THEMES

1. Rhetoric and virtue as rivals to be the superordinate craft.  Socrates argues, largely with Gorgias and Polus, that whereas rhetoric aims to flatter and please, virtue aims to benefit (462b-465d, 502e-503d).
2. The probative value of the elenchus.  If the conclusions of the elenchus depend upon the interlocutor's particular opinions and concessions, what reason is there to think that the conclusions are true?  Perhaps Socrates's interlocutors hold idiosyncratic opinions or Socrates exploits conventional views and prejudices to shame his interlocutors into concessions they shouldn't make.  What if Socrates were to meet up with a less compliant interlocutor?
3. The eudaimonist justification of justice.  The Gorgias faces overdue issues about the relationship between virtue and happiness.  In particular, it recognizes and tackles the problem of how other-regarding virtues, such as justice, can satisfy the eudaimonist constraint that virtues must benefit their possessors.

THE PROBATIVE VALUE OF DIALECTIC
What is the probative value of an interlocutor's agreement in the elenchus?  Perhaps it can't establish the objective truth of conclusions that survive particular elenctic inquiries but can only establish the consistency of a particular interlocutor's views and provide ad hominem arguments against particular assumptions of the interlocutor.  If Socratic claims do survive their criticisms, we must ask if the discussion which they survive has been fair.  When Gorgias finds himself in inconsistency, Polus interrupts and alleges that Socrates has only shamed Gorgias into contradicting himself (461bc).  And when Polus finds himself defeated, Callicles alleges that Polus has suffered the same fate as Polus alleged Gorgias suffered, viz. he has been shamed and trapped into claims he didn't believe or shouldn't have conceded (482cd).

It is not wrong in principle for Socrates to appeal to his interlocutor's sense of shame.  For attitudes of shame reflect moral or evaluative beliefs, and it is part of the elenchus to require consistency among the interlocutor's beliefs.  But an interlocutor may have accepted a belief, especially a common belief, uncritically and for the wrong reasons.  In claiming that Socrates has shamed his interlocutors into accepting claims they should have rejected, the critics may be claiming that his interlocutors maintained the wrong beliefs for the wrong reasons.  If these charges are true, Socrates's triumphs over Gorgias and Polus may seem to have little or no probative value.  This explains the importance of Callicles.  Callicles promises to be a radical critic who won't be shamed or bullied into changing or contradicting his position.  If Socrates can show that even a radical critic such as Callicles must acknowledge his claims about virtue and happiness, then Socrates's claim that elenchus aims at and can hope to produce knowledge (453b1-5) begins to look more plausible and his claims about virtue and happiness will look more defensible.

This suggests some provisional claims about the ambitions of elenctic inquiry.  We know that the immediate goal of an elenctic inquiry is the examination of the sincere beliefs of the interlocutor.  Elenctic agreement with interlocutor A is the immediate aim and is important (472b6-c3, 474a5-b3, 476a2-3).  But the importance of A's agreement depends upon how systematic the inquiry was, how compliant A was, and the reasons for A's compliance (497a1-2).  The elenchus should proceed only on the basis of claims the interlocutor sincerely accepts (495a7-9, 516c8-9); he is free to challenge or withdraw any claim that he likes (461d3-4, 462a3-5, 465a7-8, 504c5-6).  In general, a thesis is more plausible if it survives elenctic inquiry with suitably different interlocutors (e.g. with B, C, D) (508e8-509a4, 513d1-2, 527b3-4).  If all the interlocutors are compliant on exactly the same problematic point, then additional elenctic inquiries would not make the thesis in question more plausible.  (cf. Wittgenstein: "As if someone were to buy several copies of the morning paper to assure himself that what it said [was] true." Investigations sect. 265)   However, a successful elenchus with a radical critic, who is not compliant on this point, will have important evidential value (486e5-6, 487e1-3, 489a5-7).  We may wonder whether the success of the thesis even here depends upon the peculiarities of the particular radical critic whose agreement is secured.  We could avoid this worry if the argument that we use to convince a particular radical critic appeals only to claims that any (practically) rational agent must accept.  This is one interpretation of Socrates's argument against Callicles (503c-504d) (cf. Irwin. "Objectivity and Coercion in Plato's Dialectic").

POLUS
Socrates asserts that justice is necessary for happiness and that it is better to suffer injustice than to practice it (469a10-c3).  Polus denies that what is fine benefits the agent; the conventional virtues are admirable but not good for the agent (474c4-d2).  Socrates replies.

1. If something is fine or admirable, it is (most) choiceworthy.
2. Insofar as something is choiceworthy it must be to that extent beneficial or pleasant (474d4-475b4).
3. Hence, if something is fine, it must be most beneficial or most pleasant (4745b1-3).
4. Justice is fine.
5. Hence, justice is either most beneficial or most pleasant (475b8-10).
6. Justice is not (always) most pleasant (475c1-4).
7. Hence, justice is most beneficial (for the agent) (475c5-10).

CALLICLES
Callicles claims that Socrates has only shamed Polus into conceding something he doesn't believe or shouldn't have conceded (483cd) and promises to be a more formidable opponent.  However, unlike Polus, Callicles seems to accept eudaimonism.  But because he thinks that conventional virtues do not always benefit the agent, he denies that conventional virtues are genuine virtues.  His position seems to be this.

1. Justice is a virtue.
2. Virtues benefit their possessor.
3. Conventional justice often requires the agent to benefit others at her own expense.
4. Hence, conventional justice is not real justice.
5. Hence, genuine or natural justice implies that he who is naturally (individually [483b, 484a, 488d5-490a]) stronger should take what he wishes from others by force, if necessary (e.g. 488b).

CALLICLES'S EXPANSIVE CONCEPTION OF HAPPINESS
Callicles defends (3) and (5) by appeal to what we might call an expansive conception of happiness, according to which the happy person has and satisfies large and unrestrained desires (491e5-492a3, 492d6-e1).  We might distinguish three possible interpretations of Callicles's expansive conception of happiness.

1. perfectionism: expansive happiness is good, because it maximizes the agent's power and achievement (cf. Nietzsche).
2. desire-satisfaction: expansive happiness is good, because it maximizes the satisfaction of the agent's desires.
3. hedonism: expansive happiness is good, because it maximizes the agent's pleasure.

EXPANSIVE vs. ADAPTIVE CONCEPTIONS OF HAPPINESS
In opposition to Callicles's conception of happiness, Socrates argues, in part, for a more adaptive conception of happiness, according to which an agent ought to restrict and adapt her desires to fit her circumstances (439cd).

Socrates begins his attack on Callicles with the "leaky jar" argument (493b1-494d10).  He assumes that Callicles will admit that the best thing is to have a full jar.  But Callicles denies this; he thinks that this would leave one no more happy than a stone (494a8-10).   Callicles attaches value to the process of filling the jar (494c7-d9).  Thus, he should be unimpressed by the psychological profile that Socrates associates with the expansive conception -- the torrent bird (494b6-7) or the endless scratcher (494c7-d8) --  provided that the agent has access to the sort of assets and resources required to pursue and satisfy such desires.

However, Socrates can claim that if Callicles attaches value to producing pleasure and satisfying desire he must also attach value to producing anxiety and other forms of pain and frustrating desire.  Because these assets and resources are vulnerable -- they are subject to fortune -- the enlightened hedonist  -- the one concerned to maximize net happiness -- should adapt her desires to her circumstances.  This adaptive argument makes several crucial assumptions.

1. We should aim to maximize net happiness.
2. Expansive happiness depends on vulnerable external goods (assets, resources).  Achieving and maintaining expansive happiness is hard to do.  If we aim at expansive happiness and the requisite externals are lost or not forthcoming, then expansive desires go unsatisfied, which is unpleasant.  Even when the agent possesses the requisite externals, their vulnerability produces uncertainty and anxiety and requires costly protective measures.
3. We are psychologically plastic and can adapt our desires so that they depend on fewer externals.
4. Hence, we should eschew expansive happiness and aim at adaptive happiness.

• We should distinguish between the goals of undermining Callicles's expansive conception of happiness and establishing Socrates's adaptive conception of happiness.  Socrates could succeed at the first goal without succeeding at the second.  This is a serious possibility, because Socrates's adaptive argument must apparently establish complete indifference to externals if it is to show that justice is sufficient for happiness.  There might be a good prudential argument against expansive happiness that doesn't endorse complete indifference to externals.  
• Does complete indifference to externals presuppose more psychological plasticity than we possess?
• Even if one should not seek externals, mightn't they be genuine goods, which would augment the happiness of the person who acquired them?
• Doesn't enlightened hedonism counsel the cultivation of desires that require moderate amounts of externals?  Socrates assumes that expansive desires depend upon vulnerable externals.  But in calling something vulnerable we might mean that (a) it can be lost or that (b) it is likely or liable to be lost.  All externals are (a)-vulnerable, but only some are (b)-vulnerable.  If we're seeking to maximize net happiness, shouldn't we ignore (a)-vulnerability and concentrate on (b)-vulnerability?

RATIONAL AND NONRATIONAL DESIRE
One interesting feature of this account of virtue as psychic order (503e5-505c5, 506d5-507e3) is that it seems to suggest that Socrates thinks of the virtues as including noncognitive components (e.g. endurance) as well as cognitive components and that he allows for the existence of non-rational or good-independent desires.  This looks like a departure from the psychology of the Protagoras and may, as a result, threaten (a) the sufficiency of knowledge for virtue and (b) the inseparability and unity of the virtues.

THE GORGIAS'S ATTITUDE TOWARD HEDONISM
It's clear that the Gorgias contains a number of anti-hedonist claims and so appears to reject the hedonism of the Protagoras.  What's less clear is whether the arguments for these claims are successful arguments against the enlightened hedonism of the Protagoras, which identifies an agent's good with the life that is most pleasant overall (cf. Gosling and Taylor, The Greeks on Pleasure, ch. 4).
 

1. In rejecting Calliclean assumptions about happiness, Socrates claims to be rejecting hedonism (495a-d).  But we have seen that his only plausible argument -- the adaptive argument -- arguably appeals to the enlightened hedonist claim that we should maximize expected net pleasure overall.
2. In the Protagoras Socrates denies that Protagoras's recognition of good pains and bad pleasures is inconsistent with hedonism (353de), whereas in the Gorgias it is Socrates who insists, against Callicles's hedonism, that some pleasures are bad and some pains are good (494e5-495a3, 499c-e).  This anti-hedonist claim in the Gorgias seems to overlook the resources of the hedonism in the Protagoras.
3. The Gorgias contrasts the aims of virtue and rhetoric; whereas virtue aims to benefit, rhetoric aims to flatter and, hence, to please (462b-465d, 502e-503d).  But the difference between flattery and benefit could easily be explained as the difference between immediate gratification and long-term pleasure.
4. According to the hedonism of the Protagoras, good things are good insofar as they are pleasant (354 ad), whereas in the Gorgias Socrates asserts that pleasures are to be pursued insofar as they are good, not the other way around (500a, 506c-d).  This second claim certainly is incompatible with hedonism.  But why should we accept it?
5. Here is one argument Socrates makes in the Gorgias (495d-497d).
1. We can experience pleasure and pain at the same time.
2. We cannot experience good and evil at the same time.
3. Hence, pleasure is not the same as the good, and pain is not the same as evil.
Here is another argument Socrates makes (497e1 499b4).
1. Virtues are more beneficial than vices are.
2. Wisdom and courage are virtues.
3. Cowards and fools are often as pleased as the courageous and wise.
4. Hence, the good does not consist in pleasure.
The second argument is better.  But even here one might wonder if the enlightened hedonist would agree with (3).  Cowardice lacks deterrent value and might expose cowards to future predation.  If so, cowards might experience more fear, anxiety, and pain in the long-run.  But what about courage that involves wholesale sacrifice -- laying down one's life?  It's hard to see how such sacrifice assures the courageous person of more pleasure overall.  Perhaps the coward's pleasure will be claimed to be false or not worth wanting.  But this would make a pleasure's value depend upon something other than the amount of pleasure at stake.  This would vindicate Socrates's anti-hedonistic claim that pleasures are to be pursued insofar as they are good, instead of insofar as they are pleasurable.  To maintain hedonism, Socrates would apparently have to deny that courage can require wholesale sacrifice.  But that seems to be something he is not willing to say.