Review of Type-Logical Semantics

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

Review of

Type-Logical Semantics

by B.Carpenter

Riccardo Pucella Stephen Chong

Department of Computer Science

Cornell University

January30,2001

Introduction

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 for the sentence to be true.Necessarily,this approach,called truth-conditional semantics,disre-gards some relevant aspects of meaning,but has been very useful in the analysis of natural languages.Structuralist views of language(the kind held by Saussure,for instance,and later Chomsky)have typically focused on phonology, morphology,and syntax.Little progress,however,has been shown towards the structure of meaning,or content.

A common tool for the study of content,and structure in general for that matter,has been logic.During most of the20th century,an important role of logic has been to study the structure of content of mathematical languages. Many logicians have moved on to apply the techniques developed to the analysis of natural languages—Frege,Russell, Carnap,Reichenbach,and Montague.An early introduction to such classical approaches can be found in[2].

As an illustration of the kind of problems that need to be addressed,consider the following examples.The follow-ing two sentences assert the existence of a man that both walks and talks:

Some man that walks talks

Some man that talks walks

The situations with respect to which these two sentences are true are the same,and hence a truth-conditional semantics needs to assign the same meaning to such sentences.Ambiguities arise easily in natural languages: Every man loves a woman

There are at least two distinct readings of this sentence.One says that for every man,there exists a woman that he loves,and the other says that there exists a woman that every man loves.Other problems are harder to qualify. Consider the following two sentences:

Tarzan likes Jane

Tarzan wants a girlfriend

The?rst sentence must be false if there is no Jane.On the other hand,the second sentence can be true even if no woman exists.

Those examples are extremely simple,some might even say naive,but they exemplify the issues for which a theory of natural language semantics must account.A guiding principle,apocryphally due to Frege,in the study of semantics is the so-called Fregean principle.Essentially,it can be stated as“the meaning of a complex expression should be a function of the meaning of its parts.”Such a principle seems required to explain how natural languages can be learned. Since there is no arbitrary limit on both the length and the number of new sentences human beings can understand, some general principle such as the above must be at play.Moreover,since it would not be helpful to require an in?nite

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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 the?rst 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,re?ecting 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...

The?rst 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 with?rst-order logic.Roughly speaking,?rst-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 the?rst-order logic formula.The sentence Everyone likes Jane can be expressed as.This approach of using?rst-order logic to give semantics is quite straightforward. Unfortunately,for our purposes,it is also quite de?cient.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 the?rst-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 in?rst-order logic,respectively:

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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

,or equivalently,

There seems to be a discrepancy among the logical contributions of the subjects in the above sentences.There is a

distinction in the?rst-order logic translation of these sentences that is not expressed by their grammatical form.

It turns out that there is a way to solve those problems,by looking at an extension of?rst-order logic,known as higher-order logic[3].Let us give enough theory of higher-order logic to see how it can be used to assign semantics

to(a subset of)a natural language.This presentation presupposes a familiarity with both?rst-order logic and the -calculus[5].1There is a slight difference in our approach to?rst-order logic and our approach to higher-order logic. In the former,formulas,which represents properties of the world and its individuals,are the basic units of the logic.In

higher-order logics,terms are the basic units,including constants and functions,with formulas explicitly represented

as boolean-valued functions.

We start by de?ning a set of types that will be used to characterize the well-formedness of formulas,as well as

derive the models.We assume a set of basic types,where is the type of boolean values, and is the type of individuals.(In?rst-order logic,the type is not made explicit.)The set of types is the smallest set such that,and if.A type of the form is a functional(or higher-order)type,the elements of which map objects of type to objects of type.

The syntax of higher-order logic is de?ned as follows.Assume for each type a set of variables and

a set of constants of that type.The set of terms of type is de?ned as the smallest set such that:

,

,

if and,and

if,,and.

What are we doing here?We are de?ning a term language.First-order logic introduces special syntax for its logical

connectives(,,,).It turns out,for higher-order logic,that we do not need to do that,we can simply de?ne constants for those operators.(We will call these logical constants,because they will have the same interpretation in all models.)We will assume the following constants,at the following types:a constant of type and a constant of type,the interpretation of which should be clear,a family of constants each of type,which checks for equality of two elements of type,and a family of constants

each of type,used to capture universal quanti?cation.The idea is that quanti?es over objects of type.In?rst-order logic,is true if for every possible individual,replacing by in yields a true formula.Note that binds the variable in?rst-order logic.In higher-order logic,where there is only as a binder, we write the above as,which is true if is true for all objects of type.

This de?nes the syntax of higher-order logic.The models of higher-order logic are generalizations of the relational

structures used to model?rst-order logic.A(standard)frame for higher-order logic is speci?ed by giving for each basic type a domain of values of that type.These extend to functional types inductively:for a type ,,that is,the set of all functions from elements of

to elements of.Let.We also need to given an interpretation for all the constants,via a function assigning to every constant of type an object of type.(We simply write when the type is clear from the context.)Hence,a model for higher-order logic is of the form.We extend the interpretation to all the terms of the language.To deal with variables,we de?ne an assignment to be a function such that if.We denote the assignment that maps to and

to.We de?ne the denotation of the term with respect to the model and assignment as:

if,

if,

,and

such that.

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

Standard frames are subject to restrictions.For instance,the domain corresponding to boolean values must be a two-element domain,such as true false.Moreover,they must give a?xed interpretation to the logical constants(i.e.,the conjunction operator should actually behave as a conjunction operator).Hence,we require:

false if true

true if false

true if true and true

false otherwise

true if

false otherwise

true if true for all

false otherwise

One can check that if we de?ne as,it has the expected interpretation.Note that we will often use the abbreviations for,and for.

A formula of higher-order logic is a term of type.We say that a model satis?es a formula if true in the model.Two terms are said to be logically equivalent if they have the same interpretation in all models.One can check,for instance,that and are logically equivalent,as are and(that is,where every occurrence of is replaced by).

For example,consider the following simple three individual model,with constants and .

false

true

true

true

false

false

true

false

true

This model satis?es the term(Kala likes Tarzan)as

true.It also satis?es the term(There is someone Jane likes).It does not satisfy the term (Everyone likes himself/herself).

We will use higher-order logic to express our semantics.The idea is to associate with every sentence(or part of speech)a higher-order logic term.We can then use the semantics of higher-order logic to derive the truth value of the sentence.Consider the examples at the beginning of the section.We assume constants,and

of type,and a constant of type.We can translate the sentence Tarzan likes Jane as ,as in?rst-order logic.But now we can also translate the part of speech like Jane independently as.

For a more interesting example,consider the treatment of noun phrases as given at the beginning of the section. The solution to the problem of losing the grammatical structure was solved by Russell by treating all noun phrases as though they were functions over their verb phrases.This is analogous to what is already happening with the de?nition of in higher-order logic,which has type.Such generalized quanti?er takes a property of an individual(a property has type)and produces a truth value—in the case of, the truth value is true if every individual has the supplied property.A similar abstraction can be applied to a noun position.We de?ne a generalized determiner as a function taking a property stating a restriction on the quanti?ed individuals,and returning a generalized quanti?er obeying that restriction.Hence,a generalized determiner has type

.Consider the following generalized determiners,used above:

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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

One can check that the sentence An apeman likes Jane becomes,that Ev-ery apeman likes Jane becomes,and that No apeman likes Jane becomes

.The subject is interpreted as,and respectively.The verb phrase likes Jane is given the expected semantics.What about the original sentence Tarzan likes Jane.According to the above,we should be able to give a semantics to Tarzan(when used as a subject)with a type.One can check that if we interpret Tarzan as,we indeed get the required behavior.Hence,we see that the noun phrase can be given the uniform type, and that higher-order logic can be used to derive a uniform,compositional semantics.

...from syntax

We have seen in the previous section how we can associate to sentences a semantics in higher-order logic.More importantly,we have seen how we can assign a semantics to sentence extracts,in a way that does capture the intuitive meaning of the sentences.The question at this point is how to derive the higher-order logic term corresponding to a given sentence or sentence extract.

The grammatical theory we use to achieve this is categorial grammars,originally developed by Ajdukiewicz [1]and later Bar-Hillel[4].In fact,we will use a generalization of their approach due to Lambek[8].The idea behind categorial grammars is simple.We start with a set of categories,each category representing a grammatical function.For instance,we can start with the simple categories np representing noun phrases,n representing nouns, and s representing sentences.Given categories and,we can form the functor categories and.The category represents the category of syntactic units that take a syntactic unit of category to their right to form a syntactic unit of category.Similarly,the category represents the category of syntactic units that take a syntactic unit of category to their left to form a syntactic unit of category.Consider some examples.The category represents the category of prenominal modi?ers,such as adjectives:they take a noun on their right and form a noun. The category represents the category of postnominal modi?ers.The category is the category of intransitive verbs:they take a noun phrase on their left to form a sentence.Similarly,the category represents the category of transitive verbs:they take a noun phrase on their right to then expect a noun phrase on their left to form a sentence.

Before deriving semantics,let’s?rst discuss well-formedness,as this was the original goal for such grammars. The idea was to associate to every word(or complex sequence of words that constitute a single lexical entry)one or more categories.We will call this the dictionary,or lexicon.The approach described by Lambek[8]is to prescribe a calculus of categories so that if a sequence of words can be assigned a category according to the rules,then the sequence of words is deemed a well-formed syntactic unit of category.Hence,a sequence of words is a well-formed sentence if it can be shown in the calculus that it has category.As an example of reduction,we see that if has category and has category,then has category B.Schematically,.Moreover,this goes both ways,that is,if has category and can be shown to have category,then we can derive that has category.

It was the realization of van Benthem[12]that this calculus could be used to assign a semantics to terms and use the derivation of categories to derive the semantics.The semantic will be given in some higher-order logic as we saw above.We assume that to every basic category corresponds a higher-order logic type.Such a type assignment can be extended to functor categories by putting.We extend the dictionary so that we associate with every word one or more categories,and a corresponding term of higher-order logic.We stipulate that the term corresponding to a word in category should have a type corresponding to the category,i.e.

.

We will use the following notation(called a sequent)to mean that expressions of categories can be concatenated to form an expression of category.We will use capital Greek letters(,...)to represent sequences of expressions and categories.We now give rules that allow us to derive

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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

new sequents from other sequents:

In other words,if can concatenate into an expression with category,and if can concatenate into an expression with category,then can concatenate into with category.

For example,the following is a derivation of Tarzan likes Jane.

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 of?rst-order logic where quanti?cation and abstrac-tion occurs over all the entities of the language,including relations and functions.Higher-order logic is de?ned as a speci?c instance of the simply typed-calculus,with types capturing both individuals and truth values,and log-ical constants such as conjunction,negation,and universal quanti?cation.The usefulness of the resulting logic is demonstrated by showing how it can handle quanti?ers 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 more?exible 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 the?rst part to different aspects of linguistic analysis.

Chapter6,Coordination and Unbounded Dependencies,studies two well-known linguistic applications of cat-egorial grammars.The?rst,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,Quanti?ers and Scope,studies the contribution of quanti?ed noun phrases to the meaning of phrases in which they occur.Such generalized quanti?ers,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 quanti?ers 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 signi?cant approaches to quanti?ers 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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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 expressions that act locally as’s but take their semantic scope over an embedding expression of cate-gory.Generalized quanti?ers are given category,since they act like a noun phrase(category)in situ, but scope semantically to an embedding sentence(category).The Lambek calculus is extended with appropriate derivation rules.The issues of quanti?er coordination,quanti?ers within quanti?ers,and the interaction with negation are discussed.Other topics related to quanti?ers and determiners in general,such as de?nite descriptions,possessives (every kid’s toy),inde?nites(some student),generics(italians),comparatives(as tall as),and expletives(it,there)are analyzed within that context.

Chapter8,Plurals,provides a type-logical account of plurality.First,the notion of group is added to the syntax and semantics.The type is considered to be a subtype of the type and thus the domain of is a subset of the domain of.A relation linking a group to the property that de?nes membership in the group is de?ned,and restrictions are imposed to ensure that every group has a unique property that de?nes membership of that group.With this interpretation,categories for plural noun phrases and plural nouns are studied.The notions of distributors(to view a group as a set of individuals)and collectors(to view a set of individuals as a group)are de?ned,to handle,for example,verbs that apply only to individuals or only to groups.The issues of coordination and negation are examined in the context of plurals.Further topics examined include plural quanti?catives and,more generally,partitives(each, all,most,or numerical partitives such as three of,etc.),nonboolean coordination with and,comitative(the use of with in Tarzan climbed the tree with Cheetah),and mass terms such as snow and water.

Chapter9,Pronouns and Dependency,analyses the use of non-indexical pronouns such as him,she,itself,e-specially the dependent use of such pronouns.Dependent pronouns are characterized as having their interpretation depend on the interpretation of some other expression(the antecedent).For example,he in Jody believes he will be famous.A popular interpretation of pronouns in type-logical frameworks is as variables,although the treatment is subtle,at least for non-re?exive pronouns such as the he above.(Admittedly,this topic is an outstanding problem for type-logical grammars.)Re?exive pronouns,such as himself in Everyone likes himself,can be handled as quanti?ers. Topics related to pronomial forms are examined,such as reciprocals(the each other in The three kids like each other), pied piping(the which in the table the leg of which Jody broke),ambiguous verb-phrases ellipses(Jody likes himself and Brett does too),and interrogatives.

The?nal part,the last three chapters,extend the framework with modalities to account for intensional aspects of natural languages.

Chapter10,Modal Logic,introduces the logical tools required to deal with intensionality,tense and aspect.The key concept is that of a modal logic,where operators are used to qualify the truth of a statement.The chapter presents both a model theory(Kripke frames)and a proof theory for S5,a particular modal logic of necessity.A brief discussion of how the techniques of modal logic can be used to model indexicality precedes the presentation of a general modal model.First-order tense logics,which extend?rst order logics with modal operators about the truth of statements in the past and future,are presented in some depth,as they are able to provide a model of tenses in natural language.Time can be regarded as a collection of moments,or as a collection of possibly overlapping intervals.Higher order logic is extended to include modal operators by taking the domains of worlds and time to be basic types,on the same level as the domains of individuals and truth values,yielding a framework referred to as intensional logic.This approach avoids a number of problems associated with simply abstracting the model for higher order logic over possible worlds.

Chapter11,Intensionality,uses modal logic to extend the type-logical framework to cover intensional construc-tions.In particular,is added as a new basic type,and the assignment of types to basic categories is modi?ed, replacing with,i.e.truth values may be different at different worlds.This change facilitates the inclusion of many constructs,such as propositional attitudes(Frank believes Brooke cheated),modal adverbs (possibly),modal auxiliaries(should,might),and so-called control verbs(persuaded,promised),although some con-structs remain problematic.The“individual concepts”approach is considered,where the type of a noun phrase is instead of,i.e.the referent of a noun phrase may differ from world to world.Other approaches to intensionality,which do not involve possible worlds,are explained brie?y.Finally,the last section returns to the issue of giving a categorization of control verbs,and gives some problematic examples showing the need for more work in this area.

Chapter12,Tense and Aspect,extends the grammar and semantics with a theory of tense.It presents Reichen-bach’s approach to simple and perfect tenses,how this applies to discourse,and Vendler’s verb classes—a semantic classi?cation of verbs that is correlated with their syntactic use.The approach Carpenter adopts for tense and aspect is based on insights derived from these works,and on further development of these works by other authors.To extend the grammar,verbs are subcategorized by classifying them based on whether they are?nite or non-?nite,and whether they

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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

involve simple or perfect tense,resulting in several different categories for sentences.A new basic type is introduced, representing time periods,and all of the sentence categories are assigned the same type:functions from time periods to truth values.The temporal argument always corresponds to the time of the event being reported.(This is essentially similar to the intensional approach of the previous chapter,but here we distinguish time periods from possible world-s.)From this beginning,the grammar is developed to encompass many of the English constructs involving tense and aspect.Many of these constructs are very complex in their usage and generally there do not seem to be simple and complete solutions to incorporating them into the grammar.

There are some typos(potentially confusing,as they sometimes occur in the types for functions),as well as some glossing over central topics(such as the discussion of groups in Chapter8).Carpenter doesn’t generally delve into syntactic explanations,that is,explaining why the theory of syntax he develops does or does not permit certain sentences.Moreover,for linguists,it may be important to note that Carpenter does not develop a theory of morphology (the structure of words at the level of morphemes).

This book?lls a sorely void niche in the?eld of semantics of natural languages via type-logical approaches.There are some books on the subject,but the most accessible are severely limited in their development[13],while the others are typically highly mathematical and focus on the metatheory of the type-logical approach[10].

Carpenter’s book is a reasonable blend of mathematical theory and linguistic applications.Its great strength is an excellent survey of type-logical approaches applied to a great variety of linguistic phenomena.On the other hand,the preliminary chapters presenting the underlying mathematical theory are slightly confusing—not necessarily surprising considering the amount of formalism needed to account for all the linguistic phenomena studied.A background or at least exposure to ideas from both logic and programming language semantics is extremely helpful.In this sense,this book seems slightly more suited,at least as an introductory book,to mathematicians and computer scientists interested in linguistic applications,than to linguists interested in learning about applicability of type-logical approaches.(Al-though this book could nicely follow a book such as[13],or any other introductory text on type-logical grammars that focuses more on the“big picture”than on the underlying mathematical formalisms.)People that are not linguists will most likely?nd chapters9and on hard to follow,as they assume more and more knowledge of linguistic phenomena.

This book points to interesting areas of ongoing research.In particular,the later sections of the book on aspects of intensionality highlight areas where the semantics of natural languages are not clear.(This is hardly a surprise,as intensional concepts have always been problematic,leading philosophers to develop many?avors of modal logics to attempt to explain such concepts.)Another avenue of research that is worth pointing out,although not discussed in this book,is the current attempt to base semantics of natural languages not on higher-order logic as presented in this book,but rather on Martin-L¨o f constructive type theory,via categorial techniques[11].

References

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[2]J.Allwood,L.-G.Andersson,and O.Dahl.Logic in Linguistics.Cambridge Textbooks in Linguistics.Cambridge

University Press,1977.

[3]P.B.Andrews.An Introduction to Mathematical Logic and Type Theory:To Truth through Proof.Academic

Press,1986.

[4]Y.Bar-Hillel.A quasi-arithmetical notation for syntactic nguage,29:47–58,1953.

[5]H.P.Barendregt.The Lambda Calculus,Its Syntax and Semantics.Studies in Logic.North-Holland,Amsterdam,

1981.

[6]N.Chomsky.Syntactic Structures.Mouton and Co.,1957.

[7]H.B.Curry.Some logical aspects of grammatical structure.In American Mathematical Society Proceedings of

the Symposia on Applied Mathematics12,pages56–68,1961.

[8]mbek.The mathematics of sentence structure.The American Mathematical Monthly,65:154–170,1958.

[9]R.Montague.Formal Philosophy:Selected Papers of Richard Montague.Yale University Press,1974.

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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

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