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Reductio ad Absurdum - Use with Caution

You have probably heard the term, "reductio ad absurdum." It is Latin for "reduction to the absurd," and is also known as an apagogical argument, or proof by contradiction. It is form of argument that assumes a claim and then derives an absurd (incorrect) outcome in order to show that the claim is false.

For example, suppose a man says "more money always leads to better education." Using reductio ad absurdum, we assume the claim and logically conclude, "there should be a direct correlation between expenditures in school systems and academic performance." In reality, we find that the United States, which spends far more per student than most countries, has lower-scoring students, and in the world in general there is little correlation between expenditures and results.

Another example: Many people think that raising tax rates always brings in more revenue for governments. This claim can be easily shown as false using a simple "reductio" example: Imagine raising tax rates to 100% of personal and business income. Who would work if all (or even most) of their income was taken? Businesses could not or would not operate. Less production would obviously mean less to tax, and could easily mean lower total revenue.

With a couple examples like these, you can see the value in using this type of argument. In these cases, it quickly suggests that how money is spent in schools can be more important than how much is spent, and that an extreme tax rate can mean less money collected - both important points. With logical "reduction to the absurd," we often don't even need to gather evidence to show the falsity of a claim. As with the second example, the shared knowledge and logical-thinking ability of the people involved in a discussion is enough.

Reductio ad absurdum makes use of the law of non-contradiction, which says that a statement cannot be both true and false. The statement, "higher tax rates always raise more revenue," cannot be true if we can point to an example where higher tax rates cause lower revenue collection.

This type of argument is often used in a "weak" form, where a person just demonstrates that a proposition leads to a result listeners probably won't like. For example, the belief that people have a right to own any weapon can be argued against by pointing out that this would include large explosive devices. It may convince listeners, who don't want such things in the house next door, but in terms of logic it's weak, since it can be refuted by simply saying, "Yes, I'm okay with them having those weapons."

There is a common feeling that reductio ad absurdum is a "silly" or incorrect way to argue. This is partly because it's often used poorly, and to "disprove" popular ideas. For example, a person might say, "If you think welfare programs are good for a society, why don't we put everyone on welfare?" This, of course, makes assumptions not made by the proposer (like the assumption that if something is good, more is better). It is a poor use of a reductio argument, but this doesn't mean that the technique itself is flawed. Logicians will tell you that a properly constructed reductio constitutes a correct argument.

For example, suppose you agree that to steal is to "take the property of another without permission," and that to steal is always wrong. Then you tell me that you think government support of the arts is morally okay. I can, by reductio ad absurdum, point out that this involves taking property without the owner's permission. After all, the taxes used for this purpose are not taken with permission. Many of us only pay under threat of imprisonment.

Now, you may not like the conclusion that public funding of art involves stealing, but like it or not, it is a strong argument because it fits your own agreed upon definitions. At this point, in order to disagree in a logical way, you might have to redefine "stealing" or change your belief that it is always wrong. Alternately, you could work with the definitions of "permission" or "ownership."

This is what happens when we use logic in real life, where definitions are not like they are in mathematics. We have to constantly redefine or words and beliefs as reality presents us with scenarios that don't fit them. Alternately, we can accept the conclusions logic points to, even if we don't like them (I actually think it is theft to force a man to pay for another's art through taxation).

We should use reductio ad absurdum as one of our tools to get at the truth. But we should also remember that verbal formulas can easily mislead us. In fact, contrary to what the law of non-contradiction says, a statement like "John is a good man," can be both true and false. This is because there are differing definitions of "good" (and even "John").

As a result of this inevitable imprecision of words, error - or the misrepresentation of truth - is probably a hundred times more frequent in the logic of common language than in mathematics. That's something to keep in mind before we become too confident in our ideas, and too dismissive of other's.

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