Interpolation and Definability Modal and Intuitionistic Logic

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ISBN: 9780198511748 | 0198511744
Cover: Hardcover
Copyright: 7/28/2005

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This monograph is on interpolation and definability, a notion central in pure logic and with significant meaning and applicability in all areas where logic is applied, especially computer science, artificial intelligence, logic programming, philosophy of science and natural language. Suitable for researchers and graduate students in mathematics, computer science and philosophy, this is the latest in the prestigious world-renowned Oxford Logic Guides, which contains Michael Dummet's Elements of Intuitionism (Second Edition) , J.M. Dunn and G. Hardegree's Algebraic Methods in Philosophical Logic , H. Rott's Change Choice and Inference: A Study of Belief Revision and Nonmonotonic Reasoning , P.T. Johnstone's Sketches of an Elephant: A Topos Theory Compendium: Volumes 1 and 2 , and David J. Pym and Eike Ritter's Reductive Logic and Proof Search: Proof, Theory, Semantics and Control.

1 Introduction and discussion

(34)

1.1 General discussion

(13)

1.1.1 View 1: Common logical content

(1)

1.1.2 View 2: Expressive power

(2)

1.1.3 View 3: Quantifier elimination

(2)

1.1.4 View 4: Artificial intelligence

(1)

1.1.5 View 5: Proof theory

(1)

1.1.6 View 6: Consistency

(1)

1.1.7 View 7: Semantical view

(1)

1.1.8 View 8: Algebraic view

(3)

1.1.9 View 9: Definability

(1)

1.1.10 View 10: Interpolation by translation

(1)

1.1.11 View 11: Traditional studies

(1)

1.1.12 Concluding remarks

(1)

1.2 Interpolation in general logics

(11)

1.2.1 Historical background

(1)

1.2.2 General logics and interpolation

(10)

1.3 Overview of the book

(10)

2 Modal and superintuitionistic logics: basic concepts

(26)

2.1 Introduction overview

(1)

2.2 The Kripke semantics for quantified modal and intermediate logics

(15)

2.2.1 Propositional modal logics

(5)

2.2.2 Propositional intermediate logics

(2)

2.2.3 Quantified modal logics

(5)

2.2.4 Quantified superintuitionistic logics

(2)

2.3 Algebraic interpretation of propositional logics

(6)

2.3.1 Pseudoboolean algebras

(3)

2.3.2 Modal algebras

(3)

2.4 Inter-relation of relational and algebraic semantics

(5)

2.4.1 From the Kripke semantics to the algebraic one

(1)

2.4.2 Representation theorems

(3)

3 Superintuitionistic logics and normal extensions of the modal logics S4

(42)

3.1 Translation

(9)

3.1.1 Pseudoboolean and topoboolean algebras

(5)

3.1.2 Lattice of superintuitionistic logics and NE(S4)

(4)

3.2 A classification of normal extensions of S4 according to their superintuitionistic fragments

(9)

3.2.1 Characteristic formulas of pre-ordered frames

(3)

3.2.2 Some properties of the classification

(6)

3.3 Well-representable logics

(10)

3.3.1 Algebras and pre-ordered frames

(1)

3.3.2 Representing frames

(4)

3.3.3 Well-representable logics and varieties

(5)

3.4 Classification by slices

(5)

3.5 Finite pseudoboolean and topoboolean algebras

(9)

3.5.1 Finite algebras and finite frames

(3)

3.5.2 Characteristic formulas of Gödelian pseudoboolean and topoboolean algebras

(3)

3.5.3 Logics LC and KC

100

(3)

4 The interpolation theorem in intuitionistic predicate calculus

103

(26)

4.1 Interpolation in classical predicate logic

103

(7)

4.1.1 Craig's interpolation and Robinson's joint consistency

103

(3)

4.1.2 Lyndon's interpolation

106

(4)

4.2 Interpolation theorem in the intuitionistic logic

110

(11)

4.2.1 Definitions

110

(3)

4.2.2 Robinson's theorem

113

(6)

4.2.3 Equivalence of CIP and RCP

119

(2)

4.3 Propositional intermediate logics

121

(3)

4.4 Notes

124

(1)

4.5 Implicit and explicit definability

125

(4)

5 Interpolation and definability in quantified logics

129

(42)

5.1 Inter-relations between interpolation, definability, and joint consistency

129

(3)

5.2 Lyndon's interpolation in some modal systems

132

(8)

5.2.1 LIP in quantified logics

132

(7)

5.2.2 Lyndon's interpolation in propositional modal logics

139

(1)

5.3 The Craig interpolation in modal logics

140

(9)

5.3.1 Modal logics without LIP

140

(2)

5.3.2 Craig's interpolation in some modal logics

142

(7)

5.4 Failure of interpolation

149

(3)

5.5 Preserving interpolation and definability

152

(8)

5.5.1 Axioms preserving interpolation

153

(4)

5.5.2 Interpolation and intersection of logics

157

(3)

5.6 First-order logics with equality

160

(11)

5.6.1 Preliminaries

161

(1)

5.6.2 Formulas preserving interpolation

162

(3)

5.6.3 Modal logics

165

(1)

5.6.4 A counter-example

166

(2)

5.6.5 Functional symbols

168

(3)

6 Craig's theorem in superintuitionistic logics and amalgamable varieties of pseudoboolean algebras

171

(34)

6.1 Craig's theorem and amalgamation property

172

(5)

6.2 Amalgamable varieties of PBA

177

(6)

6.3 Characterization of the varieties H1-H8

183

(2)

6.4 Necessary conditions for varieties of PBA to be amalgamable

185

(14)

6.5 Logics with Craig's interpolation property

199

(4)

6.6 Positive logics

203

(2)

7 Interpolation, definability, amalgamation

205

(20)

7.1 Inter-relation of Beth's and Craig's properties in propositional logics

205

(6)

7.2 Varieties of modal algebras

211

(14)

7.2.1 Interpolation, implicit and explicit definability

211

(2)

7.2.2 The Beth property, interpolation, amalgamation in varieties

213

(9)

7.2.3 Independence of amalgamation property and the Beth property in equational theories of modal algebras

222

(3)

8 Interpolation in normal extensions of the modal logic S4

225

(40)

8.1 Interpolation and amalgamability

226

(1)

8.2 Necessary conditions for amalgamability

227

(12)

8.3 Classification of varieties of topoboolean algebras

239

(2)

8.4 Interpolation theorems in modal logics

241

(2)

8.5 Sufficient conditions for amalgamation

243

(12)

8.5.1 Well-representable logics and varieties

244

(3)

8.5.2 Sufficient conditions for amalgamability and superamalgamability

247

(1)

8.5.3 Lemmas on (α1, α2)-products

248

(4)

8.5.4 Stable and superstable classes of frames

252

(3)

8.6 Logics with interpolation in NE(S4)

255

(2)

8.7 Decidability of interpolation over S4

257

(1)

8.8 NE(S4) versus E(Int)

258

(7)

8.8.1 More on Gödel's translation

259

(2)

8.8.2 IPN in NE(S4)

261

(4)

9 Complexity of some problems in modal and intuitionistic calculi

265

(20)

9.1 Main results

265

(2)

9.2 Reducibilities

267

(2)

9.3 Complexity

269

(3)

9.4 Tabularity and related properties

272

(5)

9.5 Interpolation and amalgamation

277

(8)

10 Interpolation in modal infinite slice logics containing the logic K4

285

(26)

10.1 K4 and S4

285

(2)

10.2 Logics and varieties of infinite slice

287

(6)

10.3 Necessary condition of interpolation

293

(18)

11 An analogue of Beth's theorem in normal extensions of the modal logic K4

311

(22)

11.1 Preliminaries

311

(2)

11.2 The Replacement theorem and its corollaries

313

(2)

11.3 The main theorem

315

(12)

11.3.1 Case 1

318

(3)

11.3.2 Case 2

321

(6)

11.4 A counter-example to the Beth property

327

(1)

11.5 Explicit definitions

328

(5)

11.5.1 Logics of finite slices

329

(2)

11.5.2 Constructing explicit definitions

331

(2)

12 Extensions of the provability logic

333

(38)

12.1 Two extensions of G

333

(5)

12.2 Interpolation in infinite-slice extensions of provability logic

338

(15)

12.2.1 Definitions and notations

338

(2)

12.2.2 Description of Gγ and Gδ

340

(7)

12.2.3 Interpolation theorem

347

(6)

12.3 Continuum of extensions of the provability logic with interpolation

353

(10)

12.3.1 Interpolation and amalgamation properties

354

(1)

12.3.2 The Logic Gλ

355

(3)

12.3.3 Continuum of extensions of the logic Gλ that have the Craig interpolation property

358

(3)

12.3.4 Amalgamation and superamalgamation properties

361

(2)

12.4 Boxed formulas

363

(8)

12.4.1 Preliminaries

363

(1)

12.4.2 The main lemmas

364

(2)

12.4.3 CIP and IPB

366

(3)

12.4.4 The property B2

369

(2)

13 Syntactic proof of interpolation for the intuitionistic predicate logic

371

(10)

13.1 Formal system S

371

(2)

13.2 Proof of interpolation

373

(6)

13.3 Fragments of IntQ

379

(2)

14 Interpolation by translation

381

(22)

14.1 Introduction

381

(3)

14.2 Interpolation by quantifier elimination

384

(4)

14.2.1 SCAN: Second-order quantifier elimination

384

(1)

14.2.2 SCAN can interpolate

385

(3)

14.3 Case study: Quantified S5

388

(9)

14.3.1 Preliminary discussion

388

(2)

14.3.2 Interpolation for QS5

390

(7)

14.4 Case study: Propositional modal logic S4.3

397

(1)

14.5 Interpolation by translation: General theory

398

(5)

15 Interpolation in (intuitionistic) logic programming

403

(22)

15.1 Introduction

403

(1)

15.2 N-prolog

404

(5)

15.3 Interpolation for propositional Horn programs

409

(3)

15.4 Alternative proof

412

(2)

15.5 Controlled interpolation for propositional Horn clauses

414

(2)

15.6 Failure of interpolation for fragment of predicate intuitionistic logic

416

(2)

15.7 Weak interpolation for intuitionistic logic programs

418

(7)

16 Interpolation in goal-directed proof systems

425

(46)

16.1 Introduction

425

(7)

16.1.1 General background

425

(2)

16.1.2 Specific background

427

(5)

16.2 Interpolation for linear and for intuitionistic implication

432

(6)

16.2.1 Interpolation for linear implication

432

(3)

16.2.2 Interpolation for intuitionistic logic

435

(3)

16.2.3 Interpolation for classical logic

438

(1)

16.3 Interpolation for the Lambek calculus

438

(7)

16.4 Interpolation for strict implication

445

(19)

16.5 Concluding discussion, chain interpolation

464

(7)

16.5.1 Structural interpolation

464

(2)

16.5.2 Chain interpolation

466

(2)

16.5.3 Beth definability

468

(1)

16.5.4 Standard interpolation in classical logic

469

(1)

16.5.5 Concluding remarks

470

(1)

17 Further results and discussion

471

(12)

17.1 Introduction

471

(1)

17.2 Further results

471

(7)

17.2.1 Temporal logics

472

(1)

17.2.2 Beth properties and epimorphisms surjectivity

472

(1)

17.2.3 Projective Beth property over Int

473

(1)

17.2.4 Positive and paraconsistent logics

473

(1)

17.2.5 Modal logics and projective Beth property

474

(1)

17.2.6 Restricted interpolation and restricted amalgamation

475

(1)

17.2.7 Variable separation

476

(1)

17.2.8 Decidable properties of logics and of varieties

477

(1)

17.3 Further discussion

478

(5)

17.3.1 Interpolation and artificial intelligence

478

(1)

17.3.2 Interpolation for classical theories

478

(1)

17.3.3 A semantic/categorial engine for interpolation

478

(1)

17.3.4 Interpolation in computer science

479

(1)

17.3.5 Case study: Implementation of constant domains modal K4 in classical logic

480

(3)

Appendix

483

(1)

References

484

(19)

Index

503

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Interpolation and Definability Modal and Intuitionistic Logic

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