| Description |
1 online resource (xii, 261 pages) : illustrations |
| Series |
Progress in computer science and applied logic ; v. 25 |
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Progress in computer science and applied logic ; v. 25.
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| Contents |
Cover -- Contents -- Preface -- Chapter 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 G246;del terms of formulas -- 1.6 Proof by structural induction -- Chapter 2 Models of First-Order Languages -- 2.1 Domains and interpretations -- 2.2 Assignments and models -- 2.3 Semantics of terms -- 2.4 Semantics of logical connective symbols -- 2.5 Semantics of formulas -- 2.6 Satisfiability and validity -- 2.7 Valid formulas with -- 2.8 Hintikka set -- 2.9 Herbrand model -- 2.10 Herbrand model with variables -- 2.11 Substitution lemma -- 2.12 Theorem of isomorphism -- Chapter 3 Formal Inference Systems -- 3.1 G inference system -- 3.2 Inference trees, proof 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 -- Chapter 4 Computability & Representability -- 4.1 Formal theory -- 4.2 Elementary arithmetic theory -- 4.3 P-kernel on N -- 4.4 Church-Turing thesis -- 4.5 Problem of representability -- 4.6 States of P-kernel -- 4.7 Operational calculus of P-kernel -- 4.8 Representations of statements -- 4.9 Representability theorem -- Chapter 5 G246;del Theorems -- 5.1 Self-referential proposition -- 5.2 Decidable sets -- 5.3 Fixed point equation in 928; -- 5.4 G246;del8217;s incompleteness theorem -- 5.5 G246;del8217;s consistency theorem -- 5.6 Halting problem -- Chapter 6 Sequences of Formal Theories -- 6.1 Two examples -- 6.2 Sequences of formal theories -- 6.3 Proschemes -- 6.4 Resolvent sequences -- 6.5 Default expansion sequences -- 6.6 Forcing sequences -- 6.7 Discussions on proschemes -- Chapter 7 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 Special theory of relativity -- 7.7 Darwin8217;s theory of evolution -- 7.8 Reachability of R-calculus -- 7.9 Soundness and completeness of R-calculus -- 7.10 Basic theorem of testing -- Chapter 8 Version Sequences -- 8.1 Versions and version sequences -- 8.2 The Proscheme OPEN -- 8.3 Convergence of the proscheme -- 8.4 Commutativity of the proscheme -- 8.5 Independence of the proscheme -- 8.6 Reliable proschemes -- Chapter 9 Inductive Inference -- 9.1 Ground terms, basic sentences, and basic instances -- 9.2 Inductive inference system A -- 9.3 Inductive versions and inductive process -- 9.4 The Proscheme GUINA -- 9.5 Convergence of the proscheme GUINA -- 9.6 Commutativity of the proscheme GUINA -- 9.7 Independence of the proscheme GUINA -- Chapter 10 Workflows for Scientific Discovery -- 10.1 Three language environments -- 10.2 Basic principles of the meta-language environment -- 10.3 Axiomatization -- 10.4 Formal methods -- 10.5 Workflow of scientific research -- Appendix 1 Sets and Maps -- Appendix 2 Substitution Lemma and Its Proof -- Appendix 3 Proof of the Representability Theorem -- A3.1 Representation of the while statement in 928; -- A3.2 Representability of the P-procedure body |
| Summary |
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 |
| Bibliography |
Includes bibliographical references (pages 253-255) and index |
| Notes |
Print version record |
| Subject |
Logic, Symbolic and mathematical.
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MATHEMATICS -- Infinity.
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MATHEMATICS -- Logic.
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Informatique.
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Logic, Symbolic and mathematical
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Mathematische Logik
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| Form |
Electronic book
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| ISBN |
9783764399771 |
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3764399775 |
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