There is a theory of arguments, but I am afraid that the OP conception of argument is too idealized, and the notion of effective debate too narrow, to apply to most of them. If people argued from sets of established axioms and the only issue was whether those sets are equivalent they'd be proving mathematical theorems and meta-theorems of mathematical logic instead of having debates.
The crux of real life debates is not disagreement over axioms, but vagueness and ambiguity of translating available real life evidence into generalities, and even finding the right terms and classifications for expressing them adequately. To one person history indicates that ends justify means, to another this is a hasty generalization; to one person Napoleon is a great leader, to another he is a mass murderer; to one person soul can clearly exist apart from a body, to another this is a fanciful nonsense, etc. It is eliciting intuitions, affecting judgements, bringing out "facts", and deciding what is or is not a "fact", i.e. generating fruitful concepts and defensible "axioms", which can plausibly withstand factual objections and criticisms, that make effective debates effective. Often effective for both sides even if in the end they still do not come to an agreement. This will not be captured by a scheme that presupposes fixed concepts and axioms.
Wikipedia has a long entry on arguments, including theories of argumentation. The study of debates goes back at least to sophists and Socrates, and was known as the art of dialectic in antiquity. In recent times Toulmin's model of argumentation, developed in his book Uses of Argument, has become very influential. Here is from the abstract:
"Starting from an examination of the actual procedures in different fields of argument - the practice, as opposed to the theory, of logic - he discloses a richer variety than is allowed for by any available system. He argues that jurisprudence rather than mathematics should be the logician's model in analysing rational procedures, and that logic should be a comparative and not a purely formal study."
Toulmin models arguments based on six elements: Claim (Conclusion),
Ground (Fact, Evidence, Data), Warrant (movement from the ground to the claim), Backing (credentials certifying the ground), Rebuttal (restrictions to the claim) and Qualifier (degree of certainty for the claim). This model shifts the focus where it belongs, to inspection and genesis of claims and evaluating evidence for them rather than on piecing together deductive chains, which is often a triviality and always an afterthought.