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Revision as of 23:08, 14 June 2009
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Up to this point, we are still reading the elementary relatives of the form <math>i\!:\!j</math> in the way that Peirce read them in logical contexts: <math>i\!</math> is the relate, <math>j\!</math> is the correlate, and in our current example <math>i\!:\!j,</math> or more exactly, <math>m_{ij} = 1,\!</math> is taken to say that <math>i\!</math> is a marker for <math>j.\!</math> This is the mode of reading that we call "multiplying on the left".
Up to this point, we are still reading the elementary relatives of the form <math>i\!:\!j</math> in the way that Peirce read them in logical contexts: <math>i\!</math> is the relate, <math>j\!</math> is the correlate, and in our current example <math>i\!:\!j,</math> or more exactly, <math>m_{ij} = 1,\!</math> is taken to say that <math>i\!</math> is a marker for <math>j.\!</math> This is the mode of reading that we call "multiplying on the left".
In the algebraic, permutational, or transformational contexts of application, however, Peirce converts to the alternative mode of reading, although still calling <math>i\!</math> the relate and <math>j\!</math> the correlate, the elementary relative <math>i\!:\!j</math> now means that <math>i\!</math> gets changed into <math>j.\!</math> In this scheme of reading, the transformation <math>a\!:\!b + b\!:\!c + c\!:\!a</math> is a permutation of the aggregate <math>\mathbf{1} = a + b + c,</math> or what we would now call the set <math>\{ a, b, c \},\!</math> in particular, it is the permutation that is otherwise notated as follows:
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:
{| align="center" cellpadding="6"
|
<math>\begin{Bmatrix}
a & b & c
\\
b & c & a
\end{Bmatrix}</math>
|}
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 16==
==Note 16==
