228 The formalization of sentence-meaning

rules being their intensions) and that we were able to identify the extension of any given expression in any particular world (the domain of the function) by looking up the rule (or function) and applying it to the world. This rule, then, is a function from possible worlds to an extension: it picks out from the world that is its domain the set of entities that are being referred to; and this set is the value of the function. At this point, it should be noted that everything that has been said here, more or less loosely and semi-technically, can be fully formalized within the framework of set-theory.

But speakers of a language do not have, and in principle can­not have, a list of rules in their heads for all the expressions in a language. Apart from any other psychological considerations that would render this hypothesis implausible, there is the fact, mentioned in an earlier section of this chapter, that some, and perhaps all, natural languages contain infinitely many expres­sions. And competent speakers of such languages are able, by virtue of their competence, grammatical and semantic, to pro­duce and interpret any arbitrarily selected member of these infi­nite sets of expressions. Clearly, as Chomsky argued forcefully when he laid the foundations of generative grammar, the human brain does not, and being physical cannot, have infinite capacity for the storage of language-systems (and the processing of the products of their use in performance). What is required, then, is yet another function (or set of functions), which deter­mines the intension of composite expressions on the basis of the intension of basic expressions (lexemes), and of the syntactic rules (the rules of composition) which generate them.

All I have done so far, of course, is to reformulate Frege's prin­ciple of compositionality within an intensional framework, as Montague's predecessors, such as Rudolf Carnap, had done in the 1940s. As I said earlier, I have been taking the principle of compositionality to apply primarily to sense and denotation, and only derivatively to reference. Reference is mediated in this respect by denotation and context — in a way that is in part explained, informally, in later chapters. Most formal semanti- cists do not draw a clear distinction between reference and deno-tation. (Nor, of course, did Frege.) It is arguable, however, that

7.6 Possible worlds 229

the distinction was implicitly drawn by Montague, partly by means of his indices — which can be seen as relativizing the iden­tification of particular referents to a particular context of utter­ance - and partly by the syntactic and semantic properties which he associated with the special entity-category (e) which he introduced into one of the systems of grammar with which he operated in his analysis of quantifiers. But, in saying this, I am perhaps going beyond the evidence. And Montague's fol­lowers, in any case, have only recently begun to make explicit and to exploit more fully this feature of Montague grammar.

Let us return, therefore, to what is historically beyond dis­pute. This is that Montague sought to establish a much closer correspondence between syntax and semantics than was forma­lized in the standard theory of transformational-generative grammar of the mid-to-late 1960s. He achieved this, in so far as he was successful, by adopting a particular kind of categorial grammar and by putting the categories of syntax (roughly comparable with the major categories and subcategories of tra­ditional grammar: noun-phrases, nouns, predicates, intransitive verbs, transitive verbs, adverbs, etc.) into one-to-one correspon­dence with intensional categories.

Categorial grammar as such does not concern us directly in this book. All that needs to be said about it here is that it is a par­ticularly elegant kind of grammar, which derives all the other syntactic categories from the basic categories of name and prop­osition, or noun-phrase and sentence. (In the particularly inter­esting version of categorial grammar to which I alluded above, names and other kinds of noun-phrases are not basic, but derived, categories, formed out of the entity-category and nouns.) The term 'categorial' reflects its philosophical origins. Categorial grammar in itself is no more closely associated with one kind of ontological framework than it is with another. It does rest, however, upon the principle of categorial congruity, the violation of which results in categorial incongruity, which was mentioned earlier (and this is why I used the term 'categor- ial' in this connexion): the principle of syntactic and semantic congruity, or interdependence, with respect to the rules of composition. This notion of congruity (in Latin, 'congruitas') is emi-