[FOM] Consistent logics with non-well-founded definitions?
baford at mit.edu
Sat Sep 4 21:02:00 EDT 2004
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...
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