Making Sense of Proof by Contradiction [pdf]

For me, proof by contradiction only clicked (recently!) once I understood that logical consequence and unsatisfiability are equivalent.

Once I understood that and reframed the contradiction as a statement about unsatisfiability… I could then see directly how the positive result you get is the equivalent logical consequence.

Unfortunately, I feel like this intuition only really helps if you are pretty immersed in formal logic… otherwise it just sounds like jibberish.

Worth noting, a lot of times, what people think is proof by contradiction is in fact proving the contrapositive (i.e., if you want to prove, “if p then q”, proving “if not q then not p” will also suffice).

It's interesting that even a child can do it, but actually explaining it clearly gets confusing. One problem is that as soon as you use "Suppose A then following steps S we get not A", but a hidden, implied premise is the stipulation that the world you are reasoning about already has certain consistency properties. This premise is what trips people (students like me) up because it is not part of the rules of algebra, geometry, etc.

This reminds me of the first time I was shown this in college.

I loved this method so much that in my first formal logic test I tried to solve all of the problems via this method. It was a fun experience lol