We have said that one of the most important goals of semantic theory is to understand the compositional nature of meaning, i.e., the knowledge which allows speakers to correctly predict how word meanings will combine in complex expressions. One way of exploring this topic is to construct formal rule systems which model the abilities of speakers in this respect.
Just as syntacticians try to construct rule systems which replicate the judgments of native speakers about the grammaticality of sentences, semanticists try to construct rule systems which replicate the ability of speakers to identify the denotation of an expression in a particular context of use, and in particular, to determine the truth values of sentences in a given context. A crucial step in this kind of analysis is to describe the situation under discussion in very explicit terms, so that predictions about denotations can be easily checked. The explicit description of a situation is called a model, so this general approach to semantics is often referred to as Model Theory.1
This chapter provides a very brief introduction to the Model Theory approach to the study of compositionality. This approach, which has proven to be remarkably productive, involves stating rules of semantic interpretation for the constituents that are formed by productive syntactic processes. We mentioned two such processes in Chapter 12: the combination of subject NP with VP, and the combination of modifying adjective with head noun. In this chapter we will provide a bit more detail about how we might formulate the rules of semantic interpretation for these and other constituents.
Our goal in this chapter is not to provide detailed explanation of the Model Theory approach, but merely to give a glimpse of how it works and some sense of what the goals are. This will provide helpful context for our discussion in future chapters of topics such as quantifiers, modality, tense, etc.
§13.2 provides a brief description of the rationale behind this approach. In §13.3 we introduce some basic terms and concepts for describing sets and relations between sets, because our rules of interpretation will be stated in terms of set relations. §13.4 introduces the formal notation that is used for specifying a model, in the sense defined above, and §13.5 gives some examples of how rules of semantic interpretation might be stated for several types of syntactic constituents. The overarching goal of all these steps is to account for the ability of native speakers to determine whether the proposition expressed by a given sentence is true or false in some particular context. This, you will recall, has been our benchmark for the analysis of sentence meanings.
1 A model can also be defined an interpretation under which a given sentence or set of sentences is true (Hodges 2013). But by spelling out the denotations of the basic expressions used in the sentence(s) under discussion, the model also specifies the relevant facts about a particular situation.
