Li Mathematical Logic

Foundations for Information Science
1. Auflage 2010
ISBN: 978-3-7643-9977-1
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 25, 273 Seiten, eBook

Reihe: Progress in Computer Science and Applied Logic (PCS)

Mathematical logic is a branch of mathematics that takes axiom systems and mathematical proofs as its objects of study. This book shows how it can also provide a foundation for the development of information science and technology. The first five chapters systematically present the core topics of classical mathematical logic, including the syntax and models of first-order languages, formal inference systems, computability and representability, and Gödel's theorems. The last five chapters present extensions and developments of classical mathematical logic, particularly the concepts of version sequences of formal theories and their limits, the system of revision calculus, proschemes (formal descriptions of proof methods and strategies) and their properties, and the theory of inductive inference. All of these themes contribute to a formal theory of axiomatization and its application to the process of developing information technology and scientific theories. The book also describes the paradigm of three kinds of language environments for theories and it presents the basic properties required of a meta-language environment. Finally, the book brings these themes together by describing a workflow for scientific research in the information era in which formal methods, interactive software and human invention are all used to their advantage. This book represents a valuable reference for graduate and undergraduate students and researchers in mathematics, information science and technology, and other relevant areas of natural sciences. Its first five chapters serve as an undergraduate text in mathematical logic and the last five chapters are addressed to graduate students in relevant disciplines.

Zielgruppe

Research

Autoren/Hrsg.

Li, Wei

Inhaltsverzeichnis

1;Table of Contents ;5
2;Preface;8
3;Chapter 1 Syntax of First-Order Languages;12
3.1;1.1 Symbols of first-order languages;15
3.2;1.2 Terms;17
3.3;1.3 Logical formulas;19
3.4;1.4 Free variables and substitutions;20
3.5;1.5 Gödel terms of formulas;24
3.6;1.6 Proof by structural induction;26
4;Chapter 2 Models of First-Order Languages;30
4.1;2.1 Domains and interpretations;33
4.2;2.2 Assignments and models;35
4.3;2.3 Semantics of terms;35
4.4;2.4 Semantics of logical connective symbols;36
4.5;2.5 Semantics of formulas;38
4.6;2.6 Satisfiability and validity;41
4.7;2.7 Valid formulas with;42
4.8;2.8 Hintikka set;44
4.9;2.9 Herbrand model;46
4.10;2.10 Herbrand model with variables;49
4.11;2.11 Substitution lemma;52
4.12;2.12 Theorem of isomorphism;53
5;Chapter 3 Formal Inference Systems;56
5.1;3.1 G inference system;60
5.2;3.2 Inference trees, proof trees and provable sequents;63
5.3;3.3 Soundness of the G inference system;68
5.4;3.4 Compactness and consistency;72
5.5;3.5 Completeness of the G inference system;74
5.6;3.6 Some commonly used inference rules;77
5.7;3.7 Proof theory and model theory;79
6;Chapter 4 Computability & Representability;82
6.1;4.1 Formal theory;83
6.2;4.2 Elementary arithmetic theory;85
6.3;4.3 P-kernel on N;87
6.4;4.4 Church-Turing thesis;91
6.5;4.5 Problem of representability;92
6.6;4.6 States of P-kernel;93
6.7;4.7 Operational calculus of P-kernel;95
6.8;4.8 Representations of statements;97
6.9;4.9 Representability theorem;106
7;Chapter 5 Gödel Theorems ;108
7.1;5.1 Self-referential proposition;109
7.2;5.2 Decidable sets;111
7.3;5.3 Fixed point equation in .;115
7.4;5.4 Gödel’s incompleteness theorem;118
7.5;5.5 Gödel’s consistency theorem;120
7.6;5.6 Halting problem;123
8;Chapter 6 Sequences of Formal Theories;128
8.1;6.1 Two examples;129
8.2;6.2 Sequences of formal theories;133
8.3;6.3 Proschemes;136
8.4;6.4 Resolvent sequences;139
8.5;6.5 Default expansion sequences;141
8.6;6.6 Forcing sequences;144
8.7;6.7 Discussions on proschemes;147
9;Chapter 7 Revision Calculus;149
9.1;7.1 Necessary antecedents of formal consequences;150
9.2;7.2 New conjectures and new axioms;153
9.3;7.3 Refutation by facts and maximal contraction;154
9.4;7.4 R-calculus;156
9.5;7.5 Some examples;163
9.6;7.6 Special theory of relativity;165
9.7;7.7 Darwin’s theory of evolution;166
9.8;7.8 Reachability of R-calculus;170
9.9;7.9 Soundness and completeness of R-calculus;173
10;Chapter 8 Version Sequences;178
10.1;8.1 Versions and version sequences;180
10.2;8.2 The Proscheme OPEN;181
10.3;8.3 Convergence of the proscheme;185
10.4;8.4 Commutativity of the proscheme;187
10.5;8.5 Independence of the proscheme;189
10.6;8.6 Reliable proschemes;191
11;Chapter 9 Inductive Inference;195
11.1;9.1 Ground terms, basic sentences, and basic instances;198
11.2;9.2 Inductive inference system A;200
11.3;9.3 Inductive versions and inductive process;205
11.4;9.4 The Proscheme GUINA;205
11.5;9.5 Convergence of the proscheme GUINA;212
11.6;9.6 Commutativity of the proscheme GUINA;214
11.7;9.7 Independence of the proscheme GUINA;215
12;Chapter 10 Workflows for Scientific Discovery;217
12.1;10.1 Three language environments;217
12.2;10.2 Basic principles of the meta-language environment;221
12.2.1;1. Principle of environment;221
12.2.2;2. Principle of excluded middle;222
12.2.3;3. Principle of logical connectives;222
12.2.4;4. Church-Turing thesis;224
12.2.5;5. Principle of observability;224
12.2.6;6. Principle of Occam’s razor;224
12.3;10.3 Axiomatization;225
12.4;10.4 Formal methods;227
12.5;10.5 Workflow of scientific research;233
12.5.1;1. The Meta-language Environment L;234
12.5.2;2. The Domains;234
12.5.3;3. The Object Language;235
12.5.4;4. Formal Axiomatization;235
12.6;Appendix 1 Sets and Maps;237
12.7;Appendix 2 Substitution Lemma and Its Proof;240
12.8;Appendix 3 Proof of the Representability Theorem;244
12.8.1;A3.1 Representation of the while statement in .;244
12.8.2;A3.2 Representability of the P-procedure body;251
13;Bibliography;260
14;Index;263

Syntax of First-Order Languages.- Models of First-Order Languages.- Formal Inference Systems.- Computability & Representability.- Gödel Theorems.- Sequences of Formal Theories.- Revision Calculus.- Version Sequences.- Inductive Inference.- Workflows for Scientific Discovery.