There are two types of logic with contradictions that I know of, one is called paraconsistent logic, and was invented in Brazil, it says that when you hit a contradiction you stop. The other is relevance logic, invented in Australia, which basically allows you to prove any finite amount of things. In fact Boolean logic also deals with contradictions but says if you hit a contradiction then you prove the opposite or negation. Boolean logic is the standard logic of mathematics.
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What have you studied?
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