From Understanding Plato: The Smart Student's Guide to the Socratic Dialogues and the Republic

Part 2 Quarreling Deities

Socrates repeats his request:  “What are the pious and the impious?”  Euthyphro responds by telling Socrates that "what's loved by the gods is pious, and what's not loved by the gods is impious" (7a).  Let's call this new definition D2.

Socrates responds to D2 by saying that it implies the contradiction that one and the same action can be both pious and impious at the same time.  This implication is clearly unacceptable.  Socrates proves this by getting Euthyphro to admit that the gods quarrel and disagree about such issues as "the just and the unjust, fine and shameful, good and bad" as much as any group of human beings.  This is in marked contrast to differences about height and weight in which we can "turn to measurement and quickly put a stop to our differences" (7c).   But absent such standards, it is possible that "what you're now doing in prosecuting your father was something pleasing to Zeus but displeasing to Cronus and Uranus, or lovable to Hephaestus and displeasing to Hera, and similarly for any other gods who may differ from one another on the matter" (8b).   Hence, prosecuting your father is both pious and impious. We will label this lengthy reply R2.

Something extraordinary has happened as a result of this short back-and-forth between Socrates and Euthyphro.  Some would call this a “take-down,” using a term from the sport of wrestling to describe a humbling refutation of a claim or argument of one’s opponent. How did this happen? C.D.C. Reeve points out that Socrates’ style of questioning is called (by us, not him) an elenchus—derived from a Greek verb meaning “to examine or refute”:

[Socrates] asks what [piety] is; his interlocutor [Euthyphro] puts forward a definition he sincerely believes to be correct, at least for the moment; and Socrates refutes this definition by showing that it conflicts with other beliefs the interlocutor sincerely holds and doesn’t think he can reasonably abandon.  In the ideal situation, which is never actually portrayed in the Socratic dialogues, this process continues until a satisfactory definition emerges—one that is not inconsistent with other sincerely held beliefs, and so can withstand elenctic scrutiny (Reeve, 2012, viii).

Euthyphro puts forward D2, and Socrates refutes it with R2.   But R2 (“one and the same action cannot be both pious and impious at the same time”) is a belief that (we presume) Euthyphro “sincerely holds.”  Since Euthyphro cannot reasonably abandon R2, he must abandon D2, and the elenctic process continues.   If we put this in standard argument form, it would look like this:

1.  Hypothesis: What is loved by the gods is pious and what is not loved by the gods is impious (D2). 

2.  It is possible that some gods love what Euthyphro did to his father and some gods do not love this.

3.  If some gods love what Euthyphro did to his father and some gods do not love this, then what Euthyphro did to his father is both pious and impious (R2)

4.  It is a contradiction to say that what Euthyphro did to his father is both pious and impious (It is assumed that a contradiction is unacceptable to Euthyphro).

5.  Therefore, the hypothesis (D2) is false, that is, it is not the case that what is loved by the gods is pious and what is not loved by the gods is impious.

Euthyphro now sidesteps the problem by raising a different point: the gods all agree "that anyone who has unjustly killed another should be punished" (8b).  Socrates responds that both the gods and humans agree that the unjust should be punished, but second, he also observes that this is beside the point.  The point is that both the gods and humans disagree about what actions are unjust, not about whether the unjust should be punished (8e).  "I think that men and gods who argue...argue about actions.  It's about some action that they differ, some of them saying “that it was done justly, others unjustly" (8e).