Consider the statement (call it p):
This statement is false.
Is p true or false? If it is true, it contradicts itself. So it must be false. But that is what p states, so it must be true. Liar’s Paradox.
On the face of it, it seems that the problem with this paradox is either self-reference or bi-valued logic itself. If the problem is bi-valued logic, it seems that a lot of math is suspect, in particular proof by contradiction. Proof by contradiction is a staple of math as I know it, and giving it up would be difficult.
For a long time, I thought that the problem here was inappropriate self-reference. But not all forms of self-reference lead to contradictions. Self-referencing statements are even used in proofs of theorems, the Godel’s incompleteness theorem for example. The title of this post “This is my 200th post” is a self-referencing statement (incidentally, it is true). It does not lead to any contradiction. If the problem is inappropriate self-reference, what forms of self-reference are problematic? Change the liar’s paradox to “This statement is true.” and there is no contradiction. The form of self-reference is clearly the same. What then?
q: This statement is true.
If q is false, it contradicts itself. So q must be true. This does not lead to any contradiction. Assuming bi-valued logic is valid, q is true. But what does q mean? What is true? As I see it now, the problem with the liar’s paradox is that just like q, it is empty, devoid of any content. There is nothing in it that can be either true or false. In other words, it is not a statement at all because it does not state anything about the world. Mere adherence to rules of grammar does not produce statements.