Cover

Contents

Preface

1. The Roots of Infinity

1.1 The Roots of Infinity in Natural Language

1.2 Boolean Compounding

1.3 References

2.1 More about Sets

2.2 Sequences

2.3 Functions and Sequences

2.4 Arbitrary Unions and Intersections

2.5 Definitions by Closure (Recursion)

2.6 Bijections and the Sizes of Cross Products

2.7 A Linguistic Function

2.8 A Closing Reflection on Functions

2.10 Addendum 1: Russell's Paradox

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

2.12 Addendum 3: Some Initial Hints on Setting up Proofs

3. Boolean Phonology

3.1 Abstract Segmental Phonology

3.2 Enriching Feature Sets

3.3 Some Phonological Generalizations

3.4 The Mathematical Structure of Feature Sets

3.5 Significant Sound Segments

4. Syntax I: Trees and Order Relations

4.1 Trees

4.2 C-Command

4.3 Sameness of Structure: Isomorphism

4.4 Labeled Trees

4.5 Ordered Trees

4.6 Concluding Formal Exercises on Relations and Ordered Trees

5. Syntax II: Design for a Language

5.1 Beginning Grammar

5.2 Eng—Towards a Grammar of English

5.3 Word Order Variation in Other Languages

5.4 Three New Types of DPs

6.1 Introduction

6.2 CFGs: A Formal Definition

6.3 How Well Do CFGs Model Natural Languages?

7. Finite State Automata

7.1 Finite-State Automata

7.2 Regular Expressions and Languages

7.3 Simple Grammars

7.4 Closing the Circle

7.5 an bn is not a Regular Language

7.6 An Argument Why English is Not Regular

7.7 Two Final Observations on Finite State Automata

8.1 Compositionality and Natural Language Semantics

8.2 Sentential Logic

8.3 Interpreting a Fragment of English

9. Semantics II: Coordination, Negation and Lattices

9.1 Coordination: Syntax

9.2 Coordination: Semantics

9.3 Negation and Additional Properties of Natural Language Lattices

9.4 Properties versus Sets: Lattice Isomorphisms

9.5 Theorems on Boolean Lattices

9.6 Some (Possibly) Unexpected Boolean Lattices

9.7 A Concluding Note on Point of View

9.9 Appendix: Tarski-Knaster and Schröder-Bernstein

10.1 Translation Semantics

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

10.3 Extending the Class of VBO's: The Lambda Operator

11.1 Negative Polarity Items

11.2 Monotonicity

11.3 Semantic Generalizations

11.4 Historical Background

12. Semantics V: Classifying Quantifiers

12.1 Quantifier Types

12.2 Generalizations Concerning Det Denotations

12.3 k-Place Dets

12.4 Crossing the Frege Boundary

12.5 A Classic Syntactic Problem

12.6 Adverbial Quantification

12.7 Concluding Remarks

12.9 Appendix: Some Types of English Determiners

13. Linguistic Invariants

13.1 A Model Grammar and Some Basic Theorems

13.2 A Semantic Definition of Anaphor

13.3 A Model of Korean

13.4 Toba Batak

13.5 Some Mathematical Properties of Grammars and their Invariants

13.6 Invariants of Type 0

13.7 Invariants of Type (1)

13.8 Invariants of Type (2) and Higher

13.9 Structure Preserving Operations on Grammars

The Greek Alphabet

Bibliography

Subject Index

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