You should read
this. There really aren't any interesting philosophical questions that can be answered with deductive logic.
Really!? You should type up your proof and mail it to
Philosophical Studies for peer review. You'll be very very famous.
The correctness of rules has nothing to do with their being followed. Does my denying that molecules exist mean that chemistry is subjective?
This doesn't make a whole lot of sense. If my ideal of physics is to "concoct a theory involving lots of leprauchans" does that mean that physics is just as subjective as morality?
If a question can't be answered with logic then it can only be answered with opinion, which is worthless, or data, which is meaningless without logic to use it to support an idea.
There is no justification for a moral rule to be considered correct other than your opinion. Any justification that relies on anything other than your opinion already makes a moral assumption, which you can't do when we are trying to establish the basis of morality: for example, utilitarianism assumes that it is a moral good to act for the greatest good of everyone, even if the action harms you.
There is no possible way to justify your moral opinion as having any more moral weight than anyone else's. Thus all opinions are equal and none trumps another.
Morality is a concept/idea. It was invented by humans: it can have no existence without a mind which can conceive it. There can be no absolute morality that exists without humans and for which we are searching, just like there is no such thing as fundamental human rights.
Any other system of morality assumes a purpose and then builds morals around it, but to assume a purpose is an unwarranted assumption. Morals can be about any purpose.
Thus we conclude that morality is subjective. What did I miss that confuses people? There must be something, but I don't see it.
I'm afraid moral relativism is only one view, taken to an extreme, i.e we cannot judge other cultures because we do not know them it's moribund as a philosophy. If culture a condones the first born child being scarificed, moral realtivism would say, why should we judge? But we have a perfect right to judge such abhorent acts.
Let's take a real world example, if Sharia says that women should be subjugated, is it ethically ok to accept Sharia as a good system? Moral relativism is something I tend to follow, but not to extremes, you need to look furhter than one moral code.
There's a difference between moral relativism and cultural relativism. Cultural, as you say, seems to expect us to leave alone things that differ. Moral simply says that they do not have to conform to our ideas, just as we do not feel an imperative to conform to theirs. We may choose forcibly to impose our ideas, but moral relativism does not command or forbid this.
Logic is a means for discussion not a means to an end, we live in the real world and if logic can encompass that, then it's no logic I recognise. Philosophical arguments stand up in one moment or one time, but morallity is progressive, if philosophy isn't then it is dying or dead.
If a conclusion is reached that isn't logical, how can you justify it? What other means of discussion are there except logic? Arguments, be they philosophical or not, must be logical, or else no good as arguments.