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 bookfills a sorely void niche in thefield 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 likelyfind 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 manyflavors 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

[1]K.Ajdukiewicz.Die syntaktische Konnexit¨a t.Studia Philosophica,1:1–27,1935.

[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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