Here's a nerdy demonstration of how (in binary, mathematical logic, which is useful for some things and not very useful for others) a contradiction can be used to show that anything is true. SO: let's assume a contradiction (P and not-P), and use it to prove Q, which could stand for any proposition.
  1. P and not-P
    We just assume this! P is true and not-P is true! (Saying that not-P is true is equivalent to saying that P is false.)
  2. P
    This follows from 1. According to 1, both P and not-P are true. So P is true.
  3. P or Q
    This follows from 2. The "or" here is a disjunctive or—(P or Q) is true whenever P is true, and it's true whenever Q is true. It's only false if both P and Q are false. According to 2, P is true. And whenever P is true, (P or Q) is also true.
  4. not-P
    This also follows from 1.
  5. Q
    This follows from 3 and 4. According to 3, (P or Q) is true. And, according to 4, P is false. If Q were ALSO false, then (P or Q) would be false, (which it's not, according to 3). SO, given that (P or Q) is true and P is false.... Q has to be true. So... Q. We proved Q. We did it. We started with a contradiction that was just about P, and used mathematical logic to show that Q is true. And Q could be ANYTHING. Contradictions entail EVERYTHING. Watch out.