Peirce’s 1880 “Algebra Of Logic” Chapter 3 • Selection 5

Chapter 3. The Logic of Relatives (cont.)

§2. Relatives (cont.)

221. From the definition of a simple term given in the last section, it follows that every simple relative is the negative of an individual term. But while in non-relative logic negation only divides the universe into two parts, in relative logic the same operation divides the universe into $2^n$ parts, where $n$ is the number of objects in the system which the relative supposes; thus,

$\begin{array}{*{5}{l}} \infty & = & \mathrm{A} & + & \overline{\mathrm{A}} \end{array}$

$\begin{array}{*{9}{l}} \infty & = & \mathrm{A:B} & + & \mathrm{\overline{A}:B} & + & \mathrm{A:\overline{B}} & + & \mathrm{\overline{A}:\overline{B}} \end{array}$

$\begin{array}{*{9}{l}} \infty & = & \mathrm{A:B:C} & + & \mathrm{\overline{A}:B:C} & + & \mathrm{A:\overline{B}:C} & + & \mathrm{A:B:\overline{C}} \\[4pt] & + & \mathrm{\overline{A}:\overline{B}:\overline{C}} & + & \mathrm{A:\overline{B}:\overline{C}} & + & \mathrm{\overline{A}:B:\overline{C}} & + & \mathrm{\overline{A}:\overline{B}:C}. \end{array}$

Here, we have

$\begin{array}{*{5}{l}} \mathrm{A} & = & \mathrm{A:B} & + & \mathrm{A:\overline{B}} \\[4pt] \mathrm{\overline{A}} & = & \mathrm{\overline{A}:B} & + & \mathrm{\overline{A}:\overline{B}} \end{array}$

$\begin{array}{*{5}{l}} \mathrm{A:B} & = & \mathrm{A:B:C} & + & \mathrm{A:B:\overline{C}} \\[4pt] \mathrm{A:\overline{B}} & = & \mathrm{A:\overline{B}:C} & + & \mathrm{A:\overline{B}:\overline{C}} \\[4pt] \mathrm{\overline{A}:B} & = & \mathrm{\overline{A}:B:C} & + & \mathrm{\overline{A}:B:\overline{C}} \\[4pt] \mathrm{\overline{A}:\overline{B}} & = & \mathrm{\overline{A}:\overline{B}:C} & + & \mathrm{\overline{A}:\overline{B}:\overline{C}}. \end{array}$

It will be seen that a term which is individual when considered as $n$ -fold is not so when considered as more than $n$ -fold; but an $n$ -fold term when made $(m + n)$ -fold, is individual as to $n$ members of the system, and indefinite as to $m$ members.

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

Peirce’s 1870 Logic Of Relatives

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