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Subsection 8.3.2 Reductio ad Absurdum

The Proof by Contradiction technique that we just described is a special case of a more general reasoning strategy called reductio ad absurdum. (Translate this literally as, “reduce to absurdity”.) We can use this more general strategy in everyday rhetoric as well as in mathematics.

The earth can’t be flat; if it were, people would be falling off the edges.

But people aren’t falling off the edges. The conclusion doesn’t match our experience. It feels absurd.

We have to hire more inspectors. If we don’t, contaminated food will make it to the supermarkets.

But we can’t accept that. So we must do something to make the contaminated food conclusion false.

The square root of 2 is irrational. If it were rational, we’d have a mathematical contradiction.

We can’t accept such a contradiction, since, from False, we could then derive anything and mathematics would no longer be a useful tool.

When we use reductio ad absurdum arguments in everyday reasoning, we depend on our listeners sharing our premises, at least enough so that they agree with us that some conclusion is clearly false or clearly unacceptable.

When we use this strategy (generally then called proof by contradiction) in mathematics, we assume that we have started with some agreed upon set of axioms. So we simply reason until we find a logical contradiction, i.e., some expression that evaluates to False.

In the next several problems, we’ll present everyday arguments that we’ll evaluate. Then we’ll consider some examples from mathematics

Exercises Exercises

Exercise Group.

1. Label each of the following arguments as one of:

  1. Convincing to most people.

  2. Not convincing since the ascribed conclusion (we’ll lose our market share) isn’t particularly bad or absurd.

  3. Not convincing because there’s at least one assumption that isn’t justified.

Part 1.

We have to lower our prices. If we don’t, we’ll lose our market share because our competitors already slashed their prices.

Answer.
Correct answer is A.
Solution.
Part 2.

We have to pave over the front lawn. If we don’t, the weeds will come back in the spring.

Answer.

Correct answer is B.

Solution.

Explanation: While people don’t generally like weeds, most people would like a paved over front lawn even less.

(Part 3).

That door must be metal. If it weren’t, all those magnets would have fallen off.

Answer.
Correct answer is A.
Solution.
(Part 4).

That new soda, Flash Gulp, must be low calorie. If it weren’t, the glamorous actresses who advertise it would be fat.

Answer.
Correct answer is C
Solution.
Explanantion: There’s no reason to believe that the actresses actually drink the soda that they advertise.