کتابخانه

Title: Propositional and Predicate Calculus : A Model of Argument | Author(s): Derek Goldrei | Publisher: Springer | Year: 2005 | Edition: 1st Edition | Language: English | Pages : 315 | ISBN: 9781852339210, 1852339217 | Size: 4 MB | Extension: pdf

At the heart of the justification for the reasoning used in modern mathematics lies the completeness theorem for predicate calculus. This unique textbook covers two entirely different ways of looking at such reasoning. Topics include:
- the representation of mathematical statements by formulas in a formal language;
- the interpretation of formulas as true or false in a mathematical structure;
- logical consequence of one formula from others;
- formal proof;
- the soundness and completeness theorems connecting logical consequence and formal proof;
- the axiomatization of some mathematical theories using a formal language;
- the compactness theorem and an introduction to model theory.
This book is designed for self-study by students, as well as for taught courses, using principles successfully developed by the Open University and used across the world. It includes exercises embedded within the text with full solutions to many of these. In addition there are a number of exercises without answers so that students studying under the guidance of a tutor may be assessed on the basis of what has been taught.
Some experience of axiom-based mathematics is required but no previous experience of logic. Propositional and Predicate Calculus gives students the basis for further study of mathematical logic and the use of formal languages in other subjects.
Derek Goldrei is Senior Lecturer and Staff Tutor at the Open University and part-time Lecturer in Mathematics at Mansfield College, Oxford, UK.

Table of contents :
Preface
Contents
1 Introduction
- 1.1 Outline of the book
- 1.2 Assumed knowledge
2 Propositions and truth assignments
- 2.1 Introduction
- 2.2 The construction of propositional formulas
- 2.3 The interpretation of propositional formulas
- 2.4 Logical equivalence
- 2.5 The expressive power of connectives
- 2.6 Logical consequence
3 Formal propositional calculus
- 3.1 Introduction
- 3.2 A formal system for propositional calculus
- 3.3 Soundness and completeness
- 3.4 Independence of axioms and alternative systems
4 Predicates and models
- 4.1 Introduction: basic ideas
- 4.2 First-order languages and their interpretation
- 4.3 Universally valid formulas and logical equivalence
- 4.4 Some axiom systems and their consequences
- 4.5 Substructures and Isomorphisms
5 Formal predicate calculus
- 5.1 Introduction
- 5.2 A formal system for predicate calculus
- 5.3 The soundness theorem
- 5.4 The equality axioms and non-normal structures
- 5.5 The completeness theorem
6 Some uses of compactness
- 6.1 Introduction: the compactness theorem
- 6.2 Finite axiomatizability
- 6.3 Some non-axiomatizable theories
- 6.4 The Löwenheim–Skolem theorems
- 6.5 New models from old ones
- 6.6 Decidable theories
Bibliography
Index