Is there any three valued logic with this truth tables? T AND T = T
T AND U = T
T AND F = F
F AND T = F
F AND U = F
F AND F = F
U AND T = T 
U AND U = U 
U AND F = F

T OR T = T
T OR U = T
T OR F = T
F OR T = T
F OR U = F
F OR F = F
U OR T = T 
U OR U = U 
U OR F = F

NOT T = F
NOT U = U
NOT F = T

T =&gt; T = T
F =&gt; F = U
T =&gt; F = F
F =&gt; T = U


Add the remaining for implication:

U =&gt; U = U
T =&gt; U = U
F =&gt; U = U
U =&gt; T = U
U =&gt; F = U


I'm not very sure about the last two. But if I miss them before I guess it is because they don't add "knowledge".

The idea of U is to meant NOTHING or VOID a thing that doesn't provide any knowledge and from where we cannot obtain any information.

What is the worst paradox this logic could generate? 

Edit just to add another concrete interpretation I have when deriving the truth tables. Imaging an expression that runs forever. This espression is VOID because I'm not able to evaluate its truthness.

But probably this is total garbage. Maybe I'm just mixing inconsistents intuitions in my head. If this thing doesn't have a name I have little expectations...

Edit: After some time, my current me thinks the ultimate reason of the question is I don't like LEM standard intepretation. To divide the world in true and false and nothing else. 
Giving formal system limited powers to figure out thruth I find LEM inapropiate as an axiom.
Logically speaking, we should have an equivalence between 
$$P \implies Q $$
and
$$not  Q \implies not  P$$
but in your three valued one,

$$T \implies T ( =T)$$ is not equivalent to
$$F \implies F  (=U)$$
Logically speaking, we should have an equivalence between 
$$P \implies Q $$
and
$$not  Q \implies not  P$$
but in your three valued one,

$$T \implies T ( =T)$$ is not equivalent to
$$F \implies F  (=U)$$