It is not clear how Socrates defined virtue, exactly, not whether he believed that virtue can be taught: in the Meno he goes for a negative answer to the latter question, while in Protagoras he veers toward a positive one.
The Stoics, however, certainly believed that virtue can be taught. Indeed, they were also pretty clear about how to do so: while theoretical treatment helps clarifying one’s mind, one becomes virtuous in the same way in which one gets to Carnegie Hall: practice, practice, practice.
That’s because virtue is a type of skill that needs to be perfected. Here is how Diogenes Laertius explained it: “Virtue, in the first place, is in one sense the perfection of anything in general, say of a statue; again, it may be non-intellectual, like health, or intellectual, like prudence.” (Lives of the Eminent Philosophers, Book VIII)
Diogenes also gives us the famous list of Stoic virtues: “Amongst the virtues some are primary, some are subordinate to these. The following are the primary: wisdom, courage, justice, temperance.” (Lives of the Eminent Philosophers, Book VIII)
Moreover, Diogenes tells us that the Stoics thought that the virtues are all aspects of the same thing: “[The Stoics] hold that the virtues involve one another, and that the possessor of one is the possessor of all, inasmuch as they have common principles.” (Lives of the Eminent Philosophers, Book VIII)
So when someone asks me what is virtue, my first line of response is to say that it manifests itself in wisdom, courage, justice, and temperance. Yes, yes, my interlocutor rightly retorts, but what are “wisdom,” “courage,” “justice,” and “temperance”?
I could (and sometimes do) launch into a deeper discussion of one or more of the cardinal virtues, but at some point you have to go “Forrest Gump” on the guy: virtue is as virtue does, so to speak. To understand virtue you need to look at examples of behavior on the part of virtuous people. Or, as Marcus put it: “In the writings of the Ephesians there was this precept: constantly to think of one of the men of former times who practiced virtue.” (Meditations, XI.26)
Now, Socrates did’t like the idea of understanding concepts by listing examples. In the Theaetetus, for instance, he tells his young interlocutor (a mathematician) that one cannot simply point to examples of knowledge (the main topic of that dialogue) in order to figure out what knowledge is. Socrates seems rather to go after a definition in terms of what we today would call necessary and jointly sufficient conditions.
This is the sort of definition that is common in logic and mathematics — as a frustrated Theaetetus points out. For instance, a good definition of a triangle is that it is a geometric figure that (in Euclidean space) has the sum of its internal angles amounting to 180 degrees. Having that property is both necessary in order to be a triangle (geometric figures that don’t have it are not triangles), and sufficient (once we know that a figure has that property we also instantly know that it is a triangle).
The problem is that most interestingly complex concepts — like “knowledge,” or “virtue” — are inherently fuzzy, without sharp boundaries, and are therefore not amenable to simple necessary & sufficient type of definitions.
This, however, does not imply that we do not understand what these concepts mean. Take a look at how Ludwig Wittgenstein (not a Stoic philosopher!) tackled the similar problem represented by the definition of the deceptively simple concept of “game.”
In his Philosophical Investigations (1953), Wittgenstein begins by saying “consider for example the proceedings that we call ‘games’ … look and see whether there is anything common to all.” (§66) After mentioning a number of such examples, he says: “And we can go through the many, many other groups of games in the same way; we can see how similarities crop up and disappear. And the result of this examination is: we see a complicated network of similarities overlapping and criss-crossing: sometimes overall similarities.” Hence: “I can think of no better expression to characterize these similarities than ‘family resemblances’; for the various resemblances between members of a family: build, features, colour of eyes, gait, temperament, etc. etc. overlap and criss-cross in the same way. And I shall say: ‘games’ form a family.” (§67) Concluding: “And this is how we do use the word ‘game.’ For how is the concept of a game bounded? What still counts as a game and what no longer does? Can you give the boundary? No. You can draw one; for none has so far been drawn. (But that never troubled you before when you used the word ‘game.’)” (§68)
Similarly, virtuous characters form a family group whose boundaries cannot be drawn a priori, and yet which we usually do not have trouble identifying.