Core logic:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Oxford
Oxford University Press
2017
|
| Ausgabe: | First edition |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | Hier auch später erschienene, unveränderte Nachdrucke |
| Beschreibung: | xvii, 357 Seiten |
| ISBN: | 9780198777892 |
Internformat
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Datensatz im Suchindex
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|---|---|
| adam_text | Contents
1 Introduction and Overview 1
1.1 The aim of this book......................................... 2
1.2 Inference and proof as primary in logic...................... 4
1.3 The debate over logical reform............................... 9
1.3.1 Debate over rules governing logical operators......... 10
1.3.2 Debate over EFQ and relevance......................... 11
1.3.3 The two main lines of reform of Classical Logic .... 14
1.3.4 Structural rules in the sequent calculi............... 15
1.3.5 On inductive definition of proof-in-system- 5 ........ 16
1.3.6 Reflexive stability .................................. 17
1.4 On pluralism about logic, and the explication of deductive
validity.................................................... 18
1.5 The familiar systems M, I, and C ........................... 19
1.5.1 The system M of Minimal Logic......................... 20
1.5.2 The system I of Intuitionistic Logic ................. 22
1.5.3 The system C of Classical Logic....................... 23
1.6 The system R of Relevance Logic............................. 24
1.6.1 Axioms of R........................................... 24
1.6.2 Core proofs of the axioms of R........................ 25
2 The Road to Core Logic 29
2.1 Constructivizing and relevantizing.......................... 30
2.2 Core Logic in relation to familiar systems of logic......... 32
2.2.1 On a question of unique determination of subsystems 33
2.2.2 On the matter of relevantization...................... 34
2.2.3 The logical landscape of systems...................... 34
2.3 What is Core Logic?—a Cook’s tour........................... 35
2.3.1 Comments on the rules of Core Logic .................. 35
2.3.2 Comments on discharge rules in Core Logic............. 36
2.3.3 The admissibility of cut for Core Logic............... 37
xi
xii CONTENTS
2.3.4 Special features of Core Logic that ensure relevance . 38
2.3.5 Core Logic provides enough transitivity of deduction
for all scientific purposes........................... 38
2.3.6 Philosophical arguments for Core Logic................ 40
2.3.7 Other important advantages of Core Logic.............. 43
2.3.8 Core Logic’s eschewal of thinning and cut as rules of
the proof system......................................... 45
2.3.9 Core Logic’s shedding of other old saws.................... 46
2.4 Why ‘core’?....................................................... 48
3 The Logic of Evaluation 51
3.1 The logical operators............................................. 52
3.2 Our inferentialism delivers the truth tables...................... 52
3.3 Atomic determinations............................................. 58
3.3.1 Using the simplest kind of atomic basis ................... 58
3.3.2 Co-inductive definition of A-relative verifications and
falsifications............................................. 59
3.3.3 Rules of verification and falsification, in graphic form 64
3.4 Extension to first order: saturated terms and formulae ... 68
3.4.1 Rules of verification and of falsification at first order . 70
3.4.2 Further features of first-order verifications and
falsifications relative to the simplest kind of atomic basis 73
3.4.3 On the soundness of the evaluation rules................... 74
3.4.4 A note on nomenclature..................................... 81
3.5 More general atomic bases......................................... 82
3.5.1 Inclusions and contrarieties............................... 83
3.5.2 The notion of a determination relative to an atomic
basis...................................................... 86
3.5.3 Broadening the atomic basis affords us a new way to
verify conditionals........................................ 92
3.5.4 Rules of determination relative to an atomic basis . . 94 4
4 From the Logic of Evaluation to the Logic of Deduction 100
4.1 From evaluation to deduction.................................101
4.2 Graphic rules of Core Logic C.................................103
4.3 Reasons for preferring parallelized to serial forms of
Elimination rule..............................................109
4.4 Absurdity and contrarieties ..................................110
4.4.1 Conclusion occurrences of absurdity (_L) ........ Ill
4.4.2 Harmony between introduction and elimination rules . 112
CONTENTS
4.5 Disjunctive syllogism.......................................115
4.5.1 Disjunction revisited.................................115
4.6 The rules of natural deduction as clauses in the inductive
definition of proof.........................................117
5 Motivating the Rules of Sequent Calculus 122
5.1 Basic steps of inference....................................122
5.2 Sequent calculus for Core Logic...............................126
5.3 The sequent rules as clauses in the inductive definition of
proof.........................................................128
5.4 The isomorphism between natural deductions and sequent
proofs .......................................................131
5.5 On maintaining concentration..................................132
5.5.1 Dilution v. concentration...............................133
5.5.2 Strengthening...........................................134
5.5.3 Core logic is free of dilution..........................134
5.5.4 Diluters and non-diluters...............................136
5.5.5 A new property for rules to preserve....................138
5.5.6 Two uniqueness conjectures..............................140
6 Transitivity of Deducibility 141
6.1 Nontransitive consequence relations...........................142
6.2 Single- v. multiple-conclusion sequent calculi................144
6.3 The transitivity ‘problem’ for Core Logic.....................145
6.3.1 Terminology ............................................149
6.4 Cut...........................................................150
6.4.1 Cut as a structural rule................................150
6.4.2 Must the system contain a rule of cut?..................150
6.4.3 Cut-admissibility.......................................151
6.4.4 Cut-admissibility strengthened for the Core systems . 152
6.4.5 Cut-admissibility for Core systems of Natural
Deduction...............................................153
6.4.6 On two proofs connecting (in any system S) .....153
6.4.7 Cut-admissibility for systems S in general..............154
6.5 How admissible cuts can achieve epistemic gain................155
6.6 Rules of Minimal, Intuitionistic, and Classical Logic.........156
6.6.1 Graphic conventions for stating rules...................157
6.6.2 Rules of Minimal Logic (in our preferred format) . . . 159
6.6.3 Rules of Intuitionistic Logic (in our preferred format) 161
6.6.4 Rules of Classical Logic (in our preferred format) . . . 161
XIV
CONTENTS
6.7 Rules of Inference for the Core Systems...................161
6.7.1 Rules of Core Logic .................................161
6.7.2 Rules of Classical Core Logic C+.....................163
6.7.3 Remarks on avoiding irrelevance.....................164
6.8 Compact Polish notation for proofs........................166
6.8.1 On the 5-relativity of our notations................167
6.8.2 Notation for proofs of existentials...................167
6.8.3 Notation for proofs of universals.....................168
6.8.4 Notation for proofs from existentials.................168
6.8.5 Notation for proofs from universals...................168
6.9 Grounding cases...........................................168
6.9.1 Grounding cases for the three standard systems .... 169
6.9.2 Grounding cases for the two Core systems............170
6.10 Non-grounding IL-E proof pairs...........................172
6.11 Classification of transformations for
non-grounding cases.........................................172
6.11.1 cr does not have the cut sentence A as its MPE .... 173
6.11.2 a does have the cut sentence A as its MPE.............173
6.12 Terminology for conversions...............................173
6.12.1 Distribution conversions..............................174
6.12.2 Permutative conversions...............................174
6.13 Reductions..................................................175
6.13.1 I-reductions..........................................175
6.13.2 EFQ-reductions (only for I and C).....................175
6.13.3 Dil-reductions (only for C and C+)....................176
6.14 Table of transformations for each system,
in the terminology of natural deduction.....................177
6.15 Table of transformations for each system,
in the terminology of sequent calculus......................178
6.16 Main results................................................179
6.17 Extraction theorems ........................................180
6.18 Taking stock................................................184
7 Epistemic Gain 186
7.1 How Core Logic avoids Lewis’s First Paradox.................186
7.2 On whether A, -i Aj= B.............................. 192
7.2.1 An important dialectical qualification................195
7.3 The status of Cut, and the methodological
adequacy of the Core systems...............................195
7.3.1 The status of Cut ....................................195
CONTENTS
XV
7.3.2 Core Logic is adequate for intuitionistic mathematics . 196
7.3.3 Classical Core Logic suffices for classical mathematics 197
7.3.4 Core Logic suffices for the experimental testing of
scientific hypotheses...............................197
7.4 Replies to critics: an anticipation......................203
8 Truthmakers and Consequence 205
8.1 Defining logical consequence in terms
of truthmakers...........................................206
8.2 Truthmakers and Tarskian truth...........................211
8.3 Truthmakers as game plans ...............................212
8.4 Truthmaker transformation and classical logical consequence 216
8.5 Resisting a threatened slide to realism..................220
8.6 Any proof in Core Logic turns truthmakers for its premises
into a truthmaker for its conclusion.....................223
8.6.1 Example of preservation of truthmakers.............223
8.6.2 A distribution conversion to obtain M-relative
verifications..................................... 225
8.6.3 A distribution conversion to obtain M-relative
falsifications......................................225
8.6.4 The reduction procedure for existentials...........226
8.6.5 Back to our example.................................226
8.7 More examples of how to produce truthmakers for conclusions
of proofs, from truthmakers for their premises...........229
8.7.1 Example 1...........................................229
8.7.2 Example 2...........................................233
8.7.3 Example 3...........................................238
8.7.4 Example 4...........................................242
8.7.5 Example 5............................... .... 246
8.7.6 Example 6...........................................249
8.8 Taking stock........................................... 251
9 Transmission of Truthmakers 252
9.1 Conditional evaluations..................................252
9.2 Reducts for M-relative conditional constructs:
grounding cases..........................................254
9.3 Non-grounding LEE pairs..................................255
9.4 Transformations for reducts in non-grounding cases.......255
9.4.1 List of transformations.............................256
9.5 Results..................................................258
xvi CONTENTS
9.6 Some reflections on methodology............................259
9.7 Formal semantics reworked in terms of truthmakers.........260
10 The Relevance Properties of Core Logic 262
10.1 A reflexive irony in the motivation for R...................264
10.2 Definitions of concepts, and easy results...................265
10.2.1 On positive and negative occurrences of subsentences . 266
10.2.2 Further syntactic properties and relations defined in
terms of signed occurrences...........................268
10.2.3 The explication of relevance of premises to conclusion
of a relevantly valid sequent.........................271
10.3 Reasons why 1Z(A, ip) is a good explication of relevance . . . 275
10.3.1 Would-be strengthenings of 77(A, to which there
are counterexamples...................................275
10.4 Main result ...............................................277
10.5 Final reflections..........................................279
11 Core Logic and the Paradoxes 281
11.1 What role, if any, does the law of excluded middle play in
the derivation of the paradoxes?............................281
11.2 Excluded middle plays no essential role in the derivation
of the paradoxes............................................285
11.3 A proof-theoretic criterion of paradoxicality..............288
11.3.1 Background............................................288
11.3.2 The Ekman Problem.....................................289
11.3.3 A crucial distinction: serial v. parallelized elimination
rules in natural deduction............................289
11.4 Revisiting Russell’s Paradox...............................294
11.5 Rules of truth, and id est rules: the Liar is still paradoxical 298
11.5.1 Digression on Cut and transitivity of deduction .... 302
11.5.2 The inadmissibility of cut in languages containing
paradoxes.............................................304
11.6 Summary and conclusion.....................................305
12 Replies to Critics of Core Logic 307
12.1 Reply to Beall and Restall.................................308
12.2 Reply to Hartmann..........................................311
12.3 Interlude on perfect validity and entailment...............315
12.4 Reply to Burgess.............................................
CONTENTS
XVII
12.4.1 The distinction between prescriptive and
descriptive criticism of Classical Logic..............317
12.4.2 Correcting some misdescriptions of our brand of
perfectionism....................................... 318
12.4.3 Rebuttal of Burgess on Core-logical reformism .... 322
12.5 Reply to Friedman and Avron .................................327
12.5.1 On the empty set being a subset of every set.........328
12.5.2 Core Logic is not merely ‘partially’ complete........334
Bibliography 338
Index 346
In this book, Neil Tennant presents an original logical system. Core
Logic is the first system that ensures both relevance and adequacy for the
formalization of all scientific and constructive mathematical reasoning. It is
an elegant kernel lying deep within Classical Logic, a canon for constructive
and relevant deduction furnishing faithful formalizations of informal
constructive mathematical proofs. It provides transitivity of deduction with
potential epistemic gain. Classical Core Logic does the same for informal
non-constructive mathematical proofs.
Because of its clarity about the true internal structure of proofs, Core Logic
also affords advantages for the automation of deduction and our
appreciation of the paradoxes.
Tennant describes all the theoretically interesting aspects of Core
Logic: philosophical, metamathematical, proof-theoretic, methodological,
computational and revision-theoretic. Here for the first time they are all
examined together in a single work. A unified
and all-encompassing treatment of Core
Logic is called for because its many
and various aspects have thus far
been dealt with in relative isolation
from each other, within different
sub-specialist realms of the
literature. No single work has yet
drawn together all the different
aspects to show how they are
mutually illuminating and how
they fruitfully interconnect.
|
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| id | DE-604.BV044515655 |
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| indexdate | 2024-07-10T07:54:43Z |
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| isbn | 9780198777892 |
| language | English |
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| spelling | Tennant, Neil 1950- Verfasser (DE-588)138937672 aut Core logic Neil Tennant First edition Oxford Oxford University Press 2017 xvii, 357 Seiten txt rdacontent n rdamedia nc rdacarrier Hier auch später erschienene, unveränderte Nachdrucke Logik (DE-588)4036202-4 gnd rswk-swf Logic Logik (DE-588)4036202-4 s DE-604 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029915330&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029915330&sequence=000002&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Tennant, Neil 1950- Core logic Logik (DE-588)4036202-4 gnd |
| subject_GND | (DE-588)4036202-4 |
| title | Core logic |
| title_auth | Core logic |
| title_exact_search | Core logic |
| title_full | Core logic Neil Tennant |
| title_fullStr | Core logic Neil Tennant |
| title_full_unstemmed | Core logic Neil Tennant |
| title_short | Core logic |
| title_sort | core logic |
| topic | Logik (DE-588)4036202-4 gnd |
| topic_facet | Logik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029915330&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029915330&sequence=000002&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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