Paradox

In an logical Space S(A) based on the Axiom Set A if we find a statement x s.t.  x  =  ¬  x
or  x  ⇔ ¬  x then  x is called a paradox.

There are so many paradoxes :-

1. Self-Reference : x =  ¬ x s.t "This statement is false." :)
2. Vicious circularity, or infinite regress : 
         "What happens when Pinocchio says, 'My nose will grow now'?" [from Wikipedia ]

3. Russell paradox: S = {x| x ∉ S} is there any element in S . 

4. Proof that Almighty doesn't exist : Let A =Almighty exist and it has all power .
                                                           So it must have power P defined as who have P he will be 
                                                           destroyed .
                                                           Thus A is a paradox.

Principal Of Exclusion : We should define Our Logical Space / Language s.t. Paradox does not exist. So if you have a Language  where some statement can be a Paradox then exclude such statements from definition and you have to reduce your Language space such that no other statement can conclude by logical consequence in a paradox . 

A language with a paradox is not a logical language and it is something which can't lead us to truth . So we must see some logical language . 


Now one question is there can we describe "Paradox" in a logical language . :) 
Can we describe the illogical language in such logical language say L .
If we bind our self by logic can we tell that there is a out of logic universe or universe of error .