Residuation in Existential Graphs

Nathan Haydon*, Ahti Veikko Pietarinen

*Corresponding author for this work

Research output: Chapter in book/report/conference proceedingConference proceedingpeer-review

Abstract

Residuation has become an important concept in the study of algebraic structures and algebraic logic. Relation algebras, for example, are residuated Boolean algebras and residuation is now recognized as a key feature of substructural logics. Early work on residuation can be traced back to studies in the logic of relations by De Morgan, Peirce and Schröder. We know now that Peirce studied residuation enough to have listed equivalent forms that residuals may take and to have given a method for arriving at the different permutations. Here, we present for the first time a graphical treatment of residuation in Peirce’s Beta part of Existential Graphs (EGs). Residuation is captured by pairing the ordinary transformations of rules of EGs—in particular those concerning the cuts—with simple topological deformations of lines of identity. We demonstrate the effectiveness and elegance of the graphical presentation with several examples. While there might have been speculation as to whether Peirce recognized the importance of residuation in his later work, or whether residuation in fact appears in his work on EGs, we can now put the matter to rest. We cite passages where Peirce emphasizes the importance of residuation and give examples of graphs Peirce drew of residuals. We conclude that EGs are an effective means of enlightening this concept.

Original languageEnglish
Title of host publicationDiagrammatic Representation and Inference
Subtitle of host publication12th International Conference, Diagrams 2021, Virtual, September 28–30, 2021, Proceedings
EditorsAmrita Basu, Gem Stapleton, Sven Linker, Catherine Legg, Emmanuel Manalo, Petrucio Viana
PublisherSpringer Cham
Pages229-237
Number of pages9
Edition1st
ISBN (Electronic)9783030860622
ISBN (Print)9783030860615
DOIs
Publication statusPublished - 3 Sept 2021
Event12th International Conference on the Theory and Application of Diagrams, Diagrams 2021 - Virtual, Online
Duration: 28 Sept 202130 Sept 2021
https://link.springer.com/book/10.1007/978-3-030-86062-2

Publication series

NameLecture Notes in Computer Science
Volume12909
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349
NameLecture Notes in Artificial Intelligence
NameDiagrams: International Conference on Theory and Application of Diagrams

Conference

Conference12th International Conference on the Theory and Application of Diagrams, Diagrams 2021
CityVirtual, Online
Period28/09/2130/09/21
Internet address

Scopus Subject Areas

  • Theoretical Computer Science
  • Computer Science(all)

User-Defined Keywords

  • Charles Peirce
  • Cuts
  • Existential graphs
  • Lines of identity
  • Residuation

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