Surprisingly little is known of the virtues that Sophists taught in private when they took students under their wing for long periods (a pedagogical method referred to as "association," suneimi). Certainly skill in legal and political argument was a large part of these virtues, since the Sophists were all talented statesmen, and their students were aspiring members of the political class. Evidence suggests that the major Sophists imparted a fairly wide set of moral and intellectual virtues. Not only respect for others and justice, but euboulia, or sound political judgment, was evidently part of Protagoras' program; while Hippias taught students advanced arithmetic, astronomy, geometry, and music before arriving at anything immediately political.
An exception to this rule (besides Gorgias and other rhetoricians who eschewed the teaching of virtue) is the nefarious pair of Sophists, Euthydemus and Dionysodorus, who in Plato's dialogue Euthydemus present their crash course in "eristic" as a complete schooling in virtue. Eristic, which derives from the Greek word for strife, is a form of rapid verbal combat in which opponents are drawn into well-rehearsed philosophical paradoxes in order to be refuted or confounded. It is doubtful whether these Sophists' spurious arguments were ever taken seriously as "virtue," but they were at least philosophically interesting enough to attract the attention of Aristotle (384–322 B.C.E.), whose On Sophistical Refutations responds to them directly.