Suppose there is a discussion of the virtue of honesty, and a person is testing the words (W) "telling the truth."
The counterexample posed is: Sam tells Jane the truth that he thinks her hair looks terrible today, in order to hurt her, knowing that she is feeling especially depressed and insecure today.
One might try to respond to this counterexample by saying: "Be considerate when telling the truth."
This is another seemingly easy way to respond to a counterexample. It doesn't require trying to articulate anything difficult to articulate. "Being considerate" is a familiar concept assumed here to be clear in itself, not needing any special explanation.
But this is the very problem. Socratic questioning can show that all easy-to-understand concepts are ambiguous when it comes to true goodness. If we tried, we could come up with counterexamples showing that a person can have some kind of "being considerate" without being admirable. (I go out and kill the wife of a Mafia boss's enemy out of consideration for my boss's feelings, because that's what will make him feel better.)
So in saying "Be considerate when telling the truth," what has possibly happened is just that we have added one more virtue "being considerate" that a person needs to be an admirable person, or that a particular action needs in order to be an admirable action.
For example, in acting kindly toward one child, it is important also to act in such a way that is fair to other children. This doesn't necessarily mean that "fairness" is part of the virtue of kindness, or that fairness and kindness are the same virtue.
Another example: The very creative artist Susan needs also to be responsible in attending to her children. This doesn't mean that "being responsible" is part of the virtue of creativity, or that being responsible is the same virtue as being creative.
"Being considerate when telling the truth" potentially just adds another virtue-concept "being considerate" that would need its own discussion, a discussion that might not contribute anything relevant to the task of formulating the essence of the particular virtue "honesty." On the face of it, it is difficult to say that "honesty" and "being considerate" are the same virtue.
This is why it is important, when responding to a counterexample, to focus on answering this very particular question:
What kind of W is only connected to admirable V, and how is it connected.
In the present case when responding to the story of Sam's malicious truth-telling to hurt Jane's feelings, you would need to ask, "What kind of truth-telling (contrasted with Sam's kind) is only connected to admirable honesty, and how is it connected?" Then stay focused on the particular (W) "truth telling," and try to modify refine this concept itself, rather than just adding an entirely different concept "being considerate." This would be more difficult, because you would then have to focus on what motivates truth-telling in the ideal case (in contrast to Sam's malicious motivation for telling the truth).
It could of course be that you associate "being considerate" with the virtue of honesty itself.
In this case you could ask: What kind of 'being considerate' is only connected to admirable honesty, and how is it related? Then you would have to reflect on what differentiates that kind of being considerate connected to truth telling, from other kinds of being considerate not connected to truth-telling, and reflect also on how 'being considerate' relates to other concepts associated with truth-telling.
This is another reason for leading your discussion to a point where you can narrow down your proposal for the essence of V to a single concept. Otherwise, you could end a discussion of honesty with a list of familiar concepts -- truth-telling, courage, being considerate, being wise, being cautious, etc. -- many of which could possibly be the names of some other virtues a person or an action needs to be admirable, and which would each then need its own discussion.