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Good Math by Mark C. Chu-Carroll

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A Proof from Nothing

Now that we’ve seen all of the sequents, how can we use them?

Let’s start with a proof of a simple but fundamental rule of logic called the law of the excluded middle. It says that any statement must be either true or false. When you write that in logic, it means that if you have a statement A, then regardless of what A is, A ¬A must be true.

The law of the excluded middle is a tautology, which means that it’s a fundamental truth that must always be true, regardless of what axioms we choose. In order to show that the tautology is true, we need to build a proof that uses nothing but the statement, the logic’s rules of inference. If we can derive a proof of A ¬ A with no premises, we can show that it’s a universal truth. ...

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