First-order languages: semantics
Chapter 5 referred to first-order languages interpreted in some domain. To generate the results of the next chapter we need to be a bit more precise about just what sorts of things interpretations of L are. To prepare the following discussion, suppose that L contains a binary relation symbol R and consider the sentence (i.e. closed formula) of L. As it stands it has no truth-value, because no domain has been specified. But it automatically acquires a truth-value once we specify a domain and a binary relation defined in the domain as the interpretation of R. Now suppose L also contains a constant a. For the sentence ∃xR(x, a) to have a truth-value in that domain a must be made to refer to some individual in the domain. We can generalise these observations as follows: specifying an interpretation of a first-order language will mean specifying a domain and interpretations in that domain of the extralogical vocabulary of that language.