(1) A possible counterexample to moral relativism
Relativity is opposed to the absolute, as the subjective is to the objective. Therefore some facts are relative but objective (e.g. that a table is to my right, is relative to me, but objectively so) or absolute yet subjective (facts about the mind, for instance--although these can be parsed objectively on some level too). However, moral subjectivism easily becomes moral relativism on a certain scale, and moral relativism can support moral subjectivism by a reflection of the same ideation. To wit, if moral truths depend on my emotions or intentions, then they seem relative to me; and if moral truths are relative to me, it is probably as a function of my emotions or intentions.
OTOH, moral relativism is usually cultural relativism, that morality is relative to cultures instead of individuals. Of course, it has been argued (as I discovered in my Intro to Ethics class once upon a time) that every individual can be thought of as having his or her own unique, private culture. Indeed, the word "culture" is not entirely clear on the face of it, so one wonders about trying to define something like "morality" with reference thereto. Without detaining ourselves on minutiae, we might just go with, "Morality is relative to the standards of the society/community of its subjects," so that what is right or wrong for me, for example, will depend on the commands, desires, etc. of the local legislator and the hierarchy to which it is itself submitted.
The easiest go-to argument for moral relativism is the appearance of intractable disagreement among debaters of various moral claims. Now, genuine disagreement automatically implies the possibility of resolution, in the sense that if two people are actually disagreeing about something, there must be an objective, absolute thing for them to be disagreeing about, or else they are rather just making contrary noises while debating. So the argument-from-disagreement doesn't seem like it can really get started. However, I will allow that it might and offer a different counterargument, from what I think is a clear example of widespread cross-cultural moral agreement.
This example is:
- Self-command is a virtue.
Conclusion of this part of the OP: even if self-command has non-relative value, is it a substantial enough value to ground a further absolute ethic? For instance, if the desires to be resisted or controlled vary by person or culture, it appears that self-command might result in nevertheless a subjectively or culturally variant system of morality. The Nazis are a terribly good example of how such might play out, what with how some of them specifically advocated the action against the Jews as an example of self-denial (of feelings or desires of sympathetic bent).
(2) Can we prove the "law of non-contradiction"?
Are all assertions of the form, "So-and-so is both one way, and the opposite way, completely and at the same time," false? Can something be and not be, entirely, as one? Usually, the rejoinder goes, "No, because if that were the case, the words 'no' and 'not' would mean the same thing as words like 'yes' and 'is,' and any assertion would mean anything at all."
Sometimes people appeal to things like orthodox Christian theology or quantum descriptions of particle structure, as counterexamples to no-contradictions-in-truth. If the Son is God, and if the Father is God, and if the Son is not the Father, then God is not God (or if Christ is God and if Christ is a man and if God is not a man, then there is at least one man Who is not a man). Or if particles are waves, and if particles are also points, then particles are and are not waves and points. I don't know what to say to the argument from Christian orthodoxy, because it does seem as if the (classical) Trinity idea is self-contradictory however one tries to unravel it. But so much worse for the (classical) Trinity idea, then. As for quantum physics, well, it seems more as if particles are points while observed (hence "wave-function collapse") and waves while unobserved, which means they are not both simultaneously, which means their duality is not by any means a violation of no-contradictions-in-truth.
Because philosophers love to question as much as they know how to, I think there is a different way to ground no-contradictions-in-truth--to prove it, if you will--than appeal to "language doesn't make sense otherwise." If the axioms of philosophy ultimately are the form of our very ability to ask questions, and if the structure of closed questions (ones without wh-words defining them) corresponds to a yes-no dichotomy, e.g., "Is this blue?" means, "This is or is not blue," then we see that no-contradictions-in-truth is true because the point of asking a closed question is to get a yes-no answer, and because on the abstractmost level of these things erotetic functions define axiomatic assertoric ones.