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I can't think of any examples of ambivalent axioms in mathematics (two ideas are ambivalent if there are sometimes conflicts between them), so please let me humor you with a strange example. Suppose someone was working on a mathematical theory of economies and wanted to account for everyone's perspective and also minimize harm. So in addition to these two axioms, "harm" and "perspective" and "valid" would have to be defined:

1) Do no harm 2) All perspectives are valid

So, in the theory, suppose you come to a point where you want to apply axiom 2 but it conflicts with axiom 1, for instance, a portion of the people have a perspective such that if their will were carried out without being moderated by the other peoples' perspective, there would be harm caused to people, some other life on the planet, or the ecosystem. Then what do you do to resolve this conflict? One solution is to "split" the theory into two branches, one in which you ignore 1, and the other in which you ignore the perspectives of that harm-causing group. There are other ways to do this too, but this is just a stupid example anyway.

So with that scenario in mind, when there are ambivalent axioms, the theory branches into a tree of results in a way that is unlike conventional mathematics.

So my questions are, have I made a brain fart? If not, could ambivalent-axiom theories be transformed into conventional non-ambivalent axiom theories after all definitions given? Would this help mathematics at all or is this solely the territory of other sciences like psychology and psychohistory? Is this abuse of the beautiful, well-tested mechanisms of mathematics, and if there were some conflicting idea in a theory it would enter the stage in another way like an "if statement."

My apologies for not thinking this through myself. I'm not in a mathy perspective lately and am lazy :D

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In a logical system that obeys classical logic, if two axioms are "sometimes in conflict", i.e., if you can deduce contradictory answers from them, then you can prove anything; as such, any theory with "ambivalent axioms" would be inconsistent and hence uninteresting. There is plenty of work in studying how a "core" theory behaves under the addition of different axioms; e.g., ZF+Determinacy vs. ZF+Choice, so I don't see why you say that such a thing would be "unlike conventional mathematics" (unless you think Set Theory and Model Theory are not conventional mathematics). – Arturo Magidin Oct 2 '11 at 5:58
Can someone please re-tag this? I'm really not sure how, but at least one tag seems completely off. – Asaf Karagila Oct 2 '11 at 6:03
Psychohistory? As in, the field founded by Hari Seldon in the latter days of the First Galactic Empire?? – Pete L. Clark Oct 2 '11 at 14:03
up vote 5 down vote accepted

Mathematics which works with concrete ideas has some problem with conflicting axioms. Mostly because there is no such thing as a little bit of contradiction. You can only derive contradiction; and from it you can derive anything - which means theories in which there is a contradiction are not useful.

There are examples, however, to incompatible axioms (which is a better, and more common name than ambivalent). In set theory, there can be axioms which do not fit together as additional axioms of Zermelo-Fraenkel set theory.

Examples are:

  1. $V=L$, Goedel's Axiom of Constructibility,
  2. $AD$, The Axiom of Determinacy, which says that every game over the reals is determined.
  3. $AC+\exists \kappa$ measurable, that is the combination of both the Axiom of Choice and the existence of a measurable cardinal.

Each pair of the three above is incompatible. However each one fits just fine with the rest of the ZF axioms.

(Although to be fair we do not know if ZF itself is consistent, and thus we cannot conclude the consistency of any of the combinations above; however assertions as the existence of a measurable cardinal are strong additions to the theory, and they can provide enough power to prove the consistency of ZF. Which means they cannot prove their own consistency, or else we would have a contradiction.)

There are many similar examples of how some axioms can be incompatible. However in many places mathematicians are interested in the implications, which gives them the freedom to assume the axioms they prefer.

Lastly, there is still the idea of Boolean-valued logic. In this logic we assign truth values which are not just $0,1$ (or True, False) but rather enrich the truth with further values. In this case a theorem may have truth value $a$, and another may have truth value $b$, while $a\land b=0$.

That is to say that we can prove two theorems which are both not false (which is different than being true, in this context) but the assumption of both is indeed false.

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There have been some attempts, collectively known as paraconsistent logics, to construct logical systems that can deal with contradiction more gradually than exploding all at once. They are not intended to be used used as a basis for general mathematics, however -- but are available as one tool among many in mathematical modeling. – Henning Makholm Oct 2 '11 at 7:10
@Henning: Thanks, I was not aware of this. – Asaf Karagila Oct 2 '11 at 13:07

Asaf answered the formal part of your question well.

I think the main problem with the theory you're outlining is that your "axioms" are, in fact, not axioms as mathematics uses that word. When you construct a mathematical model of a facet of the world, your axioms are supposed to be things you know will always be true about the thing you're modeling. But your proposed axioms are not that. They are not even phrased as statements of fact, they are statements of goals or desirables, and except in ridiculously simple cases goals warrant a different mathematical treatment than objective knowledge.

Taking your goals as axioms implies that you're assuming that there is one way to fulfill all of your goals at once, but real life does not work that way. (Otherwise we could turn politics and law over to machines and re-train all of the lawyers to do something productive instead). What you want is to structure your model such that it can deal with a spectrum of outcomes of your actions, some of which you consider more or less desirable than others. There's not a single way to do this, because the structure of the model will imply some preference choices on your part -- but in general you will need representations of outcomes, and some desirability ordering among them.

One simple way to do this is if you can assign a numeric "quality" measure to each outcome and compare those. Or you can define directly how any two outcomes are to be compared. Or you can take an axiomatic approach and have axioms that go like "Outcome A is more desirable than B if such-and-such holds about them". This is the most flexible approach, but also the most dangerous one because you risk running into an inconsistency where your model says that A is better than B is better than C is better than A. In that case you need to go back to the drawing board and choose different axtioms -- or accept that your model cannot help you choose between the actions that will lead to outcomes A, B, and C respectively.

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