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

number of functions to derive the meaning of the whole from the meaning of the parts,a notion such as recursion must be at play as well.

That there is a recursive principle`a la Frege at play both in syntax and semantics is hardly contested.What is contested is the interplay between the two.The classic work by Chomsky[6]advocated essentially the autonomy of syntax with respect to semantics.Chomsky’s grammars are transformational:they transform the“surface”syntax of a sentence to extract its so-called deep structure.The semantics is then derived from the deep structure of the sentence. Some accounts of the Chomsky theory allows for a Fregean principle to apply at the level of the deep structure,while more recent accounts slightly complicate the picture.A different approach is to advocate a close correspondence between syntax and semantics.Essentially,the syntax can be seen as a map showing how the meaning of the parts are to be combined into the meaning of the whole.

The latter approach to semantics relies on two distinct developments.First,it is based on a kind of semantic anal-ysis of language originating mainly with the work of Montague[9].His was thefirst work that developed a large scale semantic description of natural languages by translation into a logical language that can be given a semantics using traditional techniques.The second development emerged from a particular syntactic analysis of language.During his analysis of logic,which led to development of the-calculus,Curry noticed that the types he was assigning to-terms could also be used to denote English word classes[7].For example,in John snores loudly,the word John has type, snores has type,and loudly has type.Independently,Lambek introduced a calculus of syntactic types, distinguishing two kinds of implication,reflecting the non-commutativity of concatenation[8].The idea was to push all the grammar into the dictionary,assigning to each English word one or more types,and using the calculus to decide whether a string of words is a grammatically well-formed sentence.This work derived in part from earlier work by Ajdukiewicz[1]and Bar-Hillel[4].

This book,“Type-Logical Semantics”by Carpenter,explores this particular approach.Essentially,it relies on techniques from type theory:we assign a type(or more than one)to every word in the language,and we can check that a sentence is well-formed by performing what amounts to type-checking.In fact,it turns out that we can take the type-checking derivation proving that a sentence has the right type,and use the derivation to derive the semantics of the sentence.In the next sections,we will introduce the framework,and give simple examples to highlight the ideas. Carpenter pushes these ideas quite far,as we shall see when we cover the table of contents.We conclude with some opinions on the book.

To semantics...

Thefirst problem we need to address is how to describe the semantics of language.We will follow in the truth-conditional tradition and model-theoretic ideas and we start withfirst-order logic.Roughly speaking,first-order logic provides one with constants denoting individuals,and predicates over such individuals.Simple example should il-lustrate this.Consider the sentence Tarzan likes Jane.Assuming constants and,and a predicate, this sentence corresponds to thefirst-order logic formula.The sentence Everyone likes Jane can be expressed as.This approach of usingfirst-order logic to give semantics is quite straightforward. Unfortunately,for our purposes,it is also quite deficient.Let us see two reasons why that is.First,recall that we want a compositional principle at work in semantics.In other words,we want to be able to derive the meaning of Tarzan likes Jane from the meaning of Tarzan and Jane,and the meaning of likes.This sounds straightforward.However, the same principle should apply to the sentence Tarzan and Kala like Jane,corresponding to thefirst-order formula

.Giving a compositional semantics seems to require giving a semantics to the extract like Jane.What is the semantics of such a part of speech?First-order logic cannot answer this easily.In-formally,like Jane should have as semantics something that expects an individual(say)and gives back the formula .A second problem is that the grammatical structure of a sentence can be lost during translation.This can lead to wild differences in semantics for similar sentences.For instance,consider the following sentences: Tarzan likes Jane.

An apeman likes Jane.

Every apeman likes Jane.

No apeman likes Jane.

These sentences can be formalized as such infirst-order logic,respectively:

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