Socrates and Thrasymachus

⊕   "Come then, Thrasymachus, go back to the beginning and answer us. You affirm that perfect and complete injustice is more profitable than justice that is complete.
  I affirm it and have told you my reasons.
  Tell me then how you would express yourself about them. You call one of them, I presume, a virtue and the other a vice" (Republic I.348b).

  "[W]riters of prose speak wrongly about men in matters of greatest moment, saying that there are many examples of men who, though unjust, are happy, and of just men who are wretched, and that there is profit in injustice if it be concealed, and that justice is the other man's good and your own loss. I presume we shall forbid them to say this sort of thing and command them to sing the opposite. Don't you think so?
  I know so, Socrates. (Republic III.392b). Socrates shows Thrasymachus that injustice is never more profitable than justice.

Argument

1.  Justice is the virtue of the soul.
2.  If (1) is true, then those who are happy are all and only those whose souls are just.
3.  If this consequent is true, injustice is never more profitable than justice.
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4.  Injustice is never more profitable than justice.

Explanation

1. ⊕ "Thrasymachus made all these admissions, not as I now narrate them [Socrates is the narrator in the Republic], but with much baulking and reluctance and prodigious sweating, it being summer, and it was then I beheld what I had never seen before—Thrasymachus blushing. But when we did reach our conclusion that justice is virtue (δικαιοσύνην ἀρετὴν εἶναι) and wisdom and injustice vice and ignorance, I said: 'Good, let this be taken as established'" (Republic I.350c).

  "The soul, has it a function (ἔργον) which you couldn't accomplish with anything else in the world, as for example, management, rule, deliberation, and the like, is there anything else than soul to which you could rightly assign these and say that they were its peculiar work?
  Nothing else.
  And again life? Shall we say that too is the function of the soul?
  Most certainly.
  And do we not also say that there is a virtue (ἀρετήν) of the soul?
  We do.
  Will the soul ever accomplish its own work well if deprived of its own virtue, or is this impossible?
  It is impossible.
  Of necessity, then, a bad soul will govern and manage things badly while the good soul will in all these things do well.
  Of necessity.
  And did we not agree that the excellence or virtue of soul is justice (δικαιοσύνην) and its defect injustice?
  Yes, we did" (Republic I.353d). Things with functions have virtues. Having its virtue is how the thing performs its function well. Part of the virtue of a knife, for example, is having a blade suitable for cutting.

Socrates says that the function of the soul is "managing things, rule, deliberation, and the like." A line later he adds that "life" is part of the function. On this picture, the soul is that in terms of which a human being lives and makes decisions about what to do. The virtue of the soul is the property whose possession allows the soul to carry out this function well.

2. If justice is the virtue of the soul, those whose souls are just function well. In their lives, they "manag[e] things, rule, deliberat[e], and the like" well. Those whose souls are unjust do not.

  "The just soul and the just man then will live well and the unjust ill?
  So it appears by your reasoning.
  But furthermore, he who lives well is blessed and happy, and he who does not the contrary.
  Of course.
  Then the just is happy and the unjust miserable.
  So be it, Socrates" (Republic I.353e).

3. If those who are happy are all and only those whose souls are just, it never profits someone (makes him better off) to have an unjust soul. This makes him κακοδαίμων or "unhappy."

  "But it surely does not pay to be miserable, but to be happy.
  Of course not, Socrates.
  Never, then, Thrasymachus, can injustice be more profitable than justice" (Republic I.354a). ⊕

Evaluation

The argument is an instance of a valid argument form. If the premises are true, the conclusion must be true. The question is whether the premises are true.

Pemises (1) and (2) seem the most questionable.

1. Even if we grant that the soul has a virtue, why is it justice?

As Plato had been portraying Socrates, he thought that wisdom is the virtue of the soul. So why now is Plato making Socrates say that justice, not wisdom, is this virtue?

We can see part of the answer in in the Gorgias.

  "And, Polus, in a soul too you believe there is a certain vice?
  Of course, Socrates.
  And do you not call this injustice, ignorance, cowardice, and so forth?
  Certainly I do.
  So now in property, body, and soul, these three, you have mentioned three vices—poverty, disease, and injustice?
  Yes.
  Then which of these vices is the foulest? Is it not injustice—in short, the vice of the soul?
  Far the foulest, Socrates" (Gorgias 477b).

⊕ Socrates had said that when we do what see fit, we act beneficially if and only if we act justly.

  "Polus, do we not admit that sometimes it is better to do those things that we were mentioning just now—to put people to death and banish them and deprive them of property—while sometimes it is not?
  To be sure, Socrates.
  Then here is a point, it seems, that is admitted both on your side and on mine.
  Yes.
  Then when do you say it is better to do these things? Tell me where you draw the line.
  No, I would rather that you answered that.
  Well then I say, Polus, if you prefer to hear it from me, that it is better when these things are done justly, and worse when they are done unjustly." (Gorgias 470b).

The same idea is in the Crito, where Crito is trying to persuade Socrates that he should escape.

  "Now see whether we still hold to this, or not, that it is not living, but living well which we ought to consider most important.
  We do hold to it.
  And that living well and living justly are the same thing, do we hold to that, or not?
  We do.
  Then we agree that the question is whether it is right for me to try to escape from here without the permission of the Athenians, or not right. And if it appears to be right, let us try it, and if not, let us give it up" (Crito 47e).

2. Why are those who are happy all and only those whose souls are just?

To know why Socrates thinks this is true, we need to know what he thinks goes on in a life in which the soul "manag[es] things, rule[s], deliberat[es]" well.

This, however, is not something Socrates tells us in Book I of the Republic.