The Power of Negation

I find it interesting how negation can be used in several different ways. For example, I am referring to double, implication, equivalence, and quantifier negation. Double negation is pretty straight forward in the sense that two negatives make a positive. ("I'm not not going" means "I'm going.") Implication negation can change an implication statement, such as "P implies Q", to a "not P or Q" statement, where there is no implication. As for equivalence negation, a statement
such as "not(P if and only if Q)" can be rewritten as "not(P implies Q) or not(Q implies P)". Although resulting in a much longer statement, the two are still equivalent. As for quantifier negation, I found this one interesting as it lets you shift between "for all" and "there exists" statements. I had some trouble with understanding this one at first, but I soon got the hang of it.


Side joke about double negation: (I just wanted to add this in for fun....)

Teacher: "A negative and a positive make a negative. Two negatives or two positives will always make a positive."
Student: "Yeah right."

(For those of you who didn't catch that, "Yeah" and "Right" are both positive, but together imply something negative. Thus counterexample. Haha.)