[FOM] Question of the Day: What is a Logic?

[Neil Tennant]

On Sat, 7 Oct 2006, Arnon Avron wrote:

> The problem is: Why should anybody want to use a paraconsistent 
> logic for mathematics if s/he thinks mathematics is consistent??

A paraconsistent logic is one in which there can be distinct inconsistent
theories (as opposed to the single inconsistent theory, i.e. the whole
language, when one uses a logic---such as intuitionistic logic or
classical logic---that is not paraconsistent).

If one values *relevance*, in proofs, of premises to conclusions one will
end up with a system of relevant logic---and it will beparaconsistent. The
main aim, when devising a relevant logic, is to avoid the first Lewis
paradox:

	A,~A : B 

(a.k.a. `ex falso quodlibet', `explosion', or `the absurdity rule').
This is the source of much irrelevance; remove it, and your logic will be
paraconsistent.

Classical Relevant logic (CR) suffices for the proof of every classical
consequence of any consistent set of premises. So, if classical
mathematics is consistent, all its theorems can be proved using only CR.
If, on the other hand, classical mathematics is inconsistent, then CR
will detect the inconsistency---indeed, will detect all such
inconsistencies (for classical mathematics might be inconsistent in more
ways than one).

Suppose that while using only CR for the deductive development of
mathematics, one experiments with various new axioms, and that some of
these turn out (severally) to be inconsistent with previous ones. Then,
with CR as one's underlying logic, one can set about studying the
different kinds of ways these inconsistencies have arisen.
*That* is the great attraction of a paraconsistent logic. Intuitively,
the user of full classical logic would do the same---thereby showing, I
think, that s/he pays only lip service to EFQ, and works instinctively
within the more insightful confines of CR.

All of the above also holds, by the way, with "I" in place of "C", and
"intuitionistic" in place of "classical".

Neil