Peirce’s 1880 “Algebra Of Logic” Chapter 3 • Selection 4

Chapter 3. The Logic of Relatives (cont.)

§2. Relatives (cont.)

220.   Every relative, like every term of singular reference, is general;  its definition describes a system in general terms;  and, as general, it may be conceived either as a logical sum of individual relatives, or as a logical product of simple relatives.  An individual relative refers to a system all the members of which are individual.  The expressions

\begin{array}{lll}  (\mathrm{A : B}) & \qquad & (\mathrm{A : B : C})  \end{array}

may denote individual relatives.  Taking dual individual relatives, for instance, we may arrange them all in an infinite block, thus,

\begin{array}{*{11}{c}}  \mathrm{A:A}&&\mathrm{A:B}&&\mathrm{A:C}&&\mathrm{A:D}&&\mathrm{A:E}&&\text{etc.}  \\[4pt]  \mathrm{B:A}&&\mathrm{B:B}&&\mathrm{B:C}&&\mathrm{B:D}&&\mathrm{B:E}&&\text{etc.}  \\[4pt]  \mathrm{C:A}&&\mathrm{C:B}&&\mathrm{C:C}&&\mathrm{C:D}&&\mathrm{C:E}&&\text{etc.}  \\[4pt]  \mathrm{D:A}&&\mathrm{D:B}&&\mathrm{D:C}&&\mathrm{D:D}&&\mathrm{D:E}&&\text{etc.}  \\[4pt]  \mathrm{E:A}&&\mathrm{E:B}&&\mathrm{E:C}&&\mathrm{E:D}&&\mathrm{E:E}&&\text{etc.}  \\[4pt]  \text{etc.}&& \text{etc.}&& \text{etc.}&& \text{etc.}&& \text{etc.}&& \text{etc.}  \end{array}

In the same way, triple individual relatives may be arranged in a cube, and so forth.  The logical sum of all the relatives in this infinite block will be the relative universe, \infty, where

x \,-\!\!\!< \infty,

whatever dual relative x may be.  It is needless to distinguish the dual universe, the triple universe, etc., because, by adding a perfectly indefinite additional member to the system, a dual relative may be converted into a triple relative, etc.  Thus, for lover of a woman, we may write lover of a woman coexisting with anything.  In the same way, a term of single reference is equivalent to a relative with an indefinite correlate;  thus, woman is equivalent to woman coexisting with anything.  Thus, we shall have

\begin{array}{*{13}{c}}  \mathrm{A} & = &  \mathrm{A:A} & + & \mathrm{A:B} & + & \mathrm{A:C} & + & \mathrm{A:D} & + & \mathrm{A:E} & + & \text{etc.}  \end{array}

\begin{array}{*{11}{c}}  \mathrm{A:B} & = &  \mathrm{A:B:A} & + & \mathrm{A:B:B} & + & \mathrm{A:B:C} & + & \mathrm{A:B:D} & + & \text{etc.}  \end{array}

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 4

  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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