Why if we assume p and ¬p we can prove any statememt?

Question

Every proof I find is a truth table, could somebody explain it to me intuitively?

In: 1

Answer 1

very intuitively: Think of proof by contradiction. assuming p and !p lets you do proofs by contradiction on demand. “Assume unicorns are not real. We know p. we know !p. This is a contradiction. Therefore unicorns are real.”

A bit more formally:

assume the statement (p AND !p) is always true. Therefore, the statement (q OR (p AND !p)) is also always true for any q. Then by boolean algebra rules, ((q OR p) AND (q or !p)) is true.

so, either one of q and p are true, and either one of q and !p are true.

but if we assume that p is true, then !p in false. But (q OR !p) is true. since either q or !p is true, and !p is false, q must be true.