[Inquiry] Re: Logic Of Relatives

[Jon Awbrey]

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LOR.  Note 41

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I return to where we were in unpacking the contents of CP 3.73.
Peirce remarks that the comma operator can be iterated at will:

| In point of fact, since a comma may be added in this way to any
| relative term, it may be added to one of these very relatives
| formed by a comma, and thus by the addition of two commas
| an absolute term becomes a relative of two correlates.
|
| So:
|
| m,,b,r
|
| interpreted like
|
| `g`'o'h
|
| means a man that is a rich individual and
| is a black that is that rich individual.
|
| But this has no other meaning than:
|
| m,b,r
|
| or a man that is a black that is rich.
|
| Thus we see that, after one comma is added, the
| addition of another does not change the meaning
| at all, so that whatever has one comma after it
| must be regarded as having an infinite number.
|
| C.S. Peirce, CP 3.73

Again, let us check whether this makes sense
on the stage of our small but dramatic model.

Let's say that Desdemona and Othello are rich,
and, among the persons of the play, only they.

With this premiss we obtain a sample of absolute terms
that is sufficiently ample to work through our example:

1    =   B +, C +, D +, E +, I +, J +, O

b    =   O

m    =   C +, I +, J +, O

r    =   D +, O

One application of the comma operator
yields the following 2-adic relatives:

1,   =   B:B +, C:C +, D:D +, E:E +, I:I +, J:J +, O:O

b,   =   O:O

m,   =   C:C +, I:I +, J:J +, O:O

r,   =   D:D +, O:O

Another application of the comma operator
generates the following 3-adic relatives:

1,,  =   B:B:B +, C:C:C +, D:D:D +, E:E:E +, I:I:I +, J:J:J +, O:O:O

b,,  =   O:O:O

m,,  =   C:C:C +, I:I:I +, J:J:J +, O:O:O

r,,  =   D:D:D +, O:O:O

Assuming the associativity of multiplication among 2-adic relatives,
we may compute the product m,b,r by a brute force method as follows:

m,b,r  =  (C:C +, I:I +, J:J +, O:O)(O:O)(D +, O)

       =  (C:C +, I:I +, J:J +, O:O)(O)

       =  O

This avers that a man that is black that is rich is Othello,
which is true on the premisses of our universe of discourse.

The stock associations of `g`'o'h lead us to multiply out the
product m,,b,r along the following lines, where the trinomials
of the form (X:Y:Z)(Y:Z)(Z) are the only ones that produce any
non-null result, specifically, of the form (X:Y:Z)(Y:Z)(Z) = X.

m,,b,r  =  (C:C:C +, I:I:I +, J:J:J +, O:O:O)(O:O)(D +, O)

        =  (O:O:O)(O:O)(O)

        =  O

So we have that m,,b,r = m,b,r.

In closing, observe that the teridentity relation has turned up again
in this context, as the second comma-ing of the universal term itself:

1,,  =  B:B:B +, C:C:C +, D:D:D +, E:E:E +, I:I:I +, J:J:J +, O:O:O.

Jon Awbrey

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