C.S. Peirce’s Logic of Relations:
Graph-theoretical and Surface-theoretical Models

William James McCurdy

Department of English and Philosophy
Idaho State University, Pocatello, Idaho, USA

Charles Sanders Peirce is, as Alfred Tarski has rightly reminded us, the father of the logic of relations. Although Augustus de Morgan pioneered investigation into the logic of dyadic relations as an outgrowth of his mathematical study of syllogistic, it was Peirce who first developed a general logic of relations, that is, a logic for relations of any adicity (valency) whatsoever. The corazon de corazon of this logic is Peirce’s so-called “Reduction Thesis” consisting of two controversial clauses. The first of these is a necessity clause stating that, besides monadic relations (one-place predicates) and dyadic relations, a relationally complete logic must also have genuine triadic relations, that is, three-place relations which cannot be analyzed into combinations of relations of lesser adicity. The second clause is a sufficiency clause, specifically, the claim that genuine triadic relations, together with monadic and dyadic relations, suffice for a relationally-complete logic. The means for composing all other (n>3)-adic relations are two logical operations, namely, the unary operation of auto-relative multiplication and the binary operation of relative multiplication. Peirce’s Reduction Thesis has been all but universally rejected, often even by scholars sympathetic to Peirce and his work in logic. This tutorial explicates his contentious thesis and subsequently presents two topological models for his logic of relations. One is a variant of topological graph theory, called Peircean Relational Graph Theory, and the other uses surface theory, called Peircean Relational Surface Theory. These two models provide justification for his remarkable contribution to a universal logic of relations including proofs of his Reduction Thesis, one in each model.


"[W]e homely thinkers believe that, considering the immense amount of disputation there has always been concerning the doctrine of logic, and especially concerning those which would otherwise be applicable to settle disputes concerning the accuracy of reasonings in metaphysics, the safest way is to appeal for our logical principles to the science of mathematics, where error can only long go unexploded on condition of not being suspected." C.S Peirce, The Regenerated Logic

Synopses of Tutorial Sessions

I. Peirce’s Logic of Relations and Peircean Relational Graph Theory
C.S. Peirce’s view that mathematics is the science of necessary reasoning about hypothetical possibilities by means diagrams will be introduced. Further, his contention that logic requires topology will be briefly examined. Peirce’s diagrammatic logic of relations will be explicated including his “Reduction Thesis,” specifically, the thesis, that a relationally complete logic requires, but only requires monadic, dyadic, and triadic relations. The fundamentals of Peircean Relational Graph Theory (PRGT), a radical variant of standard graph theory will be delineated. It will be shown that PRGT is able to represent straightforwardly both relations of one, two, and three adicities and the logical operations of auto-relative and relative multiplication.

II. Garnering the First Fruits of PRGT and Those of a Later Gleaning
The representational scope and power of PRGT will be presented via relevant combinatorial formulas as well as diagrams of relational networks. Several key theorems will be demonstrated culminating in a proof of Peirce’s Composability-of-Relations Theorem (The Reduction Thesis justified). A taxonomy of general varieties of relational networks willed be tabulated. As a preamble and a propaedeutic to the third session, surface diagrams which are two-dimensional counterparts to the one-dimensional diagrams of PRGT will be introduced. The gluing of surfaces with boundaries will be presented as the means to represent auto-relative and relative multiplication.
 


III. Peircean Relational Surface Theory
Employing some insights of such pioneers in topology as A.F. Möbius and Max Dehn, three surface models for Peirce’s logic of relations will be explored, specifically, 1.) a cap/sleeve/pair of pants model, 2.) a model of spheres with one, two, and three discs excised, and 3.) a disc/annulus/bi-annulus model. While a disc, an annulus, and a bi-annulus are homotopically distinct from each other, the above three models are homotopically equivalent. This will be diagrammatically displayed and algebraically demonstrated. A problem with using these surface models to represent Peirce’s logic of relations will be discussed and then solved, involving the use of deformation retractions of the disc, the annulus, and the bi-annulus as necessary aspects of an adequate model Peirce’s logic of relations in two dimensions.

Selected Bibliography
  • Bruno Martelli, An Introduction to Geometric Topology, University of Pisa, Italy, Version 1.0, October, 2016
  • Silvia Benvenuti and Riccardo Piergallini. “The Complex of Pant Decompositions of a Surface”, Topology and its Applications, Vol. 156, Issue 2, 1 Dec., 2008. pp. 399 - 419
  • Robert Burch, A Peircean Reduction Thesis: The Foundations of Topological Logic, Texas Tech University Press, Lubbock, Texas, 1991
  • Max Dehn, Papers on Group Theory and Topology, trans. and intro. by John Stillwell, Springer-Verlag, New York, 1987
  • Frank Harary, Graph Theory, Addison-Wesley Pub. Co., Reading, MA, 1969
  • Allen Hatcher, Algebraic Topology, Cambridge University Press, 2001
  • Kenneth Laine Ketner, "Peirce’s ‘Most Lucid and Interesting Paper’ An Introduction to Cenopythagoreanism", International Philosophical Quarterly, 25: 375- 392
  • John M. Lee, Introduction to Topological Manifolds, Springer-Verlag, New York
  • Charles Sanders Peirce, The Collected Papers of Charles Sanders Peirce, Belknap Press of Harvard University Press, Vol. I – VI., 1932
  • Charles Sanders Peirce, The Logic of Interdisciplinarity: The Monist Series, Herausgegeben van Elise Bisanz, Akademie Verlag, Berlin, 2009
  • Charles Sanders Peirce, Reasoning and the Logic Of Things, ed. by Kenneth Laine Ketner, with an introduction by Kenneth Laine Ketner and Hilary Putnam, Harvard University Press, 1992
  • Charles Sanders Peirce, Semiotics and Significs: The Correspondence between Charles S. Peirce and Victoria Lady Welby, ed. By Charles S. Hardwick with Assistance of James Cook, The Press of Arisbe Associates, Elsah, Illinois, 2001
  • Don D. Roberts, Existential Graphs of Charles S. Peirce, Mouton & Co., The Hague, 1973
  • John Stillwell, Classical Topology and Combinatorial Group Theory, Springer-Verlag, New York, 1980
  • Fernando Zalamea, The Logic of Continuity, Docent Press, Boston, MA, 2012