I'm looking for any extensive work on a framework for "arguments", that works something along these lines:

  1. When two parties are debating, they are making assertions on a particular domain, D.

  2. Those assertions are ultimately based on some axioms, A1...An of D.

  3. An argument set, ARG[] is the finite set of all axioms A1...An of a domain D, and the domain D.

  4. An effective debate is only possible if the argument set ARG[]1 of party P1 and the argument set ARG[]2 of party P2 can be shown to be equivalent.

In other words, an effective debate can be had between two parties if (and only if) they both agree on the axioms A1...An of the domain D, and the can formally define D.

  1. In the absence of any such agreement, no debate can be effectively had, because there is no way of proving or disproving assertions based on different sets of axioms of different domains that are not equivalent.

I noticed the possibility of such a thing existing after reading two debates. Both made good points, but disagreed not because the authors were not making reasonable arguments, but because they were simply talking about different things. And I've seen this several times in real-world arguments and debates where, confusion over how fundamental ideas are defined extends what turns out to be a needless debate.

  • I would argue you only need the weaker condition in point 4, that each party has a common subset of axioms they are willing to accept for the purpose of the debate. They need not hold all the same axioms on the matter, only enough commonality to prove the points used.
    – Vality
    Commented Oct 10, 2016 at 3:20
  • I realize this makes more sense as a theory of effective communication, rather than argument or debate.
    – wsgeorge
    Commented Oct 12, 2016 at 14:28

4 Answers 4


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.

  • Maybe I am using an uncommon interpretation of axioms, but some of the things you mentioned seem like axioms to me (a statement whose truth is self-evident). Napolean being a great leader seems like an axiom. If one debater had that as an self-evident truth then all his arguments and beliefs may rest on that foundational assumption and unless that assumption (axiom) matches that of the other debater it will make agreement difficult due to lack of common ground. What you describe ("vagueness and ambiguity") seems more to be the way in which we chose our axioms.
    – syntonicC
    Commented Oct 19, 2016 at 15:29
  • 1
    @syntonicC The point is that person's "axioms" aren't fixed, and are up for revision in a debate. One can make arguments to undermine person's beliefs based on their other beliefs, or on facts not known to them, or even known to them but reinterpreted. Debates are about challenging and changing "grounds", and even the name is misleading. Nothing is "self-evident" and everything is up for revision in principle, although some parts are more entrenched than others, giving the impression of "grounds". This is why axiomatic theories, with fixed and non-vague axioms, can't model debates.
    – Conifold
    Commented Oct 19, 2016 at 20:55

https://en.wikipedia.org/wiki/Proof_theory proceeds in this direction. However, outside math, and even inside some parts of it, most domains cannot be axiomatized.

The evolution of definitions is part of dialectics. From a framing like that of Wittgenstein, Lacan or deSaussure, it is the primary part. From that POV, I find it unlikely that any two parties in a real argument ever actually have the same set of axioms, as those axioms would rely on the same set of definitions. Every decent argument may well be about agreement upon terms (or axioms as their proxy).

Ultimately, such agreement would always lead straight to the death of a domain. Once the problems are all solved, there is no power to be had by participating in the struggle that constitutes the discipline. All that is left is to apply it to other aspects of the world. No one has important debates about Newtonian physics, for instance, only about its applications.


After writing that, I felt maybe you need a more basic, logically positive answer than three references to people who are widely considered inscrutable.

From a different, more classical, and perhaps more utilitarian POV, discourse takes place on five levels, related to Cicero's 'Canons' and to the Big Five theory of personality descriptions:

  1. Logic
  2. Dialectic
  3. Rhetoric
  4. Impact
  5. Artistry

One can agree on basic principles, but disagree on how some data or information is to be interpreted. One can agree on those, but disagree on the right way to frame the interpretation so that it is not misleading. One can agree on those, but not on how that framing should make one feel or act. One can agree on all of that, and still argue just to find the best and clearest way of putting that out into the world.

It may be impossible to argue if one does not, at base, have the same logic as someone else, but you seem to be talking about the next level up, where there is definitely good reason to argue when you truly disagree, even if it is unlikely you will converge on an agreement.

Ultimately, your impasse may produce alternative interpretations, and both of you may abandon your previous positions because one of those is more attractive to intuition, or because it avoids traps related to both of your original positions. This is the Hegelian notion of 'synthesis', which explains a lot of our ability to advance our thinking.


YESSSSSSSSSSS! OMG this is exactly what I've been studying in my logic class. It's so cool! There are a number of different types of logic (all of them formal systems with varying degrees of usefulness and potency). In general, though, there are three kinds of arguments: deductively strong arguments (an argument is deductively strong if it is impossible for all the assumptions to be true and the conclusion to be false - this is called truth-preservation), inductively strong arguments (basically a good argument that doesn't guarantee truth-preservation), and arguments that are complete crap. An argument is truth-functionally valid if it is deductively strong, and truth-functionally invalid otherwise. Inductively strong arguments can still be good arguments, but they're just not truth-functionally valid. Then there are sound arguments and unsound arguments. If a truth-functionally valid argument also has all-true assumptions, then it is a sound argument. If an argument is truth-functionally invalid, or if it has false assumptions, it isn't a sound argument (under the formal definition of the term). Basically it goes like this: Mathematicians are only concerned with truth-functionally valid arguments, while philosophers and regular people are often satisfied with inductively strong arguments. Lawyers try to only use deductively strong arguments, but this generally fails because (unfortunately) not all law-makers are well-versed in formal logic and mathematics. Finally there are politicians, who are people that don't care if an argument is crap.

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