Decidability of monadic first-order classical logic was established by Löwenheim in 1915. The proof made use of a semantic argument, but a purely syntactic proof has never been provided. In the present paper we introduce a syntactic proof of decidability of monadic first-order logic in innex normal form which exploits G3-style sequent calculi. In particular, we introduce a cut- and contraction-free calculus having a (complexity optimal) terminating proof-search procedure. We also show that this logic can be faithfully embedded in the (...) modal logic T. (shrink)

In contrast to Hintikka’s enormously complex distributive normal forms of first- order logic, this paper shows how to generate minimized disjunctive normal forms of first-order logic. An effective algorithm for this purpose is outlined, and the benefits of using minimized disjunctive normal forms to explain the truth conditions of propo- sitions expressible within pure first-order logic are presented.

The syllogistic figures and moods can be taken to be argument schemata as can the rules of the Stoic propositional logic. Sentence schemata have been used in axiomatizations of logic only since the landmark 1927 von Neumann paper [31]. Modern philosophers know the role of schemata in explications of the semantic conception of truth through Tarski’s 1933 Convention T [42]. Mathematical logicians recognize the role of schemata in first-order number theory where Peano’s second-order Induction Axiom is approximated by Herbrand’s Induction-Axiom (...) Schema [23]. Similarly, in first-order set theory, Zermelo’s second-order Separation Axiom is approximated by Fraenkel’s first-order Separation Schema [17]. In some of several closely related senses, a schema is a complex system having multiple components one of which is a template-text or scheme-template, a syntactic string composed of one or more “blanks” and also possibly significant words and/or symbols. In accordance with a side condition the template-text of a schema is used as a “template” to specify a multitude, often infinite, of linguistic expressions such as phrases, sentences, or argument-texts, called instances of the schema. The side condition is a second component. The collection of instances may but need not be regarded as a third component. The instances are almost always considered to come from a previously identified language (whether formal or natural), which is often considered to be another component. This article reviews the often-conflicting uses of the expressions ‘schema’ and ‘scheme’ in the literature of logic. It discusses the different definitions presupposed by those uses. And it examines the ontological and epistemic presuppositions circumvented or mooted by the use of schemata, as well as the ontological and epistemic presuppositions engendered by their use. In short, this paper is an introduction to the history and philosophy of schemata. (shrink)

Die ersten Bemerkungen Wittgensteins zur Identität stammen vom Herbst 1913; Spuren zeitweise intensiver Beschäftigung mit dem Thema finden sich indes bis in die Logisch-philosophische Abhandlung. Die vorliegende Arbeit versucht jenen Gedankenläufen Wittgensteins zu folgen, die in seine eigentümliche Schreibweise gebundener Variablen und der Streichung des Identitätszeichens aus der Begriffsschrift mündeten. Neben der Typentheorie Russells hat,was im ersten Abschnitt gezeigt wird, besonders seine Analyse definiter Beschreibungen Wittgensteins frühe Gedanken zur Identität angeregt, zumal er sich nach der Ablehnung von Russells Definition der (...) Identität der neuerlichen Frage gegenübersah, ob sich Sätze, worin das Zeichen der Identität vorkommt, weiter zerlegen lassen oder ob dieses vielmehr als Urzeichen angenommen werden müsse. Diese Frage stand für Wittgenstein in engem Zusammenhang mit dem,was er als das Grundproblem der Logik ausgemacht zu haben glaubte, d. i. mit der Frage nach der Beschaffenheit eines idealen Zeichensystems, das jede Tautologie auf ein und dieselbe Weise als Tautologie erkennen ließe. Wie in einer ausführlichen Besprechung der ab-Notation im dritten Abschnitt nachgewiesen wird, war die Lösung des Grundproblems denn auch das Ziel seines fieberhaften, letztlich aber erfolglosen Versuchs, die ab-Notation auf allgemeine Sätze - und insbesondere auf solche, worin das Identitätszeichen vorkommt - auszuweiten. Der vierte Abschnitt schließlich folgt Wittgensteins neuerlichem Anlauf zur Lösung des Grundproblems, der ausgeht von einer früheren Betrachtung zu dem bereits im zweiten Abschnitt behandelten Phänomen der Rekurrenz und deren Affinität zur Frage nach der richtigen Handhabung der Identität. Am Ende erweist sich die durch eine veränderte Variablenschreibweise ohne Verlust an Ausdruckskraft ermöglichte Streichung des Identitätszeichens gleichsam als Begleitprodukt der umfassenderen Bemühung, das Grundproblem der Logik doch noch einer Lösung zuzuführen. (shrink)

I use van Heijenoort’s published writings and manuscript materials to provide a comprehensive overview of his conception of modern logic as a first-order functional calculus and of the historical developments which led to this conception of mathematical logic, its defining characteristics, and in particular to provide an integral account, from his most important publications as well as his unpublished notes and scattered shorter historico-philosophical articles, of how and why the mathematical logic, whose he traced to Frege and the culmination of (...) its formative period in the incompleteness results of Gödel, became modern logic, as distinct from the traditional logic of Aristotle, and why and how the logistic tradition that led from Frege through Russell, rather than the algebraic tradition that led from De Morgan and Boole through Peirce and Schröder, came, in his view, to define modern logic. (shrink)

The five English words—sentence, proposition, judgment, statement, and fact—are central to coherent discussion in logic. However, each is ambiguous in that logicians use each with multiple normal meanings. Several of their meanings are vague in the sense of admitting borderline cases. In the course of displaying and describing the phenomena discussed using these words, this paper juxtaposes, distinguishes, and analyzes several senses of these and related words, focusing on a constellation of recommended senses. One of the purposes of this paper (...) is to demonstrate that ordinary English properly used has the resources for intricate and philosophically sound investigation of rather deep issues in logic and philosophy of language. No mathematical, logical, or linguistic symbols are used. Meanings need to be identified and clarified before being expressed in symbols. We hope to establish that clarity is served by deferring the extensive use of formalized or logically perfect languages until a solid “informal” foundation has been established. Questions of “ontological status”—e.g., whether propositions or sentences, or for that matter characters, numbers, truth-values, or instants, are “real entities”, are “idealizations”, or are “theoretical constructs”—plays no role in this paper. As is suggested by the title, this paper is written to be read aloud. -/- I hope that reading this aloud in groups will unite people in the enjoyment of the humanistic spirit of analytic philosophy. (shrink)