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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