[Inquiry] Re: Logic Of Relatives

[Jon Awbrey]

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

LOR.  Note 61

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

Up to this point in the LOR of 1870, Peirce has introduced the
"number of" measure on logical terms and discussed the extent
to which this measure, 'v' : S -> R such that 'v' : s ~> [s],
exhibits a couple of important measure-theoretic principles:

1.  The "number of" map exhibits a certain type of "uniformity property",
    whereby the value of the measure on a uniformly qualified population
    is in fact actualized by each member of the population.

2.  The "number of" map satisfies an "order morphism principle", whereby
    the illative partial ordering of logical terms is reflected up to a
    partial extent by the arithmetical linear ordering of their measures.

Peirce next takes up the action of the "number of" map on the two types of,
loosely speaking, "additive" operations that we normally consider in logic.

NOF 3.

| It is plain that both the regular non-invertible addition and the
| invertible addition satisfy the absolute conditions.  (CP 3.67).

The "regular non-invertible addition" is signified by "+,",
corresponding to what we'd call the inclusive disjunction
of logical terms or the union of their extensions as sets.

The "invertible addition" is signified in algebra by "+",
corresponding to what we'd call the exclusive disjunction
of logical terms or the symmetric difference of their sets,
ignoring many details and nuances that are often important,
of course.

| But the notation has other recommendations.  The conception of 'taking together'
| involved in these processes is strongly analogous to that of summation, the sum
| of 2 and 5, for example, being the number of a collection which consists of a
| collection of two and a collection of five.  (CP 3.67).

A full interpretation of this remark will require us to pick up the precise
technical sense in which Peirce is using the word "collection", and that will
take us back to his logical reconstruction of certain aspects of number theory,
all of which I am putting off to another time, but it is still possible to get
a rough sense of what he's saying relative to the present frame of discussion.

The "number of" map 'v' : S -> R evidently induces
some sort of morphism with respect to logical sums.
If this were straightforwardly true, we could write:

? 'v'(x +, y)  =  'v'x + 'v'y
?
? Equivalently:
?
? [x +, y]  =  [x] + [y]

Of course, things are just not that simple in the case
of inclusive disjunction and set-theoretic unions, so
we'd "probably" invent a word like "sub-additive" to
describe the principle that does hold here, namely:

| 'v'(x +, y)  =<  'v'x + 'v'y
|
| Equivalently:
|
| [x +, y]  =<  [x] + [y]

This is why Peirce trims his discussion of this point with the following hedge:

| Any logical equation or inequality in which no operation but addition
| is involved may be converted into a numerical equation or inequality by
| substituting the numbers of the several terms for the terms themselves --
| provided all the terms summed are mutually exclusive.  (CP 3.67).

Finally, a morphism with respect to addition,
even a contingently qualified one, must do the
right stuff on behalf of the additive identity:

| Addition being taken in this sense,
|'nothing' is to be denoted by 'zero',
| for then:
|
| x +, 0  =  x
|
| whatever is denoted by x;  and this is the definition
| of 'zero'.  This interpretation is given by Boole, and
| is very neat, on account of the resemblance between the
| ordinary conception of 'zero' and that of nothing, and
| because we shall thus have
|
| [0]  =  0.
|
| C.S. Peirce, CP 3.67

With respect to the nullity 0 in S and the number 0 in R, we have:

'v'0  =  [0]  =  0.

In sum, therefor, it also serves that only preserves
a due respect for the function of a vacuum in nature.

Jon Awbrey

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