What logics mean: from proof theory to model-theoretic semantics
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2013
|
| Edition: | 1. publ. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references Preface; 1. Introduction to model-theoretic inferentialism; 2. Deductive expression; 3. Local expression; 4. Global expression; 5. Intuitionistic semantics; 6. Conditionals; 7. Disjunction; 8. Negation; 9. Supervaluations and natural semantics; 10. Natural semantics for an open future; 11. The expressive power of sequent calculi; 12. Soundness and completeness for natural semantics; 13. Connections with proof-theoretic semantics; 14. Quantifiers; 15. Natural semantics and vagueness; 16. Modal logic; Summary |
| Physical Description: | XV, 285 S. |
| ISBN: | 9781107611962 9781107039100 |
Staff View
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Record in the Search Index
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| adam_text | What Logics Mean
From Proof Theory to Model-Theoretic Semantics
JAMES W GARSON
HI CAMBRIDGE
UNIVERSITY PRESS
Contents
Preface page xi
Acknowledgements xv
1 Introduction to model-theoretic inferentialism 1
1 1 The broader picture 1
1 2 Proof-theoretic and model-theoretic inferentialism 3
1 3 Three rule formats 7
1 4 Expressive power and models of rules 10
1 5 Deductive models 12
1 6 Global models 15
1 7 Local models 18
1 8 Three definitions of expressive power compared 19
1 9 What counts as a logical connective? 24
2 Deductive expression 25
2 1 Deductive expression defined 25
2 2 Negative results for deductive expression 26
2 3 Semantic holism 31
3 Local expression 34
3 1 Natural deduction rules 35
3 2 ND rules and sequent calculi for the conditional 37
3 3 The Local Expression Theorem 37
3 4 Local expressive power and completeness 40
3 5 Local expression evaluated 42
4 Global expression 46
4 1 Global expression and preservation of validity 47
4 2 Natural semantics 49
vu
viii Contents
4 3 The canonical model 52
4 4 Negative results for global expression 54
5 Intuitionistic semantics 57
5 1 Kripke semantics for intuitionistic logic 57
5 2 Intuitionistic models 59
5 3 Complaints against intuitionistic models 61
5 4 The Isomorphism Theorem 62
5 5 Intuitionistic models and functional semantics 65
5 6 Forcing and intuitionistic models of set theory (an aside) 68
6 Conditionals 71
6 1 Intuitionistic truth conditions for the conditional 71
6 2 Peirce s Law and Peirce s Rule 74
6 3 Intuitionistic natural semantics for equivalence 78
6 4 Summary: natural semantics for intuitionistic logic 79
7 Disjunction 81
7 1 Beth s intuitionistic truth condition for disjunction 81
7 2 What disjunction rules express 83
7 3 Do the disjunction rules express a semantics? 84
7 4 Path semantics for disjunction 86
7 5 Isomorphism for path models 88
7 6 The failure of functionality and compositionality 89
7 7 Converting natural into classical models 90
s78 Proofs of theorems in Chapter 7 93
8 Negation 105
8 1 Negation and intuitionistic semantics 106
8 2 Intuitionistic truth conditions for classical negation 108
8 3 ||LL||: what double negation expresses 109
|84 Using ||LL|| to generate classical models 110
85A structural version of ||LL| | 112
8 6 Why classical negation has no functional semantics 115
8 7 Disjunction with classical negation 116
8 8 Possibilities semantics for classical prepositional logic 121
8 9 Does classical logic have a natural semantics? 124
8 10 The primacy of natural semantics 129
1 8 11 Proofs of some results in Section 8 7 131
Contents ix
9 Supervaluations and natural semantics 134
9 1 Supervaluation semantics 134
9 2 Supervaluations and the canonical model for PL 136
9 3 Partial truth tables 137
9 4 The failure of supervaluations to preserve validity 139
9 5 Is supervaluation semantics a semantics? 141
9 6 Proofs for theorems in Chapter 9 142
10 Natural semantics for an open future 147
10 1 The open future 147
10 2 Detennination in natural semantics for PL 148
10 3 The Lindenbaum Condition revisited 150
10 4 Disjunction, choice, and Excluded Middle 151
10 5 Defeating fatalism 153
10 6 The No Past Branching condition 156
11 The expressive power of sequent calculi 162
11 1 Sequent calculi express classical truth conditions 163
11 2 The meaning of the restriction on the right-hand side 167
11 3 Sequent systems are essentially extensional 169
11 4 What counts as a logic? 171
11 5 Proofs of theorems in Section 11 2 174
12 Soundness and completeness for natural semantics 177
12 1 A general completeness theorem for natural semantics 177
12 2 Sample adequacy proofs using natural semantics 179
12 3 Natural systems and modular completeness 181
12 4 Completeness of sequent calculi using natural semantics 183
13 Connections with proof-theoretic semantics 187
13 1 Conservation and connective definition 187
13 2 Strong conservation 190
13 3 Semantical independence 193
13 4 Uniqueness 195
13 5 Harmony in the proof-theoretic tradition 197
13 6 Unity: a model-theoretic version of harmony 200
13 7 Unity and harmony compared 203
13 8 A proof-theoretic natural semantics 208
t
X Contents
14 Quantifiers 211
14 1 Syntax and natural deduction rules for the quantifiers 211
14 2 The objectual and substitution interpretations 213
14 3 Negative results for what quantifier rules express 216
14 4 The sentential interpretation 217
14 5 Showing the sentential interpretation is a semantics 220
14 6 Quantification in a classical setting 223
14 7 The prospects for referential semantics 226
14 8 The existential quantifier 229
14 9 The omega rule and the substitution interpretation 232
14 10 Hacking s program and the omega rule 235
14 11 Proofs of theorems in Chapter 14 237
15 The natural semantics of vagueness (with Joshua D K Brown) 244
15 1 Formal preliminaries 245
15 2 How vagueness is handled in natural semantics 247
15 3 How Williamson s objections are resolved 247
15 4 Why vagueness runs deep 254
16 Modal logic 255
16 1 Natural semantics for the modal logic K 255
16 2 Natural semantics for extensions of K 260
16 3 The possibility operator 263
16 4 The natural semantics of quantified modal logic 264
16 5 Variations on the definition of validity 268
Summaiy 271
References 275
Index 280
i
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| language | English |
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| spelling | Garson, James W. 1943- Verfasser (DE-588)139263128 aut What logics mean from proof theory to model-theoretic semantics James W. Garson 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2013 XV, 285 S. txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references Preface; 1. Introduction to model-theoretic inferentialism; 2. Deductive expression; 3. Local expression; 4. Global expression; 5. Intuitionistic semantics; 6. Conditionals; 7. Disjunction; 8. Negation; 9. Supervaluations and natural semantics; 10. Natural semantics for an open future; 11. The expressive power of sequent calculi; 12. Soundness and completeness for natural semantics; 13. Connections with proof-theoretic semantics; 14. Quantifiers; 15. Natural semantics and vagueness; 16. Modal logic; Summary Logic Semantics Logic, Symbolic and mathematical Metalogik (DE-588)4169642-6 gnd rswk-swf Formale Semantik (DE-588)4122144-8 gnd rswk-swf Mathematische Logik (DE-588)4037951-6 gnd rswk-swf Mathematische Logik (DE-588)4037951-6 s Metalogik (DE-588)4169642-6 s Formale Semantik (DE-588)4122144-8 s DE-604 HEBIS Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027034179&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Garson, James W. 1943- What logics mean from proof theory to model-theoretic semantics Logic Semantics Logic, Symbolic and mathematical Metalogik (DE-588)4169642-6 gnd Formale Semantik (DE-588)4122144-8 gnd Mathematische Logik (DE-588)4037951-6 gnd |
| subject_GND | (DE-588)4169642-6 (DE-588)4122144-8 (DE-588)4037951-6 |
| title | What logics mean from proof theory to model-theoretic semantics |
| title_auth | What logics mean from proof theory to model-theoretic semantics |
| title_exact_search | What logics mean from proof theory to model-theoretic semantics |
| title_full | What logics mean from proof theory to model-theoretic semantics James W. Garson |
| title_fullStr | What logics mean from proof theory to model-theoretic semantics James W. Garson |
| title_full_unstemmed | What logics mean from proof theory to model-theoretic semantics James W. Garson |
| title_short | What logics mean |
| title_sort | what logics mean from proof theory to model theoretic semantics |
| title_sub | from proof theory to model-theoretic semantics |
| topic | Logic Semantics Logic, Symbolic and mathematical Metalogik (DE-588)4169642-6 gnd Formale Semantik (DE-588)4122144-8 gnd Mathematische Logik (DE-588)4037951-6 gnd |
| topic_facet | Logic Semantics Logic, Symbolic and mathematical Metalogik Formale Semantik Mathematische Logik |
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