The primary philosophical issue that Socrates and Euthyphro are in disagreement about is what is holy and what is unholy? Socrates stands about piety that one should not even do in the response to the offence done by another; and it is inappropriate not to obey the government. On the other hand, Euthyphro thinks that it is unholy not to prosecute a wrongdoer whether he is relative or not (Cahn and Markie 105).
According to the Plato's Euthyphro, its logical mistakes are assuming that piety is defined by the act of persecuting an unjust person who has murdered or committed sacrilege or any other related crime. He cites this, by giving an example of himself making charges against his father who had killed his worker unintentionally. Although this is not entirely accurate as the worker died of exhaustion. Socrates does not consider it to be a definition, it is just an example of piety as it doesn't infer to what makes holy things holy (Cahn and Markie 138).
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Although Socrates applauds this stand he also criticizes it on the basis that gods might disagree about what is pleasing. This means that arguing about the gods would be both holy and unholy which is logically impossible (Cahn and Markie 129).
The third mistake refers to the affirmation, that what the gods approve is pious and what they don't love is not. Here Socrates argues whether the gods love something because it is holy or is it holy because the gods love