Type-Theoretical Interpretation and Generalization of Phrase Structure Grammar

In this paper, we shall present a generalization of phrase structure grammar, in which all functional categories (such as verbs and adjectives) have type restrictions, that is, their argument types are speciic domains. In ordinary phrase structure grammar, there is just one universal domain of individuals. The grammar does not make a distinction between verbs and adjectives in terms of domains of applicability. Consequently, it fails to distinguish between sentences like every line intersects every line, which is well typed, and every line intersects every point, which is ill typed. Our generalization relates to ordinary phrase structure grammar in the same way as the higher-level constructive type theory of Martin-LL of (see Nordstrr om et al. 1990, part III, or Ranta 1994, chapter 8) relates to the simple type theory of Church (Church 1940). Simple type theory has been used in linguistics and related with phrase structure grammar, especially in the tradition based on the work of Montague (1974). Our deenition of the grammar will be more formal than Montague's in the sense that we shall use a formal metalanguage, that of constructive type theory, for deening both the object language and its interpretation. The grammar, both syntax and semantics, is thus readily implementable in the type-theoretical proof system ALF (see Magnusson and Nordstrr om 1994 for ALF). Inside type theory , a distinction can be made between the object language and the model, in other words, between syntactic and semantic types. It will turn out that the object language cannot be deened independently of the model, as in ordinary Tarski semantics. This is a direct consequence of introducing the typing restrictions. The grammar presented here can be seen as a formal linguistic elaboration of the work presented in Ranta 1991 and 1994. The organization of generative grammar as categorial grammar followed by sugaring should now look more familiar to the linguist, as the grammatical formalism on which sugaring operates is no longer full type theory but a set of phrase structure trees. 1 Correspondence between rule notations Almost all contemporary grammatical theories assume some form of phrase structure grammar, consisting of rules like S ! NP VP: This rule says that a sentence (S) may consist of a noun phrase (NP) combined with a verb phrase (VP). Type-theoretically, it is an introduction rule for the category S, saying that given an expression of category NP and an expression of …