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Directory:Jon Awbrey/Papers/Differential Propositional Calculus (view source)
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===Note 13===
===Note 13===
Let us make up the model universe $1$ = ''A'' + ''B'' + ''C'' and the 2-adic relation ''n'' = "noter of", as when "''X'' is a data record that contains a pointer to ''Y''". That interpretation is not important, it's just for the sake of intuition. In general terms, the 2-adic relation ''n'' can be represented by this matrix:
<pre>
<pre>
n =
n =
| |
| |
| n_CA (C:A) n_CB (C:B) n_CC (C:C) ]
| n_CA (C:A) n_CB (C:B) n_CC (C:C) ]
</pre>
Also, let n be such that:
Also, let ''n'' be such that:
<pre>
A is a noter of A and B,
A is a noter of A and B,
B is a noter of B and C,
B is a noter of B and C,
C is a noter of C and A.
C is a noter of C and A.
</pre>
Filling in the instantial values of the "coefficients" n_ij,
Filling in the instantial values of the "coefficients" ''n''<sub>''ij''</sub>, as the indices ''i'' and ''j'' range over the universe of discourse:
as the indices i and j range over the universe of discourse:
<pre>
n =
n =
| |
| |
| 1 * (C:A) 0 * (C:B) 1 * (C:C) ]
| 1 * (C:A) 0 * (C:B) 1 * (C:C) ]
</pre>
In Peirce's time, and even in some circles of mathematics today,
In Peirce's time, and even in some circles of mathematics today, the information indicated by the elementary relatives (''i'':''j''), as ''i'', ''j'' range over the universe of discourse, would be referred to as the "umbral elements" of the algebraic operation represented by the matrix, though I seem to recall that Peirce preferred to call these terms the "ingredients". When this ordered basis is understood well enough, one will tend to drop any mention of it from the matrix itself, leaving us nothing but these bare bones:
the information indicated by the elementary relatives (i:j), as
i, j range over the universe of discourse, would be referred to
as the "umbral elements" of the algebraic operation represented
by the matrix, though I seem to recall that Peirce preferred to
call these terms the "ingredients". When this ordered basis is
understood well enough, one will tend to drop any mention of it
from the matrix itself, leaving us nothing but these bare bones:
<pre>
n =
n =
| |
| |
| 1 0 1 ]
| 1 0 1 ]
</pre>
However the specification may come to be written, this
However the specification may come to be written, this is all just convenient schematics for stipulating that:
is all just convenient schematics for stipulating that:
: ''n'' = ''A'':''A'' + ''B'':''B'' + ''C'':''C'''+ ''A'':''B'' + ''B'':''C'' + ''C'':''A''
Recognizing !1! = A:A + B:B + C:C to be the identity transformation,
Recognizing !1! = ''A'':''A'' + ''B'':''B'' + ''C'':''C'' to be the identity transformation, the 2-adic relation n = "noter of" may be represented by an element !1! + ''A'':''B'' + ''B'':''C'' + ''C'':''A'' of the so-called "group ring", all of which just makes this element a special sort of linear transformation.
the 2-adic relation n = "noter of" may be represented by an element
!1! + A:B + B:C + C:A of the so-called "group ring", all of which
just makes this element a special sort of linear transformation.
Up to this point, we are still reading the elementary relatives of
Up to this point, we are still reading the elementary relatives of the form ''i'':''j'' in the way that Peirce reads them in logical contexts: ''i'' is the relate, ''j'' is the correlate, and in our current example we read ''i'':''j'', or more exactly, ''n''<sub>''ij''</sub> = 1, to say that ''i'' is a noter of ''j''. This is the mode of reading that we call "multiplying on the left".
the form i:j in the way that Peirce reads them in logical contexts:
i is the relate, j is the correlate, and in our current example we
read i:j, or more exactly, n_ij = 1, to say that i is a noter of j.
This is the mode of reading that we call "multiplying on the left".
In the algebraic, permutational, or transformational contexts of
In the algebraic, permutational, or transformational contexts of application, however, Peirce converts to the alternative mode of reading, although still calling ''i'' the relate and ''j'' the correlate, the elementary relative ''i'':''j'' now means that ''i'' gets changed into ''j''. In this scheme of reading, the transformation ''A'':''B'' + ''B'':''C'' + ''C'':''A'' is a permutation of the aggregate $1$ = ''A'' + ''B'' + ''C'', or what we would now call the set {''A'', ''B'', ''C''}, in particular, it is the permutation that is otherwise notated as:
application, however, Peirce converts to the alternative mode of
reading, although still calling i the relate and j the correlate,
the elementary relative i:j now means that i gets changed into j.
In this scheme of reading, the transformation A:B + B:C + C:A is
a permutation of the aggregate $1$ = A + B + C, or what we would
now call the set {A, B, C}, in particular, it is the permutation
that is otherwise notated as:
<pre>
( A B C )
( A B C )
< >
< >
( B C A )
( B C A )
</pre>
This is consistent with the convention that Peirce uses in
This is consistent with the convention that Peirce uses in the paper "On a Class of Multiple Algebras" (CP 3.324–327).
the paper "On a Class of Multiple Algebras" (CP 3.324-327).
===Note 14===
===Note 14===