How relevant this is to "an abstract framework, maybe a certain kind of formal language with some extra structure" in the OP, I am not quite sure, and I am only beginning to read Halbach's works, yet I think they are *general* enough to properly belong to this thread, and they weren't yet mentioned. Very very roughly, what I gather from Halbach's work on this topic so far is that (my interpretation) 

> the 'say' in the old idea of formulae which 'say' about themselves they are not provable is *not an idea which is synthetic a priori* (like e.g. the idea of *necessity*), rather is an idea consisting of many sub-ideas. (For example, by filtering according to where the relevant formulae are placed in the arithmetical hierarchy.) 

Philosophically, the approach to self-reference I am pointing to here could be called an 'analytic' approach to self-reference, as opposed to the 'synthetic' approach of Lawvere. (Both 'analytic' and 'synthetic' to be taken in their neutral technical sense, i.e. breaking a concept into parts in the former, combining the concept with other concepts in the latter.)

Some references on Halbach's work: 

* so new that it hasn't yet appeared: [Volker Halbach](http://users.ox.ac.uk/~sfop0114/publications.html) has announced a book (with Graham Leigh) whose draft has the title *Syntax and Circularity: A Study in Self-Reference and Paradox*.

* so new that it lies after the answers given so far: 

> [Volker Halbach, Albert Visser: *Self-reference in arithmetic I.* Vol. 7(4), 2014 , pp. 671-691](https://doi.org/10.1017/S1755020314000288)
>
> [Volker Halbach, Albert Visser: *Self-reference in arithmetic II.* Vol. 7(4), 2014 , pp. 692-712](https://doi.org/10.1017/S175502031400029X)

* a relevant lecture of Halbach's with a very general title, containing explanations on the work of Halbach and Visser:  

> [V. Halbach: *Self-reference*. Talk at the Workshop in Mathematical Philosophy. Ludwig-Maximilians-Universität München. September 13, 2011](https://www.youtube.com/watch?v=xvXusC0f_mk)