The Liar Paradox is a fun piece of logic to play with, as I recently found myself doing. It is fascinatingly resistant to resolution and can easily morph to defeat attempts at resolving the conundrum into a truth value. The wikipedia article is well written, and so I will be following it’s progression.
In it’s simplest form, the Liar paradox is as follows: This statement is false.
As you can see, if you believe the statement then it is true. But if it is true, then we believe it is false, and so on. There is no way to stop the vicious circularity while maintaining the principle of bivalence, and it is the abandonment of that principle that is the typical route taken for resolving the paradox. One of the first efforts at formulating a third category in addition to true and false is the proposal that the paradox represents something that is both true and false. But this falls victim to a simple rephrasing as seen below:
This statement is not true.
This statement, in addition to supporting contradictory truth values, also rejects the possibility that it could be both. No problem, why not propose a third value that is neither true or false? If it was neither, this would be in agreement with it not being true, and so resolve the matter. But the phrase easily evolves again to deny this possibility:
This statement is only false.
Now not only is the possibility of it being both true and false denied, but since it asserts the falseness of the statement, it prevents us from claiming it to be neither true or false. What are we to make of this? The wikipedia article discusses a number of philosophical positions that have been taken in regard to the paradox, but I think perhaps we might look at what we are evaluating and not just how to evaluate it.
The liar paradox is not a statement that can be evaluated for truth claims, even if that evaluation allows for the inclusion of a multivalent solution. It is instead an oxymoron, an honest lie and a dark light. The paradox is, quite literally, falsely true. And like with any oxymoron, it is not something we evaluate for truth claims because we understand that they make no such claim, but are contradictions to be corrected if done unwittingly, and laughed at if deliberate. They are not a source of information and thus the paradox does not constitute even a real statement as such.
As for why it seems intelligible, I suspect this is because we are evaluating it as a bivalent statement and so are considering the possible outcomes serially instead of together. Similarly, if considered separately, the two terms of an oxymoron are intelligible. It is only when reckoned as a whole that they become self-contradictory nonsense. Likewise, we can easily see the failure of the paradox once we see that its truth means its falsehood.