Many people, college professors included, tell me that false statements do not imply contradictions. They instead babble on about how only necessarily false statements imply contradictions. They tell me statements such as "Babe Ruth is Governor of Massachusetts" does not imply a contradiction, even though it's false. They say a world can be conceived in which "Babe Ruth is Governor of Massachusetts," and that therefore it doesn't imply a contradiction. They will even go on to say that in order to have a contradiction, you must have two statements. Well, I disagree. If a statement is false in a world, then it also implies a contradiction in that world. These people seem to be considering the case where "Babe Ruth is Governor of Massachusetts" is true and not false. They fail to realize that I'm not doing that! I'm considering the case where it's false! When a statement is false, it will imply a contradiction, as the last two rows of a truth table show.
To put it formally:
Consider the truth table for logical implication.
Notice that for a false statement P, the last two rows of the truth table, both Q and ~Q follow. No matter what Q is, it's truth follows from false statement P, as the third row shows. We can therefore take Q to be "P is true." From here it follows that a false statement P implies it's own truth, as the third row shows.
Do false statements really imply their own truth? Do they really imply contradictions? Are false statements also true?
False. A false statement that implies a contradiction does not mean that the false statement originally asserted is true. It asserts that the negation of the false statement is true.
~P =Df. (P->contradiction)
But you are somewhat right. A false statement can imply a contradiction, but if it does then the negation of it is true.
Let us take P to mean "Babe Ruth is the governor of Massachusetts".
If P is false in a world w, then the negation of the statement ~P, is true in that world. However, P can be true at other worlds (u,v,x,....n) where its negation is false. So P-><>P. If a statement is necessarily false, then there is no world (or the given domain of worlds) in which that statement is true. So in this case if a necessarily false statement is true at a world w, then there is a contradiction.
Hope this clears things up.