One of the many roles of linguistics is to address the semantics of natural languages, that is, the meaning of sentences in natural languages. An important part of the meaning of sentences can be characterized by stating the conditions that need to hold fo

of the book,in terms of originality,compositionality,model theory and grammar fragments.Some caveats apply: he studies models of natural language itself,not models of our knowledge or ability to use language;furthermore, these models are not intended to have any metaphysical interpretation,but are only a description and approximation of natural language.

Chapter2,Simply Typed-Calculus,lays out the basic theory of the simply typed-calculus.The simply typed -calculus provides an elegant solution to the problem of giving a denotation for the basic expressions of a language in a compositional manner,as explained in Chapter3.This chapter concentrates on the basic theory,describing the language of the simply typed-calculus,along with a model theory and a proof theory for the logical language,that formalizes whether two-calculus expressions are equal(have the same denotation in all models).The standard-calculus notions of reductions,normal forms,strong normalization,the Church-Rosser theorem,and combinators are discussed.An extension of the simply typed-calculus with sums and products is described.

Chapter3,Higher-Order Logic,introduces a generalization offirst-order logic where quantification and abstrac-tion occurs over all the entities of the language,including relations and functions.Higher-order logic is defined as a specific instance of the simply typed-calculus,with types capturing both individuals and truth values,and log-ical constants such as conjunction,negation,and universal quantification.The usefulness of the resulting logic is demonstrated by showing how it can handle quantifiers in natural languages in a uniform way.The proof theory of higher-order logic is discussed.

Chapter4,Applicative Categorial Grammar,is an introduction to the syntactic theory from which the denotation of natural language terms is derived,that of categorial grammars.Categorial grammars are based on the notion of cate-gories representing syntactic functionality,and describe how to syntactically combine entities in different categories to form combined entities in new categories.The framework described in this chapter is the simplest form of applicative categorial grammar,which will be extended in later chapters.After introducing the basic categories,the chapter shows how to assign semantic domains to categories,and how to associate with every basic syntactic entity a term in the corresponding domain,creating a lexicon.The basics of how to derive the semantic meaning of a composition of basic syntactic entities based on the derivation of categories is explored.Finally,a discussion of some of the consequences of this way of assigning semantic meaning is given;mainly,it focuses on ambiguity and vagueness,corresponding respectively to expressions with multiple meanings,and expressions with a single undetermined meaning.

Chapter5,The Lambek Calculus,introduces a logical system that extends the applicative categorial grammar framework of the previous chapter.The Lambek calculus allows for a moreflexible description of the possible ways of putting together entities in different categories.The Lambek calculus is presented both in sequent form and in natural deduction form,the former appropriate for automatic derivations,the latter more palatable for humans.The Lambek calculus is decidable(i.e.,the problem of determining whether the calculus can show a given sentence grammatical is decidable).The correspondence between the Lambek calculus and a variant of linear logic is established.

The following four chapters show how to apply the machinery of thefirst part to different aspects of linguistic analysis.

Chapter6,Coordination and Unbounded Dependencies,studies two well-known linguistic applications of cat-egorial grammars.Thefirst,coordination,corresponds to the use of and in sentences.Such a coordination operator can occur on many levels,coordinating two nouns(Joe and Victoria),two adjectives(black and blue),two sentences, etc.Coordination at any level is achieved by lifting the coordination to the level of sentences,via the introduction of a polymorphic coordination operator in the semantic framework.This operator can be handled in the Lambek calcu-lus via type lifting.The resulting system remains decidable.An extension of the Lambek calculus with conjunction and disjunction is considered,to account for coordinating,for example,unlike complements of a category,such as in Jack is a good cook and always improving.The second well-known use of categorial grammars is to account for un-bounded dependencies,that is,relationships between distant expressions within an expression,the distance potentially unbounded.This is handled by introducing a new categorial combinator,an element of which can be analyzed as an with a missing somewhere within it.The appropriate derivation rules can be added to the Lambek calculus.

Chapter7,Quantifiers and Scope,studies the contribution of quantified noun phrases to the meaning of phrases in which they occur.Such generalized quantifiers,such as every kid,or some toy,are traditionally problematic because they take semantic scope around an arbitrary amount of material.For instance,every kid played with some toy has two readings,depending on the scope of the quantifiers every and some(is there a single toy with which every kid plays,or does every kid play with a possibly different toy?)Accounting for such readings is the aim of this chapter. Two historically significant approaches to quantifiers are surveyed:Montague’s quantifying in approach,and Cooper’s storage mechanism.Then,the type-logical solution of Moortgat is described.The idea is to introduce a new category

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