In order for me to concede that false statements exist, I would need clarification of what is meant by "statements" and what is meant in saying that they "exist". If the answer I receive is "they just do" or "most people accept that they do", you can argue all you want about "burden of proof", but I will not be convinced, any more than I would be convinced by the same "arguments", that "red unicorns exist".OK then. Starting with the definitions. There are various definitions of 'statement' but I will go with 'declarative sentence' for this one. As for 'exists', I will read 'false statements exist' as equivalent to 'some statements are false'. See my earlier remarks about Brentano equivalence.
So we need to demonstrate to Anthony's satisfaction that some declarative sentences are false. That is easy. The sentences "The sky is red" is a declarative sentence, and it is false. So, some declarative sentences (i.e. at least one) are false. If Anthony denies that "The sky is red" is false, there is an equally easy reply. If you deny something, you are denying that it is true. But if you are right in denying this, it must be false. So in order to make the objection at all, you have to concede that at least one declarative sentence, in this case "'The sky is red' is false" is false, and thus concede the point. More generally, to affirm "no declarative sentence is false" is to deny "some declarative sentence is false". But if that denial is right, "'some declarative sentence is false' is false" is true, and so at least one declarative sentence is false, namely "no declarative sentence is false". Slightly more formally:
(1) No declarative sentence is false (assumption)
(2) "Some declarative sentence is false" is false (E and I are contradictory opposites)
(3) "Some declarative sentence is false" is a declarative sentence (definition)
(4) Some declarative sentence is false (substitution)
(5) Contradiction (1 and 4)
I doubt this will be the end of the matter.