Peirce’s 1870 “Logic of Relatives” • Comment 12.3

Posted on June 12, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 12.3

We now have two ways of computing a logical involution raising a dyadic relative term to the power of a monadic absolute term, for example, \mathit{l}^\mathrm{w} for “lover of every woman”.

The first method applies set-theoretic operations to the extensions of absolute and relative terms, expressing the denotation of the term \mathit{l}^\mathrm{w} as the intersection of a set of relational applications.

Denotation Equation L^W

The second method operates in the matrix representation, expressing the value of the matrix \mathsf{L}^\mathsf{W} at an argument u as a product of coefficient powers.

Matrix Computation L^W

Abstract formulas like these are more easily grasped with the aid of a concrete example and a picture of the relations involved.

Involution Example 1

Consider a universe of discourse X subject to the following data.

\begin{array}{*{15}{c}} X & = & \{ & a, & b, & c, & d, & e, & f, & g, & h, & i & \} \\[6pt] W & = & \{ & d, & f & \} \\[6pt] L & = & \{ & b\!:\!a, & b\!:\!c, & c\!:\!b, & c\!:\!d, & e\!:\!d, & e\!:\!e, & e\!:\!f, & g\!:\!f, & g\!:\!h, & h\!:\!g, & h\!:\!i & \} \end{array}

Figure 55 shows the placement of W within X and the placement of L within X \times X.

Bigraph Involution L^W
\text{Figure 55. Bigraph Involution}~ \mathsf{L}^\mathsf{W}

To highlight the role of W more clearly, the Figure represents the absolute term ``\mathrm{w}" by means of the relative term ``\mathrm{w},\!" which conveys the same information.

Computing the denotation of \mathit{l}^\mathrm{w} by way of the class intersection formula, we can show our work as follows.

Class Intersection L^W

With the above Figure in mind, we can visualize the computation of \textstyle (\mathsf{L}^\mathsf{W})_u = \prod_{v \in X} \mathsf{L}_{uv}^{\mathsf{W}_v} as follows.

  1. Pick a specific u in the bottom row of the Figure.
  2. Pan across the elements v in the middle row of the Figure.
  3. If u links to v then \mathsf{L}_{uv} = 1, otherwise \mathsf{L}_{uv} = 0.
  4. If v in the middle row links to v in the top row then \mathsf{W}_v = 1, otherwise \mathsf{W}_v = 0.
  5. Compute the value \mathsf{L}_{uv}^{\mathsf{W}_v} = (\mathsf{L}_{uv} \Leftarrow \mathsf{W}_v) for each v in the middle row.
  6. If any of the values \mathsf{L}_{uv}^{\mathsf{W}_v} is 0 then the product \textstyle \prod_{v \in X} \mathsf{L}_{uv}^{\mathsf{W}_v} is 0, otherwise it is 1.

As a general observation, we know the value of (\mathsf{L}^\mathsf{W})_u goes to 0 just as soon as we find a v \in X such that \mathsf{L}_{uv} = 0 and \mathsf{W}_v = 1, in other words, such that (u, v) \notin L but v \in W.  If there is no such v then (\mathsf{L}^\mathsf{W})_u = 1.

Running through the program for each u \in X, the only case producing a non-zero result is (\mathsf{L}^\mathsf{W})_e = 1.  That portion of the work can be sketched as follows.

Matrix Coefficient L^W

Resources

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