206 The formalization of sentence-meaning

with some redundancy, several of the technical terms that are commonly employed in formal semantics. We shall not go, unnecessarily, into the technical details of formal semantics, but the limited amount of terminology introduced here will be useful later, and it will give readers with a knowledge of elementary set-theory some indication of the mathematical framework within which standard versions of formal semantics operate.

But what is the relevance of the notion of compositionality, formalized mathematically, to the semantic analysis of natural-language expressions? First of all, it should be noted that compe­tence in a particular language includes (or supports) the ability to interpret, not only lexically simple expressions, but indefi­nitely many lexically composite expressions, of the language. Since it is impossible for anyone to have learned the sense of every composite expression in the way that one, presumably, learns the sense of lexemes, formal semanticists argue that there must be some function which determines the sense of composite expressions on the basis of the sense of lexemes. Second, it is reasonable to assume that the sense of a composite expression is a function, not only of the sense of its component lexemes, but also of its grammatical structure. We have made this assumption throughout; and it can be tested empirically in a sufficient number of instances for us to accept it as valid. What is needed, then, in the ideal, is a precisely formulated procedure for the syntactic composition of all the well-formed lexically composite expressions in a language, coupled with a procedure for determining the semantic effect, if any, of each process or stage of syntactic composition. This is what formal semantics seeks to provide.

Formal linguistic semantics, as such, is not committed to any particular theory of syntax. Nor does it say anything in advance about the closeness of the correspondence between grammatical and semantic structure in natural languages. There is a wide range of options on each of these issues. That there is some degree of correspondence, or isomorphism, between grammatical and semantic structure is intuitively obvious and can be demon­strated, in particular instances, by appealing to various kinds of

7.2 Compositionality, grammatical and semantic isomorphism 207

grammatical ambiguity. For example, the ambiguity of such classic examples as:

(1) old men and women

is plausibly accounted for by saying that its two interpretations

2. "men who are old and women" and

3. "old men and old women"

reflect a difference of grammatical structure which matches semantic structure. Under one interpretation, represented in (2), 'old' is first combined with 'men' (by a rule of adjectival modifica­tion) and, then, the resultant composite expression 'old men' is combined with 'women' (by means of the co-ordinating conjunc­tion and), so that semantically, as well as grammatically, 'old' applies to 'men', but not to 'women': i.e., 'men', but not 'women', is both grammatically and semantically within the scope of 'old'. (We have already met the notion of scope in relation to nega­tion and interrogativity: see 6.5,6.7.) Under the other interpreta­tion, (3), the grammatical rules can be thought of as having operated in the reverse order, so that 'old' applies to the composite expression 'men and women': i.e., the whole phrase 'old men and women' is grammatically and semantically within the scope of 'old'. Grammatical ambiguity of this kind - so-called immedi­ate-constituent or phrase-structure ambiguity - can be handled in many different, but in this respect descriptively equivalent, sys­tems of grammatical analysis; and it is relatively easy to match the grammatical rules of adjectival modification and phrasal co­ordination (however they are formalized) with rules of semantic interpretation.

The question is whether the degree of correspondence, or iso- morphism, between grammatical and semantic structure is always as high as this. Many formal semanticists have assumed that it is and have used the so-called rule-to-rule hypothesis to guide their research. This may be formulated, for our pur- poses, non-technically as follows: (i) every rule of the grammar (and more particularly every syntactic rule) can be associated