We have noted several times that denotations, including the denotations of referring expressions and truth values of sentences, can only be evaluated relative to a particular situation of use. In order to develop and test a set of interpretive rules, which can correctly predict the denotation of a particular expression in any given situation, it is important to provide very explicit descriptions for the test situations. As stated above, this kind of description of a situation is called a model, and must include two types of information: (i) the domain, i.e., the set of all individual entities in the situation; and (ii) the denotation sets for the basic vocabulary items in the expressions being analyzed.
As a first illustration of how the system works, let us return to our simple situation containing just three individuals: King Henry VIII, Anne Boleyn, and Thomas More. Our model of this situation, which we might call Model 1, would provide the information listed in (20). We often use the name “U” as a convenient way to refer to the domain (the “universal set” of individuals). The notation ((x)) represents the denotation (or “semantic value”) of x within the current model. This notation can be used either for object language expressions or for logical formulae; so, for example, ((SNORE)) names the same set as ((snores)). By convention we use small letters for logical “constants”, e.g. proper names, and capital letters for predicates.
(20) Model 1
i. the set of individuals U = { King Henry VIII, Anne Boleyn, Thomas More }
ii. denotations:
((MAN)) = {King Henry VIII, Thomas More}
((WOMAN)) = {Anne Boleyn}
((SNORE)) = {King Henry VIII}
((a)) = Anne Boleyn
((h)) = King Henry VIII
((t)) = Thomas More
The denotation sets encode information about the current state of the world. For example, this model indicates that King Henry VIII is the only person in the current situation who snores. We can use the defined vocabulary items to build simple declarative sentences about the individuals in this situation, and then try to provide interpretations for each sentence in terms of set membership, as illustrated in Table 13.1. These interpretations express the truth conditions for each sentence. We can use them to evaluate the truth of each sentence relative to Model 1. For example, the sentence in Table 13.1a, Thomas More is a man, will be true in any situation where the individual Thomas More is a member of the denotation set of the word man. Since this is the case in Model 1, the sentence is true relative to this model.
Table 13.1: Sentence interpretation examples
The interpretations in Table 13.1b–e can be derived from the corresponding logical forms, based on the definitions of intersection, union, and subset provided in (19). For example, the or statement in Table 19b constitutes a claim that a certain individual (Anne Boleyn) is a member of the union of two sets, because the definition of A∪B involves an or statement. Once the truth conditions are stated in terms of set relations, we can determine the truth values for each sentence by inspecting the membership of the denotation sets specified in the model. The statement in (Table 13.1b) is true relative to Model 1 because the individual Anne Boleyn is a member of the set ((WOMAN)) , and thus a member of ((MAN))∪((WOMAN)).
