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Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives (view source)
Revision as of 17:32, 1 May 2009
, 17:32, 1 May 2009→Commentary Note 12.1
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{| align="center" cellspacing="6" width="90%"
| height="40" | <math>X\!</math> is a set distinguished as the ''universe of discourse''.
| height="40" | <math>X\!</math> is a set singled out in a particular discussion as the ''universe of discourse''.
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| height="40" | <math>W \subseteq X</math> is the 1-adic relation, or set, whose elements fall under the absolute term <math>\mathrm{w} = \text{woman}.\!</math> The elements of <math>W\!</math> are sometimes referred to as the ''denotation'' or the set-theoretic ''extension'' of the term <math>\mathrm{w}.\!</math>
| height="40" | <math>W \subseteq X</math> is the 1-adic relation, or set, whose elements fall under the absolute term <math>\mathrm{w} = \text{woman}.\!</math> The elements of <math>W\!</math> are sometimes referred to as the ''denotation'' or the set-theoretic ''extension'' of the term <math>\mathrm{w}.\!</math>
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It is very instructive to examine the matrix representation of <math>\mathit{l}^\mathrm{w}\!</math> at this point, not the least because it helps to dispel the mystery behind the name ''involution''.
It is very instructive to examine the matrix representation of <math>\mathit{l}^\mathrm{w}\!</math> at this point, not the least because it dispels the mystery of the name ''involution''.
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{| align="center" cellspacing="6" width="90%"
|-
|-
| height="60" | <math>(\mathfrak{L}^\mathfrak{W})_{a} ~=~ \prod_{x \in X} \mathfrak{L}_{ax}^{\mathfrak{W}_{x}}</math>
| height="60" | <math>(\mathfrak{L}^\mathfrak{W})_{a} ~=~ \prod_{x \in X} \mathfrak{L}_{ax}^{\mathfrak{W}_{x}}</math>
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To say that <math>\mathrm{J}\!</math> is a lover of every woman is to say that <math>\mathrm{J}\!</math> loves <math>\mathrm{K}\!</math> if <math>\mathrm{K}\!</math> is a woman. This can be rendered in symbols as follows:
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| <math>\mathrm{J} ~\text{loves}~ \mathrm{K} ~\Leftarrow~ \mathrm{K} ~\text{is a woman}</math>
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Interpreting the formula <math>\mathit{l}^\mathrm{w}\!</math> as <math>\mathrm{J} ~\text{loves}~ \mathrm{K} ~\Leftarrow~ \mathrm{K} ~\text{is a woman}</math> highlights the form of the converse implication inherent in it, and this in turn reveals the analogy between implication and involution that accounts for the aptness of the latter name.
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<math>
\begin{bmatrix}
0^0 & = & 1
\\
0^1 & = & 0
\\
1^0 & = & 1
\\
1^1 & = & 1
\end{bmatrix}
\qquad\qquad\qquad
\begin{bmatrix}
0\!\Leftarrow\!0 & = & 1
\\
0\!\Leftarrow\!1 & = & 0
\\
1\!\Leftarrow\!0 & = & 1
\\
1\!\Leftarrow\!1 & = & 1
\end{bmatrix}
</math>
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