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

[A.P. Hazen]

    I'm not sure if it's the kind of thing Bryan Ford had in mind (see 
his question below), but the logical treatment of "circular" 
definitions-- including  (and the paradoxes stemming from a naive 
"definition" of truth were the initial motivating examples for the 
study) pathological cases like
	Fx =df Not(Fx)
-- has been treated by Anil Gupta and Nuel Belnap in their "The 
Revision Theory of Truth" (MIT Press (Bradford Book) 1993: ISBN 
0-262-07144-4).  There's a nice discussion, and a description of a 
natural deduction system that allows consistent handling of such 
pathological definitions in Gupta's article "Remarks on definitions 
and the concept of truth," in "Proceedings of the Aristotelian 
Society," vol. 89 (1988-1989).

Ford asks:
	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...

---

Allen Hazen
Philosophy Department
University of Melbourne