[Inquiry] Re: Logic Of Relatives -- Commentary

[Jon Awbrey]

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

LOR.  Commentary Note 2

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

Peirce's remarks at CP 3.65 are so replete with remarkable ideas,
some of them so taken for granted in mathematical discourse that
they usually escape explicit mention, and others so suggestive
of things to come in a future remote from his time of writing,
and yet so smoothly introduced in passing that it's all too
easy to overlook their consequential significance, that I
can do no better here than to highlight these ideas in
other words, whose main advantage is to be a little
more jarring to the mind's sensibilities.

| Numbers Corresponding to Letters
|
| I propose to use the term "universe" to denote that class of individuals
| 'about' which alone the whole discourse is understood to run.  The universe,
| therefore, in this sense, as in Mr. De Morgan's, is different on different
| occasions.  In this sense, moreover, discourse may run upon something which
| is not a subjective part of the universe;  for instance, upon the qualities
| or collections of the individuals it contains.
|
| 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, 'Collected Papers', CP 3.65

1.  This mapping of letters to numbers, or logical terms to mathematical quantities,
    is the very core of what "quantification theory" is all about, and definitely
    more to the point than the mere "innovation" of using distinctive symbols
    for the so-called "quantifiers".  We will speak of this more later on.

2.  The mapping of logical terms to numerical measures,
    to express it in current language, would probably be
    recognizable as some kind of "morphism" or "functor"
    from a logical domain to a quantitative co-domain.

3.  Notice that Peirce follows the mathematician's usual practice,
    then and now, of making the status of being an "individual" or
    a "universal" relative to a discourse in progress.  I have come
    to appreciate more and more of late how radically different this
    "patchwork" or "piecewise" approach to things is from the way of
    some philosophers who seem to be content with nothing less than
    many worlds domination, which means that they are never content
    and rarely get started toward the solution of any real problem.
    Just my observation, I hope you understand.

4.  It is worth noting that Peirce takes the "plural denotation"
    of terms for granted, or what's the number of a term for,
    if it could not vary apart from being one or nil?

5.  I also observe that Peirce takes the individual objects of a particular
    universe of discourse in a "generative" way, not a "totalizing" way,
    and thus they afford us with the basis for talking freely about
    collections, constructions, properties, qualities, subsets,
    and "higher types", as the phrase is mint.

Jon Awbrey

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
inquiry e-lab: http://stderr.org/pipermail/inquiry/
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o