Reductios

Particular counterexamples can normally be used to refute claims only if those claims are universal, so how can we refute claims that are not univer- sal? One method is to show that the claim to be refuted implies something that is ridiculous or absurd in ways that are independent of any particular counterexample. This mode of refutation is called a reductio ad absurdum, which means a reduction to absurdity. Reductios, as they are called for short, can refute many different kinds of propositions. They are sometimes directed at a premise in an argument, but they can also be used to refute a conclusion. This method of refutation will not show exactly what is wrong with the argument for that conclusion, but it will show that something is wrong with the argument, because it cannot be sound if its conclusion is false. That might be enough in some situations.

For example, suppose someone argues that because there is a tallest mountain and a heaviest human, there must also be a largest integer. We might respond by arguing as follows: Suppose there is a largest integer. Call it N. Since N is an integer, N + 1 is also an integer. Moreover, N + 1 is larger than N. But it is absurd to think that any integer is larger than the largest integer. Therefore, our supposition—that there is a largest integer—must be false.

In this mathematical example a contradiction is derived, but absurdity also comes in other forms. Suppose a neighbor tells a parent, “The local public schools are so bad that you ought to send your kids to private school,” and the parent responds, “Do you think I’m rich?” The point of this rhetorical question is that it is absurd to think that the parent is rich, presumably because of her lifestyle or house, which the neighbor can easily see. Without being rich, the parent cannot afford a private school, so the neighbor’s advice is useless.

Often the absurdity is derived indirectly. A wonderful example occurred in the English parliamentary debate on capital punishment. One member of

3. When theologians claim that God can do anything, atheists sometimes respond that God cannot make a stone that is so large that God cannot lift it, or that God cannot make a circle with four sides. Are these really counterexamples to the theologians’ claim? Why or why not?

4. Suppose there are only two balls in a bag, and someone claims, “Most of the balls in the bag are red.” Does a single black ball in the bag refute this claim? Is it a counterexample to this claim? Is it also a counterexample to the claim that no counterexamples can refute any claim that is not universal?

5. When people today respond to counterexamples by saying, “That’s the exception that proves the rule,” they usually do not mean, “That’s the exception that tests the rule,” which was its original meaning. What do they mean?