Peirce’s 1870 “Logic of Relatives” • Comment 12.1

Posted on June 10, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 12.1

To get a better sense of why Peirce’s formulas in Selection 12 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.

X is a set singled out in a particular discussion as the universe of discourse.

W \subseteq X is the monadic relation, or set, whose elements fall under the absolute term \mathrm{w} = \text{woman}.  The elements of W are referred to as the denotation or extension of the term \mathrm{w}.

L \subseteq X \times X is the dyadic relation associated with the relative term \mathit{l} = \text{lover of}\,\underline{~~~~}.

S \subseteq X \times X is the dyadic relation associated with the relative term \mathit{s} = \text{servant of}\,\underline{~~~~}.

\mathsf{W} = (\mathsf{W}_x) = \mathrm{Mat}(W) = \mathrm{Mat}(\mathrm{w}) is the 1-dimensional matrix representation of the set W and the term \mathrm{w}.

\mathsf{L} = (\mathsf{L}_{xy}) = \mathrm{Mat}(L) = \mathrm{Mat}(\mathit{l}) is the 2-dimensional matrix representation of the relation L and the relative term \mathit{l}.

\mathsf{S} = (\mathsf{S}_{xy}) = \mathrm{Mat}(S) = \mathrm{Mat}(\mathit{s}) is the 2-dimensional matrix representation of the relation S and the relative term \mathit{s}.

A few concepts from the article on Relation Theory, touched on again in Comment 11.7, will also be useful.

The local flags of the relation L are defined as follows.

\begin{array}{lll} u \star L & = & L_{u\,@\,1} \\[6pt] & = & \{ (u, x) \in L \} \\[6pt] & = & \text{ordered pairs in}~ L ~\text{with}~ u ~\text{in the 1st place}. \\[9pt] L \star v & = & L_{v\,@\,2} \\[6pt] & = & \{ (x, v) \in L \} \\[6pt] & = & \text{ordered pairs in}~ L ~\text{with}~ v ~\text{in the 2nd place}. \end{array}

The applications of the relation L are defined as follows.

\begin{array}{lll} u \cdot L & = & \mathrm{proj}_2 (u \star L) \\[6pt] & = & \{ x \in X : (u, x) \in L \} \\[6pt] & = & \text{loved by}~ u. \\[9pt] L \cdot v & = & \mathrm{proj}_1 (L \star v) \\[6pt] & = & \{ x \in X : (x, v) \in L \} \\[6pt] & = & \text{lover of}~ v. \end{array}

Resources

cc: Cybernetics • Ontolog Forum • Structural Modeling • Systems Science
cc: FB | Peirce Matters • Laws of Form • Peirce List (1) (2) (3) (4) (5) (6) (7)

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