[Inquiry] Re: Logic Of Relatives

[Jon Awbrey]

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LOR.  Note 26

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Boole rationalized the properties of what we now dub "boolean multiplication",
roughly equivalent to logical conjunction, in terms of the laws that apply to
selective operations.  Peirce, in his turn, taking a very significant step of
analysis that has seldom been recognized for what it would lead to, much less
followed, does not consider this multiplication to be a fundamental operation,
but derives it as a by-product of relative multiplication by a comma relative.
Thus, Peirce makes logical conjunction a special case of relative composition.

This opens up a very wide field of investigation,
"the operational significance of logical terms",
one might say, but it will be best to advance
bit by bit, and to lean on simple examples.

Back to Venice, and the close-knit party
of absolutes and relatives that we were
entertaining when last we were there.

Here is the list of absolute terms that we were considering before,
to which I have thrown in 1, the universe of "anybody or anything",
just for good measure:

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

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

n   =  "noble"                =  C +, D +, O

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

Here is the list of "comma inflexions" or "diagonal extensions" of these terms:

1,  =  "anybody that is ---"  =  B:B +, C:C +, D:D +, E:E +, I:I +, J:J +, O:O

m,  =  "man that is ---"      =  C:C +, I:I +, J:J +, O:O

n,  =  "noble that is ---"    =  C:C +, D:D +, O:O

w,  =  "woman that is ---"    =  B:B +, D:D +, E:E

One observes that the diagonal extension of 1
is the same thing as the identity relation !1!.

Inspired by this identification of "1," with "!1!", and because
the affixed commas of the diagonal extensions tend to get lost
in the ordinary commas of punctuation, I will experiment with
using the alternative notations:

m,  =  !m!
n,  =  !n!
w,  =  !w!

Working within our smaller sample of absolute terms,
we have already computed the sorts of products that
apply the diagonal extension of an absolute term to
another absolute term, for instance, these products:

m,n  =  !m!n  =  "man that is noble"    =  C +, O
n,m  =  !n!m  =  "noble that is man"    =  C +, O
n,w  =  !n!w  =  "noble that is woman"  =  D
w,n  =  !w!n  =  "woman that is noble"  =  D

This exercise gave us a bit of practical insight into
why the commutative law holds for logical conjunction.

Further insight into the laws that govern this realm of logic,
and the underlying reasons why they apply, might be gained by
systematically working through the whole variety of different
products that are generated by the operational means in sight,
namely, the products indicated by {1, m, n, w}<,>{1, m, n, w}.

But before we try to explore this territory more systematically,
let us equip ourselves with the sorts of graphical and matrical
representations that we discovered to provide us with such able
assists to the intuition in so many of our previous adventures.

Jon Awbrey

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