'Contradictions do not exist' is nonsense. Contradictions exist all over the place. Any time you get two people together you'll have contradictions due to viewpoint, experience, perspectives. Denying that contradictions exist is a child-like view of the world. Working with the contradictions of life, now that's the fun and the challenge, isn't it?
Of course different people will have beliefs that contradict each other, but by the definition of a contradiction, they can't both be right. Likewise, if two of your own beliefs end up contradicting each other in the end, then one of those beliefs was wrong. That's the meaning of the quote.
Just letting a contradiction sit there is the lazy thing to do--it's the resolving of contradictions that's the fun part!
"Anyone who denies the law of non-contradiction should be beaten and burned until he admits that to be beaten is not the same as not to be beaten, and to be burned is not the same as not to be burned."--Avicenna
A human brain is but a finite machine, therefore there are only finitely many propositions which you believe. Let us label these propositions p1, p2, ..., pn, where n is the number of propositions you believe. So you believe each of the propositions p1, p2, ..., pn. Yet, unless you are conceited, you know that you sometimes make mistakes, hence not everything you believe is true. Therefore, if you are not conceited, you know that at least one of the propositions, p1, p2, ..., pn is false. Yet you believe each of the propositions p1, p2, ..., pn.
