Following is the key argument in Plato’s Euthyphro dialogue. Note that this is just exactly the logical structure of Plato’s inference, not a reconstruction. You can get the “if..then” version of this just by bracketing parts of the original text, evidence of the care with which these dialogues were constructed.

Original Euthyphro

A = “holy is the same as god-beloved”

B = “the gods love something because it is holy”

C = “something is holy because the gods love it”

D = “something is god-beloved because the gods love it”

E = “the gods love something because it is god-beloved”

Argument

P1. B

P2 not C

P3. D

P4. not E

P5. (A and B) implies E

P6. (A and D) implies C

Conclusion:: not A

The argument is certainly valid (although there is a superfluous premise).

The reasoning behind P5 is that if “holy” and “god-beloved” are identical then “god-beloved” can be substituted for “holy” in B, transforming it to E.

There are two problems with this, firstly the substitution will only work if they are otherwise identical expressions, but when we say:

“John is a bachelor because he loves his freedom”

“John is a bachelor because he is not married”

“because” does not mean the same each time. We can see this change in usage in the argument and so the substitution is not valid.

More importantly, it does not matter anyway, because nothing Euthyphro has said implies that “holy” (osion) and “god-beloved” (theophilis) should be identical (tautos). Socrates has extracted from Euthyphro a definition (horizo: define, distinguish, divide) and a definition is not a synomym, in fact a synonym would make a very poor definition.

Finally, Socrates complains that Euthyphro had promised him the essence of holiness, but has instead only told him something that happened to holiness (ie the gods loved it). But again Socrates, in full control of the dialogue, extracts a definition “what all the gods love is pious and holy, and the opposite which they all hate, impious.”

Certainly this is not a very illuminating definition and obviously of no practical use, but a definition it is. If Socrates had wanted the essence of holiness then he should have chosen his words more carefully.

I do not imagine that Euthyphro really has any idea of what holiness is, but I can imagine a more canny Euthyphro now asking Socrates to provide him with the test he will perform on any answer to his question to ensure that it actually is the essence, so that he can apply this test first to ensure that he is sure that he is really providing Socrates with the answer he wants.

In general, this seems to be the real question in many of the dialogues - what are we asking when we ask for the essence of things like “good”, “courage” etc?  I think Plato provides his answer in Phaedo, but that is another subject.