Thanks for your response. Apologies for my late acknowledgment. I have been meaning to respond "someday" when I found the time.

Here are responses to two of your points.

I do see the circle you're pointing out: "We humans say it is cold outside when the temperature is such that we feel cold. We say we feel cold when the temperature outside is that which we previously defined as cold." When considering individual preferences and keeping it on an individual level, yes, it's a circle: I feel what I feel, I like what I like, and my feelings and preferences don't have to be explained further. But what about someone who's in a position to make decisions for others? The facilities manager of a large building hears requests to turn the thermostat five degrees in either direction. How should they decide? Set the thermostat based on their own subjective feeling of a "good temperature" and ignore all feedback from everyone else? That is the problem set forward by the Euthyphro, as I understand it. Is the temperature good *just because* the facilities manager personally likes it? Or has the facilities manager chosen this temperature according to some other criteria, such as a survey of the preferences of a hundred other people in the building? Is it good [just] because God loves it, or does God love it because it is good [for other reasons]? As a regular person inside the building, I need a way to coherently express my dissatisfaction with the temperature and my belief that there is a better option out there that will work for me and/or for a number of other people. Whether I have the political power to be heard is another matter; that's outside the scope of the Euthyphro. I just mean I need to be able to say "I disagree" and have my opinion statement sound coherent (even if I never have the power to do anything about it). The response may well be "You aren't allowed to speak to the Facilities Manager," a realist constraint that doesn't cause an abstract philosophical problem, but if the response is "It's impossible to have a coherent thought that questions the Facilities Manager because that person is the Arbiter of Truth and Goodness," I am going to respond "Nah." Thus, atheism.

As for these questions:

(a) "Do you ever argue about morality without believing there is truth to your argument?" Questions framed like this are a little tricky because they tend to assume that "truth" is transcendent truth, which is a kind of truth I don't believe in. It's the very thing at issue. I believe in other kinds of truth. That said: Yes, I can argue without believing I'm pursuing truth (especially without "transcendent truth," though generally I'd be pursuing some other kind of truth). Some arguments (say, mathematical proofs, historical research, or criminal investigations) can be searches for correct answers. But other arguments (say, literary and aesthetic criticism) are attempts to create new ideas and reach agreements. Some truths are constructed socially. Sometimes this construction happens consciously, other times unconsciously (through cultural change). Ethics can mix subjective/individualist concerns with factual/collectivist concerns. It's based on agreements. If no one is being hurt or victimized, it's hard to argue that any wrong is being committed. If everyone agrees that the group is headed in the right direction, it probably is the right direction for that group (until results are achieved and someone provides evidence to the contrary). So, moral arguments are often attempts to identify consequences (who gains and who loses), to define and establish obligations, or to teach relevant virtues. That's not a search for a preexisting transcendent truth. It is an attempt to co-create social rules that attach themselves to other truths. If the moral rule in question is firmly held by one person or widely agreed upon, it attains a sort of truth, though not a transcendent truth.

(b) "Do you accept as equally moral any morality that differs from your own?" If the "morality" is on the level of habit, yes. A self-described "lifehacker" who drinks a green smoothie every morning and meditates for 45 minutes--great, that's their life ethic, good for them. Someone else doesn't want the structure--fine for them. Basically, if it's on the level of aesthetics or enjoyment of life--if it's about the degree at which someone wants the thermostat--there is no "right answer" that's universally applicable, and I just hope that people have enough autonomy to choose their best life. If it's more deeply ethical--if it's about actions that affect others and there are possible harms and victims involved-- then we may have to try to persuade each other and try to reach agreement or at least a compromise, because not all paths can be argued to be equally good, given certain relevant facts.

(c) Would the commitment to such cultural egalitarianism itself be considered a moral value/system? Sure, egalitarianism is a moral value/system, but the specifics of how we respect each other's cultures and individuality would be negotiable. It would bump up against other values/systems and have to respond and adjust to them. Egalitarianism isn't transcendentally true, any more than anything else is.