Peirce’s 1880 “Algebra Of Logic” Chapter 3 • Comment 7.2

Because it can sometimes be difficult to reconnect abstractions with their concrete instances, especially after the abstract types have become autonomous and taken on a life of their own, let us resort to a simple concrete case and examine the implications of what Peirce is saying about the relation between general relatives and individual relatives.

Suppose our initial universe of discourse has exactly two individuals, \mathrm{I} and \mathrm{J}.  Then there are exactly four individual dual relatives or ordered pairs of universe elements:

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

It is convenient arrange these in a square array:

\left( \begin{array}{rr}  \mathrm{I\!:\!I} & \mathrm{I\!:\!J} \\[4pt]  \mathrm{J\!:\!I} & \mathrm{J\!:\!J}  \end{array} \right)

There are 2^4 = 16 dual relatives in general over this universe of discourse, since each one is formed by choosing a subset of the four 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 16 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 16 dual relatives or dyadic relations over the universe \{ \mathrm{I}, \mathrm{J} \} are displayed below:

\begin{pmatrix}  0 & 0 \\  0 & 0  \end{pmatrix}  \quad  \begin{pmatrix}  1 & 0 \\  0 & 0  \end{pmatrix}  \quad  \begin{pmatrix}  0 & 0 \\  0 & 1  \end{pmatrix}  \quad  \begin{pmatrix}  1 & 0 \\  0 & 1  \end{pmatrix}

\begin{pmatrix}  0 & 1 \\  0 & 0  \end{pmatrix}  \quad  \begin{pmatrix}  1 & 1 \\  0 & 0  \end{pmatrix}  \quad  \begin{pmatrix}  0 & 1 \\  0 & 1  \end{pmatrix}  \quad  \begin{pmatrix}  1 & 1 \\  0 & 1  \end{pmatrix}

\begin{pmatrix}  0 & 0 \\  1 & 0  \end{pmatrix}  \quad  \begin{pmatrix}  1 & 0 \\  1 & 0  \end{pmatrix}  \quad  \begin{pmatrix}  0 & 0 \\  1 & 1  \end{pmatrix}  \quad  \begin{pmatrix}  1 & 0 \\  1 & 1  \end{pmatrix}

\begin{pmatrix}  0 & 1 \\  1 & 0  \end{pmatrix}  \quad  \begin{pmatrix}  1 & 1 \\  1 & 0  \end{pmatrix}  \quad  \begin{pmatrix}  0 & 1 \\  1 & 1  \end{pmatrix}  \quad  \begin{pmatrix}  1 & 1 \\  1 & 1  \end{pmatrix}

Relative to the universe \{ \mathrm{I}, \mathrm{J} \}, the individual dual relatives of the form \mathrm{A\!:\!A} are \mathrm{I\!:\!I} and \mathrm{J\!:\!J} while the individual dual relatives of the form \mathrm{A\!:\!B} are \mathrm{I\!:\!J} and \mathrm{J\!:\!I}.

Peirce assigns the name concurrents to dual relatives all whose individual aggregants are of the form \mathrm{A\!:\!A}.  There are exactly 4 of these and their matrices are shown in the top row of the above display.  All the rest are called opponents and their matrices are listed in the bottom three rows.

Peirce gives the name alio-relatives to dual relatives all whose individual aggregants are of the form \mathrm{A\!:\!B}.  There are exactly 4 of these and their matrices are shown in the first column of the above display.  All the rest are called self-relatives and their matrices are listed in the right hand three columns.

Notice that the relative {0}, represented by the matrix with all {0} entries, falls under the definitions of both a concurrent and an alio-relative.

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.

Resources

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This entry was posted in Dyadic Relations, Logic, Logic of Relatives, Mathematics, Matrix Theory, Peirce, Relation Theory, Semiotics, Sign Relations, Triadic Relations and tagged , , , , , , , , , . Bookmark the permalink.

5 Responses to Peirce’s 1880 “Algebra Of Logic” Chapter 3 • Comment 7.2

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