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Like any mathematical or other intellectual system, ethics requires a certain set of statements to be axiomatic. These must be accepted as true for later proofs to be built on. In addition, certain methods also must be able to be applied to the proofs to derive certain new proofs which are based on the axioms. Methods of doing this correctly are called 'laws of logic' and methods of doing this incorrectly are called logical fallacies.

Holding truths to be 'self-evident' is the same as declaring the truths to be axioms under the ethical system. Standard valid rhetorical techniques and the laws of logic can then be used to construct a system.

Now suppose that we have two systems that are in conflict, then you can:

1) Derive the answer from commonly accepted axioms, i.e. have a rational discussion. If this doesn't work then:

2) Determine the point of fundamental disagreement. (The inverse of an accepted axiom is held by the other, a method of proof is held to be invalid by the other etc.) Argue for your side. Try to see why others might hold their view.

3) If this doesn't work, try to agree to disagree. Each admits that each thinks the other is wrong, and defers the discussion until a later date when the truth may become more clear, or just works out an arrangement where each can tolerate the other.

4) If THAT doesn't work, clean your weapons and stock up on ammo, cause there ain't no other way to solve it. (Check your logic too. You don't want to go into battle with doubts.)

In general, a person that does their philosophical thinking well has a martial advantage, since they are also an overall better thinker. That's why really good generals also study philosophy. You will also be able to maintain more cohesion among your allies, since the rightness of the argument will maximize your ability to attract them. This doesn't mean that you won't be outnumbered or out-gunned or even lose, but it IS an advantage (see the historical record.) People that beat impossible odds are typically philosophically convinced of the rightness of their cause.

Of course, reality is the ultimate determiner and history is the set of case examples of reality. If one side is distorting, denying or ignoring history, typically it is because they know that their arguments are weak. This is an extremely good indicator of the wrong side.

A good discussion on axioms, and how to identify them is in the 'Introduction to Objectivist Epistemology'. Aristotle and Thomas Aquinas are also good. A statement is typically an axiom if it is required to even have a discussion in the first place. For instance, we know that universal truths exist, since the statement 'There are no universal truths' is an attempt to make a universally true statement. Since 'There are no universal truths' is false, then 'There are universal truths' is true.