This chapter systematically explains the way in which models allow linguistic expressions to denote abstract objects. This will give us a better insight into our theory of meaning and its relations with syntactic forms. The first step is to describe how denotations are organized in a model. Throughout Chapter 2, we usedmodels freely to describe different mathematical objects. For sentences we used truthvalues, for names like Tina we used entities, and for adjectives like tall we used sets of entities. In addition we used the membership operator for is, the intersection function for and, and the complement function for not. Using various mathematical objects in this manner was useful for expository purposes. However, in general it makes our compositionality principle hard to obtain. With each new mathematical notion we introduce, we need to see how it composes with other denotations. Toomuchmathematical freedom in the design of the denotations makes it hard to describe how they operate in different natural language expressions. Themodel structure that we introduce in this chapter helps us tomake semantic distinctions between language expressions within well-defined boundaries. In this way we gain a better understanding of denotations in general, and see more clearly how they interact with syntactic forms and with each other.

Some of the foundational themes in this chapter may seem intimidating at first glance.However, none of them is especially hard. To help you follow this chapter with ease, it is divided into four parts. Each of these parts covers a general topic that leads naturally to the topic of the next one. If you are a novice to the field, it is a good idea to solve the exercises referred to at the end of each part before reading on.