Logic and Proof
Introduction to :
One way to modify a statement is to negate it. This just means to say the statement isn't true (which, in case you were wondering, isn't the same as having an argument). For example, the negation of, "Bacon is delicious," would be, "It is not true that bacon is delicious," or more simply, "Bacon is not delicious." That's a silly example to use though, because everyone knows bacon is delicious.
Mathematical negation differs a bit from our negation in everyday English. For example, if you say to your friend in a heated argument, "I don't think Edward can even compare to that dreamy Jacob," you really mean, "I think Edward is ugly and Jacob is a were-hottie."
However, when a mathematician says, "I don't think Edward can even compare to that dreamy Jacob," she is actually saying that the two cannot be compared. Who knows what she actually thinks? She may think they are both ugly, or that neither is actually hot, but that each is attractive in their own unique way. (Actually, she is probably busy working on some groundbreaking theorem while everybody around her is pestering her with vampire gossip.)
The bottom line is that negations don't allow us to conclude very much.
Even the simple act of negation can get really complicated really quickly. Consider the statement, "This statement is not true." Is it true, or isn't it? If it's true, then we can trust what it says. But… that means that it's false. Similarly, when it's false, that means it's true. There's no way to make any sense of this madness! What ever will we do?
While this seems like a fluke, it has pestered both linguists and logicians for centuries. Modern mathematicians have developed a formal way of avoiding such conundrums by putting up restrictions on the kind of statements they can make (another victory for censorship). We should probably do the same, since statements like that just cause headaches.