The basis for the whole mathematics is a set of axioms of logic. The mathematics that we usually study is based on binary logic, its axioms can be read in the first chapter of the first book of Bourbaki. For me only conception is important. Basically, everything reduces to deductive reasoning: If (A is true) and ( (from A follows B) is true) then (B is true).

Then, axiomatic construction of any theory goes as follows: we declare what statements are true (axioms) and start to deduct about others statements. All the statements in the theory can be: true, false or such that it is impossible to conclude whether they are true or false (cf Godel theorem). The task of mathematician is to decompose all statements in these three classes. And only subclass of true/false statements is what we can learn from the theory.

This was a binary logic, it is formalization of our every-day experience. My principal question is whether we can build other types of logics? I actually afraid that I cannot formulate the question properly because English also assumes binary logic. So my question assumes an answer: yes/no/we don't know. And probably in something else than binary logic my question is gibberish!

I have some basic knowledge about other logics:

1) -ary logic: each statement can be in one of states or in "don't know" state. Then we have to define more complicated operation than deduction. In fact, it is better to represent deduction as follows: if A, B than A&B. And deduction will be consequence of A, ( (not A) & B) -> A&(not A & B)=B. So it is (not A), basic unary operation, and (A&B), basic binary operation, which have to be generalized.

2) "quantum logic": each statement is an element of Hilbert space (if you wish, each statement is a point in )

Apart of idea that these constructions exist and certainly were studied, I know nothing more about them.

Finally, here are my three more precise questions:

A. Do you know nice explicit examples where 1) or 2) are really used? Short but meaningful answers will be very appreciated (link to 100-pages text is not a short answer, link to 1-page good text is a short answer).

B. I think that 1) and 2) can be reduced to simple binary logic. Do you agree or even know a theorem?

C. If what I think in B is true, then my question: can we invent something else, essentially different from binary logic?