ONT Re: Logic Of Relatives
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
LOR. Note 58
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
I think that we have enough material on morphisms now
to go back and cast a more studied eye on what Peirce
is doing with that "number of" function, the one that
we apply to a logical term 't', absolute or relative
of any number of correlates, by writing it in square
brackets, as ['t']. It is frequently convenient to
have a prefix notation for this function, and since
Peirce reserves 'n' to signify 'not', I will try to
use 'v', personally thinking of it as a Greek 'nu',
which stands for frequency in physics, and which
kind of makes sense if we think of frequency as
it's habitual in statistics. End of mnemonics.
My plan will be nothing less plodding than to work through
all of the principal statements that Peirce has made about
the "number of" function up to our present stopping place
in the paper, namely, those that I collected once before
and placed at this location:
http://suo.ieee.org/ontology/msg04488.html
NOF 1.
| I propose to assign to all logical terms, numbers;
| to an absolute term, the number of individuals it denotes;
| to a relative term, the average number of things so related
| to one individual.
|
| Thus in a universe of perfect men ('men'),
| the number of "tooth of" would be 32.
|
| The number of a relative with two correlates would be the
| average number of things so related to a pair of individuals;
| and so on for relatives of higher numbers of correlates.
|
| I propose to denote the number of a logical term by
| enclosing the term in square brackets, thus ['t'].
|
| C.S. Peirce, CP 3.65
We may formalize the role of the "number of" function by assigning it
a local habitation and a name 'v' : S -> R, where S is a suitable set
of signs, called the "syntactic domain", that is ample enough to hold
all of the terms that we might wish to number in a given discussion,
and where R is the real number domain.
Transcribing Peirce's example, we may let m = "man" and 't' = "tooth of ---".
Then 'v'('t') = ['t'] = ['t'm]/[m], that is to say, in a universe of perfect
human dentition, the number of the relative term "tooth of ---" is equal to
the number of teeth of humans divided by the number of humans, that is, 32.
The 2-adic relative term 't' determines a 2-adic relation T c U x V,
where U and V are two universes of discourse, possibly the same one,
that hold among other things all of the teeth and all of the people
that happen to be under discussion, respectively.
A rough indication of the bigraph for T
might be drawn as follows, where I have
tried to sketch in just the toothy part
of U and the peoply part of V.
t_1 t_32 t_33 t_64 t_65 t_96 ... ...
o ... o o ... o o ... o o ... o U
\ | / \ | / \ | / \ | /
\ | / \ | / \ | / \ | / T
\|/ \|/ \|/ \|/
o o o o V
m_1 m_2 m_3 ...
Notice that the "number of" function 'v' : S -> R
needs the data that is represented by this entire
bigraph for T in order to compute the value ['t'].
Finally, one observes that this component of T is a function
in the direction T : U -> V, since we are counting only those
teeth that ideally occupy one and only one mouth of a creature.
Jon Awbrey
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o