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.
I feel like the article “c is not a low level language” is worth linking here.
Once you start to dig into how computer hardware really works, you start to realize that nothing is really O(1).
If you’ve never looked into computer hardware, you’d believe that lots of the primitive machine operations like reading and writing a value are O(1). But on modern hardware, they can be far slower. And that’s before we even get into out of order instructions etc.
So by the authors definition, I guess assembly and machine code aren’t real systems languages ;)
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.