Is Plato’s 'Republic' employing the Genealogy method for its examination of the value of justice?

Taking my definition of “genealogy” (and of “naturalism”) from Bernard Williams’s exploration of the virtues of the genealogy method as a naturalistic method in philosophy (in his Truth and Truthfulness), I argue that Plato’s argument in defence of justice, in the Republic, can be read as a vindicatory genealogy of the virtue of justice. I consider whether Plato’s method, and his answers, count as naturalistic and whether Plato’s proof of the intrinsic value of justice avoids the problematic reductionism to which genealogical accounts are prone. I consider Plato’s attempt to use the genealogy method to answer the “what is justice?” question as well as the question concerning whether justice pays, and whether he has actually found a naturalistic answer to the “what is it?” question that can satisfy the requirements of his argument.