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Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives (view source)
Revision as of 03:14, 1 May 2009
, 03:14, 1 May 2009→Commentary Note 12.1
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>
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>
To get a better sense of why the above formulas mean what they do, and to prepare the ground for understanding more complex relational expressions, it will help to assemble the following materials and definitions:
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| height="40" | <math>X\!</math> is a set distinguished 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>
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| height="40" | <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>
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| height="40" | <math>S \subseteq X \times X\!</math> is the 2-adic relation associated with the relative term <math>\mathit{s} = \text{servant of}\,\underline{~~~~}.</math>
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| height="40" | <math>\mathfrak{W} = (\mathfrak{W}_x) = \operatorname{Mat}(W) = \operatorname{Mat}(\mathrm{w})</math> is the 1-dimensional matrix representation of the set <math>W\!</math> and the term <math>\mathrm{w}.\!</math>
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| height="40" | <math>\mathfrak{L} = (\mathfrak{L}_{xy}) = \operatorname{Mat}(L) = \operatorname{Mat}(\mathit{l})</math> is the 2-dimensional matrix representation of the relation <math>L\!</math> and the relative term <math>\mathit{l}.\!</math>
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| height="40" | <math>\mathfrak{S} = (\mathfrak{S}_{xy}) = \operatorname{Mat}(S) = \operatorname{Mat}(\mathit{s})</math> is the 2-dimensional matrix representaion of the relation <math>S\!</math> and the relative term <math>\mathit{s}.\!</math>
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Recalling a few definitions, the ''local flags'' of the relation <math>L\!</math> are given as follows:
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The ''flag projections'' of the relation <math>L\!</math> are defined this way:
The ''applications'' of the relation <math>L\!</math> are defined as follows:
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| <math>\mathit{l}^\mathrm{w} ~=~ \bigcap_{x \in W} L \cdot x</math>
| <math>\mathit{l}^\mathrm{w} ~=~ \bigcap_{x \in W} \operatorname{proj}_1 (L \star x) ~=~ \bigcap_{x \in W} L \cdot x</math>
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