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.
JA: 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>:
JA: | s i
| \ //
| \//
| |||
| |||
| o
JA: 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:
JA: | s i
| " \ //"
| " \ // "
| " \ // "
| i\ \// s
| \\ ||| /
| \\____|||____/
| \==== o ====\
| / ||| \\
| / ||| \\
| s //\ \i
| " // \ "
| " // \ "
| "// \ "
| i s
JA: 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.
An essential feature of this example, as Peirce described it,
was that Sunflower 1 transmits to Sunflower 2, not just the
transient impulse to turn toward the sun, but also the law,
or the continuity of this particular form of transition,
by which it conveys all of the above to future others.
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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JA: 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?
JA: 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.
HP: Your elaborate exegesis of Peirce is interesting.
My problem is that if Peirce's meaning was as subtle
as you say, then would he not have realized how inadequate,
if not misleading, was his 40-word definition if sign. Why,
then would he repeat what was for him a crucial definition,
and claim derivations from it with Weierstrassian rigor?
HP, quoting JA:
| ... 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.
JA: Since I know you have a good imagination,
Peirce's own words would be more convincing.
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Howard,
I was on point of thanking you for your persistence
in pressing for clarity on this issue, as I believe
that it has forced me to write a couple of the most
clear and most detailed expositions of this example
of definition that I have ever written, and yet now
my gratitude is sicklied o'er with the pale cast of
gratification delayed by the circumstance that your
appreciation of my interpretation underwhelms mine.
The NEM volumes did not come out until 1976, and though it is
just possible that I might have run across them in my favorite
hide-out corner of the math library while I was starting on my
3rd cycle through math and my 2nd cycle through Peirce studies,
I am pretty sure that I did not encounter this particular text
until a good deal later, when I was already approaching my 3rd
encounter with Our Maestro. In 1989 I finished up my work on
my "Theme One" program, that integrated a few of the simpler
aspects of inductive learning and deductive reasoning over
a shared resource base of graph-theoretic data structures
and functionally-implemented algorithms. Then it struck
me that abductive reasoning was the missing piece, and
that is what brought me back to a more dedicated and
more deliberate reading of what Peirce, along with
all of the current literature, had to say on the
subject. But none of these ductions makes any
sense in isolation from the cycle of inquiry,
and so here we are ...
Anyway, I can tell you that I clearly remember that I did not
find this definition, and recognize it for what it was, until
it occured to me one day that Peirce, as a real mathematician,
just must have taken the time sometime to write out a genuine,
properly mathematical defintion of this most important notion,
and so I went out looking for it, and as I sought, I found it.
Do I need to tell you that I do not find this definition
of signs to be in the least bit inadequate or misleading?
I do find that I often have to explain to a certain class of people,
who have been trained to follow in certain lines of trickety tracks,
that our theories, however approximal to our own severe limitations,
must still be designed so as to respect the complexity and subtlety
of their intended subjects, and that the conduct of inquiring minds,
and how their inquiring minds conceivably might be embodied, is one
of the complexer and the subtler bodies of phenomena that we'll see,
but, of course, I scarcely have any cause to amuse our present fine
body of complices with that particular lecture.
The sum of it is that Peirce was the sort of person who was able to derive,
out of sheer mathematical insight, and through a remarkable logical power,
all sans hint of a clue from his contemporaries, a "theory of information"
and a version of what we today call "non-standard analysis", both of which,
as initial as he left them, have features in advance of our current renditions.
The analysis of the sign definition that I teased out here in my last couple of tries --
and I deliberately avoided bringing in the machineries of quantificational calculus --
are just the barest echoes of a form of analysis that Peirce repeated on a routine
basis, and that he developed into an exact art by means of his Existential Graphs.
I will, of course, supply examples.
The reference to Weierstrass is an allusion in part to the theory of limits,
which Peirce thought was critical to analyzing semiosis in continuous media.
What Peirce means by "formal", along with his sense of the relation
between logical and mathematical reasoning, is covered by the quote
that I posted under "Logic As Semiotics". I recur to it again here:
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| Logic, in its general sense, is, as I believe I have shown, only another name for
|'semiotic' ([Greek: semeiotike]), the quasi-necessary, or formal, doctrine of signs.
| By describing the doctrine as "quasi-necessary", or formal, I mean that we observe the
| characters of such signs as we know, and from such an observation, by a process which
| I will not object to naming Abstraction, we are led to statements, eminently fallible,
| and therefore in one sense by no means necessary, as to what 'must be' the characters
| of all signs used by a "scientific" intelligence, that is to say, by an intelligence
| capable of learning by experience. As to that process of abstraction, it is itself
| a sort of observation. The faculty which I call abstractive observation is one which
| ordinary people perfectly recognize, but for which the theories of philosophers sometimes
| hardly leave room. It is a familiar experience to every human being to wish for something
| quite beyond his present means, and to follow that wish by the question, "Should I wish for
| that thing just the same, if I had ample means to gratify it?" To answer that question, he
| searches his heart, and in doing so makes what I term an abstractive observation. He makes
| in his imagination a sort of skeleton diagram, or outline sketch, of himself, considers what
| modifications the hypothetical state of things would require to be made in that picture, and
| then examines it, that is, 'observes' what he has imagined, to see whether the same ardent
| desire is there to be discerned. By such a process, which is at bottom very much like
| mathematical reasoning, we can reach conclusions as to what 'would be' true of signs
| in all cases, so long as the intelligence using them was scientific. (CP 2.227).
|
| Charles Sanders Peirce, 'Collected Papers', CP 2.227,
| Editors' Note: From an unidentified fragment, c. 1897.
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What Peirce means by "determination" is key to his theory of information.
I began a special study of the topic a while back and have been posting
the running accumulation of source materials and other highlights here:
http://suo.ieee.org/ontology/msg02377.html
http://suo.ieee.org/ontology/msg02378.html
http://suo.ieee.org/ontology/msg02379.html
http://suo.ieee.org/ontology/msg02380.html
http://suo.ieee.org/ontology/msg02384.html
http://suo.ieee.org/ontology/msg02387.html
http://suo.ieee.org/ontology/msg02388.html
http://suo.ieee.org/ontology/msg02389.html
http://suo.ieee.org/ontology/msg02390.html
http://suo.ieee.org/ontology/msg02391.html
http://suo.ieee.org/ontology/msg02395.html
http://suo.ieee.org/ontology/msg02407.html
http://suo.ieee.org/ontology/msg02550.html
http://suo.ieee.org/ontology/msg02552.html
http://suo.ieee.org/ontology/msg02556.html
http://suo.ieee.org/ontology/msg02594.html
http://suo.ieee.org/ontology/msg02651.html
http://suo.ieee.org/ontology/msg02673.html
http://suo.ieee.org/ontology/msg02706.html
For my own part, I find that I often like
Peirce's more folksy and poetical glosses.
Here's my favorite one on "Determination":
| To determine means to make a circumstance different
| from what it might have been otherwise. For example,
| a drop of rain falling on a stone determines it to be
| wet, provided the stone may have been dry before. But
| if the fact of a whole shower half an hour previous is
| given, then one drop does not determine the stone to be
| wet; for it would be wet, at any rate. (CE 1, 245-246).
|
| Charles Sanders Peirce,
|"Harvard Lecture on Kant, 1865",
|'Writings of Charles S. Peirce:
| A Chronological Edition, Volume 1, 1857-1866',
| Peirce Edition Project, Indiana University Press,
| Bloomington, IN, 1982.
|
| http://suo.ieee.org/ontology/msg02550.html
In general, it is best to interpret Peirce's notion of determination
in an "informational" or an "instructural" sense, rather than in the
causal sense, though of course the two are occasionally linked. One
may also find it instructive to think of how the word is employed in
mathematics, as in "two points determine a line", and "a determinant
is an invariant of a linear transformation". My guess is that these
senses would not have been too far from Peirce's thought as he wrote.
I do not know what to tell you about about Peirce's use
of "correspondence" in the definition of sign relations.
I do recall seeing one or two places where he actually
uses the phrase "triple correspondence", and I could
go chasing after that, but I am getting the feeling
that it might be for nought. The fact that Peirce
is talking about genuine and irreducible 3-adic
relations as the subject matter of his theory
of signs is rife throughout everything that
he says about them, and if you find that
a matter for controversy then I just
do not see how I might address it.
But I glad to see that you do, after all,
appreciate the skills of scholasticism
beyond the arts of imagination.
Jon Awbrey
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