One aspect of deductive reasoning that may be encountered by students for the first time at the high school level is proof by contradiction. In this type of proof, the logical negation of a conjecture is taken as the premise. Then valid logical deductions from that premise are developed. If a logical contradiction, something that is both true and not true, is found, that shows that the premise is false.

Here is an example of proof by contradiction, based on a problem in Euclid's Elements.

The premise: If the angles ABC and ACD in triangle ABC are equal, then the sides AB and AC are also equal. To prove this by contradiction, assume the negative of our conjecture, namely that AB does not equal AC. Then one side must be longer; assume that the longer side is AB. If AB is longer than AC, there must be some point, D, on AB, such that DB=AC.

Now we have two triangles, ABC and DCB. Both triangles have BC in common, and since DB=AC, then angle DBC must equal angle ACB. The sides opposite these angles must be equal as well, that is DC = AB, but this is impossible, as DC cannot be equal to both AB and AC, if AB is longer than AC. A part of the whole cannot be equal to the whole. Therefore AB and AC are not unequal, they are equal.

Students should become comfortable with this type of proof, which builds in part on their earlier understanding that counterexamples can be used to show that a general conjecture does not hold. It is important for both teacher and student to keep the two concepts distinct, however. Proof by contradiction can be used to establish the validity of a conjecture. Whereas, a valid counterexample by itself simply shows that a conjecture does not hold. It disproves validity, but proves nothing.

  Inductive Reasoning