Overview

This book presents the basic principles and formal calculus of mathematical logic. It covers core contents, extensions and developments of classical mathematical logic, and it offers formal proofs and concrete examples for all theoretical results.

Full Product Details

Author:   Wei Li
Publisher:   Birkhauser Verlag AG
Imprint:   Birkhauser Verlag AG
Edition:   2010 ed.
Volume:   25
Dimensions:   Width: 16.50cm , Height: 2.00cm , Length: 23.50cm
Weight:   0.616kg
ISBN:  

9783764399764

ISBN 10:   3764399767
Pages:   273
Publication Date:   11 December 2009
Audience:   College/higher education ,  Undergraduate ,  Postgraduate, Research & Scholarly
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

Preface.- Glossary.- 1. Syntax of First-Order Languages.- 1.1 Symbols of first-order languages.- 1.2 Terms.- 1.3 Logical formulas.- 1.4 Free variables and substitutions.- 1.5 Godel terms of formulas.- 1.6 Proof by structural inductions.- 2. Models of First-Order Languages.- 2.1 Domains and interpretations.- 2.2 Assignments and models.- 2.3 Meanings of terms.- 2.4 Meanings of logical connective symbols.- 2.5 Meanings of formulas.- 2.6 Satisfiability and validity.- 2.7 Valid formulas on .- 2.8 Hintikka set.- 2.9 Herbrand model.- 2.10 Herbrand model with variables.- 2.11 Substitution lemma.- 3. Formal Inference Systems.- 3.1 G inference system.- 3.2 Proof trees, inference trees, and provable sequents.- 3.3 Soundness of the G inference system.- 3.4 Compactness and consistency.- 3.5 Completeness of the G inference system.- 3.6 Some commonly used inference rules.- 3.7 Proof theory and model theory.- 4. Computability and Representability.- 4.1 Formal theories.- 4.2 Elementary arithmetic theory .- 4.3 P-procedures on N.- 4.4 Church-Turhign thesis.- 4.5 Problem of representability.- 4.6 States of P-procedures.- 4.7 System of operational calculus of P-procedure statements.- 4.8 Representation of P-procedure statements.- 4.9 Representability theorem.- 5. Godel Theorems.- 5.1 Self-referential statements.- 5.2 Deciadable sets.- 5.3 Fixed point equation in .- 5.4 Godel incompleteness theorem.- 5.5 Godel consistency theorem.- 5.6 Halt problem.- 6. Sequences of Formal Theories.- 6.1 Two examples.- 6.2 Sequences of formal theories.- 6.3 Proxchemes.- 6.4 Resolution sequences.- 6.5 Sequences of default expansions.- 6.6 Forcing sequences.- 6.7 Discussions about proxchemes.- 7. Refutation by Facts and Revision Calculus.- 7.1 Necessary antecedents of formal consequences.- 7.2 New conjectures and new axioms.- 7.3 Refutation by facts and maximal contraction.- 7.4 R-calculus.- 7.5 Some examples.- 7.6 Reachability of theR-calculus.- 7.7 Soundness and completeness of the R-calculus.- 7.8 Basic theorem of testing.- 8. Version Sequences and Proxchemes.- 8.1 Versions and version sequences.- 8.2 OPEN proxcheme.- 8.3 Convergency of the P-proxcheme.- 8.4 Commutativity of the P-proxcheme.- 8.5 Independency of the P-proxcheme.- 8.6 Ideal proxchemes.- 9. Inductive Inference and Inductive Process.- 9.1 Basic terms, basic sentences, and basic instances.- 9.2 Inductive inference system A.- 9.3 Inductive version and inductive process.- 9.4 GUINA proxcheme.- 9.5 Convergency of the GUINA proxcheme.- 9.6 Commutativity of the GUINA proxcheme.- 9.7 Independency of the GUINA proxcheme.- 10. Metalanguage Environments of First-Order Languages.- 10.1 Environments of three kinds of languages.- 10.2 Basic principles of the environment of metalanguage.- 10.3 Axiomatization method.- 10.4 Formalization method.- 10.5 Workflow of scientific research.- Appendix 1 Sets and Mappings.- Appendix 2 Substitution Lemma and Its Proof.- Appendix 3 Proof of the Representability Theorem.- References.- Index.

Reviews

From the reviews: The book consists of two parts. The first part is written for undergraduate university students of computer science and presents the classical first-order predicate logic with set-theoretical interpretation of its formulas and a symmetrical, well-shaped, and beautiful Gentzen-type axiomatic system which describes identically true ! formulas of this logic. ! The second part may be used for a course for postgraduate students of information science and includes a definition of versions of a formal theory, version sequences and their limits. (Alex Nabebin, Zentralblatt MATH, Vol. 1185, 2010)

"From the reviews: ""The book consists of two parts. The first part is written for undergraduate university students of computer science and presents the classical first-order predicate logic with set-theoretical interpretation of its formulas and a symmetrical, well-shaped, and beautiful Gentzen-type axiomatic system which describes identically true ... formulas of this logic. ... The second part may be used for a course for postgraduate students of information science and includes a definition of versions of a formal theory, version sequences and their limits."" (Alex Nabebin, Zentralblatt MATH, Vol. 1185, 2010)"

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