# Is reductio ad absurdum a valid logical proof?

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?

• A good reductio can help clarify the (il)logical structure of an argument -- it is primarily a tool to encourage the use of reason and, generally speaking, is not intended to mock – Joseph Weissman Mar 27 '12 at 13:15
• The religion/fairy example in your question is not one of reductio ad absurdum; it's but the usual strawman. – Pacerier Sep 16 '14 at 21:59

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 absurdumreductio 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)


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].

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 Without this law reductio is no longer valid.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:

1. 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).

2. 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 is fine, and correct, except I take issue with the statement that Hilbert's formalist program died. It did die, but only because of WWII, because it was centered in Germany, and some of its major practitioners were either shunned or killed after the war. The program is alive in the ordinal analysis field of logic, which is active in Germany today. Gentzen famously proved the consistency of PA using methods that Hilbert considered finitistic, and I agree with him. The methods include transfinite induction up to a countable computable ordinal, one which can be represented on a computer. – Ron Maimon Apr 1 '12 at 8:53
• Also, the motivation Chris Isham gives for Topoi is not so great, the principle of GR does deny points, but not really strings, because strings are not exactly continuum objects, their spacetime can be reconstructed from discrete objects in Matrix Theory (and in triangulated 2D string theory too). Topoi can be motivated by formalism, as consistent universes of set theory, formulated in a way that can appeal to non-logicians. – Ron Maimon Apr 1 '12 at 8:56
• In what sense is general covariance not realizable in general relativity? Is it ment in the sense that the statements is about neighborhoods and the "realization" in (Pseudo-)Riemanngeometry really only talks about points and about dropping the first order coefficients of the metric "locally"? Is a topos theory formulation of the axioms of quantum mechanics available and accessible? Lastly, how important is it to drop points. One would expect that more people would deal with QFTs without points if it would be acknowledged that points are not the way to go! – Nikolaj-K Apr 11 '12 at 8:32
• @Nick:General covariance IS realised in GR - that is invariance under active diffeomorphism. I'm not sure what you mean by 'realisation' and dropping the first order coefficients. For a discussion by nlab of Ishams quantum theory in a topos context, see golem.ph.utexas.edu/category/2007/03/…. Removing points is an interesting move, but its too early to say how important it is. Another formalism that does without is Alain Connes Non-commutative geometry, and he does apply this machinary to physics. – Mozibur Ullah Apr 12 '12 at 16:04
• cont: An early advocate of using categorical/topos methods in physics is the physicist John Baez, his blog math.ucr.edu/home/baez/twfcontents.html covers this and much more. A good paper to look at is his arxiv.org/abs/math/0004133 where he categorifies the archetypal Quantum Oscillator and points out an explanation of its physics in terms of simple combinatorics. – Mozibur Ullah Apr 12 '12 at 16:17

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.

• Reductio doesn't "refute" God, it highlights the supposed absurdity in believing in God using a particular set of justifications. In the case of the Christian God, opponents claim there are no "real" justifications for God's existence, and FSM parodies that by invoking its own God with supposedly equally ludicrous justifications, such as its own Bible professing the truth of the Flying Spaghetti Monster, or the mere "word" of hundreds of thousands of followers. – stoicfury Mar 29 '12 at 15:46
• "The reason God has millions of followers is because the notion of God is not really magical or supernatural, but a stand in for something reasonable, which is a big, smart, agent in charge of universal ethics." [Citation required] – stoicfury Mar 29 '12 at 17:25
• God, by definition, is a supernatural being. So you should use a different term to indicate "a big, smart, agent in charge of universal ethics" because not only is that description woefully inadequate to describe God, but the very notion of God itself is "beyond nature" (i.e. supernatural). In other words, saying something like "supernatural idea of [the Abrahamic] God" is redundant; it's like saying "peanuty peanut-butter" or "watery water". – stoicfury Mar 30 '12 at 19:08
• "Nobody serious seriously believes that the Red Sea parted literally, or that water turned into wine..." No offense, but you've got a lot left to experience in the religious world if you think that's true. A lot of people believe those events actually occurred and are not simply embellished fairy tales. I can point out two dozen people standing not 30 feet from me that believe in the literal interpretation of a huge portion of the Bible, and I live in a very intellectual (and liberal) part of the US, no where near the Bible belt. But let's just agree to disagree, as this is going nowhere... – stoicfury Mar 31 '12 at 15:20
• @stoicfury: Yes, there are such people, but their opinion may be ignored, as they are deluded. – Ron Maimon Apr 1 '12 at 8:43