Appeal

An appeal, interpretation, or internal structure is an operator. More precisely, "an appeal" is short for "an appeal to ____" i.e. the most general instance of a structure of the form of "an appeal to proof", an "appeal to logic," and so on.

Why are appeals and interpretations the same thing? Recall that an interpretation, in the usual sense, associates or equates two objects according to some common (abstract) structure.

A correspondence is something that relates two structures (or theories?).