The "reductio ad absurdum" is a type of apagogic (indirect) proof which proves something true by eliminating all other possibilities. The phrase "reductio ad absurdum" means "reduce to absurdity," and refers to the process of reducing to absurdity all other possibilities until there is only one left. The reductio is used quite frequently by Euclid in his Elements and in geometry in general. It makes good sense as a tool in geometric proof, but many a philosopher has tried to co-opt it into philosophical proofs, which is not quite as air tight.

So, basically, to show how a reductio works, if you want to prove that point A falls on the circumference of circle Y, then you can use a reductio and say that point A either is inside circle Y, outside of circle Y or right on circle Y. If you demonstrate that both the possibility that point A is outside or inside circle Y lead to absurdity - namely some sort of paradox - then it follows by process of elimination, that point A must fall on circle Y. Two critical assumptions are always being employed when one uses a reductio: 1) that the correct solution is among the possibilities listed and 2) that all possibilities are included in the possibilities listed (we should note, that if condition 2) is met, then condition 1) necessarily follows, nonetheless, I think it is important to note them as two separate conditions since if condition 1) is met then one can come up with a correct answer, even if condition 2) is not met). In the case of the geometrical proof we can be confident that we have all possibilities. Within our simplified space only occupied by circle Y and point A, there can be only three possibilities. Geometry, and in particular Euclidean geometry involves a very simplified space, built from the ground up from a finite set of definitions and axioms. The geometrical space in which geometrical proofs occur is finite and circumscribed. Thus, we can say, because we are perfectly aware of all the limits and rules of this space, what are all the possibilities within a reductio proof.

But what happens when we are philosophizing on topics that concern the wider world? We can neither create a finite set of all definitions nor a finite set of all fundamental axioms for the wider world, and thus for us to confidently assert that the possibilities listed are all possibilities is questionable. It's not like reductios aren't at all possible outside the well circumscribed confines of geometric space, but it certainly is more uncertain, more difficult, and more open to skepticism.

So, I make this entry as a preface to some discussions of some reductio arguments in philosophy.

Next post, Aristotle

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