Gödel gave an example of this by making a system of arithmetic generate statements about itself, and proved that it can't generate all possible truth statements (completeness) without generating some false ones (inconsistency). So, logic as cold machinery can start from the same axioms to prove that something is true in one path and then prove it false from another path. Logic is a world with its own rules that we can map reality onto sometimes but not always. The book Gödel's Proof does a good job describing this.

A larger and more devastating argument I've heard recently is that in order to even create logical statements, you need to be arguing from a worldview that can give an account for the existence of logic that isn't arbitrary (e.g. not "it just is"). And the argument goes that if you can't justify the existence of the tool, you can't justify its usage. This is devastating because if you believe it, then you suddenly must recognize that something prior to and higher than logic must exist in order to inform you of its existence, and it is not subject to the bounds of any logical system founded arbitrarily, but becomes the means by which logic itself coheres into something meaningful.

> How do you prove that something is both true and false

By displaying two different trees, both well-formed with respect to (nodes from and leaves that are axioms of) the logic, one of which shows that the thing is true and the other which shows it false.

(we don't usually care about this possibility because if we take care to use consistent logics, whenever we have two proofs P and Q with truth values V and W respectively, then V=W)

You're missing out, the article does give an example.