ONT Re: Extension x Comprehension = Information
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
| It may be added that algebra was formerly called 'Cossic',
| in English, or the 'Rule of Cos'; and the first algebra
| published in England was called "The Whetstone of Wit",
| because the author supposed that the word 'cos' was
| the Latin word so spelled, which means a whetstone.
| But in fact, 'cos' was derived from the Italian,
| 'cosa', thing, the thing you want to find, the
| unknown quantity whose value is sought. It is
| the Latin 'caussa', a thing aimed at, a cause.
|
| CSP, NEM 2, page 50.
|
| Charles Sanders Peirce,
|'The New Elements of Mathematics',
| Four Volumes In Five Books,
| Edited by Carolyn Eisele,
| Mouton, The Hague, 1976.
JA = Jon Awbrey
JD = Jean-Luc Delatre
Back to the grinstone, as I try to work my way through
the filings that are piling up in JD's file of comments.
With my gnosis to the cossis this way, I still cannot see
the Big Picture yet, and so I will just have to continue to
continue in a opportunistically piece-meal incermental style.
The chief obstruction that I espy at present to our digging what
Peirce laid down at this site is this matter of whether we have
gotten clear yet about the rules of the "natural kinds game".
So return with me now to this exchange:
JD: So now, we are ready to restate and comment some
Peirce excerpts you mentioned in your mails:
| 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. (CSP, CE 1, 468).
JD: This is only true if the "common character" which has
been spotted is in fact common to all mammals and not a
newly discovered characteristic shared only by men, horses,
kangaroos, and whales among mammals! In this case there *is*
in fact a possible new "comprehension which is adequate to the
limitation of the extension", just add this new 'attribute' to
the 'Content' mammals and give a name to this new 'Content',
say "weirdo-mammals", if you don't do that you don't have a
name to use to replace "men, horses, kangaroos, and whales"
because you would abusively make statements about mammals
which should apply only to "weirdo-mammals".
JA: It is necessary to understand the rules of the natural kinds game.
Assume that X is the universe of discourse. everybody knows what
It would be like to work within the power set of X, the lattice
Set(X) = (Pow(X), =>). but the natural kinds lattice Nat(X)
that we are given to work with is more restricted than this.
Today we would talk of "accessible predicates" and such.
Yes, it sounds silly to speak of accessible predicates
in a finite universe, but it is merely an exposition,
not a competition. Or maybe it is. Try to imagine
that you are on a TV game show where there's a lot
of money riding on it if you can just manage to
simulate the thinking of an "ordinary thinker"
and not that of a nit-picking wiseacre. Yes,
it's easier said than done, but we must try.
On further reflection, it may be better to
say "admissible predicates" here, but the
underlying point remains exactly the same.
If the game-show host says "men, horses, kangaroos, whales",
then without hardly thinking -- they don't have to think --
the ordinary thinker is already punching the buzzer and
screaming "mammals!"
And what is our weißenheimer logic nerd thinking? Voila! --
http://math.gc.cuny.edu/Logic/MAMLS/
Now, I ask you, do "weirdo-mammals" sound like the sort of
critters that would be found in the "great chain of being",
if the great chain of being is bound to natural creation?
I think that we cannot go much further without a measure
of understanding about the nature of this natural quarry.
Jon Awbrey
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤