# How do you tell the difference between a reductio and a surprising conclusion?

There's an interesting example from mathematics that might be relevant here. The Axiom of Choice (AC) is an axiom of mathematics that was quite controversial when Zermelo introduced it in 1904. It is less controversial today, perhaps because Godel showed in 1940 that if the other axioms of mathematics are consistent, then adding AC cannot introduce a contradiction into mathematics. But AC does lead to some very surprising conclusions. One of the most famous is the Banach-Tarski Theorem , sometimes also called the Banach-Tarski Paradox because it is so surprising. The Banach-Tarski Theorem says that it is possible to decompose a ball of radius 1 into a finite number of pieces and rearrange those pieces to make two balls of radius 1. (The "pieces" are actually more like clouds of scattered points that, together, fill up the entire ball.) Should the Banach-Tarski Theorem be considered a reductio ad absurdum proof that AC is false? It certainly doesn't count as a mathematical proof that AC is...

# Is self-contradiction still the prima facie sign of a faulty argument? How do we tell an apparent contradiction from a real contradiction if the argument is in words? (Most of us don't know how to translate arguments in words into symbolic logic.)

It is perhaps worth adding that self-contradiction is not the only sign of a faulty argument. An argument can be faulty but not lead to a contradiction. For example, suppose that you know that some number x has the property that x 2 = 4. If you claim that x must be 2, you have engaged in faulty reasoning. The conclusion x = 2 does not contradict the hypothesis that x 2 = 4; the two statements are perfectly consistent. But your reasoning is faulty because you haven't taken into account the possibility that x might be -2.