o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o (01)
JA = Jon Awbrey
JFS = John F. Sowa (02)
JFS: Yes, Peirce gave very general foundations
for much, much, more than classical FOL. (03)
JA: I'm sure you must know, if you think about it, that many
of the statements that you make below are not exactly so.
Peirce's interpretation of basic logical symbols was far more
abstract, formal, or general than our conventional readings
of boolean syntax, as evidenced among many other facts by the
deep duality between the entitative interpretation and the
existential interpretation of logical graphs. (04)
JFS: The fact that Peirce did much more than classical FOL
does not contradict the fact that the FOL subset of his
1885 paper on the Algebra of Logic is exactly compatible
with the FOL subset of his existential graphs of 1906 and
exactly compatible with the FOL subset of Frege's publications. (05)
"Compatible" is a word that bears further examination, but I do not
think I could bear to begin that interrogation this late in the day. (06)
The desire to assimilate the general conception of logic that courses
through the tradition of Aristotle, Leibniz, Kant, Boole, and Peirce
to the line that was founded on Russell's Frege and Tarski's Hilbert
appears to be far stronger than I ever would have guessed a few years
ago, even though I had long been aware of the general run of its ilk. (07)
There is a natural tendency of mind to synthesize and unify, and that
is a bent that I understand, sympathize with, and share. But the wish
to assimilate even at the expense of a reductive synthesis goes farther
than I would like to see. (08)
JFS: That is a truly remarkable point that is *not* true of *any*
programming language or system that has been defined by purely
behavioral methods. It's importance cannot be overstated. (09)
There may be a particular misunderstanding on his point. Yes, I could
hear the echoes in recent exchanges of all those old wrangles between
declarative stylists and procedural stylists in programming languages,
but that is not what I personally am talking about when I refer to the
pragmatic aspects of logic. I am referring to the conception of logic
as a normative science, which is part of what Peirce meant by defining
logic as formal semiotic. This footing for logic is not a thing that
can be shoehorned into more reductive forms, not without sacrificing
a few of its toes. (010)
JA: It is also evident in the fact that Peirce gave axioms for
boolean algebra in dual symmetric form, even writing them in
the fashion that was customary to present dual axiomsets for
projective geometry. This leads to a very distinctive way of
regarding the interpretation of logical calculi, where symbols
that we once regarded as "logical constants" take on variable
meanings. (011)
JFS: Indeed, Peirce had profound insights into logic that are worth
much more study than most logicians have bothered to give them. (012)
JFS: But it is also true that the ISO standard for classical FOL with
the extensions in the CL standard is already far beyond what most
people are using today. (013)
JFS: Let's take it one step at a time. (014)
Let's just be sure they are forward steps.
Life is short, the art is long, indeed,
but what I have seen in my time so far
has mostly been a lot of bak.peddling. (015)
Jon Awbrey (016)
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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