The Socrates of Plato's Socratic dialogues was in quest of definitions because he thought they were required for living right: he supposed that in order to know whether certain actions were courageous, or pious, or admirable, one must know what the courageous, the pious, and the admirable are.

We laid out a theory of definition for Socrates: not necessarily his or Plato's own theory, but one based on the refutations of definitions in Plato's Socratic dialogues. The theory had three main components: the Substitutivity Requirement, the Paradigm Requirement, and the Explanatory Requirement.

The Explanatory Requirement, at first blush, simply demanded that one be able to use the definition for the pious in explaining why one called a given action (or person) “pious”: it was a matter of explaining content. But there turned out to be more to it than that: it was required that what one introduces and defines as the pious be itself indelibly pious. That connected the Explanatory Requirement with the Paradigm Requirement. And both were connected with the Substitutivity Requirement, construed as the demand that what is defined as the pious give necessary and sufficient conditions for something's being pious.

That theory has now turned into the Theory of Forms. Substitutivity is obvious enough: the things that partake of the Form, The Pious, have to be all and only the actions and people that are pious. The Paradigm Requirement is now the claim that the separate Form, The Pious, is through and through pious.