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Directory talk:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives (view source)
Revision as of 23:38, 30 April 2009
, 23:38, 30 April 2009→Commentary Work Area
==Commentary Work Area==
==Commentary Work Area==
===Commentary Note 12.1===
Let us make a few preliminary observations about the operation of ''logical involution'', as Peirce introduces it here:
{| align="center" cellspacing="6" width="90%" <!--QUOTE-->
|
<p>I shall take involution in such a sense that <math>x^y\!</math> will denote everything which is an <math>x\!</math> for every individual of <math>y.\!</math></p>
<p>Thus</p>
|-
| align="center" | <math>\mathit{l}^\mathrm{w}\!</math>
|-
|
<p>will be a lover of every woman.</p>
<p>(Peirce, CP 3.77).</p>
|}
In ordinary arithmetic the ''involution'' <math>x^y,\!</math> or the ''exponentiation'' of <math>x\!</math> to the power of <math>y,\!</math> is the repeated application of the multiplier <math>x\!</math> for as many times as there are ones making up the exponent <math>y.\!</math>
In analogous fashion, the logical involution <math>\mathit{l}^\mathrm{w}\!</math> is the repeated application of the term <math>\mathit{l}\!</math> for as many times as there are individuals under the term <math>\mathrm{w}.\!</math> According to Peirce's interpretive rules, the repeated applications of the base term <math>\mathit{l}\!</math> are distributed across the individuals of the exponent term <math>\mathrm{w}.\!</math> In particular, the base term <math>\mathit{l}\!</math> is not applied successively in the manner that would give something like "a lover of a lover of … a lover of a woman".
For example, suppose that a universe of discourse numbers among its contents just three women, <math>\mathrm{W}^{\prime}, \mathrm{W}^{\prime\prime}, \mathrm{W}^{\prime\prime\prime}.</math> This could be expressed in Peirce's notation by writing:
{| align="center" cellspacing="6" width="90%"
| <math>\mathrm{w} ~=~ \mathrm{W}^{\prime} ~+\!\!,~ \mathrm{W}^{\prime\prime} ~+\!\!,~ \mathrm{W}^{\prime\prime\prime}</math>
|}
Under these circumstances the following equation would hold:
{| align="center" cellspacing="6" width="90%"
| <math>\mathit{l}^\mathrm{w} ~=~ \mathit{l}^{(\mathrm{W}^{\prime} ~+\!\!,~ \mathrm{W}^{\prime\prime} ~+\!\!,~ \mathrm{W}^{\prime\prime\prime})} ~=~ (\mathit{l}\mathrm{W}^{\prime}), (\mathit{l}\mathrm{W}^{\prime\prime}), (\mathit{l}\mathrm{W}^{\prime\prime\prime}).</math>
|}
This says that a lover of every woman in the given universe of discourse is a lover of <math>\mathrm{W}^{\prime}</math> that is a lover of <math>\mathrm{W}^{\prime\prime}</math> that is a lover of <math>\mathrm{W}^{\prime\prime\prime}.</math> In other words, a lover of every woman in this context is a lover of <math>\mathrm{W}^{\prime}</math> and a lover of <math>\mathrm{W}^{\prime\prime}</math> and a lover of <math>\mathrm{W}^{\prime\prime\prime}.</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%"
|
<math>\begin{array}{lll}
u \star L
& = &
L_{u \,\text{at}\, 1}
\\[6pt]
& = & \{ (u, x) \in L \}
\\[6pt]
& = &
\text{the ordered pairs in}~ L ~\text{that have}~ u ~\text{in the 1st place}.
\\[9pt]
L \star v
& = &
L_{v \,\text{at}\, 2}
\\[6pt]
& = &
\{ (x, v) \in L \}
\\[6pt]
& = &
\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>
|}
The denotation of the term <math>\mathit{l}^\mathrm{w}\!</math> is a subset of <math>X\!</math> that can be obtained as follows: For each flag of the form <math>L \star x</math> with <math>x \in W,</math> collect the elements <math>\operatorname{proj}_1 (L \star x)</math> that appear as the first components of these ordered pairs, and then take the intersection of all these subsets. Putting it all together:
{| align="center" cellspacing="6" width="90%"
| <math>\mathit{l}^\mathrm{w} ~=~ \bigcap_{x \in W} \operatorname{proj}_1 (L \star x)</math>
|}
===Commentary Note 12.2===
===Commentary Note 12.2===