Читать книгу Formal Semantics in Modern Type Theories - Stergios Chatzikyriakidis - Страница 7

The long history of the study of semantics has produced a number of theories of meaning. For example, the referential theory adopts a Platonic viewpoint and proposes that meanings are entities in the world; the internalist theory, such as that held by Chomsky, suggests that meanings are concepts in our minds, and the use theory, which is closely related to Wittgenstein’s slogan of “meaning is use”, advocates that meanings are embodied in the ways 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.

This book studies formal semantics in modern type theories (MTT-semantics for short). We contend that MTT-semantics is both model-theoretic and proof-theoretic, a thesis that was first proposed by the second author in Luo (2014) and further elaborated in Luo (2019a), and will be explicated in this book. For natural language, both model-theoretic and proof-theoretic semantics have their own advantages and disadvantages, and it is not easy to imagine how one can combine them in a single semantic framework: in fact, up to now this has never been attempted, let alone done. We argue that MTT-semantics is the first framework with both characteristics: being model-theoretic, it provides powerful mechanisms to capture semantics for a wide range of linguistic features, and being proof-theoretic, it has a solid meaning-theoretic foundation and can be directly implemented by means of the current proof technology to support computer-assisted reasoning in natural language. This gives MTT-semantics unprecedented advantages over other semantic frameworks.

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.