Showing papers in "Notre Dame Journal of Formal Logic in 1983"

An axiomatization of predicate functor logic.

Abstract: Axiomatisation d'une logique avec foncteur de predicat inventee par Quine et dont la caracteristique principale est de fournir un equivalent de la logique elementaire libre de variables. L'A. propose une procedure d'enumeration recursive des formules de la logique avec foncteur de predicat qui ont leur equivalent en logique elementaire.

The Validity of Disjunctive Syllogism Is Not So Easily Proved

Abstract: Note a propos de l'article de J. Burgess ("Relevance: A Fallacy", ibid, 1981, pp. 97-104), portant sur la validite du syllogisme disjonctif. L'A. se situe dans le cadre des logiques de la pertinence et denie, contre Burgess, la validite universelle de ce type de syllogisme. P. 41-53, reponse de J. BURGESS: "Common Sense and Relevance". L'A. fait preceder sa reponse a l'article de C. Mortensen d'une mise au point detaillee sur les logiques de la pertinence.

On the Borel Classification of the Isomorphism Class of a Countable Model

Abstract: Introduction For p, a countable similarity type, let Xp be the space of structures of similarity type p whose universe is ω (see [13], Section 3). For any element & of Xp, let \d/\ be the set of all elements of Xp which are isomorphic to CO. Scott [10] showed that \β/\ is a Borel subset of Xp. In fact, he showed that for any such (2/ there is a sentence θ of Lωχω such that \C0\ is exactly the set of elements of Xp which are models of θ (see [ 1 ], Ch. VII, for a good write-up of Scott sentences). In [13] Vaught considerably strengthened Scott's result. There is a natural hierarchy of formulas of Lωχω. Let ΠQ = ΣQ be the quantifier-free first-order formulas. For any a > 1 the Π° formulas are those of the form:

Completeness of an ecthetic syllogistic.

Abstract: In this paper I study a formal model for Aristotelian syllogistic which includes deductive procedures designed to model the \"proof by ecthesis\" that Aristotle sometimes uses and in which all deductions are direct. The resulting system is shown to be contained within another formal model for the syllogistic known to be both sound and complete, and in addition the system is proved to have a certain limited form of completeness.

Common sense and ``relevance''.

Abstract: Introduction The "relevance" logicians of North America and the "relevant" logicians of the Antipodes have long claimed that their logical systems are in better agreement with common sense than is classical logic. The present paper is a critical examination of such claims. Though it is part of an on-going debate between classical and relevantistic logicians, I have tried to keep my discussion of the issues self-contained.

Sequential compactness and the axiom of choice.

Abstract: A theorem is effective iff it is proved in ZF°, where ZF° is ZermeloFraenkel set theory without the axioms of choice and foundation (regularity) A well known effective theorem of F Riesz states that a Hubert space is finite dimensional iff its closed unit ball is compact This may fail for sequential compactness If U is a set of urelements, equipped with the structure of 12, / is the ideal of all finite subsets of U and G is the group of unitary operators, by an argument similar to [7] In the resulting permutation model P(U,G,I), each orthonormal (= ON) system in U is finite Therefore U is locally sequentially compact, but there is no ON base for U A similar situation holds for the Dworetzky-Rogers characterisation of finite dimensional spaces ([9], Theorem Ie2) But in combination we get the effective result:

The Published Work of Kurt Godel: An Annotated Bibliography

Abstract: Bibliographie commentee et complete des oeuvres de K. Godel. Des appendices fournissent: a) un index des compte rendus d'oeuvres de Godel| b) un index de photographie| c) une table de correspondance des traductions.

KM and the finite model property.

Abstract: A partir de K. Fine "Normal forms in modal Logic" (ibid., 16, 1975, p. 35-40) l'A. etend l'attribution de la propriete de modele fini au systeme modal KM vers une preuve de completude de type classique relativement a la theorie des modeles.

Plantinga's theory of proper names.

Abstract: Plantinga also argues that his theory avoids one criticism which he takes to be very damaging to theories of proper names held by Russell and Frege. I will argue that Plantinga's theory is unsatisfactory in its handling of the puzzle presented by propositional identity in the context of propositional attitudes. In order to motivate Plantinga's theory, I will begin by giving a very brief statement of the criticism which Plantinga takes to be very damaging to Russell's and Frege's theories of naming. Then, I will state the puzzle as Plantinga renders it, presented by propositional identity in the context of propositional attitudes. Next, I will show how Plantinga's own theory avoids that criticism and at least appears to resolve the puzzle. Two objections to Plantinga's theory will then be presented. I will also consider some replies that Plantinga might reasonably make. In giving my objections, I have endeavored to present an "internal criticism" of Plantinga's view; that is, I have

Some preservation results for classical and intuitionistic satisfiability in Kripke models.

Abstract: Translations of classical into intuitionistic formal systems, as defined by Gδdel and others (for a survey see [2], Section 81 [4] p. 41 or [6]), provide among other things a method for determining which classically valid formulas are intuitionistically valid. All of the translations share the following property: if T is an intuitionistic theory and T its classical counterpart (obtained, e.g., by adding the law of excluded middle) and if φ is a formula in an appropriate language (built up with connectives v, Λ, ->, Ί , 3, V) and φ its translation then:

On the Freyd cover of a topos.

Abstract: A theory is said to have the disjunction-property (DP) if whenever a disjunction φ v φ is provable in the theory, either φ or φ must be provable. As is well-known, many theories for intuitionistic arithmetic and analysis have the DP. The DP for intuitionistic type theory was first established by Friedman. More recently, a purely topos theoretic proof has been given by Freyd. An extensive discussion of both methods can be found in [4]. Although Freyd's construction is much more elegant, A. δcedrov and P. Scott have shown that the two methods are essentially the same in [7]. A question that arises immediately is the following: If one adds new symbols and a particular set of axioms T to the logical axioms and rules, does the resulting higher-order theory still have the DPΊ Some instances of this question in which T consists of a single axiom have been considered in [5]. In this note, we will obtain a syntactic description of a class of theories that have the DP by investigating some of the logical properties of the Freyd cover, thus extending the results of [5]. The results will not cover many of the higher-order analogues of theories of intuitionistic arithmetic and analysis which are known to have the DP. One reason for this is that, from a more logical point of view, the Freyd cover lacks many nice properties. For an alternative type of cover that fills this gap, the reader is referred to [6]. In the first section of this paper, we will motivate the Freyd cover from a more logical perspective. There is probably nothing new in this, but it still is important to realize that what is really going on is a straightforward generalization of more traditional methods used in the model theory of first-order

Compactness via prime semilattices.

Abstract: Compactness is certainly one of the most fruitful conceptsof general topology. Topologically inspired notions of compactness have alsoproven useful in logic (see [3], [9]) and measure theory (see [8]). In this paperwe introduce a definition of compactness for subsets of a prime semilattice.Prime semilattices were introduced by Balbes [2] and their algebraic structureseems just right for presenting the ideas which underlie many compactnessarguments.