Algorithm for determining if a statement is a paradox
So, there are a few problems with this.


First, what is your formal definition of a paradox? Is it just a statement that can be found to be both true and false in some system? Paradox tends to be a bit of an informal term, so in CS we usually just refer to this as a contradiction, or an unsound logical system.

For example, Russel's paradox isn't actually a paradox, it's a proof by contradiction. We assume a premise i.e. "any set characterized by a logical proposition exists.", and show that we can prove falsehood from it, thus concluding that the premise must be false. (In intuitionist logic, this is actually how falsehood is defined).
How are you encoding your logic statements? Traditional propositional logic doesn't give much in the way for describing self reference, so you probably need something higher order. Starting with natural language statements and converting them into logic is a whole separate problem.
Your problem seems undecidable, since you could use it to determine if one can prove false from an arbitrary proposition, which means you can use it to determine if an arbitrary statement is true, which means you can use it to determine if a Turing Machine halts on given input.