Logic

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

Comments

The conclusion of a reduction need not be surprising. Suppose I am trying to prove that 2+3 =5. I might proceed as follows: Assume that 2+3 does not equal 5. Since 2 = 1+1, and 3 = 1+1+1, that would mean that (1+1) + (1+1+1) does not equal 5. But (1+1) + (1+1+1) does equal 5. So our original assumption must be false, in other words, 2+3=5.

This example isn't that interesting, but the point is to show that you can use the reductio strategy without ending up with a surprising conclusion.

A reductio ad absurdum argument has the following form:we assume that X is true, deduce some absurdity from it, and thenconclude that not-X must be true after all. You can view reductio as arule of inference that allows us to infer not-X from our derivation ofan absurdity from X. Why does this work? Becausededuction is a process that leads from true assumptions to trueconclusions. (See also here.) But an absurdity cannot be true.Therefore, our assumption X is not true. But either X is true or not-Xis true. Consequently, it must be not-X that's true.

I agree withAmy that the final upshot of a reductio argument needn't be surprising,so I'll interpret the question differently. Perhaps you're asking thefollowing: when we deduce our absurdity, how do we know whether it's soabsurd that we have no choice but to consider it false and thus toconclude, by reductio, not-X — or whether instead to conclude that,contrary to what one might have thought, the alleged absurdity is true after all! Youmight say, one man's reductio is another's surprising discovery.

Theconclusion that not-X must be true is only as strong as one's judgmentthat the deduced absurdity is false. Typically, in mathematics orlogic, that judgment is indubitable because the absurdityis a contradiction (and contradictions are always false); as that ruleof inference is employed in logic and mathematics, nothing short ofdeducing a contradiction would allow it to be applied. Outside thosedisciplines, however, the absurdity might fall short of acontradiction. Theabsurdity will often be a claim that is inconsistent with what theproponent of the argument takes to be obviously true. But if someonepresented with the argument chooses instead to revise his beliefs insuch a way as to permit embrace of the alleged absurdity, there'snothing in theargument itself that can stop him. More general considerations aboutwhat it is to be reasonable would have to enter into the picture atthat point.

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 false, because, as Alex says above, in mathematics nothing short of a contradiction--"P and not-P", for some P--counts as a reductio. But some might consider the Banach-Tarski Theorem so strange that it provides a good reason for deciding not to accept AC as an axiom.

A couple of good references:

Moore, G. H., Zermelo's Axiom of Choice: Its Origin, Development, and Influence, Springer-Verlag, 1982.

Wagon, S., The Banach-Tarski Paradox, Cambridge University Press, 1993.

The crucial question is which is more plausible: the premise or the negation of the conclusion. Our answer may be influenced by diverse features of our broader 'web of belief'.

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