Peircers,
Here's my 1st Comment on Selections 1 through 6 —
{ Information = Comprehension × Extension } • Comment 1
=======================================================
https://inquiryintoinquiry.com/2016/05/26/information-comprehension-x-extension-%e2%80%a2-comment-1/
At this point in his inventory of scientific reasoning,
Peirce is relating the nature of inference, information,
and inquiry to the character of the signs mediating the
process in question, a process he is presently describing
as “symbolization”.
https://inquiryintoinquiry.com/2016/05/19/information-comprehension-x-extension-%E2%80%A2-selection-1/
In the interests clarity let’s draw from Peirce’s account
a couple of quick sketches, designed to show how the examples
he gives of conjunctive terms and disjunctive terms might look
if they were cast within a lattice-theoretic frame.
Let’s examine Peirce’s example of a conjunctive term —
“spherical, bright, fragrant, juicy, tropical fruit” —
within a lattice framework. We have these six terms:
• t_1 = spherical
• t_2 = bright
• t_3 = fragrant
• t_4 = juicy
• t_5 = tropical
• t_6 = fruit
Suppose that z is the logical conjunction of the above six terms:
• z = t_1 ∙ t_2 ∙ t_3 ∙ t_4 ∙ t_5 ∙ t_6
What on earth could Peirce mean by saying that such a term
is “not a true symbol” or that it is “of no use whatever”?
https://inquiryintoinquiry.com/2016/05/20/information-comprehension-x-extension-%E2%80%A2-selection-3/
https://inquiryintoinquiry.com/2016/05/22/information-comprehension-x-extension-%E2%80%A2-selection-5/
In particular, consider the following statement:
“If it occurs in the predicate and something is said to be
a spherical bright fragrant juicy tropical fruit, since there
is nothing which is all this which is not an orange, we may say
that this is an orange at once.”
In other words, if something x is said to be z, then we may guess
fairly surely that x is really an orange, in other words, that x
has all of the additional features that would be summed up quite
succinctly in the much more constrained term y, where y means
“an orange”.
Figure 1 shows the implication ordering of
logical terms in the form of a “lattice diagram”.
[See Figure 1, attached.]
Figure 1. Conjunctive Term z, Taken as Predicate
What Peirce is saying about z not being a genuinely useful symbol can
be explained in terms of the gap between the logical conjunction z, in
lattice terms, the “greatest lower bound” (glb) of the conjoined terms,
z = glb {t_1, t_2, t_3, t_4, t_5, t_6}, and what we might regard as the
natural conjunction or natural glb of these terms, namely, y, “an orange”.
That is to say, there is an extra measure of constraint that goes into
forming the natural kinds lattice from the free lattice that logic and
set theory would otherwise impose. The local manifestations of this
global information are meted out over the structure of the natural
lattice by just such abductive gaps as the one between z and y.
Reference
=========
Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”,
Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce :
A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project,
Indiana University Press, Bloomington, IN, 1982.
Resources
=========
C.S. Peirce • Upon Logical Comprehension and Extension
http://www.iupui.edu/~peirce/writings/v2/w2/w2_06/v2_06.htm
My Notes • Information = Comprehension × Extension
http://intersci.ss.uci.edu/wiki/index.php/Information_%3D_Comprehension_%C3%97_Extension
----
Re:
{ Information = Comprehension × Extension } • Discussion 1
==========================================================
https://inquiryintoinquiry.com/2017/06/26/information-comprehension-x-extension-%e2%80%a2-discussion-1/
Peircers,
A puzzle in Peirce I have puzzled over for as long as I can remember
involves the relationship between his theory of signs, marking the
characters of icons, indices, and symbols, and his theory of inquiry,
bearing the three inferences of abduction, induction, and deduction.
I have long felt the resolution would lie in his theory of information,
epitomized by the formula “Information = Comprehension × Extension”.
Last summer looked ripe for another run at the problem, which I had
some years before begun tackling in a series of selections from and
comments on Peirce’s “Logic of Science” lectures at Harvard (1865)
and the Lowell Institute (1866).
There's a working draft of those selections and comments here:
Information = Comprehension × Extension
http://intersci.ss.uci.edu/wiki/index.php/Information_%3D_Comprehension_%C3%97_Extension
I serialized the selections and comments on my blog as I worked through them.
Introductory Comment
https://inquiryintoinquiry.com/2016/05/18/information-comprehension-x-extension/
(First Six) Selections from Peirce's Lectures
https://inquiryintoinquiry.com/2016/05/19/information-comprehension-x-extension-%e2%80%a2-selection-1/
https://inquiryintoinquiry.com/2016/05/19/information-comprehension-x-extension-%e2%80%a2-selection-2/
https://inquiryintoinquiry.com/2016/05/20/information-comprehension-x-extension-%e2%80%a2-selection-3/
https://inquiryintoinquiry.com/2016/05/21/information-comprehension-x-extension-%e2%80%a2-selection-4/
https://inquiryintoinquiry.com/2016/05/22/information-comprehension-x-extension-%e2%80%a2-selection-5/
https://inquiryintoinquiry.com/2016/05/22/information-comprehension-x-extension-%e2%80%a2-selection-6/
By September I had come to what I imagined was a new understanding of
the relationship between the types of signs and the types of inference,
at which time I put the whole matter away to cool, it being far harder
to judge a new idea when it's hot. At any rate, I think a year is long
enough to gain a cool eye or two, so I'll try sharing the new improved
analysis on the List.
Regards,
Jon
--
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