Categorial Grammar: Logical Syntax, Semantics, and by Glyn V. Morrill

By Glyn V. Morrill

This ebook offers a cutting-edge creation to categorial grammar, a kind of formal grammar which analyzes expressions as features or in accordance with a function-argument courting. The book's concentration is on linguistic, computational, and psycholinguistic elements of logical categorial grammar, i.e. enriched Lambek Calculus. Glyn Morrill opens with the heritage and notation of Lambek Calculus and its software to syntax, semantics, and processing. Successive chapters expand the grammar to a few major syntactic and semantic houses of common language. the ultimate half applies Morrill's account to numerous present concerns in processing and parsing, thought of from either a mental and a computational viewpoint. The booklet bargains a rigorous and considerate learn of 1 of the most strains of study within the formal and mathematical idea of grammar, and may be appropriate for college kids of linguistics and cognitive technological know-how from complex undergraduate point upwards.

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Sometimes lexical semantics will be unstructured, consisting simply of a non-logical constant, for example: (17) John : N : j loves : (N\S)/N : love Mary : N : m A lexical selection for a derivation of A1 , . . , An ⇒ A is a choice of lexical entries, ·1 : A1 : ˆ1 , . . , ·n : An : ˆn . A derivation plus a lexical selection determines the semantics that is the substitution of the lexical semantics into the derivational semantics. For example, recall from above the derivation and derivational semantics: N⇒N (18) N⇒N S⇒S N, N\S ⇒ S N, (N\S)/N, N ⇒ S /L \L = ((x loves x Mary ) x John ) x John , x loves , x Mary Then substituting in the lexical semantics in (17), John loves Mary is assigned semantics: (19) ((x loves x Mary ) x John ){j/x John , love/x loves , m/x Mary } = ((love m) j) In other cases lexical semantics will be represented by a structured term, encoding denotational constraints and hence logical semantic properties of the lexical expression.

4) Definition (types). The set T of types is defined on the basis of a set ‰ of basic types as follows: T ::= ‰ | T → T | T&T 26 lambek categorial grammar (5) Definition (type domains). e. e. { m1 , m2 | m1 ∈ D Ù1 & m2 ∈ D Ù2 } (6) Definition (terms). The sets ÷Ù of terms of type Ù for each type Ù are defined on the basis of a set C Ù of constants of type Ù and a denumerably infinite set V Ù of variables of type Ù for each type Ù as follows: ÷Ù ::= C Ù | V Ù | (÷Ù →Ù ÷Ù ) | 1 ÷Ù&Ù | 2 ÷Ù &Ù ÷Ù→Ù ::= ÎV Ù ÷Ù ÷Ù&Ù ::= (÷Ù , ÷Ù ) A term which does not contain constants is called pure.

An ⇒ A valid if and only if in every interpretation, [[A1 , . . , An ]] ⊆ [[A]]; otherwise we call the sequent invalid. The sequent calculus of Fig. 1 is sound with respect to these models. Consider for example the rule of \L. The first premise tells us that by hypothesis [[√]] ⊆ [[A]]. Therefore by the interpretation of under, [[√, A\C ]] ⊆ C . But the second premise tells us that by hypothesis [[ƒ(C )]] ⊆ [[D]]. Therefore, [[ƒ(√, A\C )]] ⊆ [[D]]. (11) Proposition (Soundness of L). o. semigroups.

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