I have a question...

A classical, Tarskian system standardly contains homophonic definitions as the base clauses, e.g.,

'G' denotes Gs.

This is understood to be a sentence of the metalanguage (ML) which defines a predicate of the object language (OL). And when ML contains OL, the standard assumption is that the expression 'G' used on the right-hand side is identical to the expression mentioned on the left...so that numerically one expression, 'G', is an expression of both ML and of OL.

However, if homophonic definitions are admissible, then (given the compositionality of negation) the following sentence also seems admissible into the system as a definition:

'G' denotes x iff x is ~G.

But such an interpretation suffices for paradox in OL. For instance,  'Socrates is G' will be true in OL iff it is not true in OL (contra PNC). So apparently the second definition is not admissible in a classical system. But (again by the compositionality of negation) that means the former cannot be admitted into the system either. And more broadly, it looks like ML cannot contain OL...otherwise ML would rule that these definitions are well-formed.

Gupta and Belnap 1993 take note of something like the second definition on p. 273--but their concerns at that stage of the book are not with classical logic. When I asked Anil what bearing it has for classical logic (about a year ago) he denied it showed anything noteworthy (though I wasn't clear why).

The plot now thickens. About 3 months ago, Jeremy Avigad raised the very same definition to me when commenting on one of my papers. His claim was that it reflect that a classical system cannot contain "directly self-referential" expressions. Here too, I wasn't entirely clear on the reasons given. (After all, a homophonic definition like ' 'dogs' denote dogs' seems importantly different from a self-referring definition like ' 'this very sentence' denotes this very sentence.') More broadly, however, Avigad gave the impression that the second definition indeed shows something substantive about a classical system...but also, his attitude was that all this was old news. (I didn't get any references from him however.)

So my question to you is: What is going on here? Does the latter definition show that ML cannot contain OL in a classical system? If not, what does it show? Or, can someone provide references for further reading?

Thanks,
Ted