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First-Order Logic

  • Book
  • © 1968

Overview

Authors:
  1. Raymond M. Smullyan

    1. Lehman College, City University of New York, USA

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Part of the book series: Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge (MATHE2, volume 43)

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Table of contents (17 chapters)

  1. Front Matter

    Pages I-XII
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  2. Propositional Logic from the Viewpoint of Analytic Tableaux

    1. Front Matter

      Pages 1-1
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    2. Preliminaries

      • Raymond M. Smullyan
      Pages 3-14

    3. Analytic Tableaux

      • Raymond M. Smullyan
      Pages 15-30

    4. Compactness

      • Raymond M. Smullyan
      Pages 30-40

  3. First-Order Logic

    1. Front Matter

      Pages 41-41
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    2. First-Order Logic. Preliminaries

      • Raymond M. Smullyan
      Pages 43-52

    3. First-Order Analytic Tableaux

      • Raymond M. Smullyan
      Pages 52-65

    4. A Unifying Principle

      • Raymond M. Smullyan
      Pages 65-70

    5. The Fundamental Theorem of Quantification Theory

      • Raymond M. Smullyan
      Pages 70-79

    6. Axiom Systems for Quantification Theory

      • Raymond M. Smullyan
      Pages 79-86

    7. Magic Sets

      • Raymond M. Smullyan
      Pages 86-91

    8. Analytic versus Synthetic Consistency Properties

      • Raymond M. Smullyan
      Pages 91-97

  4. Further Topics in First-Order Logic

    1. Front Matter

      Pages 99-99
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    2. Gentzen Systems

      • Raymond M. Smullyan
      Pages 101-110

    3. Elimination Theorems

      • Raymond M. Smullyan
      Pages 110-117

    4. Prenex Tableaux

      • Raymond M. Smullyan
      Pages 117-121

    5. More on Gentzen Systems

      • Raymond M. Smullyan
      Pages 121-127

    6. Craig’s Interpolation Lemma and Beth’s Definability Theorem

      • Raymond M. Smullyan
      Pages 127-133

    7. Symmetric Completeness Theorems

      • Raymond M. Smullyan
      Pages 133-141
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Keywords

About this book

Except for this preface, this study is completely self-contained. It is intended to serve both as an introduction to Quantification Theory and as an exposition of new results and techniques in "analytic" or "cut-free" methods. We use the term "analytic" to apply to any proof procedure which obeys the subformula principle (we think of such a procedure as "analysing" the formula into its successive components). Gentzen cut-free systems are perhaps the best known example of ana­ lytic proof procedures. Natural deduction systems, though not usually analytic, can be made so (as we demonstrated in [3]). In this study, we emphasize the tableau point of view, since we are struck by its simplicity and mathematical elegance. Chapter I is completely introductory. We begin with preliminary material on trees (necessary for the tableau method), and then treat the basic syntactic and semantic fundamentals of propositional logic. We use the term "Boolean valuation" to mean any assignment of truth values to all formulas which satisfies the usual truth-table conditions for the logical connectives. Given an assignment of truth-values to all propositional variables, the truth-values of all other formulas under this assignment is usually defined by an inductive procedure. We indicate in Chapter I how this inductive definition can be made explicit-to this end we find useful the notion of a formation tree (which we discuss earlier).

Authors and Affiliations

  • Lehman College, City University of New York, USA

    Raymond M. Smullyan

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