Ebook: First-Order Logic
Author: Raymond M. Smullyan (auth.)
- Tags: Mathematics general
- Series: Ergebnisse der Mathematik und ihrer Grenzgebiete 43
- Year: 1968
- Publisher: Springer-Verlag Berlin Heidelberg
- Edition: 1
- Language: English
- pdf
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).
Content:
Front Matter....Pages I-XII
Front Matter....Pages 1-1
Preliminaries....Pages 3-14
Analytic Tableaux....Pages 15-30
Compactness....Pages 30-40
Front Matter....Pages 41-41
First-Order Logic. Preliminaries....Pages 43-52
First-Order Analytic Tableaux....Pages 52-65
A Unifying Principle....Pages 65-70
The Fundamental Theorem of Quantification Theory....Pages 70-79
Axiom Systems for Quantification Theory....Pages 79-86
Magic Sets....Pages 86-91
Analytic versus Synthetic Consistency Properties....Pages 91-97
Front Matter....Pages 99-99
Gentzen Systems....Pages 101-110
Elimination Theorems....Pages 110-117
Prenex Tableaux....Pages 117-121
More on Gentzen Systems....Pages 121-127
Craig’s Interpolation Lemma and Beth’s Definability Theorem....Pages 127-133
Symmetric Completeness Theorems....Pages 133-141
Systems of Linear Reasoning....Pages 141-155
Back Matter....Pages 156-160
Content:
Front Matter....Pages I-XII
Front Matter....Pages 1-1
Preliminaries....Pages 3-14
Analytic Tableaux....Pages 15-30
Compactness....Pages 30-40
Front Matter....Pages 41-41
First-Order Logic. Preliminaries....Pages 43-52
First-Order Analytic Tableaux....Pages 52-65
A Unifying Principle....Pages 65-70
The Fundamental Theorem of Quantification Theory....Pages 70-79
Axiom Systems for Quantification Theory....Pages 79-86
Magic Sets....Pages 86-91
Analytic versus Synthetic Consistency Properties....Pages 91-97
Front Matter....Pages 99-99
Gentzen Systems....Pages 101-110
Elimination Theorems....Pages 110-117
Prenex Tableaux....Pages 117-121
More on Gentzen Systems....Pages 121-127
Craig’s Interpolation Lemma and Beth’s Definability Theorem....Pages 127-133
Symmetric Completeness Theorems....Pages 133-141
Systems of Linear Reasoning....Pages 141-155
Back Matter....Pages 156-160
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