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

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

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

  1. Pingback: Relations & Their Relatives : 10 | Inquiry Into Inquiry

  2. Pingback: Survey of Relation Theory • 2 | Inquiry Into Inquiry

  3. Pingback: Survey of Relation Theory • 3 | Inquiry Into Inquiry

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