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Paraconsistent logics 
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#1
Nov2612, 10:57 AM

P: 560

From a survey of paraconsistent logics, it appears to me that there are three main trends:
(1) Weaken implication and do away with the Axiom of Foundation in ZFC, so that the more annoying paradoxes cannot be derived. (e.g., Weber) (2) Do away with type theory, relabeling classes as "inconsistent sets", in such a way as to allow those contradictions which previously were eliminated by type theory (e.g., Carnielli), (3) Introduce a multivalued logic whereby the paradoxical statements receive a new truth value (e.g., Belnap) However, one of the reasons for interest in paraconsistent logic is not only to solve the paradoxes (which are important for Foundations but of little interest to other practicing mathematicians), but also to be able to handle information taken from humans which, for one reason or the other, ends up being contradictory. This latter style of contradiction has nothing to do with the infamous paradoxes. So it would seem that another approach is necessary than the three outlined above. Are there any? If so, I would appreciate a link that is freely accessible on the Internet. Thanks. 


#2
Nov2612, 12:27 PM

P: 159

Well, paraconsistent logic is primarily intended as a way of handling contradictory information. The motivation for Belnap's fourvalued logic is precisely that, rather than anything having to do with paradoxes. Afaik the one major paraconsistent approach that does intend to deal with the Liar paradox and such is Priest's dialetheism, embodied in the Logic of Paradox and its relatives. Other than that, paraconsistent logic has been pursued imo most prominently by Belnap, Dunn, da Costa and Béziau, so look into those.
Also, doing away with the Axiom of Foundations in ZFC doesn't seem to have anything to do with paraconsistent logic (as far as we can tell, no "paradoxes" can be derived in ZFC, so I'm not sure what you mean by "doing away with the Axiom of Foundation in ZFC, so that the more annoying paradoxes cannot be derived"), while "weakening implication" is (in the sense that A & ~A > B is not a theorem) common to all paraconsistent logics. 


#3
Nov2612, 12:49 PM

P: 560

Thanks, Preno, I will look into Dunn, da Costa and Béziau, as well as more into Belnap.
My grammar was a little off when I wrote (2), in that I meant that the weakening of implication disallowed the derivation of the paradoxes, and the elimination of the Axiom of Foundation in Weber permits sets which are members of themselves without subsequent explosion. Curious: in the email notifying me of your reply, it was mentioned that you had never heard of Weber and Carnielli, but in the Forum this was taken away, indicating that you had edited that out by the time I got to it. My guess is that you looked them up in the meantime, right? 


#4
Nov2612, 02:11 PM

P: 159

Paraconsistent logics



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