Mathematical Proof and the Principles of Mathematics/Logic/Non-classical logic

We've been requiring that mathematical statements be either true or false. But there are many statements in mathematics which we don't know whether they are true. So it seems reasonable to ask what happens if this basic assumption is really justified. Non-classical logics ask what happens if the true/false values of statements are not required and rules of inference are modified accordingly. There is quite a menagerie of these logical systems and we list some of the more important ones here.

Intuitionistic logic

This type of logic rejects the method of indirect proof. In our system this amounts to saying that the rule of inference

From not not $P$ deduce $P$

is not valid.

Fuzzy logic

In this type of logic, instead of statements have an absolute truth value they have a degree of certainty which ranges from 0 (for false) to 1 (for true). This type of logic has found applications in control theory and artificial intelligence.

Modal logic

This type of logic attempts to capture the difference between statements which are necessarily true and those which happen to be true. For instance, the statement "Donald Trump won the 2016 U.S. Presidential election," is true, but you can imagine a parallel universe where it is false, so it's not considered necessarily true. On the other can the statement "2+2=4" would be true in any parallel universe, so it's considered necessarily true.

Quantum logic

This type of attempts to model the statements of quantum physics. Certain measurements in quantum physics, such as position and momentum, are complimentary, meaning they can be made individually but not together. This means that statements such as "The momentum of the particle P is x." and "The position of the particle P is y." do not behave as expected with respect to the usual rules of logic.