6.4. Describing models

6.4.0. Overview

The grammatical variety we have been considering brings with it a greater variety of semantic values; in particular, the counterexamples to arguments that are not formally valid are now more than simple assignments of truth values.

6.4.1. Extensions and ranges
While the extensions of individual terms are (like the extensions of sentences) single values, the extensions of operators are (like the extensions of connectives) functions.

6.4.2. Building structures
The extensions of predicates (functions from reference values to truth values) can be fully specified by telling the input for which they give output T, and these cases can be presented in diagrams to which the extensions of other items of non-logical vocabulary can be added.

6.4.3. Structures as counterexamples
Although extensional interpretations are now different, it is still true that a dead-end open gap specifies the sort of counterexample that lurks in the gap.