  We consider general structures where formulas have truth values in the real
unit interval as in continuous model theory, but whose predicates and functions
need not be uniformly continuous with respect to a distance predicate. Every
general structure can be expanded to a pre-metric structure by adding a
distance predicate that is a uniform limit of formulas. Moreover, that distance
predicate is unique up to uniform equivalence. We use this to extend the
central notions in the model theory of metric structures to general structures,
and show that many model-theoretic results from the literature about metric
structures have natural analogues for general structures.
