ONT Re: Information = Comprehension x Extension -- Discussion
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
ICE. Discussion Note 5
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
AK = Antti Karttunen
JA = Jon Awbrey
AK: Then, interpreting this:
CSP: | Thus suppose a blind man to be told that
| no red things are blue. He has previously
| known only that red is a color; and that
| certain things 'A', 'B', and 'C' are red.
|
| The comprehension of red then has been for him 'color'.
| Its extension has been 'A', 'B', 'C'.
|
| But when he learns that no red thing is blue,
| 'non-blue' is added to the comprehension of red,
| without the least diminution of its extension.
|
| Its comprehension becomes 'non-blue color'.
| Its extension remains 'A', 'B', 'C'.
|
| Suppose afterwards he learns that a fourth thing 'D' is red.
| Then, the comprehension of 'red' remains unchanged, 'non-blue color';
| while its extension becomes 'A', 'B', 'C', and 'D'. Thus, the rule
| that the greater the extension of a term the less its comprehension
| and 'vice versa', holds good only so long as our knowledge is not
| added to; but as soon as our knowledge is increased, either the
| comprehension or extension of that term which the new information
| concerns is increased without a corresponding decrease of the other
| quantity.
|
| 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 when the blind man learns that 'red' is
| not-blue, 'red not-blue' becomes for him equivalent to 'red'. Before
| that, he might have thought that 'red not-blue' was a little more
| restricted term than 'red', and therefore it was so to him, but
| the new information makes it the exact equivalent of red.
| In the same way, when he learns that 'D' is red, the
| term 'D-like red' becomes equivalent to 'red'.
let's try to cast this within a finite universe of discourse X.
X = {1, 2, 3, 4, 5, 6}.
a predicate p on X is a function p : X -> B = {0, 1} = {false, true}.
therefore, the lattice of possible predicates on X
and the lattice of subsets of X are the same thing.
we could interpret "color" as a higher order predicate,
that is, color : (X -> B) -> B, where color(p) is true
if p : X -> B is color predicate, and false otherwise.
or we could interpret color as an ordinary predicate,
c : X -> B, understood as "things that have color".
then the statement "red is a color" can be read
as the simple implication "red => color".
i can't tell yet whether Peirce's example depends
critically on one interpretation or the other,
so let's try the simpler alternative first.
we start with a "state of information" (SOI)
that 1, 2, 3 are Red, and that Red => Color.
Color
|
|
|
Red
/|\
/ | \
/ | \
1 2 3
next we are told that "no red thing is blue",
which we may formulate without quantifiers as
the proposition (Red Blue), or ~[Red & Blue].
X = (Red Blue)
o
/ \
Color / \
| / \
| (Blue) \
| / \
| / \
|/ \
Red Blue
/|\ .
/ | \ .
/ | \ .
1 2 3 .
. .
. .
. .
o
0 = Red Blue
CSP: | But when he learns that no red thing is blue,
| 'non-blue' is added to the comprehension of red,
| without the least diminution of its extension.
|
| Its comprehension becomes 'non-blue color'.
| Its extension remains 'A', 'B', 'C'.
next we learn that 4 is Red.
X = (Red Blue)
o
/ \
Color / \
| / \
| (Blue) \
| / \
| / \
|/ \
Red Blue
/|\ .
/ | \ .
/ | \ .
1 2 3 4 .
. .
. .
. .
o
0 = Red Blue
CSP: | Then, the comprehension of 'red' remains unchanged, 'non-blue color';
| while its extension becomes 'A', 'B', 'C', and 'D'.
AK: So, before knowing the truth about red and non-blue,
the blind man had a conceptual graph something like this:
color
/ \
/ \
/ non-blue
/ /
red /
\ /
\ /
non-blue red
AK: But after it, the lattice changes to:
color
| \
| \
| non-blue
| /
| /
|/
red (= non-blue red),
yes, this seems to be basically the same thing as what i got.
AK: As an arrow is added between non-blue and red,
and thus the "non-blue red" = glb(red, non_blue)
happens to be red itself.
you can think of this as a quotient operation on lattices.
starting with the unconstrained power set lattice, imposing
a constraint then acts to identify nodes previously distinct.
i have worked out some examples like this that i will look up.
but i am forced to interrupret the continuum of discourse hic et nunc ...
jon awbrey
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
http://www.cs.bsu.edu/homepages/mighty/history.html
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o