Based on the explanations here  I know that 3-valued first order logic has many different versions, which differ in the definition of their operations (e.g. implication). All of these (as far as I know) are complete, i.e. any probable statement could be proven. Also these are (just as in the 2-valued case are Semi-decidable, i.e. the correctness of statement given axioms, could be proved, also wrong statements could not be disproved.) Also these are consistent, i.e. they don't contain contradiction.
Suppose I define my arbitrary 3-valued first order logic (by arbitrary, I mean, arbitrary $\Rightarrow$, AND, OR). Is it true that, it is
for any $\Rightarrow$, AND, OR operations?