204 The formalization of sentence-meaning

7.2 CO M P O S I T I O N A L I T Y, G R A M M AT I C A L

A N D S E M A N T I C I S O M O R P H I S M , AND S A V I N G

THE APPEARANCES

The principle of compositionality has been mentioned already in connexion with the sense of words and phrases. Com­monly described as Frege's principle, it is more frequently dis­cussed with reference to sentence-meaning. This is why I have left a fuller treatment of it for this chapter. It is central to formal semantics in all its developments. As it is usually formulated, it runs as follows (with 'composite' substituted for 'complex' or 'compound'): the meaning of a composite expression is a func­tion of the meanings of its component expressions. Three of the terms used here deserve attention; 'meaning', 'expression' and 'function'. I will comment upon each of them in turn and then explain, first, why the principle of compositionality is so impor­tant and, second, to what degree it is, or appears to be, valid.

(i) 'Meaning', as we have seen, can be given various interpre­tations. If we restrict it to descriptive meaning, or propositional content, we can still draw a distinction between sense and deno­tation (see 3.1). Frege's own distinction between sense and refer­ence (drawn originally in German with the terms 'Sinn' and 'Bedeutung') is roughly comparable, and is accepted in broad outline, if not in detail, by most formal semanticists. (Frege, like many formal semanticists, did not distinguish between the deno­tation of an expression and its reference on particular occasions of utterance: see section 3.1. I will pick up this point in relation to the principle of compositionality presently.) I will take the principle of compositionality to apply primarily to sense. But it may be assumed to apply also to denotation; and, as we shall see in a later section, many formal semanticists have defined sense in terms of a prior notion of denotation.

(ii) The term 'expression' is usually left undefined when it is used by linguists. But it is normally taken to include sentences and any of their syntactically identifiable constituents. I have given reasons earlier for distinguishing expressions from forms,' as far as words and phrases are concerned. More controversially perhaps, in also including sentences among the expressions of a

7.2 Compositionality, grammatical and semantic isomorphism 205

language, I have allowed that a sentence, like words and phrases, may have, not only several meanings, but also several forms. I will now assume that there is an identifiable subpart of every sentence that is the bearer of its propositional content, and that this also is an expression to which the principle of com­positionality applies. For example, if we take the view that corre­sponding interrogative and declarative sentences have the same propositional content, we shall say that what they share is an expression (which of itself is neither declarative nor interroga­tive). Some logicians who have taken this view (as did Frege) have called the expression in which the propositional content is encoded the sentence-radical; but this term has not won more general acceptance, and there is no widely used alterna­tive. I will employ instead the term sentence-kernel, or ker­nel, which has occasionally been used in linguistics, for grammatical, rather than semantic, analysis. (Indeed, the use that I am making of this term is very close to the use that Chomsky made of the term 'kernel-string' in the earliest version of transformational-generative grammar formalized by him.) The kernel of a sentence (or clause), then, is an expression, which has a form (not necessarily pronounceable) and whose meaning is (or includes) its propositional content.

(iii) The term 'function' is being employed in its mathemati­cal sense; i.e., to refer to a rule, formula or operation which assigns a single value to each member of the set of entities in its domain. (It thus establishes either a many-one-to-one or one-to-one correspondence between the members of the domain, D, and the set of values, V: it maps D either into or on to V.) For example, in standard algebras there is an arithmetical function, normally written y = x², which for any numerical value of x yields a single and determinate numerical value for x² and thus deter­mines the value of y. Similarly, in the propositional calculus there is a function which for each value of the propositional vari­ables in every well-formed expression maps that expression into the two-member domain {True, False}, or, alternatively and equivalently, {l, 0}. As we saw earlier, this is what is meant by saying that composite propositions are truth-functional. I have now spelled this out in more detail and deliberately introduced,