Proving a true statement
February 16th, 2006Have you ever tried to prove something indirectly? I am pretty sure you are doing it all day long. According to Wikipedia, see Wikipedia: Indirect Proof, it is as simple as the following:
If you have no water, you can’t make coffee.
This seems to be undoubtedly true to me, at least since adolescence. And it is because water is one of the essential ingredients for coffee.
Let us apply the nice rules of Indirect Proving to another statement that is true.
First the statement itself:
This statement is true.Now, we assume that it is wrong (meaning “we have no water”). Then we will see if this leads to a contradiction (or something as unbearable as “being unable to make more coffee”).
If the statement is wrong, that is
“This statement is true” is wrong.
it follows that This statement is wrong. Because of this, saying the statement is wrong, it follows that The statement is true. But this contradicts our assumption.
Witty readers probably see that the proof did not precisely lead to a contradiction but to a paradox. Let me suggest that for a moment, we adopt this paradoxical situation here as being as futile as a contradiction.
Well, here is another example of an often heard, and certainly true statement right for you to try out what we have just learned:
I am saying the truth.
. . .
