Q:

Possible Logical Error in Euthyphro?

There is an argument near the beginning of Euthyphro that Socrates tries to make to Euthyphro in which I believe that Socrates makes a logical mistake. It's a pretty basic mistake, so rather than assume that Plato made such a simple error, I'd like to ask everyone here what they think. All quotations are taken from G. M. A. Grube's translation of Euthyphro in Plato's Complete Works, edited by John M. Cooper.
In 7b-d, Socrates says:

"What are the subjects of difference that cause hatred and anger? ...What subject of difference would make us angry and hostile to each other if we were unable to come to a decision? Perhaps you do not have an answer ready, but examine as I tell you whether these subjects are the just and the unjust, the beautiful and the ugly, the good and the bad. Are these not the subjects of difference which, when we are unable to come to a satisfactory decision, you and I and other men become hostile to each other whenever we do?"

Now, if we let D be the statement, "We disagree about goodness, justice and beauty" and let H be the statement "We become hostile to/at odds with each other," Socrates seems to be saying here that D causes H.
I interpret D causes H to mean if D, then H.
Then, in 7e, Socrates goes on to say:

"Then according to your argument, my good Euthyphro, different gods consider different things to be just, beautiful, ugly, good, and bad, for they would not be at odds with one another unless they differed about these subjects, would they?" (Emphasis mine)

This statement, I interpret to mean not H, unless D, which is the same as if not D, then not H, which, by contraposition, is equivalent to if H, then D.
The first and second quotations here thus lead to the following two statements, respectively:
1) if D, then H
2) if H, then D
But 1) does not entail 2)! Nor does 2) entail 1), for that matter. They are converses of each other. So how can Socrates make the claim in 7e that 1), which was established earlier, entails 2)? Clearly, 1) does not entail 2).
This seems like a straightforward error to me, but I don't feel comfortable accusing Plato of such a basic mistake. Perhaps I am misreading or misinterpreting something here? What does everyone else think?

A:

I symbolize what happens in the passage roughly as follows (I've bracketed the subargument from 3-5):

Piety = Dear to gods "what is dear to the gods is holy  and what is not dear to them is unholy" (6e-7a)
gods disagree about piety (7b)

Some disagreements lead to discord (7b)
If a disagreement is empirically resolvable, it does not lead to discord (7b)
Disagreements about justice, truth, and beauty lead to discord (7d)

Ergo, since the gods are in discord, they must substantive disagreements about justice, truth, piety, etc. (7e-8a)
Finally, 6 & 1 are contradictory so something must be wrong in the argument (8a)

Starting from 6d-e, the argument is framed by Euthyphro's claim that what unifies the things we call holy (despite their differences) is

The latter half of 7b which you quote in part is the beginning of a subargument:

But what things is the disagreement about, which causes enmity and anger? Let us look at it in this way. If you and I were to disagree about number, for instance, which of two numbers were the greater, would the disagreement about these matters make us enemies and make us angry with each other, or should we not quickly settle it by resorting to arithmetic? (7b-c) ... Then, too, if we were to disagree about the relative size of things, we should quickly put an end to the disagreement by measuring?  ... weighing? (7c)

I take this line of questioning to mean that we don't waste our time disagreeing about minor things that we can simply or empirically resolve.
Conversely, Socrates and Euthyphro accept that we can wind up with 

a disagreement [that] we could not settle and which would cause us to be enemies and be angry with each other (7c)

And that this would be about questions like:

Is it not about right and wrong, and noble and disgraceful, and good and bad? (7d)

= substantive disagreements would be about things like right/wrong and notably about things that are not empirically or simply resolvable are the sort of things that can lead to discord. In other words, it's things like the Forms (which makes sense considering this dialogue is about Piety).
Note that from 7b to the middle of 7d, Socrates is not talking specifically about the gods. 

At the end of 7d, he bridges it back to his main point by saying that gods would disagree about matters of import which are not resolvable by measurement or simple math. Moving to 7e, I don't think his point is the symbolization you suggest.
Specifically, I take the part at 7e to express "discord if and only if disagreement is about substantial things" -- i.e. I take it to express that the gods aren't in discord over something minor but rather they are in discord because they disagree about the nature of piety, etc. in a fundamentally irresovable way.
Why then does Plato make this argument (since he already said they were in discord)? I take it that the point is to block out the option of just saying "god A is correct for reason of knockdown argument B or reason of empirical fact C." Instead, it's point out that there's a fundamental incapacity to define piety by appeal to the gods -- since the gods are in discord and this means by definition that they do not agree for empirically and logically non-resolvable reasons about the nature of piety.

