Quantifiers in Language and Logic

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ISBN: 9780199291250 | 019929125X
Cover: Hardcover
Copyright: 7/6/2006

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Quantification is a topic which brings together linguistics, logic, and philosophy. Quantifiers are the essential tools with which, in language or logic, we refer to quantity of things or amount of stuff. In English they include such expressions as no , some , all , both , and many . Peters and Westerstahl present the definitive interdisciplinary exploration of how they work--their syntax, semantics, and inferential role.

0. Quantification

(20)

0.1 Some quantifier expressions of natural languages

(8)

0.2 Varieties of quantification

(5)

0.2.1 Syntactic variation in quantifier expressions

(1)

0.2.2 Semantic types of NL quantification

(2)

0.2.3 Quantirelations

(2)

0.3 Explicit and implicit quantification

(1)

0.4 Monadic and polyadic quantifiers

(5)

1 THE LOGICAL CONCEPTION OF QUANTIFIERS AND QUANTIFICATION

1. A Brief History of Quantification

(32)

1.1 Early history of quantifiers

(12)

1.1.1 Aristotelian beginnings

(8)

1.1.2 The Middle Ages

(4)

1.2 Quantifiers in early predicate logic

(6)

1.2.1 Peirce

(1)

1.2.2 Peano

(1)

1.2.3 Russell

(2)

1.2.4 Frege

(2)

1.3 Truth and models

(9)

1.3.1 Absolute and relative truth

(1)

1.3.2 Uninterpreted symbols

(1)

1.3.3 Universes

(2)

1.3.4 Context sets

(3)

1.3.5 Quantifying over everything?

(2)

1.4 Postscript: Why quantifiers cannot denote individuals or sets of individuals

(4)

2. The Emergence of Generalized Quantifiers in Modern Logic

(26)

2.1 First-order logic versus first-order languages

(3)

2.2 First-order logic (FO)

(3)

2.3 Mostowski quantifiers

(3)

2.4 Lindstrom quantifiers

(4)

2.5 Branching quantifiers

(6)

2.5.1 First-order sentences making second-order claims

(2)

2.5.2 Branching generalized quantifiers

(2)

2.6 Digression: why logicians like FO

(1)

2.7 Summary

(6)

II QUANTIFIERS OF NATURAL LANGUAGE

3. Type (1) Quantifiers of Natural and Logical Languages

(40)

3.1 Preliminary concepts and distinctions

(6)

3.1.1 Global and local quantifiers

(3)

3.1.2 Quantifier expressions versus predicate expressions

(1)

3.1.3 Relational and functional views of quantifiers

(2)

3.2 Phrases denoting type (1) quantifiers

(9)

3.2.1 Examples of noun phrase denotations

(3)

3.2.2 Quantifiers living on sets

(2)

3.2.3 Boolean operations on quantifiers

(2)

3.2.4 Montagovian individuals

(2)

3.3 Isomorphism closure

(5)

3.3.1 Isom for type (1) quantifiers

(3)

3.3.2 Isom for arbitrary quantifiers

(2)

3.4 Extension

100

(12)

3.4.1 EXT as a property of quantifiers

101

(6)

3.4.2 EXT and database languages

107

(5)

3.5 How natural language quantifier expressions always denote global quantifiers

112

(7)

3.5.1 Sense and local denotation

112

(2)

3.5.2 Using global quantifiers

114

(2)

3.5.3 Examples

116

(3)

4. Type (1, 1) Quantifiers of Natural Language

119

(44)

4.1 Examples of determiners

120

(3)

4.2 On existential import and related matters

123

(7)

4.2.1 Existential import

124

(4)

4.2.2 Syntactic and semantic number

128

(2)

4.3 Boolean operations

130

(4)

4.4 Relativization

134

(3)

4.4.1 Examples

135

(2)

4.4.2 Empty universes

137

(1)

4.5 Conservativity, extension, and relativization

137

(12)

4.5.1 Conservativity

138

(1)

4.5.2 A quantirelation universal

138

(2)

4.5.3 An extension universal

140

(1)

4.5.4 The relativization characterization

141

(2)

4.5.5 Restricted quantifiers once again

143

(6)

4.6 Definiteness

149

(4)

4.7 Type (1, 1, 1) quantifiers and beyond

153

(4)

4.8 Isom and the number triangle

157

(6)

5. Monotone Quantifiers

163

(45)

5.1 Standard monotonicity

164

(4)

5.2 Monotonicity in type (1,1)

168

(4)

5.3 Monotonicity universals

172

(2)

5.4 Monotonicity under Isom

174

(4)

5.4.1 A format for monotone quantifiers over finite universes

174

(2)

5.4.2 Monotonicity and the number triangle

176

(2)

5.5 Six basic forms of monotonicity

178

(7)

5.6 Smooth quantifiers

185

(6)

5.7 Linguistic application 1: a peculiar inference scheme

191

(1)

5.8 Linguistic application 2: LAA quantifiers

192

(4)

5.9 Linguistic application 3: polarity-sensitive items in natural languages

196

(12)

5.9.1 What are NPIs and PPIs sensitive to?

197

(2)

5.9.2 What is negative about negation?

199

(2)

5.9.3 A hypothesis about licensing of NPIs and PPIs

201

(3)

5.9.4 Testing the hypothesis

204

(4)

6. Symmetry and Other Relational Properties of Type (1, 1) Quantifiers

208

(34)

6.1 Symmetry

208

(5)

6.2 On the symmetry of many and few

213

(1)

6.3 Existential-there sentences

214

(23)

6.3.1 Natural language talk about existence

214

(3)

6.3.2 Restrictions on the pivot noun phrase of existential-there sentences

217

(3)

6.3.3 What do existential-there sentences mean?

220

(5)

6.3.4 Four approaches to distinguishing between existentially acceptable determiners and existentially unacceptable ones

225

(10)

6.3.5 Some additional data and conclusions

235

(2)

6.4 Other relational properties of CONSERV and EXT type (1, 1) quantifiers

237

(5)

7. Possessive Quantifiers

242

(55)

7.1 Possessive determiners and NPs

244

(2)

7.2 Number and uniqueness

246

(1)

7.3 Universal readings and others

247

(2)

7.4 Scope ambiguities?

249

(1)

7.5 Narrowing

250

(1)

7.6 The possessor relation

251

(3)

7.7 The meaning of possessive determiners

254

(5)

7.8 Alternative accounts

259

(7)

7.8.1 Poss without narrowing

259

(2)

7.8.2 A definiteness account

261

(1)

7.8.3 A problem with theag and thepl

262

(1)

7.8.4 Narrowing versus accommodation

263

(1)

7.8.5 Summing up

264

(2)

7.9 Semantic rules for possessives

266

(6)

7.10 Iterated possessives

272

(4)

7.11 Definites and possessives

276

(6)

7.11.1 Which possessives are definite?

276

(1)

7.11.2 What makes an expression definite?

277

(3)

7.11.3 A semantic rule for the definite case

280

(2)

7.12 Closure properties of Poss

282

(6)

7.12.1 Negations

283

(1)

7.12.2 Conjunctions and disjunction

284

(2)

7.12.3 Some other determiners involving possessives

286

(1)

7.12.4 Restrictions on the (poss) rule

287

(1)

7.13 Possessives and monotonicity

288

(7)

7.14 Some remaining issues

295

(2)

8. Exceptive Quantifiers

297

(27)

8.1 Connected and free exception phrases

299

(1)

8.2 The Generality Claim

300

(1)

8.3 The Inclusion Condition

300

(1)

8.4 Exception conservativity

301

(1)

8.5 The Negative Condition

302

(2)

8.6 Entailed or implicated?

304

(1)

8.7 Other quantifiers in exception sentences

304

(1)

8.8 The classical idea of universal claims with exceptions

305

(1)

8.9 The account in von Fintel 1993

306

(4)

8.10 The account in Moltmann 1995

310

(2)

8.11 Counter-evidence to the Quantifier Constraint

312

(1)

8.12 A modest proposal

313

(3)

8.13 Quantified exception phrases

316

(6)

8.14 Further issues

322

(2)

9. Which Quantifiers are Logical?

324

(22)

9.1 Logicality and ISOM

324

(6)

9.1.1 Isom in arbitrary types

325

(2)

9.1.2 Isom is necessary for logicality

327

(1)

9.1.3 Strengthening Isom

328

(2)

9.2 Two claims about Isom and natural language quantification

330

(4)

9.2.1 Proper names

331

(1)

9.2.2 Restricted noun phrases

332

(1)

9.2.3 Possessives

332

(1)

9.2.4 Exceptive quantifiers

333

(1)

9.3 Constancy 334 9.3.1 Inference constancy

334

(9)

9.3.2 Bolzano's method

335

(2)

9.3.3 Bolzano reversed

337

(4)

9.3.4 What to interpret in models

341

(2)

9.4 Logical constants

343

(3)

9.4.1 Logic versus mathematics

343

(1)

9.4.2 ISOM + EXT

344

(2)

10. Some Polyadic Quantifiers of Natural Language

346

(29)

10.1 Iteration

346

(6)

10.2 Resumption

352

(11)

10.2.1 Quantificational adverbs and 'donkey' anaphora

354

(5)

10.2.2 Resumption, orientation, and tuple-isomorphisms

359

(4)

10.3 Branching

363

(1)

10.4 Reciprocals

364

(12)

10.4.1 What do reciprocals mean?

365

(2)

10.4.2 Type (1, 2) reciprocal quantifiers

367

(1)

10.4.3 Properties of reciprocal quantifiers

368

(1)

10.4.4 Collective predicates formed with reciprocals, and quantification of such predicates

369

(6)

III BEGINNINGS OF A THEORY OF EXPRESSIVENESS, TRANSLATION, AND FORMALIZATION

11. The Concept of Expressiveness

375

(39)

11.1 Preliminaries

376

(6)

11.1.1 Three levels of expressions

376

(4)

11.1.2 Expressibility of quantifier expressions versus predicate expressions

380

(2)

11.2 A framework for translation

382

(15)

11.2.1 Expressiveness is relative to a notion of saying the same thing

382

(1)

11.2.2 Sameness of meaning, without meanings

383

(2)

11.2.3 Sameness is a partial equivalence relation (PER)

385

(5)

11.2.4 Translation and formalization

390

(7)

11.3 Varieties of sameness

397

(17)

11.3.1 Sameness of denotation

397

(3)

11.3.2 Logical equivalence

400

(3)

11.3.3 AnalyticaUnecessary equivalence

403

(2)

11.3.4 Varieties of cognitive equivalence

405

(2)

11.3.5 Possible linguistic equivalences

407

(3)

11.3.6 The ultimate refinement: identity

410

(1)

11.3.7 Where do relations of synonymy come from?

410

(2)

11.3.8 A landscape of possible synonymies

412

(2)

12. Formalization: Expressibility, Definability, Compositionality

414

(35)

12.1 Logical equivalence revisited

414

(7)

12.1.1 Lexical mappings

415

(3)

12.1.2 Logical equivalence relative to a lexical mapping

418

(3)

12.2 Compositionality

421

(7)

12.2.1 Compositional languages

422

(2)

12.2.2 Compositional translation

424

(4)

12.3 Requirements on definitions

428

(4)

12.4 Extending lexical mappings to compositional translations

432

(5)

12.4.1 Definable grammar rules

433

(2)

12.4.2 An extension theorem

435

(2)

12.5 Why formal definability results matter for natural languages

437

(12)

12.5.1 The role of lexical mappings

438

(1)

12.5.2 Uniform expressivity

439

(2)

12.5.3 Meaning postulates and expressivity

441

(1)

12.5.4 The role of formalization

442

(2)

12.5.5 Summing up

444

(5)

IV LOGICAL RESULTS ON EXPRESSIBILITY WITH LINGUISTIC APPLICATIONS

13. Definability and Undefinability in Logical Languages: Tools for the Monadic Case

449

(16)

13.1 Logics with quantifiers

449

(2)

13.2 Definability

451

(4)

13.2.1 Examples of definability

452

(3)

13.2.2 The role of Isom

455

(1)

13.3 Undefinability

455

(2)

13.4 EF-tools for the monadic case

457

(8)

13.4.1 The structure of monadic models

457

(1)

13.4.2 A criterion for Lr-equivalence over monadic models

458

(2)

13.4.3 Proof that the criterion is correct

460

(5)

14. Applications to Monadic Definability

465

(18)

14.1 The method

465

(1)

14.2 FO-undefinability

466

(3)

14.3 Undefinability in other logics

469

(7)

14.4 Monotonicity and definability

476

(5)

14.5 Exercises

481

(2)

15. EF-tools for Polyadic Quantifiers

483

(28)

15.1 EF-games

483

(8)

15.2 Hella's bijective EF-game

491

(3)

15.3 Application to branching quantification

494

(4)

15.3.1 An instructive example

495

(1)

15.3.2 The general result

495

(3)

15.3.3 Linguistic conclusions

498

(1)

15.4 Application to Ramsey quantifiers and reciprocals

498

(2)

15.4.1 A characterization theorem

498

(1)

15.4.2 Linguistic conclusions

499

(1)

15.5 Application to resumption and adverbial quantification

500

(11)

15.5.1 Definability in terms of resumptive quantifiers

500

(2)

15.5.2 Linguistic consequences, 1

502

(1)

15.5.3 Definability of resumption

503

(5)

15.5.4 Linguistic consequences, 2

508

(3)

References

511

(10)

Index

521

(6)

Index of Symbols

527

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Quantifiers in Language and Logic

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Table of Contents

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