At Euthyphro 9e, Euthyphro claims that the pious is that which is loved by all the gods. In effect, he is claiming that the pious and the god-loved are identical. In reply to this claim, Socrates argues that “If the god-loved and the pious were the same, my dear Euthyphro, then if the pious was being loved because it was pious, the god-loved would also be being loved because it is god-loved.” Socrates is essentially arguing that if the ‘pious’ and the ‘god-loved’ are, in fact, identical, you should be able to use the terms interchangeably without changing the truth value of a particular statement.
To demonstrate this property of identity using a simpler example, imagine that I make the claim “Christopher Nolan was born on July 30, 1970.” Given the fact that Christopher Nolan is the director of Inception, I can also make the claim that “The director of Inception was born on July 30, 1970.” Regardless of whether or not Christopher Nolan was really born on July 30, 1970, notice that the truth of my original statement is not affected by interchanging ‘Christopher Nolan’ and ‘the director of Inception.’ If the statement is false, it will remain false. If it is true, it will remain true. This is because ‘Christopher Nolan’ and ‘the director of Inception’ are, in fact, identical.
Similarly, if the ‘pious’ and the god-loved’ are identical, as Euthyphro claims them to be, I should be able to demonstrate this same property with Euthyphro’s claims. For example, let’s take the claim that “The pious is loved by the gods because it is pious”. Euthyphro, at 10d, claims that this statement is true. If I interchange each occurrence of the term ‘pious’ with the term ‘god-loved,’ this brings me to the statement “The god-loved is loved by the gods because it is god-loved” –a statement that Euthyphro claims is false at 10e. By interchanging the terms ‘pious’ and ‘god-loved’ I have effectively changed the truth of Euthyphro’s claim. In this way,