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

[Arnon Avron]

Concerning your first question, I suggest that you look
at the volume:

 What is a Logical System,
 ed. by  D. Gabbay, 
 Studies in Logic and Computation, Vol.~4, 217-238, 
 Oxford University Press, 1994.

(by a strange coincidence, I happen to have a paper in this volume...)

Concerning a paraconsistent logic as a possible base for 
the foundations of mathematics, my answer is: No way. *Mathematics
should be consistent*.

  Now I myself have had several papers on paraconsistent logics
(the latest one provides simple semantics to most of the logics
developed in da-Costa school). However, in all my papers on
paraconsistent logics I am using classical logic - and so
does everybody else. What is more, if a referee would find 
contradicting propositions in a paper of mine on paraconsistent
logics, s/he will reject it immediately (or at least will demand 
to revise it so that the contradiction is eliminated). Although
that referee will be an expert on paraconsistent logics, s/he
will not see that contradiction as a wonderful opportunity
to apply paraconsistent logic from that point on... 

 To sum up: Classical logic is the primary logic. It is the
metalogic of all other logics (yes, even of intuitionistic logic!),
and it should underelie everything that is absolue and eternal,
and this includes anything that deserves the name  
mathematics (I am not saying that it is not interesting to 
find out what can be proved by some weaker means and what cannot - 
but even the formulation of this question assumes classical logic!). 

  My own approach to paraconsistent logics is that of an 
instrumentalist.  It is a fact of life that we use from 
time to time inconsistent theories in various circumstances. 
It is also a fact that when
we find out a contradiction in a theory we use 
we do not rush to infer everything that comes to our mind (though 
this would require just two steps!). Frequently we somehow manage to
continue to make good use of that  theory for a while.
As long as the theory is used only as *an instrument* for
some practical purpose, without
claims for representing truth,  there is nothing wrong with that -
provided the rules of the game are clear. The way I see it,
paraconsistent logics just try to model this types of useful games.

  By the way, the above point is relevant to the thread concerning
the Lucas-penrose argument. Some people on FOM have recently
made big fuss about the fact that while inconsistent machine
will infer everything, including 0=1, a humane being will
refrain from doing so and will remain "consistent". Those people
talked as if a machine cannot be programmed to behave similarly,
(e.g. by temporarily switching to an appropriate paraconsistent logic
until its consistency-restoring component revises 
the machine's set of beliefs).

 I know that it is hard to believe, but machines can even 
be programmed to make mistakes from time to time, like all of us! 
In fact, this happens most of the time, because (most probably)
our current machines are built and programmed by other machines.

Arnon Avron