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Socrates returns the discussion to where it had been at 334c, before the near- breakdown of dialogue. Protagoras, as Socrates reminds both him and us, had argued that the five aspects of virtue—wisdom, temperance, courage, justice and holiness—are not synonyms. Rather, they are differing, component parts of virtue. Protagoras now partly retreats from this position, probably because he feels that Socrates has exposed weaknesses in his argument. Now he argues that only courage is truly different from the other aspects of virtue. Socrates, still trying to establish that virtue is singular and indivisible, sets out to prove that courage and wisdom are merely different names for the same quality.
His first line of reasoning involves drawing a distinction between courage and boldness. Protagoras has argued that some courageous people are not wise; however, if Socrates can succeed in classifying these people as bold but not as courageous, then the way is open to show that courage and wisdom are, indeed, identical. Being courageous involves being bold, but one can be bold through madness, which, Socrates states, is clearly not a good quality and not part of virtue. Rather, courage is a function of possessing certain types of knowledge. To discredit Socrates's conclusions, Protagoras presents a parallel argument that seeks to prove the identical nature of strength and wisdom.
Socrates responds by introducing what appears to be a new subject: his argument will return to the identity of courage with the other types of virtue, but only after a long digression on the relationship between pleasure, knowledge, and the good. Pleasure, argues Socrates, is identical with the good; anything that is painful is evil. This entails that it is impossible to live pleasurably while committing evil actions. Socrates acknowledges that this is counter-intuitive; most people believe that it is only too easy to be swayed by pleasure into behaving badly. But Socrates argues that many people are unable properly to measure what will bring them most pleasure, and instead often forgo future pleasures for immediate, but less pleasurable, gratification. Wrong actions are caused by ignorance, not by a conflict between ethical duty and the desire for pleasure. What is needed is an art or science (techne) of measuring and comparing pleasures.
Socrates, having gained Protagoras's assent to this theory of virtue, returns to the question of courage. Confronting something that one believes to be dreadful is to commit an error, for it is to choose something that will probably cause pain. Courage, argues Socrates, involves a correct evaluation of the danger that is faced. Those who are bold but not courageous have confused things that will bring them pain with things that will bring them pleasure. Socrates concludes that "ignorance of what is dreadful and not dreadful" is cowardice (360c). The opposite of cowardice is courage; courage is therefore wisdom (sophia). Protagoras sullenly agrees.
Socrates then summarizes the whole dialogue. Protagoras had begun by arguing that virtue could be taught, but had ended by arguing that some forms of virtue—courage, for instance—were not techniques or forms of knowledge, and therefore could not be taught. Socrates had begun from the opposite position, but, in trying to prove that virtue was a single, unified thing, had ended by arguing that all forms of virtue are forms of knowledge. Virtue must therefore be teachable. But, states Socrates, all that has been accomplished so far has been a clarification of the ground on which an investigation of the question virtue could proceed. The arguments of the dialogue need to be reconsidered, and the unsolved questions need to be reposed.
It is possible to read this dialogue as ending with Socrates's failure: he has not succeeded in imbuing anyone with virtue. Indeed, we are still not really sure what virtue is. The discussion ends in misunderstandings and missed appointments. Socrates slights Protagoras, claiming that he stayed to argue with him only out of respect for Callias, not out of respect for Protagoras himself. The failure of this dialogue is perhaps an oblique and grim reference to Socrates's own judicial murder due to the misapprehension of what he taught. But this pessimistic conclusion is only half the story. Virtue is teachable, Socrates concludes; how we can learn to be virtuous, however, is not explained explicitly within this dialogue. Instead, that lesson comprises the dialogue form itself. While the arguments fail to reach a firm conclusion, the method of argument can help guide us in reaching our own conclusions. The ambiguities of the dialogue are not resolved in this final section, but must shape our interpretation of the dialogue as a whole. In our attempts to straddle these ambiguities, the dialogue goads us into thought. The dialogue's closure provides us with an opening for further dialogue.
This final section lays out, in some detail, one of the most important points of Socratic doctrine: the notion that the good is knowledge, and that therefore it is impossible knowingly to commit an immoral action. But in forming this argument, Socrates assumes that all different goods—health, wealth, religious devotion, and so on—can be reduced to a common scale upon which their competing claims can be judged. Such a philosophical position was at odds with Greek culture of the period. Classical tragedy—the great plays of Sophocles, for example—is most frequently understood as dramatizing the incompatibility of different ethical claims. Antigone, for instance, must decide between her duties to her family and her allegiance to her country.
If such demands could easily be compared and the greater good determined, the tragic quality of this tragedy would evaporate. The play would then comprise a simple moral lesson on how to adjudicate between apparently conflicting moral claims. Indeed, if Socrates's argument here is accepted wholly, the very idea of conflicting moral claims is an absurdity. All ethical questions could be decided simply, because any action has only one proper end: whether it brings pleasure, and how much pleasure. All that is needed to decide how to act, in any given situation, is to measure the alternatives and perform some simple addition.
But if behaving virtuously were this easy, why would it be necessary to study under a Sophist like Protagoras in order to become a virtuous person? As Socrates notes, most people assume that virtue cannot be taught, because they do not consider virtue to be a form of knowledge. Protagoras had begun by arguing that he could teach the civic virtues, but Socrates has argued—quite successfully—that Protagoras does not even know what the form of virtue really is. He does not know whether there are many virtues, or whether these are all merely different manifestations of a singular virtue.
In a sense, the question Socrates posed to Protagoras at the very start of the dialogue has been answered: virtue is teachable. But in arriving at this position, Protagoras himself has been investigated, and we must conclude that while virtue can be taught, Protagoras is not the man to teach it. But virtue is not something that can be taught simply. If virtue is a kind of knowledge, to know how it can be taught we must know what form that knowledge takes. As so often in this dialogue, the reader is forced by problematic aspects of the argument to consider questions of form. The lack of conclusion at the end of this dialogue compels the reader to ask just how he or she has been learning by reading the dialogue.
The form by which we achieve this knowledge of virtue is, of course, the dialectic, that form of thinking that unravels the logic of ignorance to arrive at fundamental questions. To approach the truth, Socrates needs an interlocutor who holds false opinions; the way to truth begins with falsity. If Protagoras does not provide us with any firm truth, this does not signify a failure, but rather a need to continue the process of thought that Socrates begins. The reader is summoned to a personal engagement with the text: there are questions left unanswered; there are characters whose responses are never given. These absences require that Protagoras be read actively. The process of reading will track and extend Socrates's dialectic method of thought. As Socrates states in conclusion, "I should like to work our way through the whole matter until at last we reach what virtue is, and then go back and consider whether it is teachable or not" (361c). Everything still remains to be done over again. It is the reader's task to engage in this philosophical labor.