Formal Semantics in Modern Type Theories. Stergios Chatzikyriakidis
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Название: Formal Semantics in Modern Type Theories

Автор: Stergios Chatzikyriakidis

Издательство: John Wiley & Sons Limited

Жанр: Языкознание

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isbn: 9781119489214

isbn:

СКАЧАТЬ that language is used in social practice. Besides being very interesting themselves, these philosophical theories have had a profound impact on the ways in which researchers think of and approach formal semantics. For example, many semanticists have been influenced by the referential theory of meaning and believed that formal semantics should be model-theoretic (see, for instance, Portner (2005)), following Tarski’s ideas in model theory for logical systems and Montague’s ideas in set-theoretical semantics for natural language (Montague 1974). On the other hand, the use theory of meaning has convinced many others and has been substantially developed more recently, both by philosophers such as Dummett (1991) and Brandom (1994, 2000) on meaning theories in general and by logicians such as Gentzen (1935), Prawitz (1973, 1974) and Martin-Löf (1984, 1996) on proof-theoretic semantics for logical systems in particular.

      In this chapter, we shall start with a brief account of the historical development of type theory for the study of the foundations of mathematics – the simple type theory for classical mathematics and dependent (modern) type theories for constructive mathematics. Simple type theory was employed by Montague and his followers as an intermediate language for the study of model-theoretic semantics of natural language, where set theory is taken as the foundational language. In MTT-semantics, on the other hand, modern type theories (MTTs) are themselves foundational languages. The new logical concepts and rich typing mechanisms in MTTs make them adequate to serve as foundational languages for formal semantics. We shall introduce MTT-semantics briefly, which will be developed further in the book, and summarize its advantages.

      Type theories are computational logical systems that were originally developed for the foundations of mathematics. At the beginning of the 20th Century, Russell (1903) developed the theory of types to solve a foundational problem of mathematics exposed as a number of well-known paradoxical contradictions in Cantor’s naive set theory that are related to self-reference. Some researchers, including Russell himself, attributed such paradoxes to “vicious circles” in formations of logical formulae (“impredicativity”, in a technical jargon), which is what Russell’s Ramified Theory of Types (Whitehead and Russell 1925) was designed to circumvent.

      In the study of formal semantics, one considers a foundational semantic language in which semantic interpretations of natural language sentences and phrases are given; in other words, it is meaning-carrying language in the sense that its sentences/phrases interpret the natural language sentences/phrases. A foundational semantic language is usually a formal mathematical language that, besides being precise and usually unambiguous, has several important salient features, including the following:

       1) the language has rich and powerful mechanisms and is hence capable of giving adequate semantic descriptions for a variety of diverse linguistic features;

       2) the language contains (or embeds) a useful logic to be used in semantic interpretations;

       3) the language is regarded as well understood (and, hence, so are the semantic interpretations).