ONT Re: Inquiry Into Symbolization
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I am going to try and explain one of the conceptual schemes
that I use to interpret what Peirce says about symbolization
and symbolizability in his First "Logic Of Science" Lectures.
To do this I need to discuss the relations between two lattices
or partial orders, one being a lattice of "arbitrary sets" (SET),
the other being a lattice of "natural kinds" (NAT). For the time
being, I limit myself to concrete, discrete, even finite universes
of discourse, where all of the sets in view are subsets of a set X.
Here is a little essay in which I first broached this subject to the
Peirce Forum last year, in what I once thought was an amusing manner.
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Subj: HOPE's & FEAR's
Date: Tue, 23 May 2000 12:20:58 -0400
From: Jon Awbrey <jawbrey@oakland.edu>
To: Peirce Discussion Forum <peirce-l@lyris.acs.ttu.edu>
Day 1
Bright and early Monday morning I woke up with the
idea of returning to a peculiar theme in Peirce's
work, one that persists in nagging me, often while
I am reading something else, or working on another
topic to which it seems fairly incidental and all
too tangential, but a theme, nevertheless, that I
can barely catch a glimmer of here and there, not
even get so much as a firm handle on it, though it
continues to plague me with that not-to-be-denied
sense of its vague imports and general importance.
Moreover, it seems like every attempt that I have
made to raise this sunken ship of a topic, whether
in conversations here or elsewhere, has soon gone
down in flames or subsided with nary a whimper --
I cannot decide which is worse, but either way it
has been a fruitless, a frustrating, and a vastly
unsatisfactory experience just to try to express it.
Now, with that sort of build-up I think that you all
must be terribly -- yes, "terribly" is precisely the
word that fits -- excited about the prospects of my
bringing it up -- whatever the heck it is -- again,
but, in spite of this all too painful suspense, on
both our parts, I woke up, as I said, on that day,
rather enthusiastic about my prospects this time.
Without further ado -- well, thank you! -- let me
just say that this is the topic of Peirce's Notes
on "A Limited Universe of Marks" (ALUOM), appearing
for the first time in the volume by Peirce and his
students entitled 'Studies in Logic', published by
Little, Brown, and Company, of Boston, MA, in 1883,
(CE 4, 405-466, specifically for this note, 450-453),
and reworked again for the "Grand Logic", in 1893,
(CP 2.517-531). The original book of studies was
republished as a kind of Centenary Edition in 1983,
but I have apparently misplaced my copy of this work
in one of my last few of many geographic relocations.
Since these two versions of the remarks on ALUOM,
as I will call it, may enjoy widely varying levels
of accessibility across the breadth of this Forum,
I think that it might benefit discussion to copy as
much of them as I think necessary into the Sources
thread. The reason why I prefer to do things this
way is to separate my own remarks and speculations
as much as possible from the texts themselves, and
further, to leave the texts available in a maximally
uncluttered fashion for use in the light of what may
well turn out to be a variety of different purposes.
To this task I will now turn, but before I leave this
bit of preamble behind for what for may well turn out,
or maybe not so well turn out, to be a lengthy interval
of time, I probably ought to say a few more words in
explanation of my current tag-line for this thread.
On what do I feel that the theme of ALUOM has a bearing?
Well, on many things, easy to collect but difficult to
classify, at least, in any thoroughly rationalized and
schematized way. Natural kinds, the logic of inquiry,
especially the teasing apart of abduction and induction,
"giving a rule to abduction" (GARTA), constraints, and
thus information, innate or acquired, on admssible hype
and on permissible hypotheses, the nature of the human,
the Pragmatic Cosmos that orders the normative sciences
into a concentrically focusing and yet climbing spiral
of ascent, aspiration, reflection, and regardedness,
from aesthetics via ethics to logic in their turns --
you will no doubt begin to think that I am merely
free-associating or spinning out topics at random
if I even begin to unroll my rigamarole shopping
list of things that I think are encompassed here.
Nevertheless, what little form of organization that I can e-spy
in this seeming chaos and this teeming cornucopia of pragmatic
commonplaces is enough to get me cranking, at least, on some
sort of exposition, no matter how chancy and risky it may
start out being at first.
In this vein, I find what appears to be two distinct ways,
perhaps a couple of dual ways to approach the instigatory
question of it all, the question of "GARTA", of what sort
of limits may exist on our admission, creation, generation,
and imagination of propositions to describe our worlds of
experience, including as a special case the propositions
that we may choose to employ as explanations of striking
phenomena.
One "way of thinking" (WOT) is the one that I will dub as
the way of "Higher Order Propositional Expressions" (HOPE's).
The other WOT that I can see, at least, in so far as I can see any
other way at all, is the one that I will dub, fittingly enough, as
the way of "Framed Extensions And Restrictions" (FEAR's).
With apologies to Pandora, I will choose to introduce
the HOPE's first, and put off the FEAR's until later.
With gratitude to Shirazad, I will choose to make these HOPE's
the story of yet another day, as who knows what any day brings?
Day 2
If one treats hypotheses as any other propositions,
as so many simple closed circles in a venn diagram,
as so many logical variables in a truth table, then
one way of talking about constraints on hypotheses
is by making use of propositions about propositions,
or "higher order propositions" (HO propositions),
which are naturally expressed in the formulas of
"higher order propositional expressions" (HOPE's),
telling what propositions, in general, hypotheses
or not, are admitted to the universe of discourse,
curtailing discussion to "a limited number of marks",
as Peirce had a habit of putting it, in his studied
and exquisitely classical way.
The dizzying hypes of these orders of abstraction
makes it advisable to begin with a concrete and
a memorable case, if a rather ridiculous example.
Here, understand that we are only concerned with
the purified form, and not the ignoble content,
of this artificially simple example.
Don't bother to try and stop me if you have already
heard this one, as I think that it is likely to be
one of Aristotle's most outlandish jokes, because
I already have in mind another end altogether that
I hope will eventually serve to redeem the evident
absurdity of it.
Consider the humorous definition of a Human Being as
a Featherless Two-legged Critter, to schematize it,
if not utterly to traumatize it, let me express the
subject matter in the following way:
| A = Apterous (featherless)
| B = Bipedal (two-legged)
| C = Critter (animal)
| H = Human (human being)
Now, I had been planning to introduce some venn diagrams
at this point, but after wasting two days of trying, and
trying the patience of all concerned, and not concerned,
I am afraid that will have to forego, for now, that brand
of diagram -- what conceivable significance could iconic
diagrams have in philosophy, anyway? -- at least, until
I can figure out a way to arrive at a non-distorted form
of representation without the occasional experiment or two,
indeed, short of a persistent, persevering, indefinite series
of experiments. Of course, if I could figure out how to do that,
in full generality and without loss of geniality, as they will say,
what need would there be for any inquiry at all, much less any brand
of theory concerned with the logic and the practice of actual inquiry?
But never mind all that. Where there is a will, there will be a way,
whether the pathways of the requistite varieties of reactions have
all of their semiotic catalysts in all of the most optimal places
or not. There is always one way or another to go forward, even
if one's active duty status as an exponent of Peirce and one's
interim role as an interpretant of Peirce must abdicate a
few of the iconic attributes that remain most fitting to
these tasks, and even if we must relegate ourselves to
symbolically talking about the kinds of diagrams that
Peirce regarded as important to actually, brutally,
crudely, deliberately, existentially, faithfully
take a modicum of trouble to draw, for all that
one can learn from the concrete and practical
process of going through the exercise to do it.
In short, short of the facilities of the graphic
medium, I will have to require you to exercise your
imagination to a somewhat greater extent than you otherwise
might have to, and I will count myself fortunate in the circumstance,
that when it comes to imagination, you folks have no shortage of that!
But I did have a bit more luck with a somewhat simpler class of diagrams --
What double-edged luck, indeed, that it should have encouraged me to go
on ahead and rush in blindly where even angles and anglers fear to leave
the marks of their treads! -- but never mind all that. These rather more
tractable diagrams, although they lack that one critically important and
crucially iconic property of continuously reminding the viewer, not only
of the conceivably-continuously-supporting extensions of human concepts,
but also of the arbitrariness of the heraldic distinctions that humans
are wont to mark upon the underlying fields of existential experience,
as if the divisions we impose in our own conceits and in our images
were capable of placing any brand of demand on Nature at its joints
that Nature could not cast off as quick as Nature can dispose of us.
But never mind all that. The sort of diagram that I have been able
to draw on these walls, at least, so far, are none other than the
"logical lattice", the "propositional partial order", or, with
a tip of the hat to Tom Gollier, the "implicative food chain"
type of diagram. And so I will satisfice with these for now.
So let me try to draw you a picture of the situation that I want to discuss,
if it must be one that requires the viewer to "connect the dots" just a bit.
Figure 1 outlines the subject matter, to wit, the category "human being" (H)
here defined as falling under the head of an "apterous biped" (G = A |^| B),
hence bound by the set-theoretic intersection G of the respective extensions
of the two concepts, "apterous" (= featherless) and "bipedal" (= two-legged).
Now the wise-cracking sort of person, that everyone among us has encountered
before, will naturally be compelled to say, ignoring the natural and implicit
constraints of the discussion to what are often described as "natural kinds",
"But what of the plucked chicken? -- a two-legged critter without feathers? --
Is that your model of a genuine Mensch?" (To get the full effect, you have
to imagine this being said in a Woody Allen voice.) And so you patiently go
about explaining, as if your interrupreter did not already know this -- such
is the role of a straightman in this genre of commodious eristic, as you know
that you'll get your turn sooner or later, hopefully in the very next bit --
all about how a "plucked chicken", along with many other hype-o-thetical and
hi-pathetical creatures that might be abduced, construed, confabulated, and
otherwise plucked from thin air, is what one calls an "artificial kind", in
"essence", or the lack thereof, not really a "kind" (Greek 'genus') at all.
| A B
| o o
| |\ /|
| |.\ /.|
| | \ / |
| | . \ / . |
| | \ / |
| | . \ / . |
| | G |
| | . / \ . |
| | / \ |
| | / . . \ |
| | / . . \ |
| | / . . \ |
| |/. .\|
| o o
| H P
|
| Figure 1. On Being Human
|
| A = Apterous (featherless animal)
| B = Bipedal (two-legged being)
| C = Critter (creature, creation)
| G = GLB = Intersection of A and B
| H = HB = Human Being
| P = PC = Plucked Chicken
Okay, I think that will do to set up the joke.
I will save the explanation and the resolution
of it -- not nearly so fun a task -- till next
we meet in this room.
Until Then,
Jon Awbrey
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