P l a n e G e o m e t r y
An Adventure in Language and Logic
PROOF BY CONTRADICTION
Book I. Proposition 6
Suppose, then, we want to conclude that statement a is true. Then we take as our hypothesis that a is false, and we see the consequences of that. We show in fact how that leads to an absurd conclusion -- that 2 is less than 1, for example. But when the conclusion is false, the hypothesis must be false. (Introduction to logic, Problem 19c.) Statement a, therefore, is not false; it is true.
Historically, proof by contradiction was known as reductio ad absurdam.
PROPOSITION 6. THEOREM
From now on, we will conclude a theorem with the traditional Q.E.D., and a problem with Q.E.F.
Hence to prove that AB is equal to AC, we show that its contradiction -- AB is not equal to AC -- leads to an absurdity. This is proof by contradiction. A direct proof of this proposition will have to wait until we can both bisect an angle and prove congruence by Side-angle-angle.
This proposition provides a third way of knowing that straight lines are equal. Do you recall the other two ways? (See Problem 4.)
Please "turn" the page and do some Problems.
Continue on to the next proposition.
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Copyright © 2012 Lawrence Spector
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