# [FOM] relevant logic and paraconsistent logic

Joseph Vidal-Rosset joseph.vidal-rosset at univ-nancy2.fr
Fri Mar 3 10:35:44 EST 2006

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On Thu, 2 Mar 2006 16:42:17 -0500 (EST)
Neil Tennant <neilt at mercutio.cohums.ohio-state.edu> wrote:

>
> On Thu, 2 Mar 2006, Arnon Avron wrote:
>
> > Personally I don't believe that paraconsistent logics in general
> > and relevance logics in particular are relevant to *mathematics*.
>
> Au contraire, Arnon!
>
> If X is a consistent set of axioms, then any classical consequence of
> X can be proved from X in classical relevant logic (CR); and
> if X is a consistent set of axioms, then any intuitionistic
> consequence of X can be proved from X in intuitionistic relevant
> logic (IR).
>
> So, just as Monsieur Jourdain discovered that he had been speaking
> prose all along, mathematicians are invited to discover that they had
> been reasoning in accordance with the canons of relevant logic all
> along.

First, many thanks to Avron, Tennant, Lance, Heck, Mares, Raatikainen
for their reply and their explanations, and Ozkural for his interest.
(A special thanks also to Neil Tennant for his reference to Moliere
and his French words. :) ).

I need here an explanation from Neil.

I understand that: if T is a consistent theory (a consistent set of
axioms), and if B is a logical consequence of T in classical logic,
then A is also a consequence of T in CR and in IR.

> This, of ocurse, is on the assumption that their chosen set of
> mathematical axioms is consistent. BUT: if their chosen set of axioms
> is *inconsistent*, then that fact too can be proved within CR (resp.
> IR).
>
> So: CR (resp. IR) is adequate for all the logical needs of the working
> classical (resp. intuitionistic) mathematician.
>
> Mathematics DOES NOT NEED ex falso quodlibet.

I am not sure to understand exactly what Neil says here.

I understand that if T is a inconsistent theory, i.e. containing both A
and ~A in the set of axioms, then B can no more be proved in CR, nor in
IR, because these relevant logics do not contain (A & ~A) |= B, unlike
classical logic. And I understand also that the inconsistency of T
("that fact") can also be proved within CR and IR.

Unfortunately your paper, Neil,  `Intuitionistic Mathematics Does Not
Need Ex Falso Quodlibet', Topoi 1994, pp. 127-133, is not online. So my
request of more explanations, even if this point will seem probably
trivial to relevant logicians.

all the best,

Jo.

- --
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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