Classical propositional logic is consistent and in conformity with human language. A formal statement is true or not true and it is possible to develope rules with which it is possible decide which statements are true or not. It is a consistent system for classifying formal statements.
Mathematical statements could be unconditionally true $(x\notin\emptyset)$ named 1, conditionally true $(x\in S)$ named 2 or unconditionally false $(x\in\emptyset)$ named 0. I'm interested to find out if there are consistent and useful systems for classifying mathematical statements with these three values with likewise good conformity with human language.
I thought there where no problems with $\wedge$, $\vee$ or $\iff$:
or 0 1 2
0 0 1 2
1 1 1 1
2 2 1 2
and 0 1 2
0 0 0 0
1 0 1 2
2 0 2 2
iff 0 1 2
0 1 0 2
1 0 1 2
2 2 2 2
But as skyking commented the combination (2,2) might result in 0,1 or 2 depending on the statements. However the result of (2,2) could maybe be $0+1+2$ ($+$ is exclusive or)?
Is it possible to define $\neg$, $\implies$ and $+$ in a consistent way using corresponding words in human language as boundary conditions? That is, so that the connectives are consistent but not contra intuitive?
As mentioned by skyking, your definition does not work. Take any statement $P$ that is not provable (Your example of "$x \in S$" is one if you allow non-sentences). Then $( P \lor \neg P )$ is unconditionally true, but you would have assigned it the 'truth-value' $2$. So your definition is self contradictory.
However, what you seem to want is Kleene's 3-valued logic, where the third truth value is intended to represent "unknown". Then your truth tables match. But this is still not about conditional truth. (You cannot allow non-sentences to be statements, otherwise you will have inconsistency.)