ONT Re: Logic Of Relatives
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LOR. Note 57
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I'm going to elaborate a little further on the subject
of arrows, morphisms, or structure-preserving maps, as
a modest amount of extra work at this point will repay
ample dividends when it comes time to revisit Peirce's
"number of" function on logical terms.
The "structure" that is being preserved by a structure-preserving map
is just the structure that we all know and love as a 3-adic relation.
Very typically, it will be the type of 3-adic relation that defines
the type of 2-ary operation that obeys the rules of a mathematical
structure that is known as a "group", that is, a structure that
satisfies the axioms for closure, associativity, identities,
and inverses.
For example, in the previous case of the logarithm map J, we have the data:
| J : R <- R (properly restricted)
|
| K : R <- R x R, where K(r, s) = r + s
|
| L : R <- R x R, where L(u, v) = u . v
Real number addition and real number multiplication (suitably restricted)
are examples of group operations. If we write the sign of each operation
in braces as a name for the 3-adic relation that constitutes or defines
the corresponding group, then we have the following set-up:
| J : {+} <- {.}
|
| {+} c R x R x R
|
| {.} c R x R x R
In many cases, one finds that both groups are written with the same
sign of operation, typically ".", "+", "*", or simple concatenation,
but they remain in general distinct whether considered as operations
or as relations, no matter what signs of operation are used. In such
a setting, our chiasmatic theme may run a bit like these two variants:
| The image of the sum is the sum of the images.
|
| The image of the product is the product of the images.
Figure 22 presents a generic picture for groups G and H.
o-----------------------------------------------------------o
| |
| G H |
| @ @ |
| /|\ /|\ |
| / | \ / | \ |
| v | \ v | \ |
| o o o o o o |
| X X X Y Y Y |
| o o o o o o |
| ^ ^ ^ / / / |
| \ \ \ / / |
| \ \ / \ / / |
| \ \ \ / |
| \ / \ / \ / |
| @ @ @ |
| J J J |
| |
o-----------------------------------------------------------o
Figure 22. Group Homomorphism J : G <- H
In a setting where both groups are written with a plus sign,
perhaps even constituting the very same group, the defining
formula of a morphism, J(L(u, v)) = K(Ju, Jv), takes on the
shape J(u + v) = Ju + Jv, which looks very analogous to the
distributive multiplication of a sum (u + v) by a factor J.
Hence another common name for a morphism: a "linear" map.
Jon Awbrey
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