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

[Erik Douglas]

I have two questions for the erudite members of this list from the
foundations of logic.  Some of the answers may extend outside the range of
topics appropriate to this list, but I would be very grateful all the same
for every response; I am a philosopher by trade, and so I am fearless if not
also a bit foolish.  My email is erik at temporality.org and I will also
eventually compile the answers in an informal discussion on the web (and
possibly more).

1. What are the qualifications on a (formal) language to be a logic? 

In general, a logic appears to belong to a subclass of formalized languages,
in a larger category of symbolic systems used to communicate.  I do not mean
to ask the rather obscure philosophical questions about what it means to
communicate, or even in what symbolic systems consist.  Rather, I accept
languages as primitive, and moreover that some are *formal* (explicit and
unambiguous in their application).  Also, it seems important that such
formal languages have both syntactic and semantic dimensions, that we can
identify something like *expressions* in the syntax, as well as some minimal
distinction between *semantic* kinds amongst those expressions (e.g., TRUE
and FALSE). 

The question is, given these minimal criteria, what further conditions
qualify, generally, a particular formal language as a logic?   There is a
surprising paucity in the literature addressing this question (I have found
some references, but I would also be grateful for any and all in your
responses).  At some level, the matter appears to be conventional or
metaphysical (depending significantly on your religious inclinations).
However, in either case, I would be very grateful to know your operative
(and/or cherished) beliefs here.

The second question (below) then turns on what is often assumed as such a
fundamental condition, the law of non-contradiction.  It seems other laws
that were at one time accepted as etched in stone, such as the law of
excluded middles, have given way to intuitionist and fuzzy logics.  In
recent years, we have seen the construction of several variety of
*paraconsistent* logics which do prima facie deny the law of
non-contradiction.  So my question is as follows, and I have in mind that
many of the folks on this list are mathematicians:

2. Suppose I present a paraconsistent logic that I claim suffices as a
foundation for mathematics (or some significant part thereof), what (other)
properties must it have for you to accept it as such?

Note on my motivation:

I am in the business of attempting to construct alternative logics, notably
paraconsistent logics, with an eye to addressing certain issues that sit
uneasily these days between philosophy, metaphysics, mathematics, and
physics, especially those that turn on how we understand *time* and *mind*.
There are other applications in the development of a universal logic that
also motivate my investigation here.

Cheers,
Erik