Review of Type-Logical Semantics(8)

...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.

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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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