Research in History and Philosophy of Mathematics pp 139-148 | Cite as

# The Influence of Arthur Cayley and Alfred Kempe on Charles Peirce’s Diagrammatic Logic

## Abstract

This paper is dedicated to the memory of Irving H. Anellis and represents joint work on the historical sources of Charles Sanders Peirce’s (1839–1914) diagrammatic logic. Arthur Cayley (1821–1895) and Alfred Bray Kempe (1849–1922) contributed to the logic of relations and its applications to geometry and foundations of geometry. This paper gives an overview of sources related to analytical trees and diagrams which were inspirational for Peirce’s development of his existential graphs. Much of the material upon which this paper draws consists of unpublished manuscripts from the Peirce Edition Project at the University of Indianapolis where for many years my collaborator Irving Anellis was a member of the research staff.

## Keywords

Algebraic Logic Diagrammatic Method Diagrammatic Logic Propositional Form Counting Tree## Notes

### Acknowledgement

The author is grateful to two unidentified referees whose comments and suggestions have improved the quality of this paper.

## References

- Cayley, A. (1857). On the theory of analytical form called trees.
*Philosophical Magazine, 13*, 172–176.Google Scholar - Cayley, A. (1859). On the theory of analytical form called trees, second part.
*Philosophical Magazine, 18*, 374–378.Google Scholar - Cayley, A. (1870). A memoir on abstract geometry.
*Philosophical Transactions of the Royal Society of London, 160*, 51–63.zbMATHCrossRefGoogle Scholar - Cayley, A. (1874). On the mathematical theory of isomers.
*Philosophical Magazine, 47*, 444–447.Google Scholar - Cayley, A. (1875). On the theory of analytical form called trees, with applications to the theory of chemical combinations.
*British Association Report*, pp. 257–305.Google Scholar - Cayley, A. (1878). The theory of groups: Graphical representation.
*American Journal of Mathematics, 1*, 174–176.MathSciNetCrossRefGoogle Scholar - Cayley, A. (1881). On the theory of analytical form called trees.
*American Journal of Mathematics, 4*, 266–268.zbMATHMathSciNetCrossRefGoogle Scholar - Dodgson, C. L. (Carroll, Lewis). (1887).
*The game of logic*. Reprinted with*Symbolic Logic, Part I*, as*The Mathematical Recreations of Lewis Carroll*. New York: Dover, 1958.Google Scholar - Grattan-Guinness, I. (2000).
*The search for mathematical roots, 1870–1940: Logics, set theories and the foundations of mathematics from Cantor through Russell to Gödel*. Princeton, NJ/London: Princeton University Press.Google Scholar - Grattan-Guinness, I. (2002). Re-interpreting ‘A’: Kempe on multisets and Peirce on graphs, 1886–1905.
*Transactions of the Charles S. Peirce Society, 38*, 327–350.MathSciNetGoogle Scholar - Hartshorne, C., & Weiss, P. (Eds.). (1933a).
*Collected papers of Charles Sanders Peirce, Vol. III, Exact logic*. Cambridge, MA: Harvard University Press.Google Scholar - Hartshorne, C., & Weiss, P. (Eds.). (1933b).
*Collected papers of Charles Sanders Peirce, Vol. IV, The simplest mathematics*. Cambridge, MA: Harvard University Press; 2nd ed., 1960.Google Scholar - Houser, N. (1997). ``Introduction: Peirce as Logician''. In Houser, N., Roberts, D.D., and Van Evra, J.W., eds.
*Studies in the Logic of Charles Sanders Peirce*. Indianapolis/Bloomington: Indiana University Press.Google Scholar - Houser, N. (2010).
*Writings of Charles S. Peirce: A chronological edition, Vol. 8: 1890–1892*(pp.~xi–xcvii). Bloomington/Indianapolis, IN: Indiana University Press.Google Scholar - Kempe, A. B. (1886). A memoir on the theory of mathematical form.
*Philosophical Transactions*252*Kempe of the Royal Society of London, 177*, 1–70. Google Scholar - Kempe, A. B. (1887). Note to a memoir on the theory of mathematical form.
*Proceedings of the Royal Society of London, 48*, 193–196.CrossRefGoogle Scholar - Kempe, A. B. (1889–1890). On the relation between the logical theory of classes and the geometrical theory of points.
*Proceedings of the London Mathematical Society, 21*, 147–182.Google Scholar - Kempe, A. B. (1890). The subject matter of exact thought.
*Nature, 43*, 156–162.zbMATHCrossRefGoogle Scholar - Kempe, A. B. (1894). Mathematics.
*Proceedings of the London Mathematical Society, 26*, 5–15.Google Scholar - Kempe, A. B. (1897). The theory of mathematical form: A correction and clarification.
*The Monist, 7*, 453–458.CrossRefGoogle Scholar - MacFarlane, A. (1879).
*Principles of the algebra of logic*. Edinburgh, UK: D. Douglas.Google Scholar - MacFarlane, A. (1881–1883). L 263. Two letters to Charles S. Peirce, March 29, 1881, and May 5, 1883. Robin catalog, Ms. #L263.Google Scholar
- Marquand, A. (1881). A logical diagram for
*n*terms.*The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, 12*, 266–270.CrossRefGoogle Scholar - Peirce, C. S. (1868). A Search for a Method. Essay VI. Robin catalog, Ms. #593, pp. 249–264.Google Scholar
- Peirce, C. S. (1869). Grounds of validity of the laws of logic: Further consequences of four incapacities.
*Journal of Speculative Philosophy, 2*, 193–208.Google Scholar - Peirce, C. S. (1870). Description of a notation for the logic of relatives, resulting from an amplification of the conceptions of Boole’s calculus of logic.
*Memoirs of the American Academy, 9*, 317–378.Google Scholar - Peirce, C. S. (1880). On the algebra of logic.
*American Journal of Mathematics, 3*, 15–57.zbMATHMathSciNetCrossRefGoogle Scholar - Peirce, C. S. (1881). On the logic of number.
*American Journal of Mathematics, 4*, 85–95.zbMATHMathSciNetCrossRefGoogle Scholar - Peirce, C. S. (1883–1884). On the algebra of logic; MS., n.p., n.d., 5 pp. of a manuscript draft; 12 pp. of a typed draft (corrected by CSP); and 2 pp. of fragments,
*ca*. 1883–84; Robin catalog, Mss. #527 and #527s.Google Scholar - Peirce, C. S. ca. (1883–1909). [Notes for Contributions to the Century Dictionary]; Robin catalog, Ms. #1170; n.p., n.d., 73 pp.Google Scholar
- Peirce, C. S. (1886). The logic of relatives: Qualitative and quantitative; Robin catalog, Ms. #584.Google Scholar
- Peirce, C. S. (1889). Notes on Kempe’s Paper on Mathematical Form; Robin catalog Ms. #714; January 15, 1889, 12 pp. Harvard University; published: [Houser 2010], 25–29.Google Scholar
- Peirce, C. S. (1891a). On the number of dichotomous divisions: A problem of permutations; Robin catalog Ms. #74; pp. 1–10 (p. 7 missing); plus 17 pp. of another draft; published: [Houser 2010], 222–228, identified as dating to Spring 1891.Google Scholar
- Peirce, C. S. (1891b(?)). A problem of trees; Robin catalog Ms. #73, n.p., n.d., 4 pp. (incomplete or unfinished).Google Scholar
- Peirce, C. S. (1896). A graphical method of logic; RC Ms. #915, 17–18 November 1896; published: [Hartshorne 1933], para. 418, 468–470.Google Scholar
- Peirce, C. S. (1897a). The logic of relatives.
*The Monist, 7*, 161–217; reprinted in part: [Hartshorne 1933a], para. 456–552.Google Scholar - Peirce, C. S. (1897b). Reply to Mr. Kempe (K); Robin catalog, Ms. #708, n.p., n.d., pp. 1–9.Google Scholar
- Peirce, C. S. (1903a). [Lecture I]; Robin catalog, Ms. #450; notebook, n.p., 1903, pp. 1–26.Google Scholar
- Peirce, C. S. (1903b). Lectures on logic, to be delivered at the Lowell Institute. Winter 1903–1904. Lecture I; Robin catalog, Ms. #454, notebook, n.p., 1903, pp. 1–26.Google Scholar
- Peirce, C. S. (1903c). [Lecture II]; Robin catalog, Ms. #455; notebook, n.p., 1903, pp. 1–31.Google Scholar
- Peirce, C. S. (1905a). The basis of pragmaticism (basis); Robin catalog, Ms. #280, n.p., [ca. 1905], pp. 1–48, plus fragments.Google Scholar
- Peirce, C. S. (1905b). The bed-rock beneath pragmaticism (bed); Robin catalog, Ms. #300; pp. 1–65; 33–40; 38–41; 37–38; 40–43.7; plus 64 pp. of fragments running brokenly from p. 1 to p. 60; published in part: [Peirce 2010], 208–210.Google Scholar
- Peirce, C. S. (n.d.(a)). Note on Kempe’s Paper in Vol. XXI of the Proceedings of the London Mathematical Society [Kempe
*1889–90*]; Robin catalog Ms. #709; n.p., n.d., pp. 1–6, plus 3 pp.Google Scholar - Peirce, C. S. (n.d.(b)). Notes on Kempe’s Paper; Robin catalog, Ms. #710; n.p., n.d., pp. 1–2, plus 7 pp.Google Scholar
- Peirce, C. S. (n.d.(c)). Notes on Kempe’s Paper; Robin catalog, Ms. #711; n.p., n.d., 4 pp.Google Scholar
- Peirce, C. S. (n.d.(d)). (Kempe); Robin catalog, Ms. #712; n.p., n.d., 1 p.Google Scholar
- Peirce, C. S. (n.d.(e)). (Kempe); Robin catalog, Ms. #713; n.p., n.d., 2 pp.Google Scholar
- Peirce, C. S. (n.d.(f)). Logic of relatives; Robin catalog, Ms. #547, n.p., n.d., 18 pp.Google Scholar
- Peirce, C. S. (n.d.(g)). Comments on Cayley’s “Memoir on Abstract Geometry” from the point of view of the logic of relatives; Robin catalog, Ms. #546; n.p., n.d., 5 pp.Google Scholar
- Peirce, C. S. (n.d.(o)). On existential graphs as an instrument of logical research; Robin catalog, Ms. #498; notebook (Harvard Cooperative), n.p., n.d.Google Scholar
- Risteen, A. D. (1895).
*Molecules and the molecular theory of matter*. Boston/London: Ginn & Co.; reprinted: 1896.Google Scholar - Risteen, A. D. (1891). Letter to Charles S. Peirce, 10 June 1891; Robin Catalog, Ms. #L376.Google Scholar
- Venn, J. (1880, July). On the diagrammatic and mechanical representations of propositions and reasonings.
*The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science (ser. 5), 10*, 1–18.Google Scholar - Von Meyer, J. L. (1864). Die modernen Theorien der Chemie, und ihre Bedeutung für die chemische Statik. Breslau: Maruschke und Berendt.Google Scholar