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

[Arnon Avron]

On Fri, Oct 06, 2006 at 03:12:29PM -0400, Neil Tennant wrote:
> 
> On Fri, 6 Oct 2006, Arnon Avron wrote:
> 
> > Concerning a paraconsistent logic as a possible base for 
> > the foundations of mathematics, my answer is: No way. *Mathematics
> > should be consistent*.
> 
> Arnon,
> 
> How can you justify to your reader the following inference, which is
> implicit in the foregoing?:
> 
> 	Mathematics should be consistent; 
>   ergo, one should not use a paraconsistent logic for mathematics.
> 
> I don't get it. Suppose mathematics is consistent. And suppose one uses a
> paraconsistent logic for mathematics. Then one can still prove all
> mathematical theorems.  Where's the problem?

Neil,

The problem is: Why should anybody want to use a paraconsistent 
logic for mathematics if s/he thinks mathematics is consistent??
As far as I know, those who play with this idea have in mind some 
inconsistent theory (like naive set theory) serving as the 
foundational theory of mathematics, and then one needs
partaconsistent logic in order for the resulting mathematics
not to be completely trivial. I have never heard any other 
motivation for using a paraconsistent logic in mathematics. 
Do you have any? If you do have some argument why it might be a good 
idea to use a paraconsistent logic for developing some 
consistent mathematical theory, then my original reply (which was
based on the assumption that the motivation is what I described above)
indeed fails. But I am really curious to know what such an
argument can possibly be (and I strongly doubt that it will
persuade me...).

Arnon Avron