On superintuitionistic logics as fragments of proof logic extensions

Coming fromI andCl, i.e. from intuitionistic and classical propositional calculi with the substitution rule postulated, and using the sign to add a new connective there have been considered here: Grzegorozyk's logicGrz, the proof logicG and the proof-intuitionistic logicI set up correspondingly by the calculiFor any calculus we denote by the set of all formulae of the calculus and by the lattice of all logics that are the extensions of the logic of the calculus, i.e. sets of formulae containing the axioms of and closed with respect to its rules of inference. In the logiclG the sign is decoded as follows: A = (A & A). The result of placing in the formulaA before each of its subformula is denoted byTrA. The maps are defined (in the definitions of x and the decoding of is meant), by virtue of which the diagram is constructedIn this diagram the maps, x and are isomorphisms, thereforex ^–1 = ; and the maps and are the semilattice epimorphisms that are not commutative with lattice operation +. Besides, the given diagram is commutative, and the next equalities take place: ^–1 = ^–1 and = ^–1 x. The latter implies in particular that any superintuitionistic logic is a superintuitionistic fragment of some proof logic extension.     

Paraconsistent logic and model theory

The object of this paper is to show how one is able to construct a paraconsistent theory of models that reflects much of the classical one. In other words the aim is to demonstrate that there is a very smooth and natural transition from the model theory of classical logic to that of certain categories of paraconsistent logic. To this end we take an extension of da Costa'sC ₁ ⁼ (obtained by adding the axiom A A) and prove for it results which correspond to many major classical model theories, taken from Shoenfield [5]. In particular we prove counterparts of the theorems of o-Tarski and Chang-o-Suszko, Craig-Robinson and the Beth definability theorem.     

The elimination of de re formulas

It is shown that de re formulas are eliminable in the modal logic S5 extended with the axiom scheme x x.     

Universality of the closure space of filters in the algebra of all subsets

In this paper we show that some standard topological constructions may be fruitfully used in the theory of closure spaces (see [5], [4]). These possibilities are exemplified by the classical theorem on the universality of the Alexandroff's cube for T ₀-closure spaces. It turns out that the closure space of all filters in the lattice of all subsets forms a generalized Alexandroff's cube that is universal for T ₀-closure spaces. By this theorem we obtain the following characterization of the consequence operator of the classical logic: If is a countable set and C: P() P() is a closure operator on X, then C satisfies the compactness theorem iff the closure space ,C is homeomorphically embeddable in the closure space of the consequence operator of the classical logic.We also prove that for every closure space X with a countable base such that the cardinality of X is not greater than 2 there exists a subset X of irrationals and a subset X of the Cantor's set such that X is both a continuous image of X and a continuous image of X.We assume the reader is familiar with notions in [5].     

On reduced matrices

It is shown that the class of reduced matrices of a logic is a 1 ^st order -class provided the variety associated with has the finite replacement property in the sense of [7]. This applies in particular to all 2-valued logics. For 3-valued logics the class of reduced matrices need not be 1 ^st order.     

Hemingway's Aging Heroes and the Concept of Phronesis

This article explores the concept of phronesis (practical wisdom) in four aging protagonists in Ernest Hemingway's works ranging from the short story A Clean Well-Lighted Place to the novella The Old Man and the Sea. Phronesis represents an understanding of the ways of the world, an acute sensitivity to a critical logic of human existence that can be attained only through extensive experience and suffering. The four aging protagonists are examples of Hemingway's definition of aging productively and profitably by purchasing an inner peace that consists of an intuitive system of continuous adjustment to the exigencies of daily living.     

Quantum Logic as a Fragment of Independence-Friendly Logic

The working assumption of this paper is that noncommuting variables are irreducibly interdependent. The logic of such dependence relations is the author's independence-friendly (IF) logic, extended by adding to it sentence-initial contradictory negation ¬ over and above the dual (strong) negation . Then in a Hilbert space turns out to express orthocomplementation. This can be extended to any logical space, which makes it possible to define the dimension of a logical space. The received Birkhoff and von Neumann quantum logic can be interpreted by taking their disjunction to be ¬(A & B). Their logic can thus be mapped into a Boolean structure to which an additional operator has been added.     

Reasoning and logic

Gilbert Harman, in Logic and Reasoning (Synthese 60 (1984), 107–127) describes an unsuccessful attempt ... to develop a theory which would give logic a special role in reasoning. Here reasoning is psychological, a procedure for revising one's beliefs. In the present paper, I construe reasoning sociologically, as a process of linguistic interaction; and show how both reasoning in the psychologistic sense and logic are related to that process.     

Context and strategy in acquiring temporal connectives

There has been considerable debate over the role of comprehension strategies in the acquisition of temporal connectives. This study examined the role of caluse logic and interpretational strategies in the acquisition of temporal words; age-related changes were also considered. Thirty-two children between 3 and 5 years of age served as subjects. Sentences with a variety of temporal words were used to tap children's comprehension of before, after, when, while, just before that, and and after that. Clause logic was found to significantly improve the understanding of these sentences. However, the order of mention and main clause first strategies were used infrequently. Apparently, when children respond to sentences that describe information in a way that is consistent with that they normally hear, these strategies are seldom applied.I wish to express my thanks to Janice Bogen, who assisted in data collection, and to Erika Hoff-Ginsberg, who assisted in some of the data analysis.     

Depression and Anxiety in Persons with Parkinson's Disease With and Without “On–Off” Phenomena

On–off phenomena in Parkinson's disease (PD) are unpredictable motor fluctuations associated with long-term levodopa use. Mood fluctuations have been found to coincide with the motor fluctuations in that depression and anxiety increase while the person with PD is in the off state. What has been relatively unexplored is whether those persons with PD who have on–off phenomena differ psychologically in fundamental ways from those who do not have on–off phenomena. In the present study, depression and anxiety symptoms were assessed in 36 persons with PD (n = 14 with on–off phenomena, n = 22 without on–off phenomena). All those with on–off phenomena were assessed in their on state. Those persons with PD with on–off phenomena had significantly higher levels of anxiety than those without on–off phenomena. However, both groups, regardless of on–off status, were mildly depressed. Neurobiological interpretations of the results implicate the locus coeruleus in the pathogenesis of both on–off phenomena and anxiety, whereas psychological interpretations of the results involve the issues of learned helplessness and control over health symptoms in PD.     

Strong Boethius’ Thesis and Consequential Implication

The paper studies the relation between systems of modal logic and systems of consequential implication, a non-material form of implication satisfying Aristotle's Thesis (p does not imply not p ) and Weak Boethius' Thesis (if p implies q, then p does not imply not q ). Definitions are given of consequential implication in terms of modal operators and of modal operators in terms of consequential implication. The modal equivalent of Strong Boethius' Thesis (that p implies q implies that p does not imply not q) is identified.     

The fallacy of fallacies

Several of the so-called fallacies in Aristotle are not in fact mistaken inference-types, but mistakes or breaches of rules in the questioning games which were practiced in the Academy and in the Lyceum. Hence the entire Aristotelian theory of fallacies ought to be studied by reference to the author's interrogative model of inquiry, based on his theory of questions and answers, rather than as a part of the theory of inference. Most of the fallacies mentioned by Aristotle can in fact be diagnosed by means of the interrogative model, including petitio principii, multiple questions, babbling, etc., and so can Aristotle's alleged anticipation of the fallacy of argumentum ad hominem. The entire Aristotelian conception of inquiry is an interrogative one. Deductive conclusions caught Aristotle's attention in the form of answers that every rational interlocutor must give, assuming only his own earlier answers. Several features of Aristotle's methodology can be understood by means of the interrogative model, including the role of endoxa in it. Theoretically, there is also considerable leeway as to whether fallacies are conceived of as mistakes in questioning or as breaches of the rules that govern questioning games.     

A note on syntactical and semantical functions

We say that a semantical function is correlated with a syntactical function F iff for any structure A and any sentence we have A F A .It is proved that for a syntactical function F there is a semantical function correlated with F iff F preserves propositional connectives up to logical equivalence. For a semantical function there is a syntactical function F correlated with iff for any finitely axiomatizable class X the class ^–1X is also finitely axiomatizable (i.e. iff is continuous in model class topology).     

Les “règles de la discussion légitime” dans la logique de Port-Royal

In the XVIIth century the conflict which opposed the jansenists to the jesuits involved the problem of the due process in theological matter. The jesuits heralded the thesis that the infallibility of the Church has to be extended from dogmatics (quaestio iuris) to the historical facts (quaestio facti). On the opposite side Arnauld maintained that such an opinion was monstruous: also in religious matters the fact has to be proved according to the principles of a due process, and not by authority. In this article the thesis pleaded by the jansenists is considered in connection with the model of argumentative procedure offered by the Port-Royal logic.The Logique ou Art de penser (1622) by Antoine Arnauld and Pierre Nicole seems to have rediscovered the classical principles of the theory of argumentation: from the burden of proof to the idea of probable truth. But really a new model of adversary-system has been introduced into the modern mind, which is very different in concept from the topical tradition. The basic metaphor of combat, implying that the truth will prevail in the fight, is compatible with the epistemological premises of the modern logic (as the separation between fact and value). Therefore the problem of the fact-finding seems to be attracted into the area of the logic of information, and not of the theory of argumentation.

     

On finite approximability of ψ-intermediate logics

The aim of this note is to show (Theorem 1.6) that in each of the cases: = {, }, or {, , }, or {, , } there are uncountably many -intermediate logics which are not finitely approximable. This result together with the results known in literature allow us to conclude (Theorem 2.2) that for each : either all -intermediate logics are finitely approximate or there are uncountably many of them which lack the property.     

Metaphors for God: Why and How Do Our Choices Matter for Humans? The Application of Contemporary Cognitive Linguistics Research to the Debate on God and Metaphor

This article explores and critiques conservative and liberal theological understandings of metaphor in light of the contemporary research in cognitive linguistics. This assessment is followed by a cognitive examination of the biblical metaphors for God, their unconscious entailments, an assessment of why certain metaphors are more effective than others, and a short discussion of the implications of this effectiveness.     

Some aspects of eloquence in Descartes's works

In Descartes's philosophy, communicating scientific and philosophical truth does not represent a problem that can be traced back to humanistic rhetoric, meant as the art of persuasion. Descartes states his belief in the eloquence of reason: a clear, precise, and adequately expressed thought cannot fail to convince the listener. This is the measure of the distance between the level of truth and the level of opinion. However, the moment of confrontation with the public is also the very moment when the truth of the new knowledge enters into conflict with other, different conceptions. Education and history influence communication with the result that the distinction between intellectual conviction and persuasion becomes less straightforward. Rational eloquence, as Descartes is well aware, must be articulated in such a way as to avoid any possible language equivocation and to adopt exposition strategies ensuring effective access to readers. The aim of this paper is to illustrate some aspects of this tension as expressed by the writer Descartes with reference to a number of texts (from the Regulae to the Meditationes) that were essential for the elaboration and dissemination of his philosophy.     

A Sound and Complete Tableau Calculus for Reasoning about only Knowing and Knowing at Most

We define a tableau calculus for the logic of only knowing and knowing at most ON, which is an extension of Levesque's logic of only knowing O. The method is based on the possible-world semantics of the logic ON, and can be considered as an extension of known tableau calculi for modal logic K45. From the technical viewpoint, the main features of such an extension are the explicit representation of "unreachable" worlds in the tableau, and an additional branch closure condition implementing the property that each world must be either reachable or unreachable. The calculus allows for establishing the computational complexity of reasoning about only knowing and knowing at most. Moreover, we prove that the method matches the worst-case complexity lower bound of the satisfiability problem for both ON and O. With respect to [22], in which the tableau calculus was originally presented, in this paper we both provide a formal proof of soundness and completeness of the calculus, and prove the complexity results for the logic ON.     

Yes,Virginia, Lemons are Yellow

This paper discusses a number of themes and arguments in The Quest for Reality: Stroud's distinction between philosophical and ordinary questions about reality; the similarity he finds between the view that coloris unreal and the view that it is subjective; his argument against thesecondary quality theory; his argument against the error theory; and the disappointing conclusion of the book.