[Inquiry] Re: Extension x Comprehension = Information

[Jon Awbrey]

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

ECI.  Commentary Note 6

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

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.
|
| CSP, CE 1, pages 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 <------------ at ------------ "P"             |
|              ^^                            ^^             |
|              | \                           | \            |
|              |  \                "S => P"  |  \  "M => P" |
|              |   \                         |   \          |
|              |    \                        |    \         |
|              |     M <------ at --------------|--- "M"       |
|              |    =                        |    ^         |
|              |   =                         |   /          |
|              |  =                          |  /  "S => M" |
|              | =                           | /            |
|              |=                            |/             |
|              S <------------ at ------------ "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

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o