Let us return now to the question of how the meaning of the simple sentence in (1) is composed from the meanings of its parts. As we noted, the sentence contains two immediate constituents: the subject NP King Henry VIII and the intransitive verb (or VP) snores. The NP King Henry VIII is a proper name, a “rigid designator”, and so always refers to the same individual; its denotation does not depend on the situation. The intransitive VP snores expresses a property which may be true of a particular individual at one time or in one situation, but not in other times or situations; so its denotation does depend on the situation in which it is used. We will refer to the set of all things which snore in the current universe of discourse as the denotation set of the predicate snores. The result of combining the subject NP with the intransitive VP is a sentence whose meaning is a proposition, and this proposition will be true if and only if the individual named King Henry VIII is a member of the denotation set of snores; i.e., if the king has the property of snoring in the time and situation being described.
This same basic rule of interpretation works for a great many simple declarative sentences: the proposition expressed by the sentence as a whole will be true if and only if the referent of the subject NP is a member of the denotation set of the VP. Of course there are many other cases for which this simple rule is not adequate; but in the present book we will touch on these only briefly.
The Principle of Compositionality also applies to complex expressions which are smaller than a sentence, including noun phrases. Even though these phrasal expressions do not have truth values, they do have denotations which are determined compositionally. In Chapter 1 we briefly discussed the compositionality of the phrase yellow submarine. Suppose we refer to the denotation set of the word yellow (i.e., the set of all yellow things in our universe of discourse) as Y, and the denotation set of the word submarine (i.e., the set of all submarines in our universe of discourse) as S. The meaning of the phrase yellow submarine is predictable from the meaning of its individual words and the way they are combined. Knowing the rules of English allows speakers to predict that the denotation set of the phrase will be the set of all things which belong both to Y and to S; in other words, the set of all things in our universe of discourse which are both yellow and submarines.
As these simple examples illustrate, our analysis of denotations and truth values will be stated in terms of set membership and relations between sets. For this reason we will introduce some basic terms and concepts from set theory at the beginning of Chapter 13. These elements of set theory will also be crucial for analyzing the meanings of quantifiers (words and phrases such as everyone, some people, most countries, etc.). Quantifiers (the focus of Chapter 14) are an interesting and important topic of study in their own right, but they are also important because certain other kinds of expressions can actually be analyzed as quantifiers (see Chapter 16, for example).
But before we proceed with a more detailed discussion of these issues, it will be helpful to review some of Frege’s insights.
