[FOM] Consistent logics with non-well-founded definitions
A.P. Hazen
a.hazen at philosophy.unimelb.edu.au
Sun Sep 5 02:45:28 EDT 2004
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
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