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Socratic philosophy is based upon Socrates' question, "What is X?" He felt that in order to teach a concept, one needed to know what it was, first. Seemingly, this would appear a logical assumption, but Socratic philosophy has several layers toward understanding the heart of what he believed. So, a Socratic definition answered the question, "What is X?"
First, a quality needed to be defined—not in terms of words, but with regard to the nature of that quality. In this case, it was not enough to define "justice" with words: Socrates wanted it defined in terms of the nature of justice. The definition needed to be objective, not based on opinion or "popularity."
He believed that there were objective moral standards; that they could be discovered; that there were right and wrong answers to moral questions that went beyond mere opinion and popular sentiment.
Socratic definitions "are fundamental for knowledge." Socrates felt that in order to teach someone a concept, that person first needed to know what it meant, and to have the "correct" answer.
Socratic definitions and knowledge are fundamental for morality. To practice moral behavior, one needed to have the correct knowledge of what morality is. Socrates believed that to have virtue was to have knowledge—based on the Socratic definition.
Virtue is knowledge: if you know what is right, you will do what is right. Knowing a Socratic definition is thus (apparently) necessary and sufficient for moral behavior.
Moral qualities deal with the essence of something—understanding its substance. E.g., popularity is based upon the number of people that like something—it is not a virtue and it's not knowledge if based on opinion.
Sung-Hoon Kang reviews Roslyn Weiss' book, The Socratic Paradox and Its Enemies. Weiss explains that Socrates believed that virtue was knowledge, and all of its parts are virtue (courage, justice, etc.) but are part of a whole. People who are virtuous choose "wisely" to live a virtuous life, based on their knowledge of what virtue is.
Socrates does not intellectualize virtue; what he means by the claim that virtue is knowledge is only that virtuous people choose virtue wisely. Socrates does not believe that the parts of virtue such as justice, temperance, courage, etc. are manifestations of a single moral craft...but that these virtues are all required for a life rightly lived.
Weiss maintains that a man knows the difference between right and wrong, and for some reason chooses to follow the wrong path believing that "injustice is to their advantage," whereas Socrates maintains that wrongdoers should fight the impulse to do what is wrong.
Socrates holds some core moral beliefs, such as that justice is an essential element of a good life...
Socrates is not opposed to justice as a virtue—he believes it is simply part of a whole of the virtue of knowledge. Plato's Republic records that Socrates debated the essence of justice many times with others, to define the true nature of "justice."
I believe that it was not justice that was in question as a virtue, but that it did not stand lone, but was a part of the greater whole: knowledge.
Hope this is of some help; this is my understanding from what I have read.