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This book studies formal semantics in modern type theories (MTTsemantics). Compared with simple type theory, MTTs have much richer type structures and provide powerful means for adequate semantic constructions. This offers a serious alternative to the traditional settheoretical foundation for linguistic semantics and opens up a new avenue for developing formal semantics that is both model-theoretic and proof-theoretic, which was not available before the development of MTTsemantics. This book provides a reader-friendly and precise description of MTTs and offers a comprehensive introduction to MTT-semantics. It develops several case studies, such as adjectival modification and copredication, to exemplify the attractiveness of using MTTs for the study of linguistic meaning. It also examines existing proof assistant technology based on MTT-semantics for the verification of semantic constructions and reasoning in natural language. Several advanced topics are also briefly studied, including dependent event types, an application of dependent typing to event semantics.

Table of Contents

  1. Cover
  2. Title page
  3. Copyright
  4. Preface
  5. 1 Type Theories and Semantic Studies
    1. 1.1. Historical development of type theories
    2. 1.2. Foundational semantic languages
    3. 1.3. Montague’s model-theoretic semantics
    4. 1.4. MTT-semantics: formal semantics in modern type theories
  6. 2 Modern Type Theories
    1. 2.1. Judgments and contextual mechanisms
    2. 2.2. Type constructors
    3. 2.3. Universes
    4. 2.4. Subtyping
    5. 2.5. Formal presentation of type theories with signatures
  7. 3 Formal Semantics in Modern Type Theories
    1. 3.1. Basic linguistic categories
    2. 3.2. Several unique features of MTT-semantics
    3. 3.3. Adjectival modification: a case study
  8. 4 Advanced Modification
    1. 4.1. The data
    2. 4.2. Gradable adjectives
    3. 4.3. Gradable nouns
    4. 4.4. Multidimensional adjectives
    5. 4.5. Adverbial modification
    6. 4.6. Final remarks on modification: vagueness
  9. 5 Copredication and Individuation
    1. 5.1. Copredication and individuation: an introduction
    2. 5.2. Dot-types for copredication: a brief introduction
    3. 5.3. Identity criteria: individuation and CNs as setoids
  10. 6 Reasoning and Verifying NL Semantics in Coq
    1. 6.1. Proof assistant technology based on MTTs
    2. 6.2. A linguist friendly introduction to Coq
    3. 6.3. MTT-semantics in Coq
  11. 7 Advanced Topics
    1. 7.1. Propositional forms of judgmental interpretations: formal treatment
    2. 7.2. Dependent event types
    3. 7.3. Dependent categorial grammars
  12. Appendices
    1. Appendix 1: Simple Type Theory C
    2. A1.1. Inference rules of C
    3. A1.2. Logical operators in C
    4. Appendix 2: Type Constructors
    5. A2.1. Π-types
    6. A2.2. Σ-types
    7. A2.3. Disjoint union types
    8. A2.4. The unit type and finite types
    9. Appendix 3: Prop and Logical Operators in Impredicative MTTs
    10. A3.1. Prop
    11. A3.2. Logical operators
    12. Appendix 4: And for Coordination
    13. Appendix 5: Formal System LFΔ
    14. A5.1. LFΔ
    15. A5.2. Σ-types in LFΔ
    16. Appendix 6: Rules for Dot-Types
    17. Appendix 7: Coq Codes
    18. A7.1. Some basic ontology and subtyping declarations
    19. A7.2. Simple homonymy by overloading in coercive subtyping
    20. A7.3. Intersective and subsective adjectives
    21. A7.4. Privative adjectives
    22. A7.5. Multidimensional adjectives
    23. A7.6. Gradable adjectives
    24. A7.7. Veridical adverbs
    25. A7.8. Manner adverbs
    26. A7.9. Individuation
  13. References
  14. Index
  15. End User License Agreement
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