The functional interpretation of logical deduction:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New Jersey [u.a.]
World Scientific
2012
|
| Series: | Advances in logic
5 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 253 - 264 |
| Physical Description: | XXXII, 266 S. |
| ISBN: | 9789814360951 9814360953 |
Staff View
MARC
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| 245 | 1 | 0 | |a The functional interpretation of logical deduction |c Ruy J.G.B. de Queiroz ; Anjolina G. de Oliveira ; Dov. M. Gabbay |
| 264 | 1 | |a New Jersey [u.a.] |b World Scientific |c 2012 | |
| 300 | |a XXXII, 266 S. | ||
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| 500 | |a Literaturverz. S. 253 - 264 | ||
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Record in the Search Index
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|---|---|
| adam_text | Contents
Preface xj
Overview
xiii
1.
Labelled Natural Deduction
1
1.1
The
rôle
of the labels
.......................
І
1.1.1
Dividing the tasks: A functional calculus
on the labels, a logical calculus on the formula
..... 4
1.1.2
Reassessing Frege s two-dimensional calculus
..... 5
1.2
Canonical proofs and normalisation
............... 8
• 1.2.1
Canonical proofs
..................... 11
1.2.2
Normalisation
...................... 13
2.
The Functional Interpretation of Implication
27
2.1
Introduction
............................ 27
2.2
Origins
.............................. 35
2.3
Types and propositions
...................... 38
2.4
A-abstractkm and implication
................... 43
2.5
Consistency proof
......................... 59
2.6
Systems of implication and
combinatore
............. 66
2.7
Finale
............................... 68
3.
The Existential Quantifier
77
3.1
Motivation
............................ 78
3.1.1
The pairing interpretation
................ 80
viii
The Functional Interpretation of Logical Deduction
3.2
Quantifiers and normalisation
.................. 84
3.2.1
Introducing variables for the Skolem
dependency functions
.................. 85
3.2.2
The hiding principle
................... 89
3.3
Other approaches to existential quantification
.......... 91
3.3.1
Systems of natural deduction based on direct
existential instantiation
................. 92
3.3.2
Axiomatic systems based on the notion
of such that
....................... 95
3.4
Model-theoretic semantics
.................... 97
3.4.1
Constants versus variables revisited
........... 98
3.4.2
Eliminability and conservative extensions
....... 100
3.5
Finale
............................... 101
3.5.1
Extensions to higher-order existentials
......... 103
3.5.2
Further connections to model-theoretic
interpretations
...................... 103
3.6
Examples of deduction
...................... 103
3.6.1
Generic examples
.................... 103
3.6.2
Specific examples
.................... 108
4.
Normalisation 111
4.1
Introduction
............................
Ill
4.2
Proof transformations in labelled deduction
........... 113
4.3
Equivalences between proofs in LND
............... 115
4.4
The term rewriting system for LND
................ 118
4.4.1
Defining the LND-TRS
.................. 118
4.4.2
The sort decreasing property
.............. 119
4.4.3
Defining an order
.................... 120
4.4.4
Proving the termination property
............ 124
4.4.5
Proving the confluence property
............. 126
4.5
Examples of transformations between proofs
........... 128
4.6
Final remarks
........................... 129
5.
Natural Deduction for Equality
147
5.1
Introduction
............................ 147
5.2
Labelled deduction
........................ 154
5.2.1
Identifiers for (compositions of) equalities
....... 160
5.2.2
The proof rales
..................... 161
5.3
Finale
............................... 169
Contents
ín
6.
Normalisation
for the Equality Fragment
171
6.1
General rules
........................... 171
6.2
The subterm substitution rule
.................. 172
6.3
The |-and
μ
-rules
........................
174
6.4
Term rewriting systems
...................... 174
6.4.1
Termination property
.................. 175
6.4.2
Confluence property
................... 175
6.4.3
The completion procedure
................ 176
6.5
The transformations between proofs in the equational
fragment of the LND
.......................176
6.5.1
Reductions on the rewrite sequence
...........177
6.5.2
Transformations on the rewrite reasons
.........182
6.6
The rewriting system for the LND equational logic
.......185
6.6.1
Termination property for the LNDbq-TRS
...... 187
6.6.2
Confluence property for the LNDeq-TRS
...... 188
6.6.3
The normalization theorems
............... 189
6.7
The normalization procedure: some examples
.......... 190
6.8
Final remarks
........................... 191
6.9
Appendix: The
β-
and
η
-reductions for the LND system
..... 191
6.9.1 ß-type
reductions
.................... 191
6.9.2
tj-type reductions
.................... 193
6.9.3
C-tyi^ transformations: The permutative
transformations
..................... 194
6.10
Termination property for LNDeq-TRS
............. 195
7.
Modal Logics
199
7.1
The functional interpretation
...................200
7.1.1
Handling assumptions with world variables
...... 202
7.1.2
Natural deduction with an extra parameter
....... 206
7.1.3
Connections with Kripke s truth definition
....... 209
7.1.4
Connections with Gentzen s sequent calculus
...... 210
7.2
Modal logics and the functional interpretation
.......... 212
7.2.1
Chellas
RM role
for standard normative
logics
..............·............ 213
7.2.2
Modal logic
К
...................... 213
7.2.3
Modal logic
D
...................... 214
7.2.4
Modal logic
Τ
...................... 216
7.2.5
Modal logic
В
...................... 217
χ
The Functional Interpretation of Logical Deduction
7.2.6
Modal logic S4 and its parallel with
intuitionistic logic
....................218
7.2.7
Grzegorczyk s extension of S4
.............218
7.2.8
Modal logic S5 and its parallel
with classical logic
................... 220
7.2.9
Non-normal modal logics
................ 221
7.2.10
Problematic cases: The scope of necessity
...... 222
7.3
Finale
............................... 223
8.
Meaning and Proofs: A Reflection on Proof-Theoretic
Semantics
225
8.1
Proof-theoretic semantics
..................... 225
8.2
Meaning, use and consequences
................. 232
8.3
Meaning and purpose
....................... 233
8.4
Meaning and use
......................... 237
8.5
Meaning and the explanation of consequences
.......... 240
8.6
Use and the explanation of consequences
............ 241
8.7
Early signs of meaning-use/
usefulness-consequences
....................244
8.8
Normalisation of proofs: the explanation
of the consequences
........................249
8.9
Concluding
remarlo
........................252
Bibliography
253
Index
265
|
| any_adam_object | 1 |
| author | Queiroz, Ruy J. G. B. de 1958- Oliveira, Anjolina G. de Gabbay, Dov M. 1945- |
| author_GND | (DE-588)173616917 (DE-588)124196314 |
| author_facet | Queiroz, Ruy J. G. B. de 1958- Oliveira, Anjolina G. de Gabbay, Dov M. 1945- |
| author_role | aut aut aut |
| author_sort | Queiroz, Ruy J. G. B. de 1958- |
| author_variant | r j g b d q rjgbd rjgbdq a g d o agd agdo d m g dm dmg |
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| bvnumber | BV040141111 |
| classification_rvk | CC 2600 SK 130 |
| ctrlnum | (OCoLC)793806745 (DE-599)BSZ358488753 |
| dewey-full | 511.3 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| 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 | Book |
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| illustrated | Not Illustrated |
| indexdate | 2024-07-10T00:17:48Z |
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| isbn | 9789814360951 9814360953 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-024998049 |
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| physical | XXXII, 266 S. |
| publishDate | 2012 |
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| publisher | World Scientific |
| record_format | marc |
| series | Advances in logic |
| series2 | Advances in logic |
| spelling | Queiroz, Ruy J. G. B. de 1958- Verfasser (DE-588)173616917 aut The functional interpretation of logical deduction Ruy J.G.B. de Queiroz ; Anjolina G. de Oliveira ; Dov. M. Gabbay New Jersey [u.a.] World Scientific 2012 XXXII, 266 S. txt rdacontent n rdamedia nc rdacarrier Advances in logic 5 Literaturverz. S. 253 - 264 Interpretation (DE-588)4072905-9 gnd rswk-swf Logischer Schluss (DE-588)4139983-3 gnd rswk-swf Logic, Symbolic and mathematical Logischer Schluss (DE-588)4139983-3 s Interpretation (DE-588)4072905-9 s DE-604 Oliveira, Anjolina G. de Verfasser aut Gabbay, Dov M. 1945- Verfasser (DE-588)124196314 aut Advances in logic 5 (DE-604)BV025293756 5 Digitalisierung UB Bamberg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024998049&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Queiroz, Ruy J. G. B. de 1958- Oliveira, Anjolina G. de Gabbay, Dov M. 1945- The functional interpretation of logical deduction Advances in logic Interpretation (DE-588)4072905-9 gnd Logischer Schluss (DE-588)4139983-3 gnd |
| subject_GND | (DE-588)4072905-9 (DE-588)4139983-3 |
| title | The functional interpretation of logical deduction |
| title_auth | The functional interpretation of logical deduction |
| title_exact_search | The functional interpretation of logical deduction |
| title_full | The functional interpretation of logical deduction Ruy J.G.B. de Queiroz ; Anjolina G. de Oliveira ; Dov. M. Gabbay |
| title_fullStr | The functional interpretation of logical deduction Ruy J.G.B. de Queiroz ; Anjolina G. de Oliveira ; Dov. M. Gabbay |
| title_full_unstemmed | The functional interpretation of logical deduction Ruy J.G.B. de Queiroz ; Anjolina G. de Oliveira ; Dov. M. Gabbay |
| title_short | The functional interpretation of logical deduction |
| title_sort | the functional interpretation of logical deduction |
| topic | Interpretation (DE-588)4072905-9 gnd Logischer Schluss (DE-588)4139983-3 gnd |
| topic_facet | Interpretation Logischer Schluss |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024998049&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV025293756 |
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