[FOM] Consistent logics with non-well-founded definitions?

[Bryan Ford]

Hi folks,

Can anyone give me pointers to any formal logic systems that have somehow 
contrived to permit arbitrary recursive logical and/or mathematical 
definitions without _any_ well-foundedness prerequisite and nevertheless 
without falling into inconsistency?  For example, I could envision a system 
that weakens the inference rules of logic enough so that you can for example 
"define" a symbol L to be equal to the negation of L (i.e., the liar 
paradox), and instead of causing an inconsistency, it might for example 
simply be impossible to prove anything interesting about the truth or untruth 
of L.  I presume someone must have studied such an idea somewhere...  Has 
anyone come up with such a system that's powerful enough to be useful?

On a similar note, can anyone give me pointers to any studies of formal logic 
systems (however weird or contrived) that can prove themselves consistent, 
but are nevertheless consistent (e.g., their consistency is provable in ZFC)?  
I'm pretty sure I've heard of such things being studied somewhere, but can't 
remember where and can't find the references...

Thanks,
Bryan