Post : Peirce's 1870 “Logic Of Relatives” • Comment 10.5
http://inquiryintoinquiry.com/2014/04/06/peirces-1870-logic-of-relatives-%e2%80%a2-comment-10-5/
Posted : April 6, 2014 at 2:30 pm
Peircers,
Some of what Peirce says in CP 3.73 has me scratching my head. And not for the first time, of
course. At any rate, here is my best current attempt to reason it out. As always, see the blog
post linked above for the properly formatted content.
⁂
We have sufficiently covered the application of the comma functor 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 commafications of dyadic relative terms and the corresponding diagonal
extensions to triadic relations.
<quote>
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,sw
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.
(Peirce, CP 3.73)
☞http://inquiryintoinquiry.com/2014/02/17/peirces-1870-logic-of-relatives-%E2%80%A2-selection-8/
</quote>
Just to plant our feet on a more solid stage, let us 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 = B +, D +, E
l = B:C +, C:B +, D:O +, E:I +, I:E +, J:D +, O:D
s = C:O +, E:D +, I:O +, J:D +, J:O
And we next 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,sw = (B:B:C +, C:C:B +, D:D:O +, E:E:I} +, I:I:E +, J:J:D +, O:O:D)
×
(C:O +, E:D +, I:O +, J:D +, J:O)
×
(B +, D +, E)
Now what are we to make of that?
If we operate in accordance with Peirce's example of goh 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 in force,
allowing us to derive this equation:
l,sw = l,s(B +, D +, E)
= l,sB +, l,sD +, l,sE
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 case (J:J:D)(J:D)(D) = J is non-null.
It follows that:
l,sw = 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.
--
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