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

[Erik Douglas]

I would like first to express my appreciation to the several persons who
responded to my solicitations for direction and references.  So, a heart
felt thank you.

Regarding the small thread that my inquiry about a putative paraconsistent
foundation to mathematics elicited: 

Arnon Avron's responded with "No way. *Mathematics should be consistent*"  

Thomas Forster's formative education maintains that "a logic is a theory
closed under uniform substitution... [and] is intended to capture validity."


I suspect these views may approximate a majority opinion on the matter,
though there remain questions: e.g., What do we really mean by consistent
any way? Or, What is uniform substitution generally and precisely anyway?
However, leaving these edges aside for the moment (though they do have a
funny way of returning, such as we can find in the myriad inquiries
surrounding the Penrose-Lucas Thesis), it would seem that for a
paraconsistent logic to provide a foundation to mathematics, it would have
first to be demonstrated that it was in some formal way commensurate with
classical laws of validity, indicating that it was not really paraconsistent
in any interesting way.  

However, Neil Tennant was more sympathetic to such an alternative black box
foundation supporting mathematics on pragmatic grounds, at least
sufficiently to consider it. Similarly, John Corcoran and Bill Abler
suggested in private correspondence a more psychological origin of logic.
And if I have understood Jean-Yves Beziau correctly, there is much to be
skeptical about with respect to more purely formalist, absolutist
approaches.  (Sincerest apologies if I have misrepresented anyone)

I do not have a paraconsistent rabbit in my hat at the moment, and I suspect
that probably insofar as we can entertain a universal framework to
characterize inference generally, the bulk of mathematics will demonstrably
derive from that part of it that is consistent in colour.  The general
problem I am really trying to address is, given a category of logics that
underpin mathematics and given another exclusive category of logics which,
say, codify our understanding of nature and anything else, what kinds of
formal or explicit conditions can we place, generally, on those other
logics.  So, a better understanding of the edges of the first would help to
delineate the second.  There is at present no well-developed discipline,
really, of "unmathematical logic" per se (at least that I am aware of) -
noting that most such alternative logical sand castles are still found on
the mathematical beach.  However, noting that my question seeks a general
characterization of the particular and perhaps singularly useful alternative
logics mentioned, it may in that capacity yet have some relevance to the
work of buttressing the mathematical castles we have constructed/discovered
on the sand bars of our consciousness.

Anyhow, further consideration of this matter probably either extends beyond
the reach of this list, or it will be found in the new threads about
formation rules.  However, I would appreciate any further views anyone taken
by the muse has to offer.  

Erik Douglas