Logic in elementary mathematics:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York (u.a.)
McGraw-Hill
1959
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XI, 274 S. |
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MARC
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Record in the Search Index
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|---|---|
| adam_text | CONTENTS
Preface v
Glossary of Special Symbols and Abbreviations xi
I. Mathematics, Formal Logic, and Names 1
1.1. Mathematics and mathematicians 1
1.2. Form and meaning 4
1.3. Use and mention 9
1.4. Statements about statements 13
II. The Statement Calculus 17
2.1. Statements and statement forms 17
2.2. Negation 20
2.3. Conjunction 22
2.4. Disjunction 23
2.5. The conditional 26
2.6. Statement formulas 30
2.7. Equivalence 33
2.8. Valid statement formulas 35
2.9. Notational conventions 38
2.10. Replacement 41
2.11. Substitution 42
2.12. Some useful valid formulas 45
2.13. A collection of valid statement formulas 48
2.14. Rules of inference 49
2.15. Collection of inference rules 55
vii
viii CONTENTS
III. Proof and Demonstration 57
3.1. Basic form of indirect proof 57
3.2. Special cases of the basic form of indirect proof 61
3.3. Proof by elimination 63
3.4. Proof by cases 68
3.5. Converses 71
3.6. Conversion in classical logic 73
3.7. Inverses 76
3.8. Demonstration and proof 78
3.9. Trees 83
3.10. Abbreviated demonstrations as proofs 83
3.11. Deduction principle 88
3.12. Conventional proofs in geometry 91
3.13. Analysis of a conventional proof 94
IV. Abstract Mathematical Systems 100
4.1. Mathematical models 100
4.2. A miniature geometry 102
4.3. Interpretations 119
V. The Restricted Predicate Calculus 126
5.1. Statement functions 126
5.2. Universal quantifier 129
5.3. Existential quantifier 135
5.4. Transformations of quantifiers 137
5.5. Free and bound variables 139
5.6. Inference rules for quantified statements 141
5.7. Inference of a statement function from a general statement 142
5.8. Inference rules of the statement calculus extended 143
5.9. Extension of the notion of valid statement formula 144
5.10. The generalization principle 145
5.11. Inference of a statement function from an existential statement: IE • 150
5.12. Introduction of the existential quantifier 153
5.13. Use of the quantification inference rules 154
5.14. Statement functions of several variables 157
5.15. Extension of quantification inference rules 162
5.16. Final forms of the quantificational inference rules 165
5.17. Use of equality 166
5.18. Formal notion of equality 173
5.19. The scope of an assumption in a demonstration 177
VI. Applications of Logic in Mathematics 182
6.1. Introduction • 182
6.2. Abstract groups • 183
6.3. Isomorphie interpretations • 189
6.4. Abstract field system 197
6.5. Interpretations of the abstract system 200
6.6. Further development of the abstract field system 201
6.7. Types of informality in proofs 202
6.8. Solution of a linear equation 205
6.9. Further theorems in the abstract field system 209
CONTENTS ix
6.10. Solution of a quadratic equation 210
6.11. Subtraction and division 213
6.12. Solution of simultaneous linear equations 214
6.13. Negation vs. negative number 218
6.14. Ordered fields 220
6.15. Absolute value 227
6.16. Inequalities and absolute value 229
6.17. Applications to notions of limit 234
6.18. Restricted quantification 238
Appendix. Symbolic Treatment of the Miniature Geometry 243
Index 271
|
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| indexdate | 2024-07-09T15:42:28Z |
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| language | English |
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| physical | XI, 274 S. |
| publishDate | 1959 |
| publishDateSearch | 1959 |
| publishDateSort | 1959 |
| publisher | McGraw-Hill |
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| spelling | Exner, Robert M. Verfasser aut Logic in elementary mathematics Robert M. Exner ; Myron F. Rosskopf* New York (u.a.) McGraw-Hill 1959 XI, 274 S. txt rdacontent n rdamedia nc rdacarrier Mathematik Philosophie Logic, Symbolic and mathematical Mathematics Philosophy Rosskopf, Myron F. Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001466944&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Exner, Robert M. Rosskopf, Myron F. Logic in elementary mathematics Mathematik Philosophie Logic, Symbolic and mathematical Mathematics Philosophy |
| title | Logic in elementary mathematics |
| title_auth | Logic in elementary mathematics |
| title_exact_search | Logic in elementary mathematics |
| title_full | Logic in elementary mathematics Robert M. Exner ; Myron F. Rosskopf* |
| title_fullStr | Logic in elementary mathematics Robert M. Exner ; Myron F. Rosskopf* |
| title_full_unstemmed | Logic in elementary mathematics Robert M. Exner ; Myron F. Rosskopf* |
| title_short | Logic in elementary mathematics |
| title_sort | logic in elementary mathematics |
| topic | Mathematik Philosophie Logic, Symbolic and mathematical Mathematics Philosophy |
| topic_facet | Mathematik Philosophie Logic, Symbolic and mathematical Mathematics Philosophy |
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