Protagoras is a prominent sophist who promises to teach virtue but who famously says that "a man is the measure of all things, of those that are, of how they are, and of those that are not, of how they are not" and that "as things appear to each of us, so they are" (Theaetetus 152a).  An important question is whether Protagorean relativism is compatible with the sort of expertise to which Protagoras lays claim.  This is the first of three main topics in the Protagoras.

Topic #1: CAN VIRTUE BE TAUGHT?
The Protagoras both begins and ends on the subject of the teachability of virtue.  Socrates and Protagoras are represented as each changing his mind, but as disagreeing initially and subsequently.  Initially, Socrates appears skeptical whether virtue can be taught (319a-320c), but later argues that it can (361a-c).  Protagoras's situation is reversed; he begins by defending the teachability of virtue and ends by questioning whether it can be taught.  It's not surprising that Protagoras would begin by claiming that virtue is teachable, because teaching virtue is his professed vocation.  And it's not surprising if Socrates should doubt that Protagoras can claim to teach virtue.  For Protagorean relativism appears to undermine the sort of objectivity and expertise required if the sophists are to justify their claim to be especially good teachers of virtue.  What would be more surprising is if Socrates himself thought virtue could not be taught.  For his inquiries and his craft analogy suggest that that is precisely what he is searching for.  Of course, he might think that virtue is not in fact taught.  After all, teaching virtue requires moral knowledge, which requires an ability to specify the form of the virtues, and so far his attempts to find someone who can satisfy these conditions have been unsuccessful.  But that virtue is not yet taught, does not entail that it cannot be taught.

Socrates offers two grounds for skepticism about the teachability of virtue.

1. On moral and political matters the Athenians let everyone speak and do not appeal to experts (319b4-e1).
2. Those who are acknowledged to be good citizens produce offspring who are not virtuous (319e1-320b4).

1. All Athenians are experts or knowledgeable.
2. Justice (virtue) is a necessary condition of political community (323a1-4, 324d8-325a3).
3. Native affective and cognitive differences enable some to benefit more from training in virtue than others (327b8-c4).
4. The Athenians' belief that virtue can be taught is evidenced by their practices of punishment (323d6-324d1) and their concern with private training in virtue (325ac5-326e5).

TOPIC #2: THE UNITY OF THE VIRTUES
At 392b5 there is a fairly abrupt shift to the topic of the relationship among the virtues.  But this new topic is connected with the old one so long as the prospects for teaching virtue depend upon the nature of virtue, especially its cognitive aspects. Socrates asks Protagoras to choose between two conceptions of the relationship among the virtues (329c6-d1).

1. The various virtues stand to virtue as parts stand to the wholes of which they are parts.
2. The various virtue terms all name the same thing.

• (1a) The face analogy
• (1b) The lump of gold analogy

Socrates's "argument from opposites" is meant to establish that the virtues other than courage (wisdom, good sense, justice, and holiness or piety) are either identical or very similar (330a1-334a3, 349d3-5).

1. The opposite of wisdom is folly (332a4-6).
2. The opposite of good sense is folly (332e4-5).
3. Properties are distinct iff they have distinct opposites (332c3-8).
4. Hence, wisdom = good sense (333b4-5).
5. The opposite of justice is folly (333b7-334a2).
6. Hence, justice = wisdom = good sense.

Rightly or wrongly, Protagoras accepts this argument.  Yet he resists inseparability and unity by resisting this cognitive conception for the virtue of courage.  At Socrates's suggestion he offers a purely noncognitive definition of courage as daring (349e2).  At this point, Socrates relies on his argument, familiar from the Laches (192cd), that knowledge is necessary for virtue (349e1-350c5).  As in the Laches, Socrates takes this argument to establish a purely cognitive conception of courage (350c).  However, Protagoras does not let Socrates get away this move here.  He rightly responds that the argument establishes only that courage requires a cognitive component, not that courage just is wisdom (350c-351b).  Protagoras insists on a noncognitive component, as well as a cognitive component (351a1-b3), the sort of affective or conative component for which Protagorean moral training would be appropriate (325c-326e).  Here, Plato seems to register a shortcoming in the argument of the Laches.

TOPIC #3: KSV AND THE DENIAL OF AKRASIA
At 351b the discussion makes another abrupt transition to the topic of hedonism  and ultimately to a discussion of akrasia or weakness of will.  But, again, there is continuity of concern if the prospects for the unity of the virtues depends on the adequacy of a purely cognitive account of virtue, which itself depends upon the possibility of akrasia.  Socrates can defend a cognitive account of the virtues, unity, and the sufficiency of knowledge for virtue (KSV) by denying the possibility of akrasia.

• Akrasia occurs when A knows (or believes) that some action y is all things considered (and not just morally) better than some alternative action x, it is in A's power and he has the opportunity to perform either x or y, and he does x.

Hedonism is the thesis that pleasure is the good and pain evil.  Protagoras and the many initially deny hedonism (351c), asserting that there are bad pleasures and good pains.  Hedonism is Socrates's thesis (351c4-5, e5, e10).  But Socrates claims that their beliefs commit them to hedonism (354c), inasmuch as bad pleasures are short-term pleasures that cause more pain or less pleasure overall and good pains are short-term pains that cause more pleasure or less pain overall.

How does hedonism support the Socxratic denial of akrasisa?  Here is one reconstruction of a central part of Socrates's argument.

1. Evaluative Hedonism: goodness = pleasantness (i.e. the general terms `goodness` and `pleasantness` express the same property).
2. If A intentionally does x rather than y, because he is overcome by the F in x, then A believes x is more F than y.
3. Sometimes A knows (believes) x is worse than y but still chooses x, because he is overcome by the pleasure in x (the many's claim).
4. Hence, sometimes A knows (believes) x is worse than y but still chooses x, because he believes that x is more pleasant than y.  (By 2 and 3)
5. Hence, sometimes A knows (believes) that x is worse than y but still chooses x, because he believes that x is better than y.  (By 1 and 4)

SO WHAT?
It is not uncommon to read the argument as offering a reductio ad absurdum of the many's hypothesis that akrasia is possible.  But this can't be quite right, because the conclusion takes the forms of contradictory beliefs, rather than an outright contradiction.  The conclusion is

• B(P) & B(not-P)

•  B(P) & not-B(P)

OH YEAH?
Is the argument sound?  One issue concerns its validity.  In many contexts, co-referential terms can be substituted for each other without changing truth values.  Would the truth of hedonism make the substitution in (5) legitimate, or does the intentional context mean that the substitution of co-referential terms is illicit?  Does it matter that hedonism is a property identity claim and not just the claim that goodness and pleasantness are coextensive?  Compare

(a) Spike drinks the whiskey in the glass because it is buzz conferring.
(b) Spike drinks the whiskey in the glass because it rots his gut.

(c) Spike likes a hot bath.
(d) Spike likes a bath with mean kinetic molecular energy x.

(e) Spike believes his bath is hot.
(f) Spike believes his bath has mean kinetic molecular energy x.

Other questions concern the plausibility of the premises.  One obvious question concerns hedonism.  Does Socrates really endorse hedonism, or is the hedonist denial of akrasia merely ad hominem or ironical?  Here are some reasons to think that Socratic hedonism is authentic, not ad hominem or ironic.

• Hedonism is introduced as Socrates's thesis, not an explicit commitment of his interlocutors or conventional Greek ethics.
• Socrates needs hedonism if he is to have an argument for his otherwise puzzling theses about the unity of the virtues, the cognitive account of virtues, and the sufficiency of knowledge for virtue..
• Hedonism promises to satisfy Socratic constraints on definition from the Euthyphro; it provides a decision procedure that eliminates disputed terms in the definition of the virtues and identifies the property that makes something good.

• completely good-independent desires
• good-dependent but non-optimizing desires

TEMPORAL BIAS
The argument we have been considering does attribute an error to putatively akratic agents, but it does not need to invoke Socrates's actual diagnosis of the error, viz. a miscalculation of the pleasurable and painful consequences of an agent's options due to the temporal proximity of pleasures and pains (356a-357e).  Socrates suggests that the putative akrates suffers from a temporal bias: the proximity of certain pleasures and pains leads the agent to an inflated estimate of their value.  On a diachronic interpretation, the putative akrates toggles back and forth between a "cool" and a "hot" judgment; however, at the time of action, he does not act contrary to his optimizing desires.  On a synchronic interpretation, the agent holds both hot and cold judgments simultaneously.

Note that this argument is compatible with hedonism, but does not seem to presuppose it; temporal proximity might distort our judgment of the magnitude of goods, whether the goods are pleasure or something else.

AKRASIA AND UNITY
Socrates takes the denial of akrasia to support the sufficiency of knowledge for virtue and to undermine Protagoras's reasons for resisting the inseparability and unity of the virtues (358d5-).  While the Protagorean conception of courage, which recognizes independent affective or conative components, would allow for akrasia, Socrates appeals to the denial of akrasia to reject this conception and thereby support unity.

1. Courage is good (indeed, best).
2. Cowardice is being overcome by pleasure.
3. Hedonism.
4. If A intentionally does x rather than y, because she is overcome by the F in x, then A believes x is more F than y.
5. Sometimes A knows that y is courageous but chooses some action x that she recognizes to be cowardly.
6. Hence, sometimes A knows that y is most pleasant, yet believes that x is more pleasant than y.
7. Hence, (5) is false.
8. Hence, knowledge is sufficient for courage.
9. Hence, courage = knowledge of good and evil (i.e. knowledge of pleasure and pain).
10. Hence (with 330a1-334a3, 349d3-5, 359a8-b1), the virtues are identical (i.e. virtue = knowledge of pleasure and pain).

• Hedonism plays a role in the denial of akrasia and, hence, in the defense of a cognitive conception of courage, and, hence in a cognitive conception of the virtues, and, hence, in the unity (and inseparability) of the virtues.
• Hedonism promises to satisfy Socratic constraints on definition from the Euthyphro.  It provides a decision procedure that eliminates disputed terms in the definition of the virtues.  The cognitive definition that virtue = knowledge of good and evil, discussed in both the Laches and Charmides, though perhaps true, would apparently not satisfy the decision procedure requirement.  Hedonism apparently does.  Moreover, hedonism might appear to provide a definition of virtue that supplies measuring procedures, as the Euthyphro requires, because it looks like we could, at least in principle, measure quantities of pleasure by measuring their duration and intensity.  Finally, hedonism, understood as the property identity claim that the good = pleasure appears to satisfy the explanatory requirement, for it purports to identify the property that makes something good.
• Hedonism promises to satisfy the demand that the craft analogy appears to make that virtue aim at some good or goal of virtue that can be specified independently of virtuous acharacter or action, namely, pleasure.  Virtuous characters or actions are those that reliably produce (maximize) net pleasure.

• Precisely because hedonism satisfies this aspect of the craft analogy so well, it appears to support the claim that virtue is valuable as an instrumental means to a form of happiness that is distinct from virtue.  Both self-regarding and other-regarding virtues are instrumental means to producing pleasure.  But, as we said before, it's hard to square this claim about the instrumental value of virtue with the strong claims that virtue is both necessary and sufficient for happiness.  Instrumental means are usually neither unique nor infallible ways of achieving their goals.  The spoon trick is an instrumental means to stopping the hiccups, but it is not an infallible remedy and there are other remedies that work.  But then the spoon trick is neither necessary nor sufficient to stop the hiccups.  Socrates would have to claim that virtue is a highly unusual instrumental means to happiness by claiming that it is a unique and infallible means for maximizing net pleasure.  This is hard to believe, especially of the other-regarding virtues, such as justice.
• Precisely because hedonism satisfies the craft analogy so well, it seems to threaten the obsolescence of Socratic method (dialectic or the elenchus).  Dialectic can be necessary to identify pleasure as the conception of happiness at which virtue aims.  But once this is established, the practice and teaching of virtue should just consist in working out the science of pleasure and pain.  But it's not clear that Socratic method is an especially useful way of discovering this empirical information, just as it would not appear to be an especially useful way of practicing or teaching medicine or engineering.  Perhaps this conclusion is otherwise acceptable, but it is hard to with the essential role Socrates seems to assign the elenchus in the best of all possible lives, viz. the examined life (Ap 38a).  Doesn't socrates assume that dialectic has an important ongoing role in a happy life?