I have this half memory of an article about the car guys, Tom and Ray Magliozzi, in which they talked about their experience at MIT. One of them had an assignment to show that something was possible. The answer was intended to be theoretical. Although the professor hadn't been explicit, the assumption was they the students would go fiddle with equations and axioms until they had proved that this particular thing was physically possible. Instead, the Magliozzi who was taking this class went home and built a physical object that performed the action in question.
Problem: Prove that X is possible.
Solution: Build a device that performs X.
Evaluation: But you were asked to prove X is possible.
Response: It's a proof by construction. That thing right in front of you does what you asked. X must be possible or our device couldn't do it.
At the time I just shook my head. How could this not be a convincing argument? Proof is a funny thing, but an existence proof can be satisfied by showing that at least one exists.
Of course, there are other systems of logic. Some of them seem silly to us. We think that we live in a Boolean world. We think we live in a world where a double-negative implies a positive. I scratched my head for days after hearing Atish explain the world he lived in. It was a world in which, as he would say convincingly, "if it is not true that that animal is not a cow, it does not mean that the animal must be a cow."
A dozen years later and I'm still trying to work that one out.
Kim and I are having a great deal of difficulty making sense of things. The usual laws of logic don't seem to apply. Yesterday was three months since Elena died. You'd think things would be getting easier. Some things are. Some things aren't. Kim was cautioned that this is the point where some aspects will get even harder to deal with.
Logic is one of them. You stand so firmly on logic only to find the foundation is shaky. Maybe our world isn't so Boolean. It was not true that Elena was not a healthy child. It seems though that this was not enough for it to be true that she was a healthy child. This excluding the excluded middle seems to be leaving an awful lot out.
We're having a harder time, though, with proof. It just doesn't seem possible for a healthy and happy six year old to die so suddenly. The fact that it happened still doesn't convince us that there is any way in which it could have. The existence proof is unconvincing.
Argument: It can't happen.
Reply: It did happen.
Evaluation: Somehow that doesn't prove that it could happen.