[FOM] relevant logic and paraconsitent logic

Edwin Mares Edwin.Mares at vuw.ac.nz
Wed Mar 1 15:05:25 EST 2006

Hi Joseph,
A paraconsistent logic is any logic that rejects as valid the rule scheme: A, ~A => B. One need not accept any contradiction as true to think that paraconsistent logic is "right" (whatever that means) or useful. The is a wide spectrum of views on paraconsistent logic. On the right are those that think it is useful in order to formalise inconsistent theories, people's beliefs, systems of norms, and so on. On the left, there are those (such as Priest and Tanaka) who hold that there are true contradictons (these people are known as "dialetheists").
Relevant logic arose out of a desire to avoid the socalled paradoxes of material and strict implication and provide a more robust implication than is found in classical or modal logic. The resulting logics that do not contain (A&~A)->B as a theorem scheme (i.e. some instances of this scheme are not theorems of the logic), nor do they contain the rule scheme given above. As such, relevant logics are paraconsistent logics. But practioners of relevant logic need not be on the left of the spectrum given above. For example, I am a relevant logician but not a dialetheist (see my book, Relevant Logic: A Philosophical Interpretation, Cambridge U Press, 2004). I believe that Francois Rivenc's view is similar (in this respect) in his Introduction a la logique pertinente (PUF, 2005). I hold my view for semantic reasons and Rivenc holds his for proof theoretic reasons.
Ed Mares


From: fom-bounces at cs.nyu.edu on behalf of Joseph Vidal-Rosset
Sent: Wed 1/03/2006 11:25 p.m.
To: Foundations of Mathematics
Subject: [FOM] relevant logic and paraconsitent logic

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Reading this paper from Priest and Tanaka :
I would be glad if experts in Logic of this list would be kind enough
to develop on the list the links between paraconsistent logic and
relevant logic.

My request is mainly motivated by a logico-philosophical position:

(1) I feel uneasy with logical systems accepting "true contradictions"
(paraconsistent logics), but

(2) I willingly accept the idea of a system like IR where neither
(p & ~p)-> q nor (p & ~p) -> ~q are not valid deductions and where
paradoxes of material and strict implication are avoided.

Apart from the clarification I still need about paraconsitent logics, I
would be thankful to f.o.m. subsrcibers if they could help me to reply
technically to this logical question: if I accept relevant logic am I
involved to accept paraconsitent logic? If (p & ~p) -> q is not valid,
hence (p & ~p) must be true, or can I avoid this conclusion with the
rejection of the universality of Bivalence principle or is there
another way out?

I need help. Of course I believe that logico-philosophical polemics on
this topic will be welcome, because we are on the FOM list. :)


- --
Joseph Vidal-Rosset
Université de Nancy 2
Département de philosophie
Bd Albert 1er
F-54000 Nancy

page web: http://jvrosset.free.fr
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