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Overview
This completely self-contained study, widely considered the best book in the field, 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. Impressed by the simplicity and mathematical elegance of the tableau point of view, the author focuses on it here.
After preliminary material on tress (necessary for the tableau method), Part I deals with propositional logic from the viewpoint of analytic tableaux, covering such topics as formulas or propositional logic, Boolean valuations and truth sets, the method of tableaux and compactness.
Part II covers first-order logic, offering detailed treatment of such matters as first-order analytic tableaux, analytic consistency, quantification theory, magic sets, and analytic versus synthetic consistency properties.
Part III continues coverage of first-order logic. Among the topics discussed are Gentzen systems, elimination theorems, prenex tableaux, symmetric completeness theorems, and system linear reasoning.
Raymond M. Smullyan is a well-known logician and inventor of mathematical and logical puzzles. In this book he has written a stimulating and challenging exposition of first-order logic that will be welcomed by logicians, mathematicians, and anyone interested in the field.
Product Details
| ISBN-13: | 9780486683706 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 01/30/1995 |
| Series: | Dover Books on Mathematics Series |
| Pages: | 176 |
| Sales rank: | 334,664 |
| Product dimensions: | 5.50(w) x 8.50(h) x (d) |
About the Author
Raymond Smullyan received his PhD from Princeton University and taught at Dartmouth, Princeton, Indiana University, and New York's Lehman College. Best known for his mathematical and creative logic puzzles and games, he was also a concert pianist and a magician. He wrote over a dozen books of logic puzzles and texts on mathematical logic.
Raymond Smullyan: The Merry Prankster
Raymond Smullyan (1919–2017), mathematician, logician, magician, creator of extraordinary puzzles, philosopher, pianist, and man of many parts. The first Dover book by Raymond Smullyan was First-Order Logic (1995). Recent years have brought a number of his magical books of logic and math puzzles: The Lady or the Tiger (2009); Satan, Cantor and Infinity (2009); an original, never-before-published collection, King Arthur in Search of His Dog and Other Curious Puzzles (2010); and Set Theory and the Continuum Problem (with Melvin Fitting, also reprinted by Dover in 2010). More will be coming in subsequent years.
In the Author's Own Words:
"Recently, someone asked me if I believed in astrology. He seemed somewhat puzzled when I explained that the reason I don't is that I'm a Gemini."
"Some people are always critical of vague statements. I tend rather to be critical of precise statements: they are the only ones which can correctly be labeled 'wrong.'" — Raymond Smullyan
Table of Contents
Part I. Propositional Logic from the Viewpoint of Analytic TableauxChapter I. Preliminaries
0. Foreword on Trees
1. Formulas of Propositional Logic
2. Boolean Valuations and Truth Sets
Chapter II. Analytic Tableaux
1. The Method of Tableaux
2. Consistency and Completeness of the System
Chapter III. Compactness
1. Analytic Proofs of the Compactness Theorem
2. Maximal Consistency: Lindenbaum's Construction
3. An Analytic Modification of Lindenbaum's Proof
4. The Compactness Theorem for Deducibility
Part II. First-Order Logic
Chapter IV. First-Order Logic. Preliminaries
1. Formulas of Quantification Theory
2. First-Order Valuations and Models
3. Boolean Valuations vs. First-Order Valuations
Chapter V. First-Order Analytic Tableaux
1. Extension of Our Unified Notation
2. Analytic Tableaux for Quantification Theory
3. The Completeness Theorem
4. The Skolem-Löwenheim and Compactness Theorems for First-Order Logic
Chapter VI. A Unifying Principle
1. Analytic Consistency
2. Further Discussion of Analytic Consistency
3. Analytic Consistency Properties for Finite Sets
Chapter VII. The Fundamental Theorem of Quantification Theory
1. Regular Sets
2. The Fundamental Theorem
3. Analytic Tableaux and Regular Sets
4. The Liberalized Rule D
Chapter VIII. Axiom Systems for Quantification Theory
0. Foreword on Axiom Systems
1. The System Q subscript 1
2. The Systems Q subscript 2, Q* subscript 2
Chapter IX. Magic Sets
1. Magic Sets
2. Applications of Magic Sets
Chapter X. Analytic versus Synthetic Consistency Properties
1. Synthetic Consistency Properties
2. A More Direct Construction
Part III. Further Topics in First-Order Logic
Chapter XI. Gentzen Systems
1. Gentzen Systems for Propositional Logic
2. Block Tableaux and Gentzen Systems for First-Order Logic
Chapter XII. Elimination Theorems
1. Gentzen's Hauptsatz
2. An Abstract Form of the Hauptsatz
3. Some Applications of the Hauptsatz
Chapter XIII. Prenex Tableaux
1. Prenex Formulas
2. Prenex Tableaux
Chapter XIV. More on Gentzen Systems
1. Gentzen's Extended Hauptsatz
2. A New Form of the Extended Hauptsatz
3. Symmetric Gentzen Systems
Chapter XV. Craig's Interpolation Lemma and Beth's Definability Theorem
1. Craig's Interpolation Lemma
2. Beth's Definability Theorem
Chapter XVI. Symmetric Completeness Theorems
1. Clashing Tableaux
2. Clashing Prenex Tableaux
3. A Symmetric Form of the Fundamental Theorem
Chapter XVII. Systems of Linear Reasoning
1. Configurations
2. Linear Reasoning
3. Linear Reasoning for Prenex Formulas
4. A System Based on the Strong Symmetric Form of the Fundamental Theorem
References; Subject index