Ok since @littmath is currently obsessed... I'm going to tell a story about the cube root of true.

So there a well-known trick in propositional logic called arithmetization. Here is how it works:

You convert propositions into polynomials over FF_2 and observe that true values are respected:

T --- 1
F --- 0
!X --- x+1
X & Y --- xy
X or Y --- xy + x + y

You can do general propositions now in many variables now. This is actually a polynomial time process.

Lets look at an example:
X and (! X)
this becomes
x(x+1)
and you can notice that when you plug in 0 you get 0 and when you plug in 1 you get 0 (we are working in characteristic 2). This proposition is not satisfiable.

Asking for a satisfiability of the proposition X & (!X) is equivalent to looking for solutions of the polynomial equation x(x+1)=1. So satisfiability of this proposition is equivalent to looking for solutions of the polynomial equation x^2+x+1=0 over FF_2. (and now you see)

We all know this doesn't have any solutions over FF_2. But it does have solutions over
FF_4 =FF_2[zeta]=FF_2[x]/(x^2+x+1).

I propose that we make zeta the new truth value. This is the primitive cube root of true.

Now you can go blue in the face having fun with this.

A couple things...

1) Why was I thinking about this?

Maybe you could convert 3SAT into polynomial arithmetic over FF_2 and then use Grobner basis algorithms to prove P = NP. Grobner bases were sold to me as the solution to all of my computational problems at the time...

...all this ends up doing is proving that the polynomial ideal membership problem is NP complete; it provides another good example of why understanding the division algorithm is so fundamental in separating the complexity classes.

2) Well 4 truth values... that sounds a little bit like quantum mechanics with two entangled states 00 01 10 11.

I've never been able to make sense of this and sat down with a couple quantum information people in grad school and this never amounted to anything for me.

3) This is *not* a Heyting algebra. (so this isn't really a intuitionist/ no law of excluded middle thing)

4) It turns out that there is a branch of logic called paraconsistent logic which does this sort of stuff (Tom Scanlon pointed this out to me when I was a grad student). I never really followed it up after this. Maybe someone can say more about his.

5) I have no idea what the algebraic closure of FF_2 means logically.

I also have no idea if this idea is good for anything but it is sort of fun.