After Euthyphro's definition of holiness as persecuting religious offenders has been dismissed by Socrates, Euthyphro posits a second definition: that holiness is what is agreeable to the gods. Socrates is quite pleased with the generality allowed by this definition, and is eager to investigate whether or not it is accurate. First, he points out that the gods themselves often quarrel, as is recounted in the tales of Hesiod and Homer that Euthyphro believes in so literally. Socrates points out that quarrels do not arise over questions of fact, since agreements can be reached through calculation or investigation, but over questions of value, such as what is just and what is good. Socrates points out that if the gods quarrel over what is just and what is good, then there is clearly no agreement among them on these questions. And if they have different opinions as to what is just and what is good, they must approve of different things. It follows that there must be certain things that are approved of by some gods and disapproved of by other gods. But according to Euthyphro's definition, that would mean that those things are both holy and unholy, since they are approved of by some gods and disapproved of by others. For instance, Zeus might approve of punishing one's father, while Kronos or Uranus might not.
Euthyphro replies that surely the gods all agree that a person who kills someone unjustly should be punished. Socrates points out that disputes do not arise as to whether an acknowledged wrongdoer should be punished, but as to whether or not the person has in fact acted unjustly. For Euthyphro's argument to have any weight, he must show not that all the gods agree that someone who kills unjustly should be punished, but that all the gods agree that a certain killing was unjust. Socrates presses Euthyphro to prove that all the gods would agree that Euthyphro is acting justly in prosecuting his father for unintentionally having left a man to die of exposure when that man had killed someone in a drunken rage.
Euthyphro assures Socrates that he could convince both him and the jury at the trial that his actions are just. He shifts his definition of what is holy slightly, arguing that it is not simply what is agreeable to the gods, since the gods tend to disagree, but that it is what is approved of by all the gods.
Euthyphro's original definition was unsatisfying to Socrates because it only identified a certain class of actions that Euthyphro considers holy without providing an overarching definition. This definition, that what is holy is what is agreeable to the gods, works far better for Socrates because of its generality. We can ask of any action whether or not the gods find it agreeable, and then classify all those actions that are agreeable to the gods as holy. If Euthyphro is correct, this definition should help us determine precisely what is holy and what is not. The problem, of course, is that Euthyphro is not correct, as we shall see. The first step in dismantling Euthyphro's argument is to have him state his position in a clear manner. Now it will be easier for Socrates to examine his position.
In this section, we see Socrates beginning to apply the familiar method of elenchus, or cross-examination. Now that Euthyphro's position is more clear, Socrates begins to examine it more closely, asking Euthyphro to draw out the consequences of what he has asserted. This method of questioning is very much in line with Plato's theories of teaching and knowledge, and with the dialogue form in general. In middle period dialogues, such as the Meno and the Republic, Plato asserts the distinction between knowledge and true belief--that knowledge requires justification, but true belief does not. That is, if I tell you that the sum of the squares of the legs of a right triangle are equal to the square of the hypotenuse, I have only given you a true belief. For you to actually come to know the Pythagorean theorem, I must also give you an explanation and proof for why this is so, so that you can reason it through on your own.
Similarly here, Socrates does not want simply to point out the flaws in Euthyphro's argument, but to lead Euthyphro to recognize and understand why his arguments are flawed. To do this, Socrates must only question Euthyphro, leaving him to recognize on his own the flaws in his reasoning. Thus, Socrates merely urges Euthyphro to look more carefully at the consequences of what he has claimed. For instance, Euthyphro has claimed both that what is holy is what is agreeable to the gods, and that the gods often disagree as to what is just. Socrates leaves it Euthyphro to sort out how these two positions can be reconciled.
It is worth noting also how closely linked the Socratic method of elenchus is to Socratic irony. In order to properly examine Euthyphro's claims, Socrates must appear to accept them wholeheartedly, and see where they lead. Socrates and the reader know perfectly well that he will ultimately find Euthyphro's definition unsatisfactory, but it is necessary for the inquiry to succeed that Socrates at least nominally accept the definition at the outset.
Socrates is treating Euthyphro as the teacher when in fact Socrates is teaching Euthyphro