Imaginary Number in Logic The equation $x^2 = -1$ was once said to have no solution. Then the number $i$ was discovered (or invented?) and our number system got richer. In particular, in this new wonderful world of complex numbers, we can prove the fundamental theorem of algebra, a consequence of which is that every polynomial is solvable in the complex domain.
In a similar vein, the logical statement $P = \lnot P$ has no solution in the set $\{True, False\}$. This is the well known liar's paradox, and has appeared in various forms throughout logic e.g. Russel's paradox, Godel's incompleteness theorem.
Now say we invent a new logical value $iTrue$, an imaginary one if you like, that is defined as the solution of the equation $P = \lnot P$. Our propositions would then range over $\{True, False, iTrue\}$.
My Question: Could this new system, or one like it, free us from such paradoxical, unsolvable logical statements in an analogous way to the way complex numbers freed us from unsolvable polynomials?
My Thoughts: I am skeptical of the above system. I feel that it may help us "solve" some paradoxes but not all. Furthermore it is not obvious what these solutions would mean. On the other hand, it would be lovely if the analogy with complex numbers actually worked on a more rigorous footing.
 A: There is a vast literature on many-valued logics. Lukasiewicz's original 3-valued logic is perhaps the simplest such logic and the extra truth value $P$ satisfies your formula $P \iff \lnot P$. Lukasiewicz developed this into what are now called Lukasiewicz logics that have been intensively studied and generalised over the years. One of these generalisations is the subject of fuzzy logic, which has practical applications, e.g. to model situations where knowledge is imperfect.
See  http://plato.stanford.edu/entries/lukasiewicz/ for more information about Lukasiewicz's work and http://plato.stanford.edu/entries/logic-manyvalued/ for a survey of many-valued logics. 
A: You may be underestimating what was involved in the creation of the complex number system. It's not just a matter of introducing a symbol $i$ and declaring that $i^2=-1$. Rather, it's important that much of what people knew about real numbers also applies to complex numbers; specifically, $\mathbb C$ is a field of characteristic zero (though not an ordered field). So people could continue to manipulate equations (though not inequalities) involving complex numbers just as they did with real numbers.  It turned out that the properties of real numbers that persist when one passes to complex numbers are enough to give an interesting and useful theory (and later, people found additional properties of $\mathbb C$, like algebraic closure, that make it  even more interesting and useful).
Analogously, it would do little good to introduce a new truth value and declare it to be equal to its negation. One would need to show that the new system of truth values retains enough of the properties of the traditional system so that people can continue to reason more or less as they are accustomed to do.
A: As Rob Arthan says, there's a vast literature on this topic, only some of it in application to the paradoxes.
As a solution to paradoxes like the liar, though, the introduction of a third truth-value meets an immediate difficulty, the so-called "strengthened liar":
This sentence is not true.
A: Your introduction to the question seems to be very similar than the thoughts of L. H. Kauffman in Virtual Logic and Imaginary Values. Either you were inspired by them or it could be worth checking them.
Regarding your question, it seems that you are looking for a logic where the Law of Excluded Middle is not an axiom. So, I would suggest to check the field of paraconsitent logic.
