Is it possible for there to be a true contradiction? According to Aristotle, "one cannot say of something that it is and that it is not in the same respect and at the same time." This is known as the law of noncontradiction. In other words, given proposition P, P^~P (P and not P) will always be false. Accordingly, contradictions cannot be true.
I will contradict the law of noncontradiction below the fold.
Normally, I do think to question such a thing as the
law of noncontradiction. Like the law of identity, it just makes sense. So, imagine my surprise when I
came across this gem of a sentence: "This sentence is false." This is known as the
liar paradox.
The proposition is a dialetheia; for if the sentence is true, it is at the same time false.
The phrase I provided is inherently a dialetheia, but further thought lead me to other considerations. Nonspecific propositions can be both true and false at the same time. If I say "Shadowin is weak" and someone else said that "Shadowin is not weak," both of us could be correct. Shadowin could be weak in one sense but strong in another.
It seems clear that in a case where something is either absolutely true or absolutely false, then the law of noncontradiction holds up. However, if any ambiguity is introduced then we can throw that law out the door. This indicates to me that the law of noncontradiction needs to be defined in such a way that accounts for ambiguity.
As far as I can tell, classical logic is not set up to handle ambiguity. In fact, it is this weakness that allows rhetoric to destroy an otherwise well formulated argument. Non-axiom propositions are naturally weak, and are subject to attack.
However, there is no ambiguity in the liar paradox. It is a very clear, very certain contradiction. How can this be resolved?
I don't believe such a thing can be handled in a Boolean logic system. Therefore, I offer two potential solutions: reject the statement and others like it or deny the principle of bivalence (that everything is either true or false).
Since rejecting the statement is akin to rejecting pieces of our reality that disagree with our formal systems, I would rather reject principles that restrict my ability to analyze data if those principles are hindering me.
Luckily for us, the principle is already abandoned by other logic systems that contain more values than true or false. With this abandonment, we can state that the phrase, "this sentence is false" is neither true nor false. We can also state that the phrase "Shadowin is weak" is both truth and false. This usage has the advantage of making sense to our reasoning facilities.
***UPDATE***
It was pointed out to me by buzzsaw that the law of noncontradiction does in fact account for ambiguity by the "same respect" portion. It is only useful to point out ambiguous statements having the potential to be dialetheias when someone is arguing whether or not dialetheias exist.