Synopses & Reviews
Considered the best book in the field, this completely self-contained study is both an introduction to quantification theory and an exposition of new results and techniques in "analytic" or "cut free" methods. The focus in on the tableau point of view. Topics include trees, tableau method for propositional logic, Gentzen systems, more. Includes 144 illustrations.
Synopsis
This self-contained study is both an introduction to quantification theory and an exposition of new results and techniques in "analytic" or "cut free" methods. The focus is on the tableau point of view. Includes 144 illustrations.
Synopsis
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. Presented in tableau format, the material covers propositional and first-order logic. 144 illustrations.
Description
Includes bibliographical references (p. [155]) and index.
About the Author
Born in New York City in 1919, Raymond Smullyan is a philosopher and magician as well as a famous mathematician and logician. His career as a stage magician financed his undergraduate studies at the University of Chicago as well his doctoral work at Princeton. The author of several imaginative books on recreational mathematics, Smullyan is also a classical pianist.
Raymond Smullyan: The Merry Prankster
Raymond Smullyan (1919- ), 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
Critical Acclaim for The Lady or the Tiger:
"Another scintillating collection of brilliant problems and paradoxes by the most entertaining logician and set theorist who ever lived." — Martin Gardner
Table of Contents
Part I. Propositional Logic from the Viewpoint of Analytic Tableaux
Chapter 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