wolframscience.com

A New Kind of Science: The NKS Forum : Powered by vBulletin version 2.3.0 A New Kind of Science: The NKS Forum > NKS Way of Thinking > Information = Comprehension x Extension -- Commentary
  Last Thread   Next Thread
Author
Thread Post New Thread    Post A Reply
Jon Awbrey


Registered: Feb 2004
Posts: 557

Information = Comprehension x Extension

ICE. Commentary Note 1

Peirce's incipient theory of information, that he appears
to have developed by sheer force of logical insight from
his early understanding of signs and scientific inquiry,
is not an easy subject to grasp in its developing state.
I will now attempt to follow its reasoning step by step.

| Let us now return to the information.
|
| The information of a term is the measure of its superfluous comprehension.

Today we say that information has to do with constraint, law, redundancy.
I think that Peirce is talking about more or less the same thing under
the theme of "superfluous comprehension", where the comprehension of
a term or expression is the collection of properties, also known as
"intensions", that it implies about the things to which it applies.

Jon Awbrey

---------------------------------------------------------------------------------

Re: ICE 2. http://stderr.org/pipermail/inquiry...ber/001914.html
In: ICE. http://stderr.org/pipermail/inquiry...hread.html#1913

Re: ICE 2. http://forum.wolframscience.com/sho...d=1996#post1996
In: ICE. http://forum.wolframscience.com/sho...hp?threadid=609

Last edited by Jon Awbrey on 11-22-2004 at 02:14 PM

Report this post to a moderator | IP: Logged

Old Post 11-22-2004 02:36 AM
Jon Awbrey is offline Click Here to See the Profile for Jon Awbrey Click here to Send Jon Awbrey a Private Message Visit Jon Awbrey's homepage! Edit/Delete Message Reply w/Quote
Jon Awbrey


Registered: Feb 2004
Posts: 557

Information = Comprehension x Extension

ICE. Commentary Note 2

| For instance, you and I are men because we possess those attributes --
| having two legs, being rational, &c. -- which make up the comprehension
| of 'man'. Every addition to the comprehension of a term lessens its
| extension up to a certain point, after that further additions increase
| the information instead.
|
| Thus, let us commence with the term 'colour'; add to the comprehension
| of this term, that of 'red'. 'Red colour' has considerably less extension
| than 'colour'; add to this the comprehension of 'dark'; 'dark red colour'
| has still less [extension]. Add to this the comprehension of 'non-blue' --
| 'non-blue dark red colour' has the same extension as 'dark red colour',
| so that the 'non-blue' here performs a work of supererogation; it tells
| us that no 'dark red colour' is blue, but does none of the proper business
| of connotation, that of diminishing the extension at all.

When we set about comprehending the comprehension of a sign, say,
a term or expression, we run into a very troublesome issue as to
how many intensions (predicates, properties, qualities) an object
of that sign has. For how do we quantify the number of qualities
a thing has? Without some more or less artificial strait imposed
on the collection of qualities, the number appears without limit.

Let's pass this by, as Peirce does, for now, and imagine
that we have fixed on some way of speaking sensibly about
"the" comprehension of a sign in a particular set of signs,
the collection of which we may use as a language or a medium.

Then we can begin to talk about the amount of redundancy,
the superfluidity of comprehension, if you will, as Peirce
does, that belongs to a given sign, and thus to its object.

Jon Awbrey

---------------------------------------------------------------------------------

Re: ICE 2. http://stderr.org/pipermail/inquiry...ber/001914.html
In: ICE. http://stderr.org/pipermail/inquiry...hread.html#1913

Re: ICE 2. http://forum.wolframscience.com/sho...d=1996#post1996
In: ICE. http://forum.wolframscience.com/sho...hp?threadid=609

Last edited by Jon Awbrey on 11-22-2004 at 02:14 PM

Report this post to a moderator | IP: Logged

Old Post 11-22-2004 12:44 PM
Jon Awbrey is offline Click Here to See the Profile for Jon Awbrey Click here to Send Jon Awbrey a Private Message Visit Jon Awbrey's homepage! Edit/Delete Message Reply w/Quote
Jon Awbrey


Registered: Feb 2004
Posts: 557

Information = Comprehension x Extension

ICE. Commentary Note 3

| Thus information measures the superfluous comprehension.
| And, hence, whenever we make a symbol to express any thing
| or any attribute we cannot make it so empty that it shall
| have no superfluous comprehension. I am going, next, to
| show that inference is symbolization and that the puzzle
| of the validity of scientific inference lies merely in
| this superfluous comprehension and is therefore entirely
| removed by a consideration of the laws of 'information'.

In a sense of primal innocence, logical laws bind the
vacuum state of any medium that is capable of bearing,
delivering, nurturing, and preserving signal meanings.
In other words, when we use symbols, not simple signs,
in a channel, language, or medium that is constrained
by logical laws, these laws do more than strain, they
also exact the generation of symbols upon symbols to
fill the requisite logical forms, and so there will
always be lots more ways than one to say any given
thing you might choose to say.

So to speak ...

Jon Awbrey

---------------------------------------------------------------------------------

Re: ICE 2. http://stderr.org/pipermail/inquiry...ber/001914.html
In: ICE. http://stderr.org/pipermail/inquiry...hread.html#1913

Re: ICE 2. http://forum.wolframscience.com/sho...d=1996#post1996
In: ICE. http://forum.wolframscience.com/sho...hp?threadid=609

Report this post to a moderator | IP: Logged

Old Post 11-22-2004 02:10 PM
Jon Awbrey is offline Click Here to See the Profile for Jon Awbrey Click here to Send Jon Awbrey a Private Message Visit Jon Awbrey's homepage! Edit/Delete Message Reply w/Quote
Jon Awbrey


Registered: Feb 2004
Posts: 557

Information = Comprehension x Extension

ICE. Commentary Note 4

| For this purpose, I must call your attention to
| the differences there are in the manner in which
| different representations stand for their objects.
|
| In the first place there are likenesses or copies -- such as
| 'statues', 'pictures', 'emblems', 'hieroglyphics', and the like.
| Such representations stand for their objects only so far as they
| have an actual resemblance to them -- that is agree with them in
| some characters. The peculiarity of such representations is that
| they do not determine their objects -- they stand for anything
| more or less; for they stand for whatever they resemble and
| they resemble everything more or less.
|
| The second kind of representations are such as are set up
| by a convention of men or a decree of God. Such are 'tallies',
| 'proper names', &c. The peculiarity of these 'conventional signs'
| is that they represent no character of their objects. Likenesses
| denote nothing in particular; 'conventional signs' connote nothing
| in particular.
|
| The third and last kind of representations are 'symbols' or general
| representations. They connote attributes and so connote them as to
| determine what they denote. To this class belong all 'words' and
| all 'conceptions'. Most combinations of words are also symbols.
| A proposition, an argument, even a whole book may be, and
| should be, a single symbol.

In order to speak of the meandering channel, the abdundancy of language,
the superfluidity of media, the play in the wheel of symbolism, then,
it is necessary to classify the different kinds of signs, the varied
ways that signs up to and including symbols, namely, those that are
interpretive by dint of their very essence, can be interpreted as
being referential to their objects.

On running through this familiar yet ever strange refrain for another time,
I see that I have scarcely begun to trace the sinews of the linkages among
the three types of signs, "the differences there are in the manner in which
different representations stand for their objects", the matter of extension
and comprehension, and the whole life-cycle of inquiry that engages me most.

Jon Awbrey

---------------------------------------------------------------------------------

Re: ICE 3. http://stderr.org/pipermail/inquiry...ber/001915.html
In: ICE. http://stderr.org/pipermail/inquiry...hread.html#1913

Re: ICE 3. http://forum.wolframscience.com/sho...d=1997#post1997
In: ICE. http://forum.wolframscience.com/sho...hp?threadid=609

Last edited by Jon Awbrey on 11-22-2004 at 03:42 PM

Report this post to a moderator | IP: Logged

Old Post 11-22-2004 03:06 PM
Jon Awbrey is offline Click Here to See the Profile for Jon Awbrey Click here to Send Jon Awbrey a Private Message Visit Jon Awbrey's homepage! Edit/Delete Message Reply w/Quote
Jon Awbrey


Registered: Feb 2004
Posts: 557

Information = Comprehension x Extension

ICE. Commentary Note 5

Da Capo Al Segno ...

Signs, inquiry, and information.
Let's focus on that for a while.

To put Peirce's examples more in line with the order of his
three categories, let us renumber them in the following way:


    1. The conjunctive term "spherical bright fragrant juicy tropical fruit".

    2.1. The disjunctive term "man or horse or kangaroo or whale".

    2.2. The disjunctive term "neat or swine or sheep or deer".

Peirce suggests an analogy or a parallelism between
the corresponding elements of the following triples:

    1. Conjunctive Term : Iconical Sign : Abductive Case

    2. Disjunctive Term : Indicial Sign : Inductive Rule

Here is an overview of the two patterns of reasoning, along
with the first steps of an analysis in sign-theoretic terms:

1. Conjunctive term "spherical bright fragrant juicy tropical fruit".

o-----------------------------o-----------------------------o
| ` ` Objective Framework ` ` | ` Interpretive Framework` ` |
o-----------------------------o-----------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `t_1` t_2 `...` t_5 `t_6` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` `o` ` ` ` `o` ` o ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` * ` * ` ` ` * ` * ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` * `*` ` `*` * ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` * * ` * * ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` **`** ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` z o ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |\` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | \ `Rule ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | `\ y=>z ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` \ ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `Fact | ` `\` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `x=>z | ` ` o y ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` `/` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` / ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | `/ Case ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | / `x=>y ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |/` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` x o ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| Conjunctive Predicate z, Abduction to the Case x => y ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| !S! `=` !I! `=` {t_1, t_2, t_3, t_4, t_5, t_6, x, y, z} ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| t_1 `=` "spherical" ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| t_2 `=` "bright"` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| t_3 `=` "fragrant"` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| t_4 `=` "juicy" ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| t_5 `=` "tropical"` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| t_6 `=` "fruit" ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| x ` `=` "subject" ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| y ` `=` "orange"` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| z ` `=` "spherical bright fragrant juicy tropical fruit"` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

| A similar line of thought may be gone through
| in reference to hypothesis. In this case we
| must start with the consideration of the term:
|
| 'spherical, bright, fragrant, juicy, tropical fruit'.
|
| Such a term, formed by the sum of the comprehensions of several terms,
| is called a conjunctive term. A conjunctive term has no extension
| adequate to its comprehension. Thus the only spherical bright
| fragrant juicy tropical fruit we know is the orange and that
| has many other characters besides these. Hence, such a term
| is of no use whatever. 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. On the other hand, if the conjunctive term is
| subject and we know that every spherical bright fragrant
| juicy tropical fruit necessarily has certain properties,
| it must be that we know more than that and can simplify the
| subject. Thus a conjunctive term may always be replaced by
| a simple one. So if we find that light is capable of producing
| certain phenomena which could only be enumerated by a long conjunction
| of terms, we may be sure that this compound predicate may be replaced
| by a simple one. And if only one simple one is known in which the
| conjunctive term is contained, this must be provisionally adopted.
|
| C.S. Peirce, 'Chronological Edition', CE 1, 470.

2. Disjunctive term "neat or swine or sheep or deer".

o-----------------------------o-----------------------------o
| ` ` Objective Framework ` ` | ` Interpretive Framework` ` |
o-----------------------------o-----------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` w o ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |\` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | \ `Rule ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | `\ v=>w ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` \ ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `Fact | ` `\` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `u=>w | ` ` o v ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` `/` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` / ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | `/ Case ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | / `u=>v ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |/` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` u o ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` **`** ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` * * ` * * ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` * `*` ` `*` * ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` * ` * ` ` ` * ` * ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` `o` ` ` ` `o` ` o ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `s_1` s_2 ` ` ` s_3 `s_4` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| Disjunctive Subject u, Induction to the Rule v => w ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| !S! `=` !I! `=` {s_1, s_2, s_3, s_4, u, v, w} ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| s_1 `=` "neat"` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| s_2 `=` "swine" ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| s_3 `=` "sheep" ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| s_4 `=` "deer"` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| u ` `=` "neat or swine or sheep or deer"` ` ` ` ` ` ` ` ` |
| v ` `=` "cloven-hoofed" ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| w ` `=` "herbivorous" ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

| Hence if we find out that neat are herbivorous, swine are herbivorous,
| sheep are herbivorous, and deer are herbivorous; we may be sure that
| there is some class of animals which covers all these, all the members
| of which are herbivorous. Now a disjunctive term -- such as 'neat swine
| sheep and deer', or 'man, horse, kangaroo, and whale' -- is not a true
| symbol. It does not denote what it does in consequence of its connotation,
| as a symbol does; on the contrary, no part of its connotation goes at all
| to determine what it denotes -- it is in that respect a mere accident if it
| denote anything. Its 'sphere' is determined by the concurrence of the four
| members, man, horse, kangaroo, and whale, or neat swine sheep and deer as
| the case may be.
|
| C.S. Peirce, 'Chronological Edition', CE 1, 468-469.

Jon Awbrey

---------------------------------------------------------------------------------

Re: ICE 4. http://stderr.org/pipermail/inquiry...ber/001917.html
Re: ICE 5. http://stderr.org/pipermail/inquiry...ber/001918.html
Re: ICE 6. http://stderr.org/pipermail/inquiry...ber/001919.html
In: ICE. http://stderr.org/pipermail/inquiry...hread.html#1913

Re: ICE 4. http://forum.wolframscience.com/sho...d=1999#post1999
Re: ICE 5. http://forum.wolframscience.com/sho...d=2000#post2000
Re: ICE 6. http://forum.wolframscience.com/sho...d=2002#post2002
In: ICE. http://forum.wolframscience.com/sho...hp?threadid=609

Last edited by Jon Awbrey on 11-22-2004 at 04:48 PM

Report this post to a moderator | IP: Logged

Old Post 11-22-2004 04:34 PM
Jon Awbrey is offline Click Here to See the Profile for Jon Awbrey Click here to Send Jon Awbrey a Private Message Visit Jon Awbrey's homepage! Edit/Delete Message Reply w/Quote
Jon Awbrey


Registered: Feb 2004
Posts: 557

Information = Comprehension x Extension

ICE. Commentary Note 6

Before we return to Peirce's description of a near duality
between icons and indices, involving a reciprocal symmetry
between intensions and extensions of concepts that becomes
perturbed to the breaking and yet the growing point by the
receipt of a fresh bit of information, I think that it may
help to recall a few pieces of technical terminology that
Peirce introduced into this discussion.

Passage 1 --

| It is important to distinguish between the two functions of a word:
| 1st to denote something -- to stand for something, and 2nd to mean
| something -- or as Mr. Mill phrases it -- to 'connote' something.
|
| What it denotes is called its 'Sphere'.
| What it connotes is called its 'Content'.
| Thus the 'sphere' of the word 'man' is for
| me every man I know; and for each of you it
| is every man you know. The 'content' of 'man'
| is all that we know of all men, as being two-
| legged, having souls, having language, &c., &c.
| It is plain that both the 'sphere' and the
| 'content' admit of more and less. ...
|
| Now the sphere considered as a quantity is called the Extension;
| and the content considered as quantity is called the Comprehension.
| Extension and Comprehension are also termed Breadth and Depth. So that
| a wider term is one which has a greater extension; a narrower one is
| one which has a less extension. A higher term is one which has a
| less Comprehension and a lower one has more.
|
| The narrower term is said to be contained under the wider one;
| and the higher term to be contained in the lower one.
|
| We have then:
|
| o-----------------------------o-----------------------------o
| | ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| | `What is 'denoted'` ` ` ` ` | `What is 'connoted' ` ` ` ` |
| | ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| | `Sphere ` ` ` ` ` ` ` ` ` ` | `Content` ` ` ` ` ` ` ` ` ` |
| | ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| | `Extension` ` ` ` ` ` ` ` ` | `Comprehension` ` ` ` ` ` ` |
| | ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| | ` ` ` ` ` ( wider ` ` ` ` ` | ` ` ` ` ( lower ` ` ` ` ` ` |
| | `Breadth` < ` ` ` ` ` ` ` ` | `Depth` < ` ` ` ` ` ` ` ` ` |
| | ` ` ` ` ` ( narrower` ` ` ` | ` ` ` ` ( higher` ` ` ` ` ` |
| | ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| | `What is contained 'under'` | `What is contained 'in' ` ` |
| | ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| o-----------------------------o-----------------------------o
|

| The principle of explicatory or deductive reasoning then is that:
|
| Every part of a word's Content belongs to
| every part of its Sphere,
|
| or:
|
| Whatever is contained 'in' a word belongs to
| whatever is contained under it.
|
| Now this maxim would not be true if the Extension and Comprehension
| were directly proportional to one another; this is to say if the
| Greater the one the greater the other. For in that case, though
| the whole Content would belong to the whole Sphere; yet only
| a particular part of it would belong to a part of that Sphere
| and not every part to every part. On the other hand if the
| Comprehension and Extension were not in some way proportional
| to one another, that is if terms of different spheres could
| have the same content or terms of the same content different
| spheres; then there would be no such fact as a content's
| 'belonging' to a sphere and hence again the maxim would
| fail. For the maxim to be true, then, it is absolutely
| necessary that the comprehension and extension should
| be inversely proportional to one another. That is
| that the greater the sphere, the less the content.
|
| Now this evidently true. If we take the term 'man' and increase
| its 'comprehension' by the addition of 'black', we have 'black man'
| and this has less 'extension' than 'man'. So if we take 'black man'
| and add 'non-black man' to its sphere, we have 'man' again, and so
| have decreased the comprehension. So that whenever the extension
| is increased the comprehension is diminished and 'vice versa'.
|
| C.S. Peirce, 'Chronological Edition', CE 1, 459-460.
| ICE 12. http://stderr.org/pipermail/inquiry...ber/001926.html

Passage 2 --

| The highest terms are therefore broadest and
| the lowest terms the narrowest. We can take
| a term so broad that it contains all other
| spheres under it. Then it will have no
| content whatever. There is but one
| such term -- with its synonyms --
| it is 'Being'. We can also take a
| term so low that it contains all other
| content within it. Then it will have no
| sphere whatever. There is but one such term --
| it is 'Nothing'.
|
| o------------------------o------------------------o
| | ` ` ` ` ` ` ` ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` `|
| | `Being` ` ` ` ` ` ` ` `| `Nothing` ` ` ` ` ` ` `|
| | ` ` ` ` ` ` ` ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` `|
| | `All breadth` ` ` ` ` `| `All depth` ` ` ` ` ` `|
| | ` ` ` ` ` ` ` ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` `|
| | `No depth ` ` ` ` ` ` `| `No breadth ` ` ` ` ` `|
| | ` ` ` ` ` ` ` ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` `|
| o------------------------o------------------------o
|

| We can conceive of terms so narrow that they are next to nothing,
| that is have an absolutely individual sphere. Such terms would be
| innumerable in number. We can also conceive of terms so high that
| they are next to 'being', that is have an entirely simple content.
| Such terms would also be innumerable.
|
| o------------------------o------------------------o
| | ` ` ` ` ` ` ` ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` `|
| | `Simple terms ` ` ` ` `| `Individual terms ` ` `|
| | ` ` ` ` ` ` ` ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` `|
| o------------------------o------------------------o
|

| But such terms though conceivable in one sense --
| that is intelligible in their conditions --
| are yet impossible.
| You never can narrow down to an individual.
| Do you say Daniel Webster is an individual?
| He is so in common parlance,
| but in logical strictness he is not.
| We think of certain images in our memory --
| a platform and a noble form uttering convincing and patriotic words --
| a statue --
| certain printed matter --
| and we say that which
| that speaker and the
| man whom that statue
| was taken for and the
| writer of this speech --
| that which these are in
| common is Daniel Webster.
| Thus, even the proper name
| of a man is a general term or
| the name of a class, for it names
| a class of sensations and thoughts.
| The true individual term the absolutely
| singular 'this' & 'that' cannot be reached.
| Whatever has comprehension must be general.
|
| In like manner, it is impossible to find any simple term.
| This is obvious from this consideration. If there is
| any simple term, simple terms are innumerable for in
| that case all attributes which are not simple are
| made up of simple attributes. Now none of these
| attributes can be affirmed or denied universally
| of whatever has any one. For let 'A' be one
| simple term and 'B' be another. Now suppose
| we can say All 'A' is 'B'; then 'B' is
| contained in 'A'. If, therefore, 'A'
| contains anything but 'B' it is
| a compound term, but 'A' is
| different from 'B', and is
| simple; hence it cannot
| be that All 'A' is 'B'.
| Suppose No 'A' is 'B', then
| not-'B' is contained in 'A';
| if therefore 'A' contains anything
| besides not-'B' it is not a simple term;
| but if it is the same as not-'B', it is not a
| simple term but is a term relative to 'B'. Now it is a
| simple term and therefore Some 'A' is 'B'. Hence if we take
| any two simple terms and call one 'A' and the other 'B' we have
|
| Some 'A' is 'B'
|
| and Some 'A' is not 'B'
|
| or in other words the universe will contain every possible kind of thing
| afforded by the permutation of simple qualities. Now the universe does not
| contain all these things; it contains no 'well-known green horse'. Hence the
| consequence of supposing a simple term to exist is an error of fact. There
| are several other ways of showing this besides the one that I have adopted.
| They all concur to show that whatever has extension must be composite.
|
| C.S. Peirce, 'Chronological Edition', CE 1, 460-461.
| ICE 13. http://stderr.org/pipermail/inquiry...ber/001927.html
| ICE 14. http://stderr.org/pipermail/inquiry...ber/001928.html
| ICE 15. http://stderr.org/pipermail/inquiry...ber/001929.html

Passage 3 --

| The moment, then, that we pass from nothing and the vacuity of being
| to any content or sphere, we come at once to a composite content and
| sphere. In fact, extension and comprehension -- like space and time --
| are quantities which are not composed of ultimate elements; but
| every part however small is divisible.
|
| The consequence of this fact is that when we wish to enumerate the
| sphere of a term -- a process termed 'division' -- or when we wish
| to run over the content of a term -- a process called 'definition' --
| since we cannot take the elements of our enumeration singly but must
| take them in groups, there is danger that we shall take some element
| twice over, or that we shall omit some. Hence the extension and
| comprehension which we know will be somewhat indeterminate. But
| we must distinguish two kinds of these quantities. If we were to
| subtilize we might make other distinctions but I shall be content
| with two. They are the extension and comprehension relatively to
| our actual knowledge, and what these would be were our knowledge
| perfect.
|
| Logicians have hitherto left the doctrine of extension
| and comprehension in a very imperfect state owing to the
| blinding influence of a psychological treatment of the
| matter. They have, therefore, not made this distinction
| and have reduced the comprehension of a term to what it
| would be if we had no knowledge of fact at all. I mention
| this because if you should come across the matter I am now
| discussing in any book, you would find the matter left in
| quite a different state.
|
| C.S. Peirce, 'Chronological Edition', CE 1, 462.
| ICE 16. http://stderr.org/pipermail/inquiry...ber/001930.html

Jon Awbrey

---------------------------------------------------------------------------------

In: ICE 00. http://forum.wolframscience.com/sho...hp?threadid=609
Re: ICE 12. http://forum.wolframscience.com/sho...d=2008#post2008
Re: ICE 13. http://forum.wolframscience.com/sho...d=2009#post2009
Re: ICE 14. http://forum.wolframscience.com/sho...d=2010#post2010
Re: ICE 15. http://forum.wolframscience.com/sho...d=2011#post2011
Re: ICE 16. http://forum.wolframscience.com/sho...d=2012#post2012

Report this post to a moderator | IP: Logged

Old Post 11-22-2004 06:00 PM
Jon Awbrey is offline Click Here to See the Profile for Jon Awbrey Click here to Send Jon Awbrey a Private Message Visit Jon Awbrey's homepage! Edit/Delete Message Reply w/Quote
Jon Awbrey


Registered: Feb 2004
Posts: 557

Information = Comprehension x Extension

ICE. Commentary Note 7

I find one more patch of material from Peirce's early
lectures that we need to cover the subject of indices.
I include a piece of the context, even if it overlaps
a bit with fragments that still live in recent memory.

Passage 4 --

| Yet there are combinations of words and combinations of conceptions
| which are not strictly speaking symbols. These are of two kinds
| of which I will give you instances. We have first cases like:
|
| 'man and horse and kangaroo and whale',
|
| and secondly, cases like:
|
| 'spherical bright fragrant juicy tropical fruit'.
|
| The first of these terms has no comprehension which is adequate to the
| limitation of the extension. In fact, men, horses, kangaroos, and whales
| have no attributes in common which are not possessed by the entire class of
| mammals. For this reason, this disjunctive term, man and horse and kangaroo
| and whale, is of no use whatever. For suppose it is the subject of a sentence;
| suppose we know that men and horses and kangaroos and whales have some common
| character. Since they have no common character which does not belong to the
| whole class of mammals, it is plain that 'mammals' may be substituted for
| this term. Suppose it is the predicate of a sentence, and that we know
| that something is either a man or a horse or a kangaroo or a whale; then,
| the person who has found out this, knows more about this thing than that it
| is a mammal; he therefore knows which of these four it is for these four have
| nothing in common except what belongs to all other mammals. Hence in this case
| the particular one may be substituted for the disjunctive term. A disjunctive
| term, then, -- one which aggregates the extension of several symbols, -- may
| always be replaced by a simple term.
|
| Hence if we find out that neat are herbivorous, swine are herbivorous,
| sheep are herbivorous, and deer are herbivorous; we may be sure that there
| is some class of animals which covers all these, all the members of which are
| herbivorous. Now a disjunctive term -- such as 'neat swine sheep and deer',
| or 'man, horse, kangaroo, and whale' -- is not a true symbol. It does not
| denote what it does in consequence of its connotation, as a symbol does;
| on the contrary, no part of its connotation goes at all to determine what
| it denotes -- it is in that respect a mere accident if it denote anything.
| Its 'sphere' is determined by the concurrence of the four members, man,
| horse, kangaroo, and whale, or neat swine sheep and deer as the case
| may be.
|
| Now those who are not accustomed to the homologies of the conceptions of
| men and words, will think it very fanciful if I say that this concurrence
| of four terms to determine the sphere of a disjunctive term resembles the
| arbitrary convention by which men agree that a certain sign shall stand
| for a certain thing. And yet how is such a convention made? The men
| all look upon or think of the thing and each gets a certain conception
| and then they agree that whatever calls up or becomes an object of that
| conception in either of them shall be denoted by the sign. In the one
| case, then, we have several different words and the disjunctive term
| denotes whatever is the object of either of them. In the other case,
| we have several different conceptions -- the conceptions of different
| men -- and the conventional sign stands for whatever is an object of
| either of them. It is plain the two cases are essentially the same,
| and that a disjunctive term is to be regarded as a conventional sign
| or index. And we find both agree in having a determinate extension
| but an inadequate comprehension.
|
| C.S. Peirce, 'Chronological Edition', CE 1, 468-469.

Jon Awbrey

---------------------------------------------------------------------------------

Re: ICE 4. http://stderr.org/pipermail/inquiry...ber/001917.html
In: ICE. http://stderr.org/pipermail/inquiry...hread.html#1913

Re: ICE 4. http://forum.wolframscience.com/sho...d=1999#post1999
In: ICE. http://forum.wolframscience.com/sho...hp?threadid=609

Report this post to a moderator | IP: Logged

Old Post 11-22-2004 06:28 PM
Jon Awbrey is offline Click Here to See the Profile for Jon Awbrey Click here to Send Jon Awbrey a Private Message Visit Jon Awbrey's homepage! Edit/Delete Message Reply w/Quote
Jon Awbrey


Registered: Feb 2004
Posts: 557

Information = Comprehension x Extension

ICE. Commentary Note 8

I'm going to make yet another try at following the links that Peirce makes
among conventions, disjunctive terms, indexical signs, and inductive rules.
For this purpose, I'll break the text up into smaller pieces, and pick out
just those parts of it that have to do with the indexical aspect of things.

Before I can get to this, though, I will need to deal with the uncertainty
that I am experiencing over the question as to whether a "connotation" is
just another "notation", and thus belongs to the interpretive framework,
that is, the SI-plane, or whether it is an objective property, a quality
of objects of terms. I have decided to finesse the issue by forcing my
own brand of interpretation on the next text, where the trouble starts:

| It is important to distinguish between the two functions of a word:
| 1st to denote something -- to stand for something, and 2nd to mean
| something -- or as Mr. Mill phrases it -- to 'connote' something.
|
| What it denotes is called its 'Sphere'.
| What it connotes is called its 'Content'.
|
| Thus the 'sphere' of the word 'man' is for me every man
| I know; and for each of you it is every man you know.
|
| The 'content' of 'man' is all that we know of all men,
| as being two-legged, having souls, having language, &c., &c.
|
| It is plain that both the 'sphere' and the 'content' admit of more and less. ...
|
| Now the sphere considered as a quantity is called the Extension;
| and the content considered as quantity is called the Comprehension.
|
| Extension and Comprehension are also termed Breadth and Depth.
|
| So that a wider term is one which has a greater extension;
| a narrower one is one which has a less extension.
|
| A higher term is one which has a less Comprehension
| and a lower one has more.
|
| The narrower term is said to be contained under the wider one;
| and the higher term to be contained in the lower one.
|
| We have then:
|
| o-----------------------------o-----------------------------o
| | ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| | `What is 'denoted'` ` ` ` ` | `What is 'connoted' ` ` ` ` |
| | ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| | `Sphere ` ` ` ` ` ` ` ` ` ` | `Content` ` ` ` ` ` ` ` ` ` |
| | ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| | `Extension` ` ` ` ` ` ` ` ` | `Comprehension` ` ` ` ` ` ` |
| | ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| | ` ` ` ` ` ( wider ` ` ` ` ` | ` ` ` ` ( lower ` ` ` ` ` ` |
| | `Breadth` < ` ` ` ` ` ` ` ` | `Depth` < ` ` ` ` ` ` ` ` ` |
| | ` ` ` ` ` ( narrower` ` ` ` | ` ` ` ` ( higher` ` ` ` ` ` |
| | ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| | `What is contained 'under'` | `What is contained 'in' ` ` |
| | ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| o-----------------------------o-----------------------------o
|

| The principle of explicatory or deductive reasoning then is that:
|
| Every part of a word's Content belongs to
| every part of its Sphere,
|
| or:
|
| Whatever is contained 'in' a word belongs to
| whatever is contained under it.
|
| Now this maxim would not be true if the Extension and Comprehension
| were directly proportional to one another; this is to say if the
| Greater the one the greater the other. For in that case, though
| the whole Content would belong to the whole Sphere; yet only
| a particular part of it would belong to a part of that Sphere
| and not every part to every part. On the other hand if the
| Comprehension and Extension were not in some way proportional
| to one another, that is if terms of different spheres could
| have the same content or terms of the same content different
| spheres; then there would be no such fact as a content's
| 'belonging' to a sphere and hence again the maxim would
| fail. For the maxim to be true, then, it is absolutely
| necessary that the comprehension and extension should
| be inversely proportional to one another. That is
| that the greater the sphere, the less the content.
|
| Now this evidently true. If we take the term 'man' and increase
| its 'comprehension' by the addition of 'black', we have 'black man'
| and this has less 'extension' than 'man'. So if we take 'black man'
| and add 'non-black man' to its sphere, we have 'man' again, and so
| have decreased the comprehension. So that whenever the extension
| is increased the comprehension is diminished and 'vice versa'.
|
| C.S. Peirce, 'Chronological Edition', CE 1, 459-460.

I am going to treat Peirce's use of the "quantity consideration"
as a significant operator that transforms its argument from the
syntactic domain !S! |_| !I! to the objective domain !O!.

| Now the sphere considered as a quantity is called the Extension;
| and the content considered as quantity is called the Comprehension.

Taking this point of view, then, I will consider the Extensions of terms
and the Comprehensions of terms, to be "quantities", in effect, objective
formal elements that are subject to being compared with one another within
their respective domains. In particular, I will view them as elements of
partially ordered sets. On my reading of Peirce's text, the word "content"
is still ambiguous from context of use to context of use, but I will simply
let that be as it may, hoping that it will suffice to fix the meaning of the
more technical term "comprehension".

This is still experimental -- I'll just have to see how it works out over time.

Jon Awbrey

---------------------------------------------------------------------------------

Re: ICE 12. http://stderr.org/pipermail/inquiry...ber/001926.html
In: ICE. http://stderr.org/pipermail/inquiry...hread.html#1913

Re: ICE 12. http://forum.wolframscience.com/sho...d=2008#post2008
In: ICE. http://forum.wolframscience.com/sho...hp?threadid=609

Report this post to a moderator | IP: Logged

Old Post 11-22-2004 07:52 PM
Jon Awbrey is offline Click Here to See the Profile for Jon Awbrey Click here to Send Jon Awbrey a Private Message Visit Jon Awbrey's homepage! Edit/Delete Message Reply w/Quote
Jon Awbrey


Registered: Feb 2004
Posts: 557

Information = Comprehension x Extension

ICE. Commentary Note 9

2. Conventions, Disjunctive Terms, Indexical Signs, Inductive Rules

2.1. "man and horse and kangaroo and whale" (intensional conjunction).

Nota Bene. In this particular choice of phrasing, Peirce is using the
intensional "and", meaning that the compound term has the intensions that
are shared by all of the component terms, in this way producing a term that
bears the "greatest common intension" of the terms that are connected in it.
This is formalized as the "greatest lower bound" in a lattice of intensions,
dual to the union of sets or "least upper bound" in a lattice of extensions.

It is perhaps more common today to use the extensional "or"
in order to express the roughly equivalent compound concept:

2.1. "men or horses or kangaroos or whales" (extensional disjunction).

| Yet there are combinations of words and combinations
| of conceptions which are not strictly speaking symbols.
|
| These are of two kinds of which I will give you instances.
|
| We have first cases like: "man and horse and kangaroo and whale" ...
|
| [This term] has no comprehension which is
| adequate to the limitation of the extension.
|
| In fact, men, horses, kangaroos, and whales have no attributes
| in common which are not possessed by the entire class of mammals.
|
| For this reason, this disjunctive term,
| "man and horse and kangaroo and whale",
| is of no use whatever.
|
| For suppose it is the subject of a sentence; suppose we know that
| men and horses and kangaroos and whales have some common character.
|
| Since they have no common character which does not belong to
| the whole class of mammals, it is plain that "mammals" may be
| substituted for this term.
|
| Suppose it is the predicate of a sentence, and that we know that
| something is either a man or a horse or a kangaroo or a whale;
|
| then, the person who has found out this, knows
| more about this thing than that it is a mammal;
|
| he therefore knows which of these four it is
| for these four have nothing in common except
| what belongs to all other mammals.
|
| Hence in this case the particular one may
| be substituted for the disjunctive term.
|
| A disjunctive term, then, -- one which aggregates the extension
| of several symbols, -- may always be replaced by a simple term.
|
| C.S. Peirce, 'Chronological Edition', CE 1, 468.

Let us first assemble a minimal syntactic domain !S!
that is sufficient to begin discussing this example:


    !S! = {"m", "h", "k", "w", "S", "M", "P"}

Here, I have introduced the abbreviations:

    "m" = "man"
    "h" = "horse"
    "k" = "kangaroo"
    "w" = "whale"

    "S" = "man or horse or kangaroo or whale"
    "M" = "Mammal"
    "P" = "Predicate shared by man, horse, kangaroo, whale"

Let's attempt to keep tabs on things by using
angle brackets for the comprehension of a term,
and square brackets for the extension of a term.

For brevity, let x = ["x"], in general.

Here is an initial picture of the situation, so far as I can see it:

o-----------------------------o-----------------------------o
| ` ` Objective Framework ` ` | ` Interpretive Framework` ` |
o-----------------------------o-----------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `P <------------o------------ "P" ` ` ` ` ` ` |
| ` ` ` ` ` ` = \ ` ` ` ` ` ` ` ` ` ` ` ` ` `|\ ` ` ` ` ` ` |
| ` ` ` ` ` `=` `\` ` ` ` ` ` ` ` ` ` ` ` ` `|`\` ` ` ` ` ` |
| ` ` ` ` ` = ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` `|` \ ` ` ` ` ` |
| ` ` ` ` `=` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` `|` `\` ` ` ` ` |
| ` ` ` ` = ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` `|` ` \ ` ` ` ` |
| ` ` ` `P` ` ` ` ` `M <------o--------------|--- "M" ` ` ` |
| ` ` ` ` \ ` ` ` ` = ` ` ` ` ` ` ` ` ` ` ` `|` ` / ` ` ` ` |
| ` ` ` ` `\` ` ` `=` ` ` ` ` ` ` ` ` ` ` ` `|` `/` ` ` ` ` |
| ` ` ` ` ` \ ` ` = ` ` ` ` ` ` ` ` ` ` ` ` `|` / ` ` ` ` ` |
| ` ` ` ` ` `\` `=` ` ` ` ` ` ` ` ` ` ` ` ` `|`/` ` ` ` ` ` |
| ` ` ` ` ` ` \ = ` ` ` ` ` ` ` ` ` ` ` ` ` `|/ ` ` ` ` ` ` |
| ` ` ` ` ` ` `S <------------o------------ "S" ` ` ` ` ` ` |
| ` ` ` ` ` `** **` ` ` ` ` ` ` ` ` ` ` ` `** **` ` ` ` ` ` |
| ` ` ` ` `*`*` `*`*` ` ` ` ` ` ` ` ` ` `*`*` `*`*` ` ` ` ` |
| ` ` ` `*` * ` ` * `*` ` ` ` ` ` ` ` `*` * ` ` * `*` ` ` ` |
| ` ` `*` `*` ` ` `*` `*` ` ` ` ` ` `*` `*` ` ` `*` `*` ` ` |
| ` `o` ` o ` ` ` ` o ` `o` ` ` ` `o` ` o ` ` ` ` o ` `o` ` |
| ` `m` ` h ` ` ` ` k ` `w` ` ` ` "m" `"h"` ` ` `"k"` "w" ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| Disjunctive Subject "S" and Inductive Rule "M => P" ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| !S! `=` !I! `= `{"m", "h", "k", "w", "S", "M", "P"} ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| "m" `=` "man" ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| "h" `=` "horse" ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| "k" `=` "kangaroo"` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| "w" `=` "whale" ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| "S" `=` "man or horse or kangaroo or whale" ` ` ` ` ` ` ` |
| "M" `=` "Mammal"` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| "P" `=` "Predicate shared by man, horse, kangaroo, whale" |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

In effect, relative to the lattice of natural (non-phony) kinds,
any property P, predicated of S, can be "lifted" to a mark of M.

Jon Awbrey

---------------------------------------------------------------------------------

Re: ICE 4. http://stderr.org/pipermail/inquiry...ber/001917.html
In: ICE. http://stderr.org/pipermail/inquiry...hread.html#1913

Re: ICE 4. http://forum.wolframscience.com/sho...d=1999#post1999
In: ICE. http://forum.wolframscience.com/sho...hp?threadid=609

Report this post to a moderator | IP: Logged

Old Post 11-22-2004 08:45 PM
Jon Awbrey is offline Click Here to See the Profile for Jon Awbrey Click here to Send Jon Awbrey a Private Message Visit Jon Awbrey's homepage! Edit/Delete Message Reply w/Quote
Jon Awbrey


Registered: Feb 2004
Posts: 557

Information = Comprehension x Extension

ICE. Commentary Note 10

2. Conventions, Disjunctive Terms, Indexical Signs, Inductive Rules (cont.)

| We come next to consider inductions. In inferences of this kind
| we proceed as if upon the principle that as is a sample of a class
| so is the whole class. The word 'class' in this connection means
| nothing more than what is denoted by one term, -- or in other words
| the sphere of a term. Whatever characters belong to the whole sphere
| of a term constitute the content of that term. Hence the principle of
| induction is that whatever can be predicated of a specimen of the sphere
| of a term is part of the content of that term. And what is a specimen?
| It is something taken from a class or the sphere of a term, at random --
| that is, not upon any further principle, not selected from a part of
| that sphere; in other words it is something taken from the sphere
| of a term and not taken as belonging to a narrower sphere. Hence
| the principle of induction is that whatever can be predicated of
| something taken as belonging to the sphere of a term is part of
| the content of that term. But this principle is not axiomatic
| by any means. Why then do we adopt it?
|
| To explain this, we must remember that the process of induction is a
| process of adding to our knowledge; it differs therein from deduction --
| which merely explicates what we know -- and is on this very account called
| scientific inference. Now deduction rests as we have seen upon the inverse
| proportionality of the extension and comprehension of every term; and this
| principle makes it impossible apparently to proceed in the direction of
| ascent to universals. But a little reflection will show that when our
| knowledge receives an addition this principle does not hold. ...
|
| The reason why this takes place is worthy of notice. Every addition to
| the information which is incased in a term, results in making some term
| equivalent to that term. ...
|
| Thus, every addition to our information about a term is an addition
| to the number of equivalents which that term has. Now, in whatever
| way a term gets to have a new equivalent, whether by an increase in
| our knowledge, or by a change in the things it denotes, this always
| results in an increase either of extension or comprehension without
| a corresponding decrease in the other quantity.
|
| C.S. Peirce, 'Chronological Edition', CE 1, 462-464.

2.1. "man and horse and kangaroo and whale" (aggregarious animals).

It seems to me now that my previous explanation of the use of "and" in
this example was far too complicated and contrived. So let's just say
that the conjunction "and" is being used in its "aggregational" sense.

I will also try an alternate style of picture for the "lifting" property,
by means of which, relative to the lattice of natural (non-ad-hoc) kinds,
a property P, naturally predicated of S, can be "elevated" to apply to M.

o-----------------------------o-----------------------------o
| ` ` Objective Framework ` ` | ` Interpretive Framework` ` |
o-----------------------------o-----------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `P <------------o------------ "P" ` ` ` ` ` ` |
| ` ` ` ` ` ` `|\ ` ` ` ` ` ` ` ` ` ` ` ` ` `|\ ` ` ` ` ` ` |
| ` ` ` ` ` ` `|`\` ` ` ` ` ` ` ` ` ` ` ` ` `|`\` ` ` ` ` ` |
| ` ` ` ` ` ` `|` \ ` ` ` ` ` ` ` ` ` ` ` ` `|` \ ` ` ` ` ` |
| ` ` ` ` ` ` `|` `\` ` ` ` ` ` ` ` ` ` ` ` `|` `\` ` ` ` ` |
| ` ` ` ` ` ` `|` ` \ ` ` ` ` ` ` ` ` ` ` ` `|` ` \ ` ` ` ` |
| ` ` ` ` ` ` `|` ` `M <------o--------------|--- "M" ` ` ` |
| ` ` ` ` ` ` `|` ` = ` ` ` ` ` ` ` ` ` ` ` `|` ` / ` ` ` ` |
| ` ` ` ` ` ` `|` `=` ` ` ` ` ` ` ` ` ` ` ` `|` `/` ` ` ` ` |
| ` ` ` ` ` ` `|` = ` ` ` ` ` ` ` ` ` ` ` ` `|` / ` ` ` ` ` |
| ` ` ` ` ` ` `|`=` ` ` ` ` ` ` ` ` ` ` ` ` `|`/` ` ` ` ` ` |
| ` ` ` ` ` ` `|= ` ` ` ` ` ` ` ` ` ` ` ` ` `|/ ` ` ` ` ` ` |
| ` ` ` ` ` ` `S <------------o------------ "S" ` ` ` ` ` ` |
| ` ` ` ` ` `** **` ` ` ` ` ` ` ` ` ` ` ` `** **` ` ` ` ` ` |
| ` ` ` ` `*`*` `*`*` ` ` ` ` ` ` ` ` ` `*`*` `*`*` ` ` ` ` |
| ` ` ` `*` * ` ` * `*` ` ` ` ` ` ` ` `*` * ` ` * `*` ` ` ` |
| ` ` `*` `*` ` ` `*` `*` ` ` ` ` ` `*` `*` ` ` `*` `*` ` ` |
| ` `o` ` o ` ` ` ` o ` `o` ` ` ` `o` ` o ` ` ` ` o ` `o` ` |
| ` `m` ` h ` ` ` ` k ` `w` ` ` ` "m" `"h"` ` ` `"k"` "w" ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| Disjunctive Subject "S" and Inductive Rule "M => P" ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| !S! `=` !I! `= `{"m", "h", "k", "w", "S", "M", "P"} ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| "m" `=` "man" ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| "h" `=` "horse" ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| "k" `=` "kangaroo"` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| "w" `=` "whale" ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| "S" `=` "man or horse or kangaroo or whale" ` ` ` ` ` ` ` |
| "M" `=` "Mammal"` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| "P" `=` "Predicate shared by man, horse, kangaroo, whale" |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

I believe that we can now begin to see the linkage to inductive rules.
When a sample S is "fairly" or "randomly" drawn from the membership M
of some population and when every member of S is observed to have the
property P, then it is naturally rational to expect that every member
of M will also have the property P. This is the principle behind all
of our more usual statistical generalizations, giving us the leverage
that it takes to lift predicates from samples to a membership sampled.

Now, the aggregate that is designated by "man, horse, kangaroo, whale",
even if it's not exactly a random sample from the class of mammals, is
drawn by design from sufficiently many and sufficiently diverse strata
within the class of mammals to be regarded as a quasi-random selection.
Thus, it affords us with a sufficient basis for likely generalizations.

Jon Awbrey

---------------------------------------------------------------------------------

Re: ICE 18. http://stderr.org/pipermail/inquiry...ber/001932.html
Re: ICE 19. http://stderr.org/pipermail/inquiry...ber/001933.html
In: ICE. http://stderr.org/pipermail/inquiry...hread.html#1913

Re: ICE 18. http://forum.wolframscience.com/sho...d=2014#post2014
Re: ICE 19. http://forum.wolframscience.com/sho...d=2015#post2015
In: ICE. http://forum.wolframscience.com/sho...hp?threadid=609

Report this post to a moderator | IP: Logged

Old Post 11-22-2004 09:20 PM
Jon Awbrey is offline Click Here to See the Profile for Jon Awbrey Click here to Send Jon Awbrey a Private Message Visit Jon Awbrey's homepage! Edit/Delete Message Reply w/Quote
Jon Awbrey


Registered: Feb 2004
Posts: 557

Information = Comprehension x Extension

ICE. Commentary Note 11

At this point it will help to jump ahead a bit in time,
and to take in the more systematic account of the same
material from Peirce's "New List of Categories" (1867).

| I shall now show how the three conceptions of reference to a ground,
| reference to an object, and reference to an interpretant are the
| fundamental ones of at least one universal science, that of logic.
|
| C.S. Peirce, 'Collected Papers', CP 1.559.
|
|"On a New List of Categories" (1867),
|'Chronological Edition', CE 2, 49-59,
|'Collected Papers', CP 1.545-567.

We will have occasion to consider this paragraph in detail later,
but for the present purpose let's hurry on down to the end of it.

| In an argument, the premisses form a representation of the conclusion,
| because they indicate the interpretant of the argument, or representation
| representing it to represent its object. The premisses may afford a likeness,
| index, or symbol of the conclusion. In deductive argument, the conclusion is
| represented by the premisses as by a general sign under which it is contained.
| In hypotheses, something 'like' the conclusion is proved, that is, the premisses
| form a likeness of the conclusion. Take, for example, the following argument:
|
| [Abduction of a Case]
|
| M is, for instance, P_1, P_2, P_3, and P_4;
|
| S is P_1, P_2, P_3, and P_4:
|
| Therefore, S is M.
|
| Here the first premiss amounts to this, that "P_1, P_2, P_3, and P_4"
| is a likeness of M, and thus the premisses are or represent a likeness
| of the conclusion. That it is different with induction another example
| will show:
|
| [Induction of a Rule]
|
| S_1, S_2, S_3, and S_4 are taken as samples of the collection M;
|
| S_1, S_2, S_3, and S_4 are P:
|
| Therefore, All M is P.
|
| Hence the first premiss amounts to saying that "S_1, S_2, S_3, and S_4"
| is an index of M. Hence the premisses are an index of the conclusion.

1. Abductive Inference and Iconic Signs --

Peirce's analysis of the patterns of abductive argument
can be understood according to the following paraphrase:


  • Abduction of a Case:

    Fact: S => P_1, S => P_2, S => P_3, S => P_4
    Rule: M => P_1, M => P_2, M => P_3, M => P_4
    -------------------------------------------------
    Case: S => M

  • If X => each of A, B, C, D, ...,

    then we have the following equivalents:

  • 1. X => the "greatest lower bound" (GLB) of A, B, C, D, ...

  • 2. X => the logical conjunction A & B & C & D & ...

  • 3. X => Q = A & B & C & D & ...

More succinctly, letting Q = P_1 & P_2 & P_3 & P_4,
the argument is summarized by the following scheme:

  • Abduction of a Case:

    Fact: S => Q
    Rule: M => Q
    --------------
    Case: S => M

In this piece of Abduction, it is the GLB or the conjunction
of the ostensible predicates that is the operative predicate
of the argument, that is, it is the predicate that is common
to both the Fact and the Rule of the inference.

Finally, the reason why one can say that Q is an iconic sign
of the object M is that Q can be taken to denote M by virtue
of the qualities that they share, namely, P_1, P_2, P_3, P_4.

Notice that the iconic denotation is symmetric, at least in principle,
that is, icons are icons of each other as objects, at least potentially,
whether or not a particular interpretive agent is making use of their
full iconicity during a particular phase of semeiosis.

The situation is diagrammed in Figure 11.1.

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` P_1 ` P_2 ` ` ` ` P_3 ` P_4 ` ` ` ` ` |
| ` ` ` ` ` `o` ` `o` ` ` ` ` `o` ` `o` ` ` ` ` ` |
| ` ` ` ` ` ` `*` ` * ` ` ` ` * ` `*` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `*` `*` ` ` `*` `*` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `*` * ` ` * `*` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `* *` `* *` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `** **` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `Q o` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|\ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|`\` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` \ ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` `\` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` \ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` `o M` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|/ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `S o` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| Figure 1. `Abduction of the Case S => M ` ` ` ` |
o-------------------------------------------------o

In a diagram like this, even if one does not bother to
show all of the implicational or the subject-predicate
relationships by means of explicit lines, then one may
still assume the "transitive closure" of the relations
that are actually shown, along with any that are noted
in the text that accompanies it.

2. Inductive Inference and Indexic Signs --

Peirce's analysis of the patterns of inductive argument
can be understood according to the following paraphrase:

  • Induction of a Rule:

    Case: S_1 => M, S_2 => M, S_3 => M, S_4 => M
    Fact: S_1 => P, S_2 => P, S_3 => P, S_4 => P
    -------------------------------------------------
    Rule: M => P

  • If X <= each of A, B, C, D, ...,

    then we have the following equivalents:

  • 1. X <= the "least upper bound" (LUB) of A, B, C, D, ...

  • 2. X <= the logical disjunction A v B v C v D v ...

  • 3. X <= L = A v B v C v D v ...

More succinctly, letting L = P_1 v P_2 v P_3 v P_4,
the argument is summarized by the following scheme:

  • Induction of a Rule:

    Case: L => M
    Fact: L => P
    --------------
    Rule: M => P

In this bit of Induction, it is the LUB or the disjunction
of the ostensible subjects that is the operative subject
of the argument, to wit, the subject that is common
to both the Case and the Fact of the inference.

Finally, the reason why one can say that L is an indexical sign
of the object M is that L can be taken to denote M by virtue of
the instances that they share, namely, S_1, S_2, S_3, S_4.

Notice that the indexical denotation is symmetric, at least in principle,
that is, indices are indices of each other as objects, at least potentially,
whether or not a particular interpretive agent is making use of their full
indiciality during a particular phase of semeiosis.

The situation is diagrammed in Figure 11.2.

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `P o` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|\ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` \ ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` \ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` `o M` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|`/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|/ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `L o` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `** **` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `* *` `* *` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `*` * ` ` * `*` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `*` `*` ` ` `*` `*` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `*` ` * ` ` ` ` * ` `*` ` ` ` ` ` ` |
| ` ` ` ` ` `o` ` `o` ` ` ` ` `o` ` `o` ` ` ` ` ` |
| ` ` ` ` ` S_1 ` S_2 ` ` ` ` S_3 ` S_4 ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| Figure 2. `Induction of the Rule M => P ` ` ` ` |
o-------------------------------------------------o

Jon Awbrey

---------------------------------------------------------------------------------

Last edited by Jon Awbrey on 11-23-2004 at 03:14 AM

Report this post to a moderator | IP: Logged

Old Post 11-23-2004 03:00 AM
Jon Awbrey is offline Click Here to See the Profile for Jon Awbrey Click here to Send Jon Awbrey a Private Message Visit Jon Awbrey's homepage! Edit/Delete Message Reply w/Quote
Jon Awbrey


Registered: Feb 2004
Posts: 557

Information = Comprehension x Extension

ICE. Commentary Note 12

Let's redraw the "New List" pictures of Abduction and Induction
in a way that is a little less cluttered, availing ourselves of
the fact that logical implications or lattice subsumptions obey
a transitive law to leave unmarked what is thereby understood.

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` P_1 ` ... ` P_k ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `o` ` `o` ` `o` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` `|` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` `|` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ `|` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\`|`/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \|/ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `Q o` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|\ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|`\` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` \ ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` `\` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` \ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` `o M` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ^ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|`/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|/ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `S o` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| Icon Q of Object M, Abduction of Case "S is M"` |
o-------------------------------------------------o

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `P o` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|^ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|`\` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` \ ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` `\` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` \ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` `o M` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|`/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|/ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `L o` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` /|\ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `/`|`\` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` / `|` \ ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `/` `|` `\` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` / ` `|` ` \ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `o` ` `o` ` `o` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` S_1 ` ... ` S_k ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| Index L of Object M, Induction of Rule "M is P" |
o-------------------------------------------------o

The main problem that I have with these pictures in their
present form is that they do not sufficiently underscore
the distinction in roles between signs and objects, and
thus we may find it a bit jarring that the middle term
of a syllogistic figure is described as an "object"
of iconic and indexic signs.

I will try to address that issue when
I return to Peirce's earlier lectures.

Jon Awbrey

---------------------------------------------------------------------------------

Report this post to a moderator | IP: Logged

Old Post 11-23-2004 03:44 AM
Jon Awbrey is offline Click Here to See the Profile for Jon Awbrey Click here to Send Jon Awbrey a Private Message Visit Jon Awbrey's homepage! Edit/Delete Message Reply w/Quote
Jon Awbrey


Registered: Feb 2004
Posts: 557

Information = Comprehension x Extension

ICE. Commentary Note 13

In the process of rationalizing Peirce's account of induction to myself
I find that I have now lost sight of the indexical sign relationships,
so let me go back to the drawing board one more time to see if I can
get the indexical and the inductive aspects of the situation back
into the very same picture. Here is how we left off last time:

o-----------------------------o-----------------------------o
| ` ` Objective Framework ` ` | ` Interpretive Framework` ` |
o-----------------------------o-----------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `P <------------------------- "P" ` ` ` ` ` ` |
| ` ` ` ` ` ` `|\ ` ` ` ` ` ` ` ` ` ` ` ` ` `|\ ` ` ` ` ` ` |
| ` ` ` ` ` ` `|`\` ` ` ` ` ` ` ` ` ` ` ` ` `|`\` ` ` ` ` ` |
| ` ` ` ` ` ` `|` \ ` ` ` ` ` ` ` ` ` ` ` ` `|` \ ` ` ` ` ` |
| ` ` ` ` ` ` `|` `\` ` ` ` ` ` ` ` ` ` ` ` `|` `\` ` ` ` ` |
| ` ` ` ` ` ` `|` ` \ ` ` ` ` ` ` ` ` ` ` ` `|` ` \ ` ` ` ` |
| ` ` ` ` ` ` `|` ` `M <---------------------|--- "M" ` ` ` |
| ` ` ` ` ` ` `|` ` = ` ` ` ` ` ` ` ` ` ` ` `|` ` / ` ` ` ` |
| ` ` ` ` ` ` `|` `=` ` ` ` ` ` ` ` ` ` ` ` `|` `/` ` ` ` ` |
| ` ` ` ` ` ` `|` = ` ` ` ` ` ` ` ` ` ` ` ` `|` / ` ` ` ` ` |
| ` ` ` ` ` ` `|`=` ` ` ` ` ` ` ` ` ` ` ` ` `|`/` ` ` ` ` ` |
| ` ` ` ` ` ` `|= ` ` ` ` ` ` ` ` ` ` ` ` ` `|/ ` ` ` ` ` ` |
| ` ` ` ` ` ` `S <------------------------- "S" ` ` ` ` ` ` |
| ` ` ` ` ` `** **` ` ` ` ` ` ` ` ` ` ` ` `** **` ` ` ` ` ` |
| ` ` ` ` `*`*` `*`*` ` ` ` ` ` ` ` ` ` `*`*` `*`*` ` ` ` ` |
| ` ` ` `*` * ` ` * `*` ` ` ` ` ` ` ` `*` * ` ` * `*` ` ` ` |
| ` ` `*` `*` ` ` `*` `*` ` ` ` ` ` `*` `*` ` ` `*` `*` ` ` |
| ` `o` ` o ` ` ` ` o ` `o` ` ` ` `o` ` o ` ` ` ` o ` `o` ` |
| ` `m` ` h ` ` ` ` k ` `w` ` ` ` "m" `"h"` ` ` `"k"` "w" ` |
| ` S_1 `S_2` ` ` `S_3` S_4 ` ` "S_1" `"S_2"` ` "S_3" "S_4" |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| Disjunctive Subject "S" and Inductive Rule "M => P" ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| !S! `=` !I! `= `{"m", "h", "k", "w", "S", "M", "P"} ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| "m" `=` "man" ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| "h" `=` "horse" ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| "k" `=` "kangaroo"` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| "w" `=` "whale" ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| "S" `=` "man or horse or kangaroo or whale" ` ` ` ` ` ` ` |
| "M" `=` "Mammal"` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| "P" `=` "Predicate shared by man, horse, kangaroo, whale" |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

In this double-entry account we are more careful to distinguish
the signs that belong to the "interpretive framework" (IF) from
the objects that belong to the "objective framework" (OF). One
benefit of this scheme is that it immediately resolves many of
the conceptual puzzles that arise from confusing the roles of
objects and the roles of signs in the relevant sign relation.

For example, we observe the distinction between
the objects S, M, P and the signs "S", "M", "P".
The objects may be regarded as extensive classes
or as intensive properties, as the context demands.
The signs may be regarded as sentences or as terms,
in accord with the application and the ends in view.

It is as if we collected a stratified sample S of the disjoint
type "man, horse, kangaroo, whale" from the class M of mammals,
and observed the property P to hold true of each of them. Now
we know that this could be a statistical fluke, in other words,
that S is just an arbitrary subset of the relevant universe of
discourse, and that the very next M you pick from outside of S
might not have the property P. But that is not very likely if
the sample was "fairly" or "randomly" drawn. So the objective
domain is not a lattice like the power set of the universe but
something more constrained, of a kind that makes induction and
learning possible, a lattice of "natural kinds", you might say.
In the natural kinds lattice, then, the LUB of S is close to M.

Now that I have this much of the picture assembled in one frame,
it occurs to me that I might be confusing myself about what are
the sign relations of actual interest in this situation. After
all, samples and signs are closely related, as evidenced by the
etymological connection between them that goes back at least as
far as Hippocrates.

I need not mention any further the more obvious sign relations
that we use just to talk about the objects in the example, for
the signs and the objects in these relations of denotation are
organized according to their roles in the diptych of objective
and interpretive frames. But there are, outside the expressly
designated designations, the ways that samples of species tend
to be taken as signs of their genera, and these sign relations
are discovered internal to the previously marked object domain.

Let us look to Peirce's "New List" of the next year for guidance:

| In an argument, the premisses form a representation of
| the conclusion, because they indicate the interpretant
| of the argument, or representation representing it to
| represent its object. The premisses may afford a
| likeness, index, or symbol of the conclusion. ...
|
| [Induction of a Rule]
|
| S_1, S_2, S_3, and S_4 are taken as samples of the collection M;
|
| S_1, S_2, S_3, and S_4 are P:
|
| Therefore, All M is P.
|
| Hence the first premiss amounts to saying that "S_1, S_2, S_3, and S_4"
| is an index of M. Hence the premisses are an index of the conclusion.
|
| C.S. Peirce, 'Collected Papers', CP 1.559.
|
|"On a New List of Categories" (1867),
|'Chronological Edition', CE 2, 49-59,
|'Collected Papers', CP 1.545-567.

There we see an abstract example with the same logical structure
and almost precisely the same labeling. It is a premiss of this
argument that "S_1, S_2, S_3, S_4" is an index of M. But we are
left wondering if he means the objective class M or the sign "M".
If we take the quotation marks of "S_1, S_2, S_3, S_4" as giving
the disjunctive term equal to "S", then we have the next picture:

o-----------------------------o-----------------------------o
| ` ` Objective Framework ` ` | ` Interpretive Framework` ` |
o-----------------------------o-----------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `P <------------------------- "P" ` ` ` ` ` ` |
| ` ` ` ` ` ` `|\ ` ` ` ` ` ` ` ` ` ` ` ` ` `|\ ` ` ` ` ` ` |
| ` ` ` ` ` ` `|`\` ` ` ` ` ` ` ` ` ` ` ` ` `|`\` ` ` ` ` ` |
| ` ` ` ` ` ` `|` \ ` ` ` ` ` ` ` ` ` ` ` ` `|` \ ` ` ` ` ` |
| ` ` ` ` ` ` `|` `\` ` ` ` ` ` ` ` ` ` ` ` `|` `\` ` ` ` ` |
| ` ` ` ` ` ` `|` ` \ ` ` ` ` ` ` ` ` ` ` ` `|` ` \ ` ` ` ` |
| ` ` ` ` ` ` `|` ` `M <---------------------|--- "M" ` ` ` |
| ` ` ` ` ` ` `|` ` = $ ` ` ` ` ` ` ` ` ` ` `|` ` / % ` ` ` |
| ` ` ` ` ` ` `|` `=` ` ` ` ` ` ` ` ` ` ` ` `|` `/` ` ` ` ` |
| ` ` ` ` ` ` `|` = ` ` ` ` ` ` ` ` ` ` ` ` `|` / ` ` ` ` ` |
| ` ` ` ` ` ` `|`=` ` `$` ` ` ` ` ` ` ` ` ` `|`/` ` `%` ` ` |
| ` ` ` ` ` ` `|= ` ` ` ` ` ` ` ` ` ` ` ` ` `|/ ` ` ` ` ` ` |
| ` ` ` ` ` ` `S <------------------------- "S" ` ` ` ` ` ` |
| ` ` ` ` ` `** **` ` ` $ ` ` ` ` ` ` ` `$ ** *%` ` ` % ` ` |
| ` ` ` ` `* *` `* *` ` ` ` ` ` ` ` `$` `* *` `* %` ` ` ` ` |
| ` ` ` `*` * ` ` * `*` ` ` ` ` `$` ` `*` * ` ` * `%` ` ` ` |
| ` ` `*` `*` ` ` `*` `* $` `$` ` ` `*` `*` ` ` `*` `% %` ` |
| ` `o` ` o ` ` ` ` o ` `o` ` ` ` `o` ` o ` ` ` ` o ` `o` ` |
| ` `m` ` h ` ` ` ` k ` `w` ` ` `"m"` `"h"` ` ` `"k"` `"w"` |
| ` S_1 `S_2` ` ` `S_3` S_4 ` ` "S_1" "S_2" ` ` "S_3" "S_4" |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| Disjunctive Subject "S" and Inductive Rule "M => P" ` ` ` |
o-----------------------------------------------------------o

So we have two readings of what Peirce is saying:


  1. The interpretation where "S" is an index of M
    by virtue of "S" being a property of each S_j,
    literally a generic sign of each of them, and
    by virtue of each S_j being an instance of M.
    The "S" to S_4 to M linkage is painted $ $ $.

  2. The interpretation where "S" is an index of "M"
    by virtue of "S" being a property of each "S_j",
    literally an implicit sign of each of them, and
    by dint of each "S_j" being an instance of "M".
    The "S" to "S_4" to "M" link is drawn as % % %.

On third thought, there is still the possibility
of a sense in which S is literally an index of M,
that is, we might regard a fair sample from S as
nothing less than a representative sample from M.

Jon Awbrey

---------------------------------------------------------------------------------

Report this post to a moderator | IP: Logged

Old Post 11-23-2004 11:34 AM
Jon Awbrey is offline Click Here to See the Profile for Jon Awbrey Click here to Send Jon Awbrey a Private Message Visit Jon Awbrey's homepage! Edit/Delete Message Reply w/Quote
Jon Awbrey


Registered: Feb 2004
Posts: 557

Information = Comprehension x Extension

ICE. Commentary Note 14

With the clarity afforded by a reflective interval, my third thought,
the relatively ultimate, more perfect interpretant of the intervening
struggle toward that final, hopefully neither dying nor raging light,
begins to look ever more like the fitting icon of my first impression.

I was trying to understand the things that Peirce said and wrote about
the conventional, disjunctive, indexical, inductive complex of notions
in the period 1865-1867. And I was focused for the moment on this bit:

| In an argument, the premisses form a representation of
| the conclusion, because they indicate the interpretant
| of the argument, or representation representing it to
| represent its object. The premisses may afford a
| likeness, index, or symbol of the conclusion. ...
|
| [Induction of a Rule]
|
| S_1, S_2, S_3, and S_4 are taken as samples of the collection M;
|
| S_1, S_2, S_3, and S_4 are P:
|
| Therefore, All M is P.
|
| Hence the first premiss amounts to saying that "S_1, S_2, S_3, and S_4"
| is an index of M. Hence the premisses are an index of the conclusion.
|
| C.S. Peirce, 'Collected Papers', CP 1.559.

I've gotten as far as sketching this picture of the possible readings:

o-----------------------------o-----------------------------o
| ` ` Objective Framework ` ` | ` Interpretive Framework` ` |
o-----------------------------o-----------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `P <------------------------- "P" ` ` ` ` ` ` |
| ` ` ` ` ` ` `|\ ` ` ` ` ` ` ` ` ` ` ` ` ` `|\ ` ` ` ` ` ` |
| ` ` ` ` ` ` `|`\` ` ` ` ` ` ` ` ` ` ` ` ` `|`\` ` ` ` ` ` |
| ` ` ` ` ` ` `|` \ ` ` ` ` ` ` ` ` ` ` ` ` `|` \ ` ` ` ` ` |
| ` ` ` ` ` ` `|` `\` ` ` ` ` ` ` ` ` ` ` ` `|` `\` ` ` ` ` |
| ` ` ` ` ` ` `|` ` \ ` ` ` ` ` ` ` ` ` ` ` `|` ` \ ` ` ` ` |
| ` ` ` ` ` ` `|` ` `M <---------------------|--- "M" ` ` ` |
| ` ` ` ` ` ` `|` ` = * ` ` ` ` ` ` ` ` ` ` `|` ` / % ` ` ` |
| ` ` ` ` ` ` `|` `=` ` ` ` ` ` ` ` ` ` ` ` `|` `/` ` ` ` ` |
| ` ` ` ` ` ` `|` = ` ` ` ` ` ` ` ` ` ` ` ` `|` / ` ` ` ` ` |
| ` ` ` ` ` ` `|`=` ` `*` ` ` ` ` ` ` ` ` ` `|`/` ` `%` ` ` |
| ` ` ` ` ` ` `|= ` ` ` ` ` ` ` ` ` ` ` ` ` `|/ ` ` ` ` ` ` |
| ` ` ` ` ` ` `S <------------------------- "S" ` ` ` ` ` ` |
| ` ` ` ` ` `** **` ` ` * ` ` ` ` ` ` ` `$ ** *%` ` ` % ` ` |
| ` ` ` ` `* *` `* *` ` ` ` ` ` ` ` `$` `* *` `* %` ` ` ` ` |
| ` ` ` `*` * ` ` * `*` ` ` ` ` `$` ` `*` * ` ` * `%` ` ` ` |
| ` ` `*` `*` ` ` `*` `* *` `$` ` ` `*` `*` ` ` `*` `% %` ` |
| ` `o` ` o ` ` ` ` o ` `o` ` ` ` `o` ` o ` ` ` ` o ` `o` ` |
| ` `m` ` h ` ` ` ` k ` `w` ` ` `"m"` `"h"` ` ` `"k"` `"w"` |
| ` S_1 `S_2` ` ` `S_3` S_4 ` ` "S_1" "S_2" ` ` "S_3" "S_4" |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| Disjunctive Subject "S" and Inductive Rule "M => P" ` ` ` |
o-----------------------------------------------------------o

In order of increasing "objectivity", here are three alternatives:


  1. The interpretation where "S" is an index of "M"
    by virtue of "S" being a property of each "S_j",
    literally an implicit sign of each of them, and
    by dint of each "S_j" being an instance of "M".
    The "S" to "S_4" to "M" link is drawn [% % % %].

  2. The interpretation where "S" is an index of M
    by virtue of "S" being a property of each S_j,
    literally a generic sign of each of them, and
    by virtue of each S_j being an instance of M.
    The "S" to S_4 to M link is a 2-tone [$ $ * *].

  3. The interpretation where S is an index of M
    by virtue of S being a property of each S_j,
    literally a supersample of each of them, and
    by virtue of each S_j being an instance of M.
    The S to S_4 to M link is shown as [* * * *].

Perhaps it is the nature of the sign situation that all three
interpretations will persevere and keep some measure of merit.
At the moment I am leaning toward the third interpretation as
it manifests the possibility of a higher grade of objectivity.

Jon Awbrey

---------------------------------------------------------------------------------

Report this post to a moderator | IP: Logged

Old Post 11-23-2004 12:18 PM
Jon Awbrey is offline Click Here to See the Profile for Jon Awbrey Click here to Send Jon Awbrey a Private Message Visit Jon Awbrey's homepage! Edit/Delete Message Reply w/Quote
Jon Awbrey


Registered: Feb 2004
Posts: 557

Information = Comprehension x Extension

ICE. Commentary Note 15

I am going to stick with the Index-Induction side of the problem
until I feel like I understand what's going on with this linkage
between the faces of the sign relation and the phases of inquiry.

The New List (1867) account of the relationship between
the kinds of signs and the kinds of arguments says this:


| In an argument, the premisses form a representation of
| the conclusion, because they indicate the interpretant
| of the argument, or representation representing it to
| represent its object.

In general, if one takes the components of an Argument
to be its Conclusion, its Premisses taken collectively,
and its Interpretant, then they can be seen to take up
the following sign relational duties:

  • <Conclusion, Premisses, Interpretant> = <Object, Sign, Interpretant>

This generality may be broken down according to the role of the premisses:

| The premisses may afford a likeness,
| index, or symbol of the conclusion.

In the case of the inductive argument,
we have the following role assigments:

  • <Conclusion, Premisses, Interpretant> = <Object, Index, Interpretant>

Marked out in greater detail, we have the following role assignments:

Premisses (Index):

| S_1, S_2, S_3, and S_4 are taken as samples of the collection M.
|
| S_1, S_2, S_3, and S_4 are P.

Conclusion (Object):

| All M is P.

Remark:

| Hence the first premiss amounts to saying that "S_1, S_2, S_3, & S_4"
| is an index of M. Hence the premisses are an index of the conclusion.

One of the questions that I have at this point is whether Peirce
is speaking loosely or strictly when he refers to the conclusion
and the premisses of the argument in question. Strictly speaking,
the conclusion has the form M => P and the premisses have the forms
S_j => M and S_j => P. But taken more loosely, as often happens in
contexts where the antecedent of a conditional statement is already
assumed to hold true, people will sometimes refer to the consequent
of a conditional conclusion as the conclusion and the consequents
of conditional premisses as the premisses. In the present case,
such a practice would lead to speaking of the predicate M as
one of the premisses and the predicate P as the conclusion.
So let us keep that interpretive option in mind as we go.

Jon Awbrey

---------------------------------------------------------------------------------

Report this post to a moderator | IP: Logged

Old Post 11-23-2004 02:00 PM
Jon Awbrey is offline Click Here to See the Profile for Jon Awbrey Click here to Send Jon Awbrey a Private Message Visit Jon Awbrey's homepage! Edit/Delete Message Reply w/Quote
Post New Thread    Post A Reply
  Last Thread   Next Thread
Show Printable Version | Email this Page | Subscribe to this Thread


 

wolframscience.com  |  wolfram atlas  |  NKS online  |  Wolfram|Alpha  |  Wolfram Science Summer School  |  web resources  |  contact us

Forum Sponsored by Wolfram Research

© 2004-14 Wolfram Research, Inc. | Powered by vBulletin 2.3.0 © 2000-2002 Jelsoft Enterprises, Ltd. | Disclaimer | Archives