Paraconsistent Logics: 3 Trends & Another Approach?

[nomadreid]

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 multi-valued 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.

 

[Preno]

Well, paraconsistent logic is primarily intended as a way of handling contradictory information. The motivation for Belnap's four-valued 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.

 

[nomadreid]

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?

 

[Preno]

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?

Yes. Carnielli seems to be a student of Newton da Costa, while Weber seems to belong to the Australasian school like Graham Priest. I removed that because my personal ignorance need not reflect on the actual state of affairs in the field of paraconsistent logic. (However, speaking as an outsider, it is true that Weber and Carnielli to me are not the most visible members of the paraconsistent community.)

 

[blue_raver22]

As a scientist, it is important to consider not only the theoretical implications of paraconsistent logics, but also their practical applications. While the three trends mentioned have certainly made progress in addressing the infamous paradoxes and redefining sets and classes, it is important to also consider the potential use of paraconsistent logics in handling real-world contradictory information. This is a crucial aspect that sets paraconsistent logics apart from traditional logic systems.

In terms of another approach, there is ongoing research in the field of paraconsistent logics that focuses on the development of dynamic or interactive logics. These logics aim to capture the reasoning processes of human agents, including their ability to handle contradictory information and revise their beliefs accordingly. One example of such a logic is the Logic of Paradox (LP), which allows for the representation of changing beliefs and inconsistent knowledge. Other examples include the Paraconsistent Logic of Action (PLA) and the Paraconsistent Logic of Induction (PLI).

Furthermore, there is also research being done on the application of paraconsistent logics in fields such as artificial intelligence and cognitive science. These logics have the potential to improve the ability of AI systems to handle complex and contradictory data, and to better model human reasoning processes.

In conclusion, while the three trends mentioned in the content have made valuable contributions to paraconsistent logics, it is important to also consider the practical applications of these logics in handling contradictory information. The development of dynamic and interactive logics, as well as their potential use in various fields, shows promise for a more comprehensive approach to paraconsistent reasoning. I recommend exploring the work of researchers such as Newton da Costa and Dov Gabbay for further information on these developments.