ONT Re: Inquiry Into Inquiry
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Our Text For Today:
| No. 12. 'On the Definition of Logic'.
|
| Logic is 'formal semiotic'. A sign is something, 'A', which brings
| something, 'B', its 'interpretant' sign, determined or created by it,
| into the same sort of correspondence (or a lower implied sort) with
| something, 'C', its 'object', as that in which itself stands to 'C'.
| This definition no more involves any reference to human thought than
| does the definition of a line as the place within which a particle lies
| during a lapse of time. It is from this definition that I deduce the
| principles of logic by mathematical reasoning, and by mathematical
| reasoning that, I aver, will support criticism of Weierstrassian
| severity, and that is perfectly evident. The word "formal" in
| the definition is also defined. (CSP, NEM 4, 54).
|
| Charles Sanders Peirce,
|'The New Elements of Mathematics',
| Four Volumes In Five Books,
| Edited by Carolyn Eisele,
| Mouton, The Hague, 1976.
HP: Howard wrote [trying to understand Peirce's language]:
|
| A brings B to correspond with C as A corresponds to C.
| So we can diagram it as a triangle:
|
| A (sign)
| / \
| / \
| / \
| / \
| C - - - - - B
| (object) (interpretant sign)
HP: So, A creates B and also brings B into the relation BC which is
the same (or lower?) relation as AC. (Have I got that right?
JA: No, the "correspondence" that is indicated here is a "triple correspondence",
what might be called a "3-place transaction" in database terms. Let us name
this 3-place relation L. Accordingly, to say that "A brings B to correspond
with C as A corresponds to C" is simply to say that A brings B into the same
3-place relation L with something else B' and C as A occupies in the 3-place
relation L with B and C.
HP: I don't see how Peirce's words say what you say they say.
Howard, my first time around in Peirce studies
was off and on all throughout my undergrad years,
a desulkory decade of dropping in and dropping out,
during which time I stayed mostly within the bounds
of CP 3&4, due to my interest in logic and math and
all things graphical. I spent the last couple of
years of this period writing my Senior Thesis on
the puzzles arising out of one little paragraph,
CP 4.306. The pursuit of enlightenment on this
problem, as it turned on Peirce's use of matrix
representations over B = {off, on} for logical
operators, led me back into mathematics proper,
from which I had been vacationing for a while
in philosophy. I believe that it was toward
the end of this phase that I figured out what
Peirce was talking about when he wrote about
sign relations, but it was a cumulative grasp,
no sudden epiphany, no definitive succincture,
nor had I seen anything called a "definition"
that I would have regarded, nor imagined that
Peirce intended, as anything but a partially
illustrative gloss on the notion. The best
clue that I had was the "Sunflower" example,
the text of which I cannot find at the moment,
but I can remember drawing pictures like this
to represent 3-tuples of the form <s, i, o>:
| s i
| \ //
| \//
| |||
| |||
| o
What struck me as Peirce's most striking illustration
was a story about a sunflower, that in turning toward
the sun, and by virtue of that very act alone, brings
another sunflower to turn toward the sun. The object
is the sun, the sign is the orienting of that initial
sunflower, and the interpretant sign is the orienting
of that next instructed sunflower, turning to the sun.
To illustrate this example of semiosis or sign-action
I vividly remember drawing a picture of the following
form, the best that I can render it in ASCII, anyway:
| s i
| " \ //"
| " \ // "
| " \ // "
| i\ \// s
| \\ ||| /
| \\____|||____/
| \==== o ====\
| / ||| \\
| / ||| \\
| s //\ \i
| " // \ "
| " // \ "
| "// \ "
| i s
Here, I was able to manage drawing only four 3-tuples,
and I had to use a line of ditto marks (") to indicate
that their termini are to be treated as identical nodes,
whereas as I was accustomed to draw it I would have had
eight or more 3-tuples, arranged in the form of a flower,
and I would have then been able simply to fuse the nodes
that were to be identified. The point of the picture is,
of course, that s_1 -> i_1 = s_2 -> i_2 = s_3 -> i_3, ...,
and so on.
CSP: | A sign is something, 'A', which brings something, 'B',
| its 'interpretant' sign, determined or created by it,
| into the same sort of correspondence (or a lower implied
| sort) with something, 'C', its 'object', as that in which
| itself stands to 'C'.
HP, interpreting CSP:
| A creates B and also brings B into the relation BC
| which is the same (or lower?) relation as AC.
HP: What, exactly, did my shorter version leave out that essentially alters Peirce's meaning?
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Using the acronym "SOC" for "sort of correspondence",
let us try the following paraphrase of the criterion:
| A brings something into the same SOC with C
| as the SOC in which A stands to C.
In order to expand the expression correctly, one has to ask:
0. What is the SOC in which A stands to C?
The SOC in which A stands to C is given by Formula 1.
1. A brings something into some SOC with C,
to wit, the SOC in which A stands to C.
If B enters into the same SOC with C as A stands in, that is to say,
if B takes up the same role that A had in Formula 1, then we obtain:
2. B brings something into some SOC with C,
to wit, the SOC in which A stands to C.
Let us give the name B' to the something in Formula 2.
Now, B' can either be something old or something new.
If old, it's in A, B, C. If new, let's leave it B'.
In any case, we have:
3. B brings B'into some SOC with C,
the SOC in which A stands to C.
In sum, the bringing of a new, possibly old, thing
into the relation is part of being in the relation.
And so it goes ...
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There are three important ideas that are involved in Peirce's definition
of a sign that are not themselves defined local to its immediate context:
"Correspondence", "Determination", "Formal". It is common for thoroughly
modern readers to simply supply their own meanings without pausing to ask
if there could be other senses, in which case they read mush -- typically
hashing together a mess'o'misapprehensions about the 2-adic mirror images
of a 2-adic correspondence theory of truth, some enchanted doctrine about
a semiotic causal determinism imputed to Peirce, plus who knows what when
it comes to their pro forma reading of "formal". But should anybody want
to know what the guy was really saying, all a body has to do is look into
the many other places where he says exactly what he means by these words.
HP: In other words, I fail to see how, from Peirce's definition alone, I would
be justified in interpreting it as the "3-place transaction, L" you describe
(whatever L is). Is this called an elliptical definition?
Yes, in the sense that every word in the dictionary is defined elliptically,
if not in the orbits of another conic section, in terms of other words that
one must trace elsewhere, in their own proper spheres.
Jon Awbrey
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