[Inquiry] Re: Logic Of Relatives

[Jon Awbrey]

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LOR.  Note 35

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We have sufficiently covered the application of the comma functor,
or the diagonal extension, to absolute terms, so let us return to
where we were in working our way through CP 3.73, and see whether
we can validate Peirce's statements about the "commifications" of
2-adic relative terms that yield their 3-adic diagonal extensions.

| But not only may any absolute term be thus regarded as
| a relative term, but any relative term may in the same
| way be regarded as a relative with one correlate more.
| It is convenient to take this additional correlate as
| the first one.
|
| Then:
|
| 'l','s'w
|
| will denote a lover of a woman
| that is a servant of that woman.
|
| The comma here after 'l' should not be considered
| as altering at all the meaning of 'l', but as only
| a subjacent sign, serving to alter the arrangement
| of the correlates.
|
| C.S. Peirce, CP 3.73

Just to plant our feet on a more solid stage,
let's apply this idea to the Othello example.

For this performance only, just to make the example more interesting,
let us assume that Jeste (J) is secretly in love with Desdemona (D).

Then we begin with the modified data set:

 w   =  "woman"           =  B +, D +, E

'l'  =  "lover of ---"    =  B:C +, C:B +, D:O +, E:I +, I:E +, J:D +, O:D

's'  =  "servant of ---"  =  C:O +, E:D +, I:O +, J:D +, J:O

And next we derive the following results:

'l',  =  "lover that is --- of ---"

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

'l','s'w  =  (B:B:C +, C:C:B +, D:D:O +, E:E:I +, I:I:E +, J:J:D +, O:O:D)

          x  (C:O +, E:D +, I:O +, J:D +, J:O)

          x  (B +, D +, E)

Now what are we to make of that?

If we operate in accordance with Peirce's example of `g`'o'h
as the "giver of a horse to an owner of that horse", then we
may assume that the associative law and the distributive law
are by default in force, allowing us to derive this equation:

'l','s'w  =  'l','s'(B +, D +, E)

          =  'l','s'B +, 'l','s'D +, 'l','s'E

Evidently what Peirce means by the associative principle,
as it applies to this type of product, is that a product
of elementary relatives having the form (R:S:T)(S:T)(T)
is equal to R but that no other form of product yields
a non-null result.  Scanning the implied terms of the
triple product tells us that only the following case
is non-null:  J = (J:J:D)(J:D)(D).  It follows that:

'l','s'w  =  "lover and servant of a woman"

          =  "lover that is a servant of a woman"

          =  "lover of a woman that is a servant of that woman"

          =  J

And so what Peirce says makes sense in this case.

Jon Awbrey

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