Peirce’s 1880 “Algebra Of Logic” Chapter 3 • Comment 7.5

Suppose we add another individual to our initial universe of discourse, arriving at a three-point universe $\{ \mathrm{I}, \mathrm{J}, \mathrm{K} \}.$

It might be thought that adding one more element to the universe of discourse would allow slightly more complicated relations to be compounded from its basic ingredients, but the truth is that crossing the threshold from a two-point universe to a three-point universe occasions a steep ascent in the complexity of relations generated.

Looking back from the ascent we see that the two-point universe $\{ \mathrm{I}, \mathrm{J} \}$ manifests a type of formal degeneracy (loss of generality) compared with the three-point universe $\{ \mathrm{I}, \mathrm{J}, \mathrm{K} \}.$ This is due to the circumstance that the number of “diagonal” pairs, those of the form $\mathrm{A\!:\!A},$ equals the number of “off-diagonal” pairs, those of the form $\mathrm{A\!:\!B},$ so the two-point case exhibits symmetries that will be broken as soon as one adds another element to the universe.

A universe of three individuals $\mathrm{I, J, K}$ yields exactly nine individual dual relatives or ordered pairs of universe elements:

$\begin{matrix} \mathrm{I\!:\!I}, & \mathrm{I\!:\!J}, & \mathrm{I\!:\!K}, & \mathrm{J\!:\!I}, & \mathrm{J\!:\!J}, & \mathrm{J\!:\!K}, & \mathrm{K\!:\!I}, & \mathrm{K\!:\!J}, & \mathrm{K\!:\!K}. \end{matrix}$

It is convenient arrange the pairs in a square array:

$\left( \begin{matrix} \mathrm{I\!:\!I} & \mathrm{I\!:\!J} & \mathrm{I\!:\!K} \\[4pt] \mathrm{J\!:\!I} & \mathrm{J\!:\!J} & \mathrm{J\!:\!K} \\[4pt] \mathrm{K\!:\!I} & \mathrm{K\!:\!J} & \mathrm{K\!:\!K} \end{matrix} \right)$

There are $2^9 = 512$ dual relatives over this universe of discourse, since each one is formed by choosing a subset of the nine ordered pairs and then “aggregating” them, forming their logical sum, or simply regarding them as a subset. Taking the square array of ordered pairs as a backdrop, any one of the $512$ dual relatives may be represented by a square matrix of binary values, a value of $1$ occupying the place of each ordered pair that belongs to the subset and a value of $0$ occupying the place of each ordered pair that does not belong to the subset in question.

The matrix representations of the $512$ dual relatives or dyadic relations over the universe $\{ \mathrm{I}, \mathrm{J}, \mathrm{K} \}$ are tabulated below according to the following plan:

Since the diagonal and off-diagonal components of the matrices vary independently of each other, the whole set of matrices factors as a product of two smaller sets, making the break along the lines of the corresponding cardinalities, $2^9 = 2^3 \times 2^6.$
The eight diagonal matrices are shown in the first row of the display, omitting the off-diagonal zeroes for ease of reading and pattern recognition.
The $64$ off-diagonal matrices are shown in the rest of the display, arranged in rank order by increasing numbers of $\text{1's}$ and decreasing numbers of $\text{0's},$ suppressing the fixed zeroes along the diagonals to make the changing patterns of $\text{0's}$ and $\text{1's}$ easier to follow.
The number of off-diagonal matrices of rank $k$ is equal to the binomial coefficient $\tbinom{6}{k}$ or $\mathrm{C}(6, k).$ The values of $\mathrm{C}(6, k)$ are given by the row of Pascal’s Triangle that contains the sequence ${1, 6, 15, 20, 15, 6, 1}.$

$\begin{pmatrix} 0 & ~ & ~ \\ ~ & 0 & ~ \\ ~ & ~ & 0 \end{pmatrix} \, \begin{pmatrix} 1 & ~ & ~ \\ ~ & 0 & ~ \\ ~ & ~ & 0 \end{pmatrix} \, \begin{pmatrix} 0 & ~ & ~ \\ ~ & 1 & ~ \\ ~ & ~ & 0 \end{pmatrix} \, \begin{pmatrix} 0 & ~ & ~ \\ ~ & 0 & ~ \\ ~ & ~ & 1 \end{pmatrix} \, \begin{pmatrix} 1 & ~ & ~ \\ ~ & 1 & ~ \\ ~ & ~ & 0 \end{pmatrix} \, \begin{pmatrix} 1 & ~ & ~ \\ ~ & 0 & ~ \\ ~ & ~ & 1 \end{pmatrix} \, \begin{pmatrix} 0 & ~ & ~ \\ ~ & 1 & ~ \\ ~ & ~ & 1 \end{pmatrix} \, \begin{pmatrix} 1 & ~ & ~ \\ ~ & 1 & ~ \\ ~ & ~ & 1 \end{pmatrix}$

$\times$

$\begin{pmatrix} ~ & 0 & 0 \\ 0 & ~ & 0 \\ 0 & 0 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 1 & 0 \\ 0 & ~ & 0 \\ 0 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 1 \\ 0 & ~ & 0 \\ 0 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 0 \\ 1 & ~ & 0 \\ 0 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 0 \\ 0 & ~ & 1 \\ 0 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 0 \\ 0 & ~ & 0 \\ 1 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 0 \\ 0 & ~ & 0 \\ 0 & 1 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 1 & 1 \\ 0 & ~ & 0 \\ 0 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 0 \\ 1 & ~ & 0 \\ 0 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 0 \\ 0 & ~ & 1 \\ 0 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 0 \\ 0 & ~ & 0 \\ 1 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 0 \\ 0 & ~ & 0 \\ 0 & 1 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 0 & 1 \\ 1 & ~ & 0 \\ 0 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 1 \\ 0 & ~ & 1 \\ 0 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 1 \\ 0 & ~ & 0 \\ 1 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 1 \\ 0 & ~ & 0 \\ 0 & 1 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 0 & 0 \\ 1 & ~ & 1 \\ 0 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 0 \\ 1 & ~ & 0 \\ 1 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 0 \\ 1 & ~ & 0 \\ 0 & 1 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 0 & 0 \\ 0 & ~ & 1 \\ 1 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 0 \\ 0 & ~ & 1 \\ 0 & 1 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 0 & 0 \\ 0 & ~ & 0 \\ 1 & 1 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 1 & 1 \\ 1 & ~ & 0 \\ 0 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 1 \\ 0 & ~ & 1 \\ 0 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 1 \\ 0 & ~ & 0 \\ 1 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 1 \\ 0 & ~ & 0 \\ 0 & 1 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 1 & 0 \\ 1 & ~ & 1 \\ 0 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 0 \\ 1 & ~ & 0 \\ 1 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 0 \\ 1 & ~ & 0 \\ 0 & 1 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 1 & 0 \\ 0 & ~ & 1 \\ 1 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 0 \\ 0 & ~ & 1 \\ 0 & 1 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 1 & 0 \\ 0 & ~ & 0 \\ 1 & 1 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 0 & 1 \\ 1 & ~ & 1 \\ 0 & 0 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 0 & 1 \\ 1 & ~ & 0 \\ 0 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 1 \\ 1 & ~ & 0 \\ 1 & 0 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 0 & 1 \\ 0 & ~ & 0 \\ 1 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 1 \\ 0 & ~ & 1 \\ 0 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 1 \\ 0 & ~ & 1 \\ 1 & 0 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 0 & 0 \\ 0 & ~ & 1 \\ 1 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 0 \\ 1 & ~ & 0 \\ 1 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 0 \\ 1 & ~ & 1 \\ 0 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 0 \\ 1 & ~ & 1 \\ 1 & 0 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 1 & 1 \\ 1 & ~ & 1 \\ 0 & 0 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 1 & 1 \\ 1 & ~ & 0 \\ 0 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 1 \\ 1 & ~ & 0 \\ 1 & 0 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 1 & 1 \\ 0 & ~ & 0 \\ 1 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 1 \\ 0 & ~ & 1 \\ 0 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 1 \\ 0 & ~ & 1 \\ 1 & 0 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 1 & 0 \\ 0 & ~ & 1 \\ 1 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 0 \\ 1 & ~ & 0 \\ 1 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 0 \\ 1 & ~ & 1 \\ 0 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 0 \\ 1 & ~ & 1 \\ 1 & 0 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 0 & 0 \\ 1 & ~ & 1 \\ 1 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 1 \\ 0 & ~ & 1 \\ 1 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 1 \\ 1 & ~ & 0 \\ 1 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 1 \\ 1 & ~ & 1 \\ 0 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 1 \\ 1 & ~ & 1 \\ 1 & 0 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 0 & 1 \\ 1 & ~ & 1 \\ 1 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 0 \\ 1 & ~ & 1 \\ 1 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 1 \\ 0 & ~ & 1 \\ 1 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 1 \\ 1 & ~ & 0 \\ 1 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 1 \\ 1 & ~ & 1 \\ 0 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 1 \\ 1 & ~ & 1 \\ 1 & 0 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 1 & 1 \\ 1 & ~ & 1 \\ 1 & 1 & ~ \end{pmatrix}$

References

Peirce, C.S. (1880), “On the Algebra of Logic”, American Journal of Mathematics 3, 15–57. Collected Papers (CP 3.154–251), Chronological Edition (CE 4, 163–209).
Peirce, C.S., Collected Papers of Charles Sanders Peirce, vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958. Volume 3 : Exact Logic, 1933.
Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–. Volume 4 (1879–1884), 1986.

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Peirce’s 1870 Logic Of Relatives

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