[FOM] re relevant logic and paraconsistent

[Mark Lance]

on 2 mar 2006 neil tenant wrote:

"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."

Just a small clarification -- which is surely consistent with what  
Neil meant.  If X is a consistent set, then any classical consequence  
of X can be proved in RL.  But not "proved from X" depending, at  
least on what this means.  That is, for any logical truth P, CL let's  
us derive P from Q in the strong sense of proving the conditional.    
Of course RL doesn't.  It avoids irrelevancy as much as it avoids  
deriving anything from a contradiction.  But this is why mathematical  
practice is arguably closer to RL than to CL: No mathematician would  
be happy with a proof that had extraneous and random premises.  (A  
point made in the original discussion of this by Anderson and Belnap.)

As I say, Neil knows this, and the proper interpretation of what he  
wrote is clear enough.  I only mention it since some in this  
discussion are self-avowedly new to the issues.

Mark Lance