Claims of fact can sometimes be logically absurd, or can have implications that reasonable people should reject. And claims that don't quite contradict themselves can sometimes have logical implications that the person making the claim would never swallow. Exposing logical contradictions or otherwise unacceptable implications of certain claims thus offers another method of showing that those claims are false.
In classical logic, the method of "Reductio ad Absurdum" is an elegant way of showing that some particular claim is false by showing that it leads to a logically self-contradictory conclusion. This is based on the valid deductive form of modus tollens, in which a claim can be proved to be false by showing it implies something that is known to be false. Here, I'm also using the term "reductio" as a more-or-less descriptive term for any argument that tries to show that some claim is false by showing that, if it was true, some other, insanely far fetched claim would also have to be true. The idea is that, since insanely far fetched claims pretty much cannot be true, the original claim pretty much cannot be true either. I realize that "insanely far fetched" is a little different from "self-contradictory," so this might be easier to understand if you think about the similarites between deductive and inductive reductio arguments.
The main difference between deductive reductio and inductive arguments is that a good deductive reductio proves that the claim it is attacking is absolutely certainly to be false, whereas the good inductive reductio proves that the claim it is attacking is very, very likely to be false. (Of course, you should also consider all the arguments offered for that claim. Reductios only really definitively kick claims in the head when those claims turn out to be unsupported.) In this chapter I will deal mostly with deductive reductios, mainly because they're easier to find. If I come across an in ductive one, I will let you know.
Example 1
The classic "Reductio ad Absurdum" is a deductive argument that works by showing that some claim has mutually contradictory implications. For instance, suppose someone claims to have drawn a "squircle," which is a closed plane figure exactly following the edge of an exactly circular area with a line consiting of four straight segments of identical length joined by four right angles. Against this we can argue:
1. If a squircle exactly follows the edge of an exactly circular area, it is a circle.
2. If a squircle consits of four straight segments of identical length joined by four right angles, it is a square.
3. A squircle does both things, so a squircle is both a circle and a square.
4. Nothing can be both a circle and a square.
C. Squircles cannot exist.
Here, the reductio ad absurdum of the concept of a squircle starts by examining the deductive implications of the definition. First, one part of the definition is shown to imply that a squircle is a kind of circle. Second, another part of the definition is shown to imply that a squircle is a kind of square. Third, it is pointed out that a squircle, as defined, is both a circle and a square. Finally, it is pointed out that nothing can be both a circle and a square, so the conclusion, that squircles can't exist, naturally follows. Here it is as a version of Modus Tollens.
1. If squircles exist then square circles exist.
2. Square circles do not exist.
C. Squircles do not exist.
Example 2
Now suppose someone claims that his deity Vuntag is both perfectly good and will punish non-believers with eternal pain. From this, we can develop the following argument against the existance of Vuntag.
1. Vuntag is defined as being perfectly good and punishing non-believers with eternal pain.
2. Punishing people for not believing in you is morally wrong.
3. A perfectly good being never does anything morally wrong.
4. Vuntag does something morally wrong and never does anything morally wrong.
5. Nothing can both do something and never do it.
C. Vuntag doesn't exist.
The thing that makes this, just barely, an inductive argument is that morality is somewhat debatable. While I cannot imagine anyone believing that it is morally okay to beat up people who don't believe you exist, it is just barely possible, well conceivable, that some half-way reasonable person could believe this. (In which case he would believe that I had the right to beat him up if he didn't believe I existed.) So this is an inductive argument because it's not logically impossible for me to have the moral right to beat up people who don't believe in me. It's just extremely unlikely.
Reductio by Analogy
Sometimes, the easiest way to show that a claim is logically absurd or stupid, or that it is stupid or has logically absurd implications is to come up with an analogy that illustrates the stupidity or logical absurdity.
I call these arguments deductive analogies because they are fundementally about logical relationships between concepts. If one situation is logicically impossible, then every situation that is logically identical to it is also logically impossible. If a certain idea is self-contradictory, then every idea that has exactly the same logical structure will also be self-contradictory.
The utility of these arguments stems from a combination of two facts.
1. People occasionally believe things that are either incredibly unlikely or flat out logically impossible. People who believe absurd things are often quite unable to recognize the absurdity of their belief, even when presented with rationally compelling arguments.
2. The absurdity of a belief can sometimes be made apparant by putting the exact same logical structure into a totally different context, wherein the absurdity will stick out like a sore thumb at a pinkie convention.
To put it another way, someone believes something that is logically absurd, but cannot see the absurdity of his logic because he is used to believing that particular claim. You may be able to get through to him by showing how his logic will have absurd results when applied to other situations. So, if we can show that a certain claim is logically analogous to a some other, logically impossible claim, then we will have shown that the origical claim is itself logically impossible.
Suppose again that someone claims that his deity Vuntag is both perfectly good and will punish non-believers with eternal pain. Against this, we can develop the following analogy argument. First, we imagine that there exist three people: Albert, Brian and Clarisse. Clarisse is Brian's girlfriend, but Albert has never met her and, after six months of never meeting her, does not believe she exists. Then one day they meet, Albert discloses that he did not believe in Clarisse, and Clarisse commences to beat the holy living $&@% out of Albert. With this in mind, we can construct the following reductio.
1. Vuntag's definition implies it is okay for him to punish unbelievers.
2. If it's okay for Vuntag, it's okay for everyone, including Clarisse.
C. Either it is okay for Clarisse to beat up Albert, or Vuntag's definition is wrong.
Or as a classic Modus Tollens.
1. If it's okay for Vuntag to punish unbelievers, it's okay for Clarisse to punish unbelievers.
2. It's not okay for Clarisse to punish unbelievers.
C. It's not okay for Vuntag to punish unbelievers.
Notice that the motor of the argument here is the analogy. Once that is set up, the argument can be phrased a number of different ways. For this reason, the analogy between Clarisse and Vuntag must be taken as part of the argument, so a full standardization of the argument might look something like this:
1. Vuntag punishes people merely for not believing in him.
2. Clarisse punishes people merely for not believing in her.
C. Vuntag punishes unbelievers in exactly the same circumstances that Clarisse punishes unbelievers.
1. Vuntag punishes unbelievers in exactly the same circumstances that Clarisse punishes unbelievers.
2. Moral rules are universal. What applies to one being applies to all beings.
C. If it's okay for Vuntag to punish unbelievers, it's okay for Clarisse to punish unbelievers.
1. If it's okay for Vuntag to punish unbelievers, it's okay for Clarisse to punish unbelievers
2. It's not okay for Clarisse to punish unbelievers.
C. It's not okay for Vuntag to punish unbelievers.
Remember that the rule is, if expanding the argument makes it easier for you to evaluate it, then it might be a good idea. If expanding the argument doesn't make evaluation easier, then it might be a waste of time.
Wriggling On The Hook
Um, I mean "arguing against a deductive analogy." Okay, if you can show that the two situations are not logically isomorphic in the relevant way, then you will have shown that the analogy doesn't work. But beware! Just showing that the situations are different isn't enough. You have to show that they're different in a way that gets you off the hook. Otherwise you're just wriggling, wriggling like an impaled nightcrawler. Ew! Ew! Wrong metaphor, Ew.
And if you change the definition of the conclusion thingy, there's nothing to stop your opponant making the exact same changes to the premise thingy.
Jordon. That "Clarisse" argument is no good because Vuntag has authority to punish people, and Clarisse doesn't.
Mohamed. Okay then, lets make Clarisse a judge. Judges have authority to punish people. So by your logic Judge Clarisse would be morally entitled to punish people for not believing in her.
Jordon. What? No! I mean Vuntag has the authority to decide who should be punished and for what!
Mohamed. And he gets that authority from?
Jordon. He just has it!
Mohamed. Okay then, let's say that Clarissa just has the same authority, so ...
Jordon. No no no, your imaginary Clarissa can't have that authority!
Mohamed. If Vuntag can have it, she can have it.
Jordon. But Vuntag is the creator of the world, with the power to inflict eternal torment!
Mohamed. And Clarissa has the power to inflict a pretty good ass-kicking.
Jordon. That's not enough! It has to be eternal torment.
Mohamed. So then if Clarissa could inflict eternal torment on Albert, then it would be okay for her to do it to him for not believing in her.
Just so long as each version of the Clarisse situation is logically isomorphic to the relevant version of the Vuntag situation, the deductive analogy argument will keep working. The reason it works this way is that Mohamed is giving a series of direct arguments against the idea that a being can punish people merely for disbelieving and still be a perfectly good being. If any of Mohamed's arguments are good, then Jordon's belief about Vuntag is false. Not just unproven but just plain false.
Cutting Your Own Head Off
It is also possible to wriggle yourself into a position where you make yourself even worse off. This doesn't happen often, but when it does, it's sweet!
Jordon. Look, Vuntag is defined as a perfectly good being who punishes people who don't believe in him.
Mohamed. So, if vuntag exists, he absolutely has to be a perfectly good being who punishes people who don't believe in him?
Jordon. You've got it now!
Mohamed. So since the Clarisse example shows it's not logically possible to be a perfectly good being who punishes people who don't believe in him, and Vuntag is defined as such a being, your argument proves that Vuntag cannot possibly exist.
Jordon. What?
Mohamed. Logically impossible things can't exist. Vuntag is a logically impossible thing, so Vuntag can't possibly exist.
Noticing that by insisting that his definition has to be correct, Jordon has put himself in the position of implying that Vuntag is logically impossible, like a square circle. His definition of Vuntag thus amounts to a deductive proof that Vuntag doesn't exist.
Resolving the conflict
Jordon can escape the dilemma, but only at the cost of changing his definition of Vuntag.
Jordon. Okay, I guess Vuntag isn't perfectly good. He's just almost perfectly good. Vuntag is now defined as a being who is perfectly good except that he punishes people who don't believe in him.
Mohamed. And I guess that that version of Vuntag logically could possibly exist. So maybe I'll believe he exists, and ignore him otherwise.
Jordon. Oh no, you can't just ignore him, because Vuntag punishes people who break his laws.
Mohamed. Did I mention that Clarisse demands unquestioning obedience from Brian, and punishes him whenever he breaks her laws?.
And so it goes.
There are two points I want to make about this. The first is that applying this kind of argument constitutes a sort of moral courage test. Many people believe things that have bizarre logical implications, and these people sustain their peace of mind by studiously ignoring the bizarre implications of the things that they believe. When such people are confronted with a reductio of one of these beliefs, they can choose to accept and understand that their belief is not true, or at least that believing it commits them to believing something they hitherto found bizarre. Such acceptance usually requires a great deal of moral courage, and the more usual, and cowardly response, is to make excuses or to ignore the reductio entirely. While such moral cowardice can be harmless, it can also result in enormous amounts of completely unnecesssary suffering.
Secondly, reductio should not be confused with ridicule. Ridicule is the fallacy of thinking that just because some idea can be made fun of, that idea itself must be false. Reductio is the argument of showing that because a claim, if taken seriously, will logically turn out to have ridiculous implications. In order to make a reductio work, you have to show that the claim, as stated by it's believers, logically implies something absurd or stupid. Ridicule merely mocks the claim, and doesn't trace out any logical implications, so ridicule always fails to be reductio.
Reductio arguments are based on the fact that if a claim is true, then everything that claim logically implies will also be true. This is an ironclad law of logic. From this it also follows that if one of those implications turns out to be false, the claim that implies it will be false. Again, this is an ironclad law of logic, enshrined in the valid Modus Tollens argument form. So, if you can prove that claim A implies claim B, and prove that claim B is false, then you will have proved that claim A is false too. No "ifs," "ands" or buts." If it's true that A implies B, and that B is false, then A can only be false. Nyah, nyah, nyah!
Reductio arguments work because claims have to be logically possible in order to be true. If a claim implies a logical contradiction, then it is false, period. If a particular claim has logical implications that reasonable people have very good reason to reject, then reasonable people have very good reason to reject that particular claim as well.
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by Martin C. Young
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