[FOM] relevant logic and paraconsistent logic

[Neil Tennant]

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.

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.

Neil Tennant