> One of those is the law of the excluded middle, which says something can't be true and false at the same time.

That would be the law of non-contradiction (LNC). The law of the excluded middle (LEM) says that for every proposition it is true or its negation is true.

LEM: For all p, p or not p.

LNC: For all p, not (p and not p).

Classical logic satisfies both, intuitionistic logic only satisfies LNC.