It strikes me that atheists often in the religion debate will try to characterize religion in a funny or silly way, often comparing religion to belief in fairies or unicorns or flying spaghetti monsters.
Is this a valid debating technique? Does reductio ad absurdum have any logical justification behind it?
Yes, it is valid. How strongly it refutes a particular claim will depend on usage (there are several "types" of reductio ad absurdum) and context, but strictly speaking it is simply the process of logically following a conclusion to its extreme and thereby revealing an absurd consequence of such a belief.
The wikipedia article is somewhat sparse, but IEP provides good coverage of it, here's an excerpt:
In its most general construal, reductio ad absurdum – reductio for short – is a process of refutation on grounds that absurd – and patently untenable consequences would ensue from accepting the item at issue. This takes three principal forms according as that untenable consequence is:
1. a self-contradiction (ad absurdum) 2. a falsehood (ad falsum or even ad impossibile) 3. an implausibility or anomaly (ad ridiculum or ad incommodum)The first of these is reductio ad absurdum in its strictest construction and the other two cases involve a rather wider and looser sense of the term... Reductio argumentation is a special case of demonstrative reasoning [indirect proof].
Reductio ad absurdum is a valid argument form and inference rule in classical logic. It says: given some background assumptions, Γ, to show that some proposition P is false, we can show that Γ and P lead to a contradiction.
Symbolically, for any propositions P and Q, and any set of propositions Γ (including the empty set):
If Γ, P ⊨ Q ∧ ¬Q, then Γ ⊨ ¬P
("⊨" means logical implication.)
What you describe in your first paragraph is not a reductio ad absurdum, but a straw man fallacy. It's a fallacy because you're not characterizing your opponent's position properly; you're caricaturing their position in an obviously absurd way (that even they would probably say was false) to give the false impression of having refuted it. The straw man fallacy is the "moral opposite" of the principle of charity, which says we should try to earnestly understand what our opponent is saying, so that we can attack the strongest form of their argument, and not attack straw men.
In formal classical logic, reductio is acknowledged as valid move, but in intuitionistic logic it is not. This logic was espoused by Brouwer as a rival to Hilberts formalist programme to reduce logic to mathematics via set theory, in fact Brouwer correctly predicted that Hilberts programme would end in contradiction (as eventually shown by Godel).
Intuitionistic logic proceeds by denying the classical law of the excluded middle, that is for any proposition, it must either be true or false. Without this law reductio is no longer valid. Brouwer believed that truth must be justified by a constructive proof, that it actually produces what it claims is true (I imagine he considered that closer to our own intuition). In a sense, he's replacing the idea of truth with the idea of justification.
Though Hilberts formalist programme died, this didn't mean that intuitionistic logic prospered, rather it led something of an underground existance, but recently it has been establishing itself as an important part of mainstream mathematics & logic, via what is called Topos Theory, which is a generalised Set Theory built on structuralist principles. There isn't only a single unique Topos theory, there are many, and each has a so-called internal language/logic built within it, and this logic is intuitionistic; further, and this is important, each topos has a geometry.
A good historical analogy to think about is, how dropping the parallel axiom from euclidean geometry opened up a new world of non-euclidean geometries. Whereas, here we're talking about dropping the excluded middle, and getting a brave new world of non-classical logic, entangled with non-classical set theory & non-classical geometry.
But pure theory is all very well, after all non-euclidean geometry established itself with the success of General Relativity. Some applications I can point to would be:
Smooth Synthetic Geometry, which makes rigorous the idea of the infinitesimal, to be used in pretty much the way Newton & Leibniz did (and so dismissed by Bishop Berkely).
Recent papers by Chris Isham, a respected physicist working in Quantum Gravity, which looks at quantum theory in the context of topos theory. He makes the audacious point, that taking the principle of General Covariance seriously in General Relativity appears to deny the existence of points, that Quantum Field Theory is/has been plagued with nonsense answers for what appears to be the same reason (one of the motivations for String Theory is that point particles are expanded to strings), and his motivation for looking at Topoi, is that in their geometric incarnation, points are also denied.
This has nothing to do with reductio ad absurdum. The comparison is made to show that the claim of the existence of god cannot be falsified. I.e. one cannot disprove its existence, similarly as one cannot disprove the existence of fairies, unicorns, the flying spaghetti monster, the many world interpretation and the fact that you only exist in the imagination of someone else or of some computer code.
But reductio ad absurdum is a completely valid logical proof.
The technique is valid, but the application to this case is not. The problem is that the notion of God in religion is often presented as a magical prayer-answering entity which acts in arbitrary ways, inconsistent with the laws of physics, and undecidable by any type of verifiable experience. This notion is refuted by the flying spaghetti monster.
But the notion of God that religions use in day-to-day practice is a different more subtle notion of a law-giver God. This notion, the law-giver god, can't be refuted with such arguments, because it does not make magical claims, and it is not an arbitrary entity either meddling with or divorced from the universe, but a meta-entity which is partially and approximately formed by human beings behaving ethically.
Because of this, you can define the existence of God in a way that should be acceptable to a logical positivist, and makes no claims about creating the universe, or standing outside the universe meddling, or anything like that (although it explains why these pictures are used to illustrate the concept).
I wrote something about this on the Christianity website, although I am not sure if it will stay there: https://christianity.stackexchange.com/questions/6338/does-the-bible-espouse-moral-absolutism-or-moral-relativism/6660#6660
The terse version of the link is that monotheistic God is the extension to nonsymmetric games of Hofstadter's superrational strategy. God is the agent whose utility is maximized when the players play using the perfect superrational strategy for all games. It is a bit of an act of faith to believe such a strategy can be formulated, and it clearly requires super-human intelligence to find this strategy in all circumstances, but it is a qualitatively different leap of faith than assuming there is somebody listening to your prayers and doing magic to answer them.
