In Chapter 3 we saw that a Tarski-style theory of truth for a language diagrams the semantic structure of the language. As with any mathematical theory, however, that structure can be fitted on to any system of objects that satisfies the conditions expressed by its axioms. Consider, for example, a simple theory K that one might set up to describe the pattern of relations among the lengths of assorted objects. K would contain an axiom of the form,

(1) If Rxy, then not Ryx,

which, when we assign the interpretation “is longer than” to “R”, says that if some object x is longer than an object y, then y is not longer than x; under this assignment, (1) expresses that length is asymmetric. Notice, however, that at least in so far as it contains (1), K could have been a theory of time rather than length, if we had assigned to “R” the interpretation “is earlier than” and taken “x” and “y” to range over events; under this assignment, (1) says that if an event x is earlier than an event y, then y is not earlier than x. Time, like length, is asymmetric.

We begin to convert a mere formal diagram into a theory of truth when we assign semantic concepts (truth, satisfaction and reference) to its predicate constants, but unless we show that the theory's T-sentences correctly assign truth-conditions to sentences from a language that someone actually speaks, we have failed to supply the theory with an empirical interpretation.