## 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

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