Intuitionism vs. classicism: a mathematical attack on classical logic
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| Main Author: | |
|---|---|
| Format: | Thesis Book |
| Language: | English |
| Published: |
Frankfurt am Main
Vittorio Klostermann
[2015]
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| Series: | Studies in theoretical philosophy
vol. 2 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XVI, 269 Seiten 235 mm x 155 mm |
| ISBN: | 9783465039068 |
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CONTENTS
PREFACE XI
INTRODUCTION XIII
1 ARGUING ABOUT CLASSICAL LOGIC 1
1.1 LOGICAL THEORIES 1
1.1.1 THE BEARERS OF LOGICAL PROPERTIES 2
1.1.2 TOTAL AND PARTIAL LOGICAL THEORIES 3
1.1.3 THE SHAPE OF A LOGICAL THEORY 4
1.2 CLASSICAL AND INTUITIONISTIC LOGIC 6
1.2.1 THE FORMAL SYNTACTIC PART 6
1.2.2 A MODEL-THEORETIC LOGICAL THEORY 8
1.2.3 TWO DERIVATIONAL LOGICAL THEORIES 10
1.2.4 THE TRANSFORMATIONAL PART 13
1.2.5 LOGICALLY VALID RULES 16
1.3 ARGUING AGAINST A LOGICAL THEORY 19
1.3.1 TYPES OF SENTENCES OF THE LANGUAGE OF T 19
1.3.2 FOUR WAYS OF ARGUING AGAINST A LOGICAL THEORY 21
1.4 RELEVANCE 24
1.4.1 CONDITIONALLY, IMPLICATION, AND VALIDITY 25
1.4.2 RELEVANTISM AND CLASSICAL MATHEMATICS 27
1.5 EMPTY SINGULAR TERMS 29
1.5.1 THEORIES OF FREE LOGIC 30
1.5.2 FREE LOGIC AND NON-STANDARD CLASSICAL LOGIC 31
2 A PURELY VERBAL DISPUTE? 35
2.1 MATHEMATICAL LANGUAGES 36
2.1.1 NATURAL MATHEMATICAL LANGUAGES 36
2.1.2 FORMAL MATHEMATICAL LANGUAGES 38
2.2 LOGICAL VOCABULARY 39
2.2.1 INCLUSIVE MONISM AND THE BURDEN OF PROOF 41
2.2.2 META-LINGUISTIC UNIQUENESS RESULTS 42
2.2.3 RULE-BASED UNIQUENESS RESULTS 46
2.2.4 PHILOSOPHICAL CONSEQUENCES OF UNIQUENESS 51
2.3 MATHEMATICAL VOCABULARY 57
2.3.1 THEORIES ABOUT NUMBERS 58
2.3.2 THEORIES ABOUT SETS OF NUMBERS 63
HTTP://D-NB.INFO/1064044689
VIII CONTENTS
2.3.3 THEORIES ABOUT SEQUENCES OF NUMBERS 66
2.3.4 SECOND-ORDER THEORIES 68
2.4 THE SIMPLICITY OF UNDERSTANDING 72
2.4.1 AN ARGUMENT AGAINST SHARED UNDERSTANDING 73
2.4.2 A CAUSAL THEORY OF UNDERSTANDING 75
2.4.3 LOW REQUIREMENTS FOR UNDERSTANDING 77
3 COUNTEREXAMPLES TO CLASSICAL LOGIC 81
3.1 A MATHEMATICAL CONFLICT 82
3.1.1 THE UNIFORMITY PRINCIPLE (UP) 82
3.1.2 THE CONTINUITY PRINCIPLE (CP) 84
3.2 JUSTIFICATIONS OF UP AND CP 86
3.2.1 A JUSTIFICATION OF UP 87
3.2.2 A JUSTIFICATION OF CP 96
3.3 WEAK COUNTEREXAMPLES 100
3.3.1 WHAT ARE WEAK COUNTEREXAMPLES? 101
3.3.2 ESTABLISHING WEAK COUNTEREXAMPLES 103
3.4 A LOGICAL BASIS OF MATHEMATICS? 106
3.4.1 BROUWERIAN MATHEMATICS 107
3.4.2 THE NORMATIVITY OF LOGIC AND MATHEMATICS 109
3.4.3 PRAWITZ' ACCOUNT OF CORRECT INFERENCES 114
4 THE MANIFESTATION ARGUMENT 121
4.1 A SKETCH OF THE MANIFESTATION ARGUMENT 122
4.1.1 A BASIC VERSION OF THE ARGUMENT 123
4.1.2 THREE VARIANTS OF THE ARGUMENT 126
4.2 TRUTH CONDITIONAL SEMANTICS 131
4.2.1 PRELIMINARIES 132
4.2.2 A PROBLEM FOR TRUTH CONDITIONAL SEMANTICS 133
4.2.3 TRUTH-THEORY 134
4.2.4 THE SEMANTIC SIGNIFICANCE OF THEORIES OF TRUTH 138
4.2.5 REFINED TRUTH-THEORY 139
4.3 UNDERSTANDING AND KNOWLEDGE 142
4.3.1 IDEAL UNDERSTANDING 142
4.3.2 UNDERSTANDING AND KNOWLEDGE OF TRUTH CONDITIONS . 146
4.4 UNDERSTANDING AND USE 149
4.4.1 DUMMETTIAN ACCOUNTS OF UNDERSTANDING 150
4.4.2 KINDS OF MANIFESTING BEHAVIOUR 154
4.4.3 DESCRIPTIONS OF MANIFESTING BEHAVIOUR 155
4.4.4 MANIFESTABILITY OF KNOWLEDGE OF TRUTH CONDITIONS . 156
4.5 THE SIMPLICITY OF MANIFESTABILITY 160
4.5.1 COMPOSITIONAL MANIFESTATION 161
4.5.2 MANIFESTATION AND EPISTEMIC CONSTRAINTS 167
5 PROOF-THEORETIC ARGUMENTS 174
5.1 PROOF-THEORETIC PRELIMINARIES 175
5.1.1 GENERAL PROOF-THEORETIC CONCEPTS 175
CONTENTS IX
5.1.2 CALCULI OF NATURAL DEDUCTION 179
5.1.3 SCHEMATA FOR RULES AND STANDARD CALCULI 183
5.2 ACCOUNTS OF MEANING AND LOGICAL FRAMEWORKS 186
5.2.1 THE GENERAL FORM OF PROOF-THEORETIC ARGUMENTS 186
5.2.2 PERMISSIBLE LOGICAL FRAMEWORKS 190
5.3 TENNANT'S PROOF-THEORETIC ARGUMENT 195
5.3.1 SEPARABILITY, PURITY, SIMPLICITY, AND DIRECTNESS 196
5.3.2 FAILURE AND IMPROVEMENT 202
5.4 HARMONY 206
5.4.1 THE IDEA OF DUMMETT'S PROOF-THEORETIC ARGUMENT . 206
5.4.2 INTRODUCTION AND ELIMINATION RULES 209
5.4.3 THE RELATION OF HARMONY 213
5.5 DUMMETT'S PROOF-THEORETIC ARGUMENT 218
5.5.1 THE FUNDAMENTAL IDEA OF DUMMETT'S ARGUMENT 220
5.5.2 THE FIRST VERSION OF DUMMETT'S ARGUMENT 224
5.5.3 THE SECOND VERSION OF DUMMETT'S ARGUMENT 226
5.6 AGAINST DUMMETT'S PROOF-THEORETIC ARGUMENT 229
5.6.1 REFLECTION AND CONJUGATION 229
5.6.2 CLASSICAL LOGIC AND NON-CIRCULAR CLASSICAL RULES 236
6 CONCLUSION 240
A APPENDIX TO CHAPTER 2 244
A.L ARITHMETICAL UNIQUENESS 244
A.2 SET-THEORETIC UNIQUENESS 253
A.3 FUNCTION-THEORETIC UNIQUENESS 255
BIBLIOGRAPHY 257
INDEX 268 |
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| author | Haverkamp, Nick |
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| dewey-ones | 511 - General principles of mathematics |
| dewey-raw | 511.3 |
| dewey-search | 511.3 |
| dewey-sort | 3511.3 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Philosophie |
| format | Thesis Book |
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| spelling | Haverkamp, Nick Verfasser aut Intuitionistic arguments against classical logic Intuitionism vs. classicism a mathematical attack on classical logic Nick Haverkamp Frankfurt am Main Vittorio Klostermann [2015] © 2015 XVI, 269 Seiten 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Studies in theoretical philosophy vol. 2 Dissertation Universität Hamburg 2013 Mathematische Logik (DE-588)4037951-6 gnd rswk-swf Intuitionistische Logik (DE-588)4162199-2 gnd rswk-swf (DE-588)4113937-9 Hochschulschrift gnd-content Mathematische Logik (DE-588)4037951-6 s Intuitionistische Logik (DE-588)4162199-2 s DE-604 Studies in theoretical philosophy vol. 2 (DE-604)BV042196788 2 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=5113967&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027722654&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Haverkamp, Nick Intuitionism vs. classicism a mathematical attack on classical logic Studies in theoretical philosophy Mathematische Logik (DE-588)4037951-6 gnd Intuitionistische Logik (DE-588)4162199-2 gnd |
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| title | Intuitionism vs. classicism a mathematical attack on classical logic |
| title_alt | Intuitionistic arguments against classical logic |
| title_auth | Intuitionism vs. classicism a mathematical attack on classical logic |
| title_exact_search | Intuitionism vs. classicism a mathematical attack on classical logic |
| title_full | Intuitionism vs. classicism a mathematical attack on classical logic Nick Haverkamp |
| title_fullStr | Intuitionism vs. classicism a mathematical attack on classical logic Nick Haverkamp |
| title_full_unstemmed | Intuitionism vs. classicism a mathematical attack on classical logic Nick Haverkamp |
| title_short | Intuitionism vs. classicism |
| title_sort | intuitionism vs classicism a mathematical attack on classical logic |
| title_sub | a mathematical attack on classical logic |
| topic | Mathematische Logik (DE-588)4037951-6 gnd Intuitionistische Logik (DE-588)4162199-2 gnd |
| topic_facet | Mathematische Logik Intuitionistische Logik Hochschulschrift |
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