Logicism renewed: logical foundations for mathematics and computer science
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Wellesley, Mass.
A. K. Peters
2005
|
| Series: | Lecture notes in logic
23 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVII, 230 S. |
| ISBN: | 1568812752 |
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| adam_text | LECTURE NOTES IN LOGIC 23 LOGICISM RENEWED LOGICAL FOUNDATIONS FOR
MATHEMATICS AND COMPUTER SCIENCE PAUL C. GILMORE DEPARTMENT OF COMPUTER
SCIENCE UNIVERSITY OF BRITISH COLUMBIA, VANCOUVER ASSOCIATION FOR
SYMBOLIC LOGIC A K PETERS, LTD. * WELLESLEY, MASSACHUSETTS CONTENTS
PREFACE IX CHAPTER 1. ELEMENTARY LOGIC 1 1.1. LANGUAGE AND LOGIC 1 1.2.
TYPES, TERMS, AND FORMULAS 2 1.3. LOGICAL CONNECTIVES 5 1.3.1. CHOICE OF
CONNECTIVES 6 1.4. VALUATIONS 7 1.4.1. DOMAINS, VALUATIONS, AND MODELS 9
1.5. THE LAMBDA ABSTRACTION OPERATOR 11 1.5.1. SYNTAX FOR QUANTIFICATION
12 1.5.2. SEMANTICS FOR ABSTRACTION TERMS 13 1.5.3. FREE AND BOUND
VARIABLES 15 1.5.4. SUBSTITUTIONS AND CONTRACTIONS 16 1.6. SYNTAX AND
SEMANTICS FOR EL 19 1.7. ELEMENTARY PROOF THEORY 20 1.7.1. SEMANTIC
RULES FOR EL 22 1.7.2. DERIVATION OF A SEQUENT 23 1.7.3. TERMINOLOGY FOR
AND PROPERTIES OF DERIVATIONS 24 1.7.4. REUSING DERIVATIONS ...: 26
1.7.5. AN ALTERNATIVE PROOF THEORY 27 1.7.6. DERIVATIONS AS BINARY TREES
OF SEQUENTS 28 1.7.7. OTHER FORMULATIONS OF FIRST ORDER LOGIC 31 1.8.
CONSISTENCY OF EL 32 1.9. COMPLETENESS OF EL 33 1.9.1. A SYSTEMATIC
SEARCH FOR A DERIVATION 35 1.9.2. A DESCENDING CHAIN DEFINES A
COUNTER-EXAMPLE 37 CHAPTER 2. TYPE THEORY 43 2.1. THE TYPE THEORY TT 43
2.1.1. THE TERMS OF TT 44 2.2. POLYMORPHIC TYPING OF TT 46 2.2.1. TYPING
STRINGS OF CHARACTERS 46 VI CONTENTS 2.2.2. TYPE EXPRESSIONS AND THEIR
RANGES 47 2.2.3. STRING ASSIGNMENTS AND UNIFICATION 48 2.2.4. POLYTYPING
TT 51 2.3. LAMBDA CONTRACTIONS 55 2.4. THE PROOF THEORY 60 2.5.
INTENSIONAL AND EXTENSIONAL IDENTITY 61 2.5.1. EXTENSIONAL AND
INTENSIONAL PREDICATES 63 2.5.2. IDENTITY AND STRING IDENTITY 65 2.6.
SEMANTICS FOR TT 66 2.6.1. AN EXTENSIONAL SEMANTICS FOR TT 66 2.6.2. AN
INTENSIONAL SEMANTICS FOR TT 68 CHAPTER 3. AN INTENSIONAL TYPE THEORY 71
3.1. THE TERMS OF ITT 71 3.1.1. MOTIVATION FOR SECONDARY TYPING 71
3.1.2. SECONDARY TYPING 74 3.1.3. THE NORMAL FORM OF A TERM OF ITT 75
3.2. POLYTYPING ITT 76 3.2.1. A POLYTYPING ALGORITHM FOR ITT 76 3.3. AN
INTENSIONAL SEMANTICS FOR ITT 79 3.3.1. DOMAINS AND VALUATIONS 79 3.3.2.
SEMANTIC BASIS FOR THE PROOF THEORY 84 3.3.3. SEQUENTS AND
COUNTER-EXAMPLES 84 3.4. PROOF THEORY FOR ITT 85 3.4.1. AN ESSENTIAL
RESTRICTION 86 3.4.2. DERIVABLE RULES FOR IDENTITY 86 3.4.3.
RELATIONSHIP BETWEEN THE IDENTITIES 87 3.4.4. PROPERTIES OF DERIVATIONS
87 3.5. COMPLETENESS OF ITT 88 3.5.1. A COUNTER-EXAMPLE THAT IS AN
INTENSIONAL MODEL 90 3.5.2. DENUMERABLE MODELS 93 3.6. SITT 93 3.6.1.
TYPES AND TERMS OF SITT 94 3.6.2. LAMBDA CONTRACTIONS 95 3.6.3.
SEMANTICS OF SITT 96 3.6.4. PROOF THEORY FOR SITT 96 3.6.5. RUSSELL S
SET 96 CHAPTER 4. RECURSIONS 99 4.1. INTRODUCTION 99 4.2. LEAST AND
GREATEST PREDICATE OPERATORS 101 4.3. MONOTONIC RECURSION GENERATORS 103
4.3.1. PROPERTIES OF MONOTONIC RECURSION GENERATORS 104 CONTENTS VII
4.4. ZERO, SUCCESSOR, AND NATURAL NUMBERS 106 4.4.1. A ZERO AND
SUCCESSOR 106 4.4.2. NIL AND ORDERED PAIR 107 4.4.3. PROJECTIONS OF
ORDERED PAIR 109 4.4.4. A RECURSION GENERATOR FOR THE S-SEQUENCE 110
4.4.5. UNDERSTANDING GT(RN) ILL 4.5. CONTINUOUS RECURSION GENERATORS 115
4.6. POSITIVE OCCURRENCES OF A VARIABLE 116 4.6.1. SUBFORMULA PATH 116
4.6.2. POSITIVE AND E-POSITIVE OCCURRENCES OF VARIABLES 116 4.6.3.
MONOTONICITY AND CONTINUITY 118 4.7. HORN SEQUENTS AND RECURSION
GENERATORS 122 4.7.1. HORN SEQUENTS 122 4.7.2. SIMULTANEOUS HORN
SEQUENTS 123 4.8. DEFINITION BY ITERATION 127 4.8.1. DEFINING LT(RG) BY
ITERATION 128 4.8.2. DEFINING GT(RG) BY ITERATION 130 4.9. POTENTIALLY
INFINITE PREDICATES 131 4.9.1. CHARACTERS AND STRINGS 131 4.9.2. LISTS
132 4.9.3. UNIVERSAL QUANTIFICATION 133 CHAPTER 5. CHOICE AND FUNCTION
TERMS 137 5.1. INTRODUCTION 137 5.1.1. A FUNCTIONAL NOTATION FROM CHOICE
TERMS 138 5.2. INTRODUCING CHOICE TERMS 140 5.2.1. THE RELATION E ON
TERMS OF ITTE 141 5.3. PROOF THEORY FOR ITTE 143 5.4. CONSISTENCY AND
COMPLETENESS 145 5.4.1. COMPLETENESS OF ITTE 147 5.5. FUNCTION TERMS 148
5.6. PARTIAL FUNCTIONS AND DEPENDENT TYPES 149 CHAPTER 6. INTUITIONIST
LOGIC 151 6.1. INTUITIONIST/CONSTRUCTIVE MATHEMATICS 151 6.2. A SEQUENT
CALCULUS FORMULATION ITTG OF ITT 152 6.3. AN INTUITIONIST FORMULATION
HITTG OF ITTG 154 6.3.1. DERIVABILITY IN ITTG AND HITTG 155 6.4.
SEMANTIC TREE FORMULATION HITT OF HITTG 159 6.4.1. PROPERTIES OF HITT
DERIVATIONS 162 6.4.2. EQUIVALENCE OF HITT AND HITTG 164 6.5. SEMANTICS
FOR HITT 167 6.5.1. FORESTS OF SEMANTIC TREES 167 6.6. RECURSIONS IN
HITT 170 VIII CONTENTS CHAPTER 7. LOGIC AND MATHEMATICS 171 7.1.
INTRODUCTION 171 7.2. SELF-APPLIED LOGIC 172 7.3. LOGIC AND CATEGORY
THEORY 173 7.3.1. ABELIAN SEMI-GROUPS 173 7.3.2. DEFINITIONS 174 7.3.3.
TYPING 175 7.3.4. DERIVATION OF (SG) 175 7.4. SET THEORY AND LOGIC 179
7.4.1. A SET THEORY FORMALIZED IN ITT 180 7.4.2. CANTOR S DIAGONAL
ARGUMENT IN ITT 181 7.5. WHY IS MATHEMATICS APPLICABLE? 183 CHAPTER 8.
LOGIC AND COMPUTER SCIENCE 185 8.1. INTRODUCTION 185 8.2. DEFINITION,
DERIVATION, AND COMPUTATION 189 8.3. SEMANTICS FOR RECURSIVE COMMANDS
191 8.3.1. PRIMITIVE SYNTAX 191 8.3.2. EXPRESSIONS 192 8.3.3. EXPRESSION
SEMANTICS 193 8.3.4. COMMAND SEMANTICS 196 8.3.5. EXAMPLE THEOREM 201
8.4. RECURSIVE DOMAINS 203 8.4.1. PRIMITIVE DOMAINS 204 8.4.2. FINITE
FUNCTIONS AND DOMAIN CONSTRUCTORS 204 8.4.3. DOMAIN CONSTRUCTORS 205
8.4.4. SOLVING DOMAIN EQUATIONS: AN EXAMPLE 210 8.5. LOGICAL SUPPORT FOR
SPECIFICATION LANGUAGES 213 REFERENCES 215 INDEX 225
|
| adam_txt |
LECTURE NOTES IN LOGIC 23 LOGICISM RENEWED LOGICAL FOUNDATIONS FOR
MATHEMATICS AND COMPUTER SCIENCE PAUL C. GILMORE DEPARTMENT OF COMPUTER
SCIENCE UNIVERSITY OF BRITISH COLUMBIA, VANCOUVER ASSOCIATION FOR
SYMBOLIC LOGIC A K PETERS, LTD. * WELLESLEY, MASSACHUSETTS CONTENTS
PREFACE IX CHAPTER 1. ELEMENTARY LOGIC 1 1.1. LANGUAGE AND LOGIC 1 1.2.
TYPES, TERMS, AND FORMULAS 2 1.3. LOGICAL CONNECTIVES 5 1.3.1. CHOICE OF
CONNECTIVES 6 1.4. VALUATIONS 7 1.4.1. DOMAINS, VALUATIONS, AND MODELS 9
1.5. THE LAMBDA ABSTRACTION OPERATOR 11 1.5.1. SYNTAX FOR QUANTIFICATION
12 1.5.2. SEMANTICS FOR ABSTRACTION TERMS 13 1.5.3. FREE AND BOUND
VARIABLES 15 1.5.4. SUBSTITUTIONS AND CONTRACTIONS 16 1.6. SYNTAX AND
SEMANTICS FOR EL 19 1.7. ELEMENTARY PROOF THEORY 20 1.7.1. SEMANTIC
RULES FOR EL 22 1.7.2. DERIVATION OF A SEQUENT 23 1.7.3. TERMINOLOGY FOR
AND PROPERTIES OF DERIVATIONS 24 1.7.4. REUSING DERIVATIONS .: 26
1.7.5. AN ALTERNATIVE PROOF THEORY 27 1.7.6. DERIVATIONS AS BINARY TREES
OF SEQUENTS 28 1.7.7. OTHER FORMULATIONS OF FIRST ORDER LOGIC 31 1.8.
CONSISTENCY OF EL 32 1.9. COMPLETENESS OF EL 33 1.9.1. A SYSTEMATIC
SEARCH FOR A DERIVATION 35 1.9.2. A DESCENDING CHAIN DEFINES A
COUNTER-EXAMPLE 37 CHAPTER 2. TYPE THEORY 43 2.1. THE TYPE THEORY TT 43
2.1.1. THE TERMS OF TT 44 2.2. POLYMORPHIC TYPING OF TT 46 2.2.1. TYPING
STRINGS OF CHARACTERS 46 VI CONTENTS 2.2.2. TYPE EXPRESSIONS AND THEIR
RANGES 47 2.2.3. STRING ASSIGNMENTS AND UNIFICATION 48 2.2.4. POLYTYPING
TT 51 2.3. LAMBDA CONTRACTIONS 55 2.4. THE PROOF THEORY 60 2.5.
INTENSIONAL AND EXTENSIONAL IDENTITY 61 2.5.1. EXTENSIONAL AND
INTENSIONAL PREDICATES 63 2.5.2. IDENTITY AND STRING IDENTITY 65 2.6.
SEMANTICS FOR TT 66 2.6.1. AN EXTENSIONAL SEMANTICS FOR TT 66 2.6.2. AN
INTENSIONAL SEMANTICS FOR TT 68 CHAPTER 3. AN INTENSIONAL TYPE THEORY 71
3.1. THE TERMS OF ITT 71 3.1.1. MOTIVATION FOR SECONDARY TYPING 71
3.1.2. SECONDARY TYPING 74 3.1.3. THE NORMAL FORM OF A TERM OF ITT 75
3.2. POLYTYPING ITT 76 3.2.1. A POLYTYPING ALGORITHM FOR ITT 76 3.3. AN
INTENSIONAL SEMANTICS FOR ITT 79 3.3.1. DOMAINS AND VALUATIONS 79 3.3.2.
SEMANTIC BASIS FOR THE PROOF THEORY 84 3.3.3. SEQUENTS AND
COUNTER-EXAMPLES 84 3.4. PROOF THEORY FOR ITT 85 3.4.1. AN ESSENTIAL
RESTRICTION 86 3.4.2. DERIVABLE RULES FOR IDENTITY 86 3.4.3.
RELATIONSHIP BETWEEN THE IDENTITIES 87 3.4.4. PROPERTIES OF DERIVATIONS
87 3.5. COMPLETENESS OF ITT 88 3.5.1. A COUNTER-EXAMPLE THAT IS AN
INTENSIONAL MODEL 90 3.5.2. DENUMERABLE MODELS 93 3.6. SITT 93 3.6.1.
TYPES AND TERMS OF SITT 94 3.6.2. LAMBDA CONTRACTIONS 95 3.6.3.
SEMANTICS OF SITT 96 3.6.4. PROOF THEORY FOR SITT 96 3.6.5. RUSSELL'S
SET 96 CHAPTER 4. RECURSIONS 99 4.1. INTRODUCTION 99 4.2. LEAST AND
GREATEST PREDICATE OPERATORS 101 4.3. MONOTONIC RECURSION GENERATORS 103
4.3.1. PROPERTIES OF MONOTONIC RECURSION GENERATORS 104 CONTENTS VII
4.4. ZERO, SUCCESSOR, AND NATURAL NUMBERS 106 4.4.1. A ZERO AND
SUCCESSOR 106 4.4.2. NIL AND ORDERED PAIR 107 4.4.3. PROJECTIONS OF
ORDERED PAIR 109 4.4.4. A RECURSION GENERATOR FOR THE S-SEQUENCE 110
4.4.5. UNDERSTANDING GT(RN) ILL 4.5. CONTINUOUS RECURSION GENERATORS 115
4.6. POSITIVE OCCURRENCES OF A VARIABLE 116 4.6.1. SUBFORMULA PATH 116
4.6.2. POSITIVE AND E-POSITIVE OCCURRENCES OF VARIABLES 116 4.6.3.
MONOTONICITY AND CONTINUITY 118 4.7. HORN SEQUENTS AND RECURSION
GENERATORS 122 4.7.1. HORN SEQUENTS 122 4.7.2. SIMULTANEOUS HORN
SEQUENTS 123 4.8. DEFINITION BY ITERATION 127 4.8.1. DEFINING LT(RG) BY
ITERATION 128 4.8.2. DEFINING GT(RG) BY ITERATION 130 4.9. POTENTIALLY
INFINITE PREDICATES 131 4.9.1. CHARACTERS AND STRINGS 131 4.9.2. LISTS
132 4.9.3. UNIVERSAL QUANTIFICATION 133 CHAPTER 5. CHOICE AND FUNCTION
TERMS 137 5.1. INTRODUCTION 137 5.1.1. A FUNCTIONAL NOTATION FROM CHOICE
TERMS 138 5.2. INTRODUCING CHOICE TERMS 140 5.2.1. THE RELATION E ON
TERMS OF ITTE 141 5.3. PROOF THEORY FOR ITTE 143 5.4. CONSISTENCY AND
COMPLETENESS 145 5.4.1. COMPLETENESS OF ITTE 147 5.5. FUNCTION TERMS 148
5.6. PARTIAL FUNCTIONS AND DEPENDENT TYPES 149 CHAPTER 6. INTUITIONIST
LOGIC 151 6.1. INTUITIONIST/CONSTRUCTIVE MATHEMATICS 151 6.2. A SEQUENT
CALCULUS FORMULATION ITTG OF ITT 152 6.3. AN INTUITIONIST FORMULATION
HITTG OF ITTG 154 6.3.1. DERIVABILITY IN ITTG AND HITTG 155 6.4.
SEMANTIC TREE FORMULATION HITT OF HITTG 159 6.4.1. PROPERTIES OF HITT
DERIVATIONS 162 6.4.2. EQUIVALENCE OF HITT AND HITTG 164 6.5. SEMANTICS
FOR HITT 167 6.5.1. FORESTS OF SEMANTIC TREES 167 6.6. RECURSIONS IN
HITT 170 VIII CONTENTS CHAPTER 7. LOGIC AND MATHEMATICS 171 7.1.
INTRODUCTION 171 7.2. SELF-APPLIED LOGIC 172 7.3. LOGIC AND CATEGORY
THEORY 173 7.3.1. ABELIAN SEMI-GROUPS 173 7.3.2. DEFINITIONS 174 7.3.3.
TYPING 175 7.3.4. DERIVATION OF (SG) 175 7.4. SET THEORY AND LOGIC 179
7.4.1. A SET THEORY FORMALIZED IN ITT 180 7.4.2. CANTOR'S DIAGONAL
ARGUMENT IN ITT 181 7.5. WHY IS MATHEMATICS APPLICABLE? 183 CHAPTER 8.
LOGIC AND COMPUTER SCIENCE 185 8.1. INTRODUCTION 185 8.2. DEFINITION,
DERIVATION, AND COMPUTATION 189 8.3. SEMANTICS FOR RECURSIVE COMMANDS
191 8.3.1. PRIMITIVE SYNTAX 191 8.3.2. EXPRESSIONS 192 8.3.3. EXPRESSION
SEMANTICS 193 8.3.4. COMMAND SEMANTICS 196 8.3.5. EXAMPLE THEOREM 201
8.4. RECURSIVE DOMAINS 203 8.4.1. PRIMITIVE DOMAINS 204 8.4.2. FINITE
FUNCTIONS AND DOMAIN CONSTRUCTORS 204 8.4.3. DOMAIN CONSTRUCTORS 205
8.4.4. SOLVING DOMAIN EQUATIONS: AN EXAMPLE 210 8.5. LOGICAL SUPPORT FOR
SPECIFICATION LANGUAGES 213 REFERENCES 215 INDEX 225 |
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| index_date | 2024-07-02T13:49:35Z |
| indexdate | 2024-07-09T20:34:50Z |
| institution | BVB |
| isbn | 1568812752 |
| language | English |
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| publisher | A. K. Peters |
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| series | Lecture notes in logic |
| series2 | Lecture notes in logic |
| spelling | Gilmore, Paul C. 1925-2015 Verfasser (DE-588)14035977X aut Logicism renewed logical foundations for mathematics and computer science Paul C. Gilmore Wellesley, Mass. A. K. Peters 2005 XVII, 230 S. txt rdacontent n rdamedia nc rdacarrier Lecture notes in logic 23 Logic, Symbolic and mathematical Mathematische Logik (DE-588)4037951-6 gnd rswk-swf Mathematische Logik (DE-588)4037951-6 s DE-604 Lecture notes in logic 23 (DE-604)BV008909514 23 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014610954&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gilmore, Paul C. 1925-2015 Logicism renewed logical foundations for mathematics and computer science Lecture notes in logic Logic, Symbolic and mathematical Mathematische Logik (DE-588)4037951-6 gnd |
| subject_GND | (DE-588)4037951-6 |
| title | Logicism renewed logical foundations for mathematics and computer science |
| title_auth | Logicism renewed logical foundations for mathematics and computer science |
| title_exact_search | Logicism renewed logical foundations for mathematics and computer science |
| title_exact_search_txtP | Logicism renewed logical foundations for mathematics and computer science |
| title_full | Logicism renewed logical foundations for mathematics and computer science Paul C. Gilmore |
| title_fullStr | Logicism renewed logical foundations for mathematics and computer science Paul C. Gilmore |
| title_full_unstemmed | Logicism renewed logical foundations for mathematics and computer science Paul C. Gilmore |
| title_short | Logicism renewed |
| title_sort | logicism renewed logical foundations for mathematics and computer science |
| title_sub | logical foundations for mathematics and computer science |
| topic | Logic, Symbolic and mathematical Mathematische Logik (DE-588)4037951-6 gnd |
| topic_facet | Logic, Symbolic and mathematical Mathematische Logik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014610954&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV008909514 |
| work_keys_str_mv | AT gilmorepaulc logicismrenewedlogicalfoundationsformathematicsandcomputerscience |