[Inquiry] Re: Extension x Comprehension = Information

[Jon Awbrey]

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ECI.  Commentary Note 5

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2.  Disjunctive Terms

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

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

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