Showing papers in "History and Philosophy of Logic in 1993"
TL;DR: The authors describes the evolution and transformation of the compactness theorem during the period 1930-1970, with special attention to the roles of Kurt Godel, A. I. Maltsev, Leon Henkin, Abraham Robinson, and Alfred Tarski.
Abstract: Though regarded today as one of the most important results in logic, the compactness theorem was largely ignored until nearly two decades after its discovery. This paper describes the vicissitudes of its evolution and transformation during the period 1930-1970, with special attention to the roles of Kurt Godel, A. I. Maltsev, Leon Henkin, Abraham Robinson, and Alfred Tarski
69 citations
TL;DR: Second-order logic has been considered as a logical framework for set theory as mentioned in this paper, and it performs its role in the underlying logic of set theory, which is the basis of the set theory of this paper.
Abstract: Because of its capacity to characterize mathematical concepts and structures—a capacity which first-order languages clearly lack—second-order languages recommend themselves as a convenient framework for much of mathematics, including set theory. This paper is about the credentials of second-order logic:the reasons for it to be considered logic, its relations with set theory, and especially the efficacy with which it performs its role of the underlying logic of set theory
18 citations
TL;DR: The concept of formal ontology was first developed by Husserl and it concerns problems relating to the notions of object, substance, property, part, whole, predication, nominalization, etc as discussed by the authors.
Abstract: The concept of formal ontology was first developed by Husserl. It concerns problems relating to the notions of object, substance, property, part, whole, predication, nominalization, etc. The idea of formal ontology is present in many of Husserl’s works, with minor changes. This paper provides a reconstruction of such an idea. Husserl’s proposal is faced with contemporary logical orthodoxy and it is presented also an interpretative hypothesis, namely that the original difference between the general perspective of usual model theory and formal ontology is grounded in the fact that this latter starts from an intended interpretation and not from the set of all the possible interpretations
14 citations
TL;DR: Thomas Solly's A syllabus of logic (1839) is the first English tract where symbolical representation and mathematical methods are introduced to explain the nature of abstract conceptions and exhibit properties of syllogistic laws.
Abstract: Thomas Solly’s A syllabus of logic (1839) is the first English tract where symbolical representation and mathematical methods are introduced to explain the nature of abstract conceptions and exhibit properties of syllogistic laws. Solly’s innovations had no effect on the development of algebraic logic, and his work is basically unknown in our century. This paper rescues from oblivion an interesting attempt at the mathematization of logic, investigating its mathematical and logical origins, as well as connecting it with the work of his successors. Two unpublished letters to A. De Morgan are used, and Solly’s possible contacts with other contemporaries, especially D. F. Gregory, are considered
9 citations
TL;DR: Tarski 1968 makes a move in the course of providing an account of "definitionally equivalent" classes of algebras with a businesslike lack of fanfare and commentary, the significance of which may accordingly be lost on the casual reader as discussed by the authors.
Abstract: Tarski 1968 makes a move in the course of providing an account of ’definitionally equivalent’ classes of algebras with a businesslike lack of fanfare and commentary, the significance of which may accordingly be lost on the casual reader. In §1 we present this move as a response to a certain difficulty in the received account of what it is to define a function symbol (or ’operation symbol’). This difficulty, which presents itself as a minor technicality needing to be got around especially for the case of symbols for zero-place functions (for ’distinguished elements’), has repercussions—not widely recognised—for the account of functional completeness in sentential logic. A similarly stark comment in Church 1956 reveals an appreciation of this difficulty, though not every subsequent author on the topic has taken the point. We fill out this side of the picture in §2. The discussion of functional completeness in §2 is supplemented by some remarks on what is involved in defining a connective, which have been in...
8 citations
TL;DR: In this article, the authors show that the falsification of the Law of the Excluded Middle (LEM) lies on confusion or can be circumvented, but a few of them cannot.
Abstract: Most claims for the falsification of the Law of the Excluded Middle (LEM) rest on confusion or can be circumvented. A few of them, however, cannot. I concentrate on two of those, (a) cases involving conflicting criteria(border line cases) and (b) cases imposed by quantum discontinuity(no real state admitted between any two consecutive states). Nevertheless, despite the authenticity of both, closer analysis reveals these two modes of violating LEM to be direct opposites In (a) LEM fails in the relation between language and the world, and not in the world when viewed independently of this relation; in (b) LEM fails in the world itself, if at all. In (a) LEM fails because the Law of Non-Contradiction fails first (mutually exclusive classifications are equally warranted). But in (b) LEM fails because The Law of Non-Contradiction holds and not otherwise Finally, ‘a statement is neither true nor false’ and ‘two contradictory assertions are both false’ should be equivalent ways of expressing LEM’s failure. Howev...
4 citations
TL;DR: Aristotle's treatment of mixed, first-figure, problematic-assertoric syllogisms has generated a good deal of controversy among modern commentators as discussed by the authors, and they argue that W.D.Ross's criticism of A.Becker's treatment is unsuccessful.
Abstract: Aristotle’s treatment of mixed, first-figure, problematic-assertoric syllogisms has generated a good deal of controversy among modern commentators.I argue that W.D.Ross’s criticism of A.Becker’s criticism of Aristotle’s treatment is unsuccessful.I then attempt to justify Aristotle’s treatment, employing some ideas found in Alexander of Aphrodisias
3 citations
TL;DR: The authors argue that far from contributing directly to oral instruction, the axiomatic account of demonstrative science is a model for the written expression of science, and show how this interpretation accords with related theories in the Organon, including the theories of dialectic in Topics and of deduction in Prior analytics.
Abstract: To meet a dilemma between the axiomatic theory of demonstrative science in Posterior analyticsand the non-aximatic practice of demonstrative science in the physical treatises, Jonathan Barnes has proposed that the theory of demonstration was not meant to guide scientific research but rather scientific pedagogy. The present paper argues that far from contributing directly to oral instruction, the axiomatic account of demonstrative science is a model for the written expression of science.The paper shows how this interpretation accords with related theories in the Organon, including the theories of dialectic in Topicsand of deduction in Prior analytics
2 citations
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TL;DR: In this article, the main result of the project will be the publication of a book on Pieri, in which we will mainly study his contributions to the foundations of geometry and translate into English his major contributions.
Abstract: In this paper we briefly expose a project which could be summed up as ‘doing justice to Mario Pieri’. The main result of the project will be the publication of a book on him, in which we will mainly study his contributions to the foundations of geometry and translate into English his major contributions to the field
1 citations