Mathematical Structures in Languages

Mathematical Structures in Languages

by Edward L. Keenan, Lawrence S. Moss

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Overview

Mathematical Structures in Languages by Edward L. Keenan, Lawrence S. Moss

Mathematical Structures in Languages introduces a number of mathematical concepts that are of interest to the working linguist. The areas covered include basic set theory and logic, formal languages and automata, trees, partial orders, lattices, Boolean structure,  generalized quantifier theory, and linguistic invariants, the last drawing on Edward L. Keenan and Edward Stabler’s Bare Grammar: A Study of Language Invariants, also published by CSLI Publications. Ideal for advanced undergraduate and graduate students of linguistics, this book contains numerous exercises and will be a valuable resource for courses on mathematical topics in linguistics. The product of many years of teaching, Mathematic Structures in Languages is very much a book to be read and learned from.

Product Details

ISBN-13: 9781575868479
Publisher: Center for the Study of Language and Inf
Publication date: 11/15/2016
Series: Lecture Notes Series
Pages: 250
Product dimensions: 5.90(w) x 8.90(h) x 1.10(d)

About the Author

Edward L. Keenan is professor of linguistics at the University of California, Los Angeles. Lawrence S. Moss is professor of mathematics; director of the Program in Pure and Applied Logic; an adjunct professor of computer science, informatics, linguistics, and philosophy; and a member of the Programs in Cognitive Science and Computational Linguistics, all at Indiana University, Bloomington. With Jon Barwise, he is coauthor of Vicious Circles, also published by CSLI Publications.

Table of Contents

Preface ix

1 The Roots of Infinity 1

1.1 The Roots of Infinity in Natural Language 2

1.2 Boolean Compounding 23

1.3 References 25

2 Some Mathematical Background 27

2.1 More about Sets 27

2.2 Sequences 34

2.3 Functions and Sequences 40

2.4 Arbitrary Unions and Intersections 42

2.5 Definitions by Closure (Recursion) 43

2.6 Bisections and the Sizes of Cross Products 47

2.7 A Linguistic Function 50

2.8 A Closing Reflection on Functions 52

2.9 Suggestions for Further Study 53

2.10 Addendum 1: Russell's Paradox 53

2.11 Addendum 2: An Equivalent, Useful, Definition of Closure 54

2.12 Addendum 3: Some Initial Hints on Setting up Proofs 56

3 Boolean Phonology 61

3.1 Abstract Segmental Phonology 63

3.2 Enriching Feature Sets 68

3.3 Some Phonological Generalizations 69

3.4 The Mathematical Structure of Feature Sets 71

3.5 Significant Sound Segments 72

4 Syntax I: Trees and Order Relations 81

4.1 Trees 82

4.2 C-Command 95

4.3 Sameness of Structure: Isomorphism 99

4.4 Labeled Trees 105

4.5 Ordered Trees 110

4.6 Concluding Formal Exercises on Relations and Ordered Trees 121

5 Syntax II: Design for a Language 129

5.1 Beginning Grammar 130

5.2 Eng- Towards a Grammar of English 132

5.3 Word Order Variation in Other Languages 137

5.4 Three New Types of DPs 139

6 A Taste of Formal Language Theory 147

6.1 Introduction 147

6.2 CFGs: A Formal Definition 149

6.3 How Well Do CFGs Model Natural Languages? 158

7 Finite State Automata 183

7.1 Finite-State Automata 184

7.2 Regular Expressions and Languages 191

7.3 Simple Grammars 197

7.4 Closing the Circle 201

7.5 Anbn is not a Regular Language 206

7.6 An Argument Why English is Not Regular 208

7.7 Two Final Observations on Finite State Automata 209

8 Semantics I: Compositionality and Sentential Logic 215

8.1 Compositionality and Natural Language Semantics 215

5.1 Sentential Logic 225

8.1 Interpreting a Fragment of English 239

9 Semantics II: Coordination, Negation and Lattices 257

9.1 Coordination: Syntax 259

9.2 Coordination: Semantics 260

9.3 Negation and Additional Properties of Natural Language Lattices 270

9.4 Properties versus Sets: Lattice Isomorphisms 275

9.5 Theorems on Boolean Lattices 279

9.6 Some (Possibly) Unexpected Boolean Lattices 280

9.7 A Concluding Note on Point of View 281

9.8 Further Reading 283

9.9 Appendix: Tarski-Knaster and Schröder-Bernstein 283

10 Semantics III: Logic and Variable Binding Operators 285

10.1 Translation Semantics 285

10.2 Some General Linguistic Properties of First Order Logic (FOL) 304

10.3 Extending the Class of VBO's: Tire Lambda Operator 316

11 Semantics IV: DPs, Monotonicity and Semantic Generalizations 335

11.1 Negative Polarity Items 335

11.2 Monotonicity 340

11.3 Semantic Generalizations 348

11.4 Historical Background 353

12 Semantics V: Classifying Quantifiers 357

12.1 Quantifier Types 359

12.2 Generalizations Concerning Det Denotations 370

12.3 k-Place Dets 375

12.4 Crossing the Frege Boundary 379

12.5 A Classic Syntactic Problem 381

12.6 Adverbial Quantification 383

12.7 Concluding Remarks 387

12.8 Historical Background 388

12.9 Appendix: Some Types of English Determiners 388

13 Linguistic Invariants 391

13.1 A Model Grammar and Some Basic Theorems 393

13.2 A Semantic Definition of Anaphor 407

13.3 A Model of Korean 411

13.4 Toba Batak 416

13.5 Some Mathematical Properties of Grammars and their Invariants 420

13.6 Invariants of Type 0 422

13.7 Invariants of Type (1) 423

13.8 Invariants of Type (2) and Higher 426

13.9 Structure Preserving Operations on Grammars 431

List of Symbols 435

The Greek Alphabet 435

Bibliography 437

Author Index 457

Language Index 463

Subject Index 465

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