TITLE: Imaginary Number in Logic
QUESTION [11 upvotes]: 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.

REPLY [4 votes]: 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.