Axiomatic Systems: Does Boolean Logic Apply?

[Manchot]

I was just thinking about something recently, and I'm now posing a question. It seems to me that in all mathematical proofs I've seen, there is an implicit assumption that the rules of Boolean logic are what are "logical." That is, if you have two mathematical statements P and Q, you assume that ~(P and Q) is the same thing as ~P or ~Q, and that P and (P implies Q) is the same thing as Q, etc. Is it possible to create other "mathematics" where these standard Boolean rules do not apply?

 

[DeadWolfe]

Yes. For example:

http://en.wikipedia.org/wiki/Constructive_mathematics