SUO: *Date 24 Apr 2002 -- Theory Query
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Theory Query
John,
I have not had an easy time wrestling with the issues that developed,
in ways that were often surprising to me, as I pursued this question.
I have collected here a few of your comments that seemed to resonate,
sometimes positively and sometimes negatively, with my own concerns
along these lines. I have taken this idea seriously, and these days
that means I ask myself what sorts of algorithms and data structures
would be needed to support a competency in negotiating the proposed
architecture, problem space, or other sort of terrain.
JS = John Sowa
JS: However, most authors of current textbooks on logic are
abysmally ignorant of the history of their own subject
and of the best available axiomatizations for logic.
I have never seen a single textbook (other than mine)
that adopts Peirce's rules of inference, which every
logician who has ever seen them admits are the most
general and elegant rules of inference ever proposed
for FOL. (And they also have generalizations to other
logics as well.)
JS: In Peirce's system, the same three rules are used for both propositional
and predicate logic. And none of those three rules use substitution
for propositional variables -- because the system has no variables.
So the distinction you are proposing depends on features of just
one rather clunky system of inference.
JS: But as I have said many times, most current textbooks are woefully
deficient in the way they teach logic -- and the current distaste
for logic among significant numbers of programmers is just one
symptom of that deficiency. (Of course, the current use of
"tautology" is a rather trivial issue, but it arises from
an outmoded way of teaching proof theory.)
And not just programmers. It is amazing to me how many times
over the years some mathematician or physicist or other person
like that has said to me "I don't have much use for logic", and
when I inquire, it always turns out they mean that same brand of
rote manipulation of ritual formulas that they learned in their
sophomore "symbolic logic" course or something else of that ilk.
Four thirds of a century after Peirce first lectured
at Harvard "On the Logic of Science" we still find the
established doctrines of logic today in a state of near
total disconnection from the real life of inquiry in the
sciences, the humanities, and the practical professions.
For my part, I believe that if we really want to get serious about
using use logical languages to describe the real world, and to solve
a few of the outstanding problems that this real world presents to us,
then we will have to take a better look at the kinds of mathematical
models that have been carrying the load all this time, while logic
has languished in its monkish retreats.
One of the first things that I noticed when I took a look at the styles
of information storage and transformation that are part and parcel of
these mathematical models is that their computational forms are almost
always as near to "equational" as possible, preserving as much of the
initial and accumulated information as can be, right up until the
final stage, when an information-reducing "projection" is taken.
That is the ideal, of course, but it is a very distinct ideal
from the pattern that one sees in the styles of deduction
that are based on modus ponens or resolution.
I believe that deductive argument, or explicative reasoning,
is critically important in its place, and its place is right
in the middle of the cycle of inquiry, but it makes no sense
in isolation from the abductive and inductive phases that
are needed to connect it, in its current approximation,
to the real world beyond its immediate scope.
This has implications for the way we do logic,
consequences that we only lately acquired the
computational power to explore systematically.
It's a sign that you've been spending way too much time
talking to proof theorists when you start to think that
the purpose of a model is to establish the consistency
of a theory. That is just plain backward. The models
are the reason why we have theories. It is the aim of
the theory to say something worthwhile about the model.
And not just formal models, but those real objects and
complex systems of compelling interest in the world of
which every formal model is but the pallid reflection.
Jon Awbrey
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