeater deflector in conjunction with teleosemantics.     

Resolution of Algebraic Systems of Equations in the Variety of Cyclic Post Algebras

There is a constructive method to define a structure of simple k-cyclic Post algebra of order p, L _p,k , on a given finite field F(p ^k ), and conversely. There exists an interpretation ??₁ of the variety ${\mathcal{V}(L_{p,k})}$ generated by L _p,k into the variety ${\mathcal{V}(F(p^k))}$ generated by F(p ^k ) and an interpretation ??₂ of ${\mathcal{V}(F(p^k))}$ into ${\mathcal{V}(L_{p,k})}$ such that ??₂??₁(B) =  B for every ${B \in \mathcal{V}(L_{p,k})}$ and ??₁??₂(R) =  R for every ${R \in \mathcal{V}(F(p^k))}$ . In this paper we show how we can solve an algebraic system of equations over an arbitrary cyclic Post algebra of order p, p prime, using the above interpretation, Gröbner bases and algorithms programmed in Maple.     

Semantics for Higher Level Attacks in Extended Argumentation Frames Part 1: Overview

In 2005 the author introduced networks which allow attacks on attacks of any level. So if a → b reads a attacks b, then this attack can itself be attacked by another node c. This attack itself can attack another node d. This situation can be iterated to any level with attacks and nodes attacking other attacks and other nodes. In this paper we provide semantics (of extensions) to such networks. We offer three different approaches to obtaining semantics.

1. The translation approach This uses the methodology of ‘Logic by translation’. We translate faithfully the new networks into ordinary Dung networks with more nodes and extract the semantics from the translation.
2. The labelling approach This method regards the arrows as additional entities to be attacked and to mount attacks and applies a variation of the usual machinery of Camindada like labelling to the network. The new concept we need to employ here is that of ‘joint attacks’.
3. The logic programming approach We translate the higher level network into a logic program and obtain semantics for it through known semantics for logic programs.

We then compare our methods with those of S. Modgil and P. M. Dung et al.     

Logical consequence: A defense of Tarski

In his classic 1936 essay “On the Concept of Logical Consequence”, Alfred Tarski used the notion of satisfaction to give a semantic characterization of the logical properties. Tarski is generally credited with introducing the model-theoretic characterization of the logical properties familiar to us today. However, in his book, The Concept of Logical Consequence, Etchemendy argues that Tarski's account is inadequate for quite a number of reasons, and is actually incompatible with the standard model-theoretic account. Many of his criticisms are meant to apply to the model-theoretic account as well. In this paper, I discuss the following four critical charges that Etchemendy makes against Tarski and his account of the logical properties:

1. Tarski's account of logical consequence diverges from the standard model-theoretic account at points where the latter account gets it right.
2. Tarski's account cannot be brought into line with the model-theoretic account, because the two are fundamentally incompatible.
3. There are simple counterexamples (enumerated by Etchemendy) which show that Tarski's account is wrong.
4. Tarski committed a modal fallacy when arguing that his account captures our pre-theoretical concept of logical consequence, and so obscured an essential weakness of the account.
5. Tarski's account depends on there being a distinction between the “logical terms” and the “non-logical terms” of a language, but (according to Etchemendy) there are very simple (even first-order) languages for which no such distinction can be made. Etchemendy's critique raises historical and philosophical questions about important foundational work. However, Etchemendy is mistaken about each of these central criticisms. In the course of justifying that claim, I give a sustained explication and defense of Tarski's account. Moreover, since I will argue that Tarski's account and the model-theoretic account really do come to the same thing, my subsequent defense of Tarski's account against Etchemendy's other attacks doubles as a defense against criticisms that would apply equally to the familiar model-theoretic account of the logical properties.

     

On Gentzen Relations Associated with Finite-valued Logics Preserving Degrees of Truth

When considering m-sequents, it is always possible to obtain an m-sequent calculus VL for every m-valued logic (defined from an arbitrary finite algebra L of cardinality m) following for instance the works of the Vienna Group for Multiple-valued Logics. The Gentzen relations associated with the calculi VL are always finitely equivalential but might not be algebraizable. In this paper we associate an algebraizable 2-Gentzen relation with every sequent calculus VL in a uniform way, provided the original algebra L has a reduct that is a distributive lattice or a pseudocomplemented distributive lattice. We also show that the sentential logic naturally associated with the provable sequents of this algebraizable Gentzen relation is the logic that preserves degrees of truth with respect to the original algebra (in contrast with the more common logic that merely preserves truth). Finally, for some particular logics we obtain 2-sequent calculi that axiomatize the algebraizable Gentzen relations obtained so far.     

Why objects exist but events occur

I distinguish between sentences like

1. Last Thursday we drove from Wellington to Waikanae and
2. Last Thursday my copy of Aspects of the Theory of Syntax remained on my bookshelf.

Sentence (2) has the subinterval property. If it is true at an interval t it is true at every subinterval of t. (1) lacks this property. (1) reports an event. (2) reports a state. Events do not have the subinterval property but states do have it, and so do objects. If something is a linguist at an interval t then that person is a linguist at all subintervals of t. I argue that ‘exists’ applies to things which have the subdinterval property, and ‘occurs’ applies to things which lack it.