Virtue is knowledge, and therefore virtue can be taught.
To the doctrine that virtue is knowledge, Plato has been constantly tending in the previous Dialogues.
He would not have preferred the poet or man of action to the philosopher, or the virtue of custom to the virtue based upon ideas.
Characteristic also of the temper of the Socratic enquiry is, (4) the proposal to discuss the teachableness of virtue under an hypothesis, after the manner of the mathematicians; and (5) the repetition of the favourite doctrine which occurs so frequently in the earlier and more Socratic Dialogues, and gives a colour to all of them--that mankind only desire evil through ignorance; (6) the experiment of eliciting from the slave-boy the mathematical truth which is latent in him, and (7) the remark that he is all the better for knowing his ignorance.
His definition of virtue as 'the power and desire of attaining things honourable,' like the first definition of justice in the Republic, is taken from a poet.
In the Republic the relation of knowledge to virtue is described in a manner more consistent with modern distinctions.
For instance, the name given to the runner or boxer, who is so called in virtue of an inborn capacity, is not derived from that of any quality; for lob those capacities have no name assigned to them.
The things, also, which are said to be such and such in virtue of these qualities, may be contrary the one to the other; for that which is unjust is contrary to that which is just, that which is white to that which is black.
However that may be, it is an incontrovertible fact that the things which in virtue of these qualities are said to be what they are vary in the degree in which they possess them; for one man is said to be better versed in grammar, or more healthy or just, than another, and so on.
One thing is like another only with reference to that in virtue of which it is such and such; thus this forms the peculiar mark of quality.
The knowledge of grammar is not relative to anything external, nor is the knowledge of music, but these, if relative at all, are relative only in virtue of their genera; thus grammar is said be the knowledge of something, not the grammar of something; similarly music is the knowledge of something, not the music of something.
Those particular branches, therefore, of knowledge, in virtue of which we are sometimes said to be such and such, are themselves qualities, and are not relative.