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Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives (view source)
Revision as of 19:45, 30 April 2009
, 19:45, 30 April 2009→Commentary Note 12.1
Given a universe of discourse <math>X,\!</math> suppose that <math>W \subseteq X</math> is the 1-adic relation, that is, the set, associated with the absolute term <math>\mathrm{w} = \text{woman}\!</math> and suppose that <math>L \subseteq X \times X\!</math> is the 2-adic relation associated with the relative term <math>\mathit{l} = \text{lover of}\,\underline{~~~~}.</math>
Given a universe of discourse <math>X,\!</math> suppose that <math>W \subseteq X</math> is the 1-adic relation, that is, the set, associated with the absolute term <math>\mathrm{w} = \text{woman}\!</math> and suppose that <math>L \subseteq X \times X\!</math> is the 2-adic relation associated with the relative term <math>\mathit{l} = \text{lover of}\,\underline{~~~~}.</math>
Recalling a few definitions, the ''local flags'' of the relation <math>L\!</math> are given as follows:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
\\[6pt]
\\[6pt]
& = &
& = &
\text{the set of ordered pairs in}~ L ~\text{that have}~ u ~\text{in the 1st place}.
\text{the ordered pairs in}~ L ~\text{that have}~ u ~\text{in the 1st place}.
\\[9pt]
\\[9pt]
L \star v
L \star v
\\[6pt]
\\[6pt]
& = &
& = &
\text{the set of ordered pairs in}~ L ~\text{that have}~ v ~\text{in the 2nd place}.
\text{the ordered pairs in}~ L ~\text{that have}~ v ~\text{in the 2nd place}.
\end{array}</math>
|}
The ''flag projections'' of the relation <math>L\!</math> are defined this way:
{| align="center" cellspacing="6" width="90%"
|
<math>\begin{array}{lll}
u \cdot L
& = &
\operatorname{proj}_2 (u \star L)
\\[6pt]
& = &
\{ x \in X : (u, x) \in L \}
\\[6pt]
& = &
\text{loved by}~ u.
\\[9pt]
L \cdot v
& = &
\operatorname{proj}_1 (L \star v)
\\[6pt]
& = &
\{ x \in X : (x, v) \in L \}
\\[6pt]
& = &
\text{lover of}~ v.
\end{array}</math>
\end{array}</math>
|}
|}