Stergios Chatzikyriakidis
Zhaohui Luo
First published 2020 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
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Library of Congress Control Number: 2020943853
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ISBN 978-1-78630-128-4
Preface
To communicate linguistically is to convey meaning. Semantics is the academic discipline that analyzes meaning. In this book, we study natural language semantics formally in the framework of modern type theories (MTTs). Examples of MTTs include Martin-Löf’s predicative type theory (Martin-Löf 1984; Nordström et al. 1990) and the impredicative type theories – unifying theory of dependent types UTT (Luo 1994) and pCIC (Coq 2010). MTTs offer a foundational framework that was not available in the study of formal semantics until Ranta’s work in the early 1990s when he applied Martin-Löf’s type theory to formal semantics (Ranta 1994). In the past decade, the development of formal semantics in MTTs (MTT-semantics for short) has shown that MTTs are powerful alternatives to, and arguably more advantageous than, Church’s simple type theory (Church 1940) as employed in Montague semantics (Montague 1974).
MTTs have rich type structures and, at the same time, are specified by means of proof-theoretic natural deduction rules. Therefore, they provide powerful tools in formal semantics, on the one hand, and have the appeal of being supported by a proof-theoretic theory of meaning, on the other hand. The former allows one to study a wide range of linguistic features, some of which have proved difficult to describe adequately in the Montagovian setting. The latter not only provides a solid proof-theoretic foundation for MTT-semantics but also leads to practical computer-assisted reasoning tools supported by the existing proof technology. These two aspects can be summarized by saying that MTT-semantics is both model-theoretic and proof-theoretic, a thesis that was put forward by the second author in Luo (2014) and further elaborated in Luo (2019a). This makes MTTs a promising and attractive semantic framework that is unique: such a framework was not available in traditional logical settings and the MTT-framework offers new perspectives on natural language semantics.
Learning MTTs can be laborious. This is partly because MTTs incorporate several new concepts that even traditional logicians may find difficult to comprehend at the beginning, let alone the linguists who apply logics to formal semantics. Therefore, in this book, besides studying MTT-semantics, we shall try our best to introduce the basics of MTTs informally: the explanations will be rigorous with examples from linguistic semantics. It is our hope that this will make it easier to understand and study MTT-semantics.
This book consists of the following seven chapters, where Chapters 1, 2, 3, sections 7.1 and 7.2 are mainly written by Luo, and Chapters 4, 5, 6 and section 7.3 are mainly written by Chatzikyriakidis.
– Chapter 1, Type Theories and Semantic Studies, starts with an overview of the historical development of type theories and a discussion on how to use a type theory as a foundational language for formal semantics. We then briefly describe Montague semantics, which is based on simple type theory,1 and MTT-semantics, and compare them by means of some simple examples. This serves to introduce the basics of MTT-semantics in a very sketchy manner, together with some summary notes on its historical development, and allows us to outline several important merits of employing MTTs as a foundational semantic framework.
– Chapter 2, Modern Type Theories, consists of an informal but rigorous and precise introduction to MTTs, mainly for linguistic semanticists. In order to introduce the basic concepts in MTTs, we use explanatory examples to show how these may be employed in semantic constructions and, in most cases, we present their formal rules in appendices (mainly for completeness). Special attention is paid to several mechanisms that are particularly important for MTT-semantics, including coercive subtyping (Luo 1997), signatures (Luo 2014) and linguistic universes (Luo 2011b; Chatzikyriakidis and Luo 2012). We shall also give a complete formal presentation of the type theories used in this book, which is based on various studies (Luo 2014, 2019a), both for completeness and for reference in future work.
– Chapter 3, Formal Semantics in Modern Type Theories, is a central chapter that explicates several important features of MTT-semantics and shows that the MTT-semantic approach is viable and coherent, on the one hand, and provides flexible and powerful mechanisms for adequate semantic interpretations, on the other hand. Some features are discussed in depth: the CNs-as-types paradigm (Ranta 1994; Luo 2012a), subtyping in MTT-semantics (Luo 2009c, 2012b) and propositional forms of judgmental interpretations (Xue et al. 2018). As a case study for MTT-semantics, we study semantics of adjectival modification according to a traditional classification of adjectives into intersective, subsective, privative and non-committal
