I started with the intepretation of my third logical value as "True or false, but it's not certain which". For negation, I could extend the normal table for true or false, and reasoned, The negation of "true or false" would be "false or true": If I don't know whether or not a statement is true, I also don't know whether or not its negation is true. So, if P = U, then ~P = U.
The table for "or", I decided "if P is true, and Q is either true or false, the value of P or Q doesn't depend on the truth of Q", so if P = T and Q = U, P v Q = U, and likewise with P and Q interchanged. U or U should be U, U or F should be U.
Similarly with the table for "and".
Actually, as I found out later, this turns to be an unfortunate interpretation, but the truth tables still work. There is another interpretation that works better, though.