Socrates versus Thrasymachus
Socrates deploys four arguments against Thrasymachus :
There are four main arguments that he advances against Thrasymachus, directed against four of Thrasymachus' contentions: his
definition of justice as the advantage of the stronger (338a-347a),
his claim that the unjust man is better and more intelligent than
the just man (348b-350d), his claim that the unjust man is
stronger (350d-352d), and his claim that the unjust man is happier (352d-354a - although this comprehends the two preceding proofs as well. ( Kenneth Dorter, 'Socrates' Refutation of Thrasymachus and Treatment of Virtue', Philosophy & Rhetoric, Vol. 7, No. 1 (Winter, 1974), pp. 25-46 : 29.
Your interest centres on the fourth argument : Socrates' denial that the unjust man is happier (352d-354a).
Thrasymachus' claim that the unjust man is happier
'Happier' is not really the correct word. The question raised by Socrates against Thrasymachus is whether the just live better (ameinon) and are more fortunate or, better, flourishing (eudaimonesteroi) than the unjust (Rep. I. 352 d 1-2). Eudaimonia from which eudaimonesteroi derives does not mean happiness : its sense is closer to that of the true and absolute human good or, as I have expressed it, flourishing (as a healthy plant flourishes).
Nor is 'justice' the right word to render Plato's dikaiosune. The latter has rather the sense of 'righteousness or in Vlastos's phrase, 'all social
conduct that is morally right' (Gregory Vlastos, 'The Argument in the Republic that "Justice Pays"', The Journal of Philosophy, Vol. 65, No. 21, Sixty-Fifth Annual Meeting of the American Philosophical Association Eastern Division (Nov. 7, 1968), pp. 665-674 : 665.)
Socrates argument goes as follows :
The function of a thing is to do that which nothing else could accomplish equally well, I can see only with my eye; so seeing is the function [ergon] of the eye. The soul [psuche], too, has a function : to superintend, command, deliberate. Living, too, is its function. Since a thing which has a function has an excellence [arete] pecular to itself- the excellence of the eye is to see clearly and well - and since justice is the excellence of the soul, the soul will perform its function well only in accordance with justice. The just sou, then, will live well just as the eye, when it posseses its proper excellence, sees well.
The assumptions on which this argument rests are, to say the least, questionable. Is it perhaps proper to ascribe a function to a thing only if it was intentionally made to do a particular job ? Can the excellence of the soul kn performing its function (if it has one) be unquestionably identified as the specific moral virtue, justice ? Socrates acknowledges that the case against Thrasymachus has not been proved. We have not discovered, he says in the last words of the book [354b], what justice is; until we know both what it is and whether or not it is a virtue, we cannot decide whether the just man is happy [eudaimon] or not. (R.M. Hare & D.A. Russell, 'Introduction to Republic', The Dialogues of Plato, 4, London : Sphere Books, 1970, 40.)
Terence Irwin, Plato's Ethics, SBN 10: 0195086457 / ISBN 13: 9780195086454
Published by Oxford University Press, USA, 1995.
Parry, Richard D., Plato's Craft of Justice (S U N Y Series in Ancient Greek Philosophy), ISBN 10: 0791427323 / ISBN 13: 9780791427323
Published by State University of New York Press, 1995.
Julia Annas, An Introduction To Plato's Republic, ISBN 10: 0198274297 / ISBN 13: 9780198274292
Published by Oxford University Press, U.S.A. 1998-11-12, 1998.