This is a question about selfreference: Has anyone established an abstract framework, maybe a certain kind of formal language with some extra structure, which makes it possible to define what is a selfreferential statement?

I am not quite sure if it fits the bill but you can also check out: N. Yanofsky  A Universal Approach to SelfReferential Paradoxes, Incompleteness and Fixed Points 


For pleasure only I can at least give you the shortest definition of self reference. You need only to look in a good dictionary ( from Borges world of course) it says: Selfreference : see selfreference. 


Perhaps the right question to ask is if the statement is expressible in any system whose prooftheoretic ordinal is smaller than the Feferman–Schütte ordinal. 


Raymond Smullyan, "Diagonalization and SelfReference", 1994 


You might also be interested in Graham Priest's article "The Structure of the Paradoxes of SelfReference", Mind 103 (1994) pp. 2534. (Journal page ; JStor) and similar work by Priest. He has a general framework that he argues captures the various selfreferential paradoxes. I believe he also discusses this in some of his other work and monographs. 


I found another one: John Bell, “Incompleteness in a General Setting”. Bulletin of Symbolic Logic 13, 2007. It's paper number 66 here 


I have heard that Kapranov once said that he really wants to understand what is selfreference (i.e. is/was working on the question). 


Have you checked Craig Smorynski's work such as "Modal Logic and SelfReference" (Google Books)? 


There's a bunch of writing on this topic by the philosopher of mathematics Charles Chihara, includng a book called "Ontology and the Vicious Circle Principle". I haven't read that one but he also discusses the topic in his later book "Constructability and Mathematical Existence". You can probably find reviews of these books online that would help you decide if they are relevant to your interests. 

