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.
Supported by (Haydon) the ESF funded Estonian IT Academy research measure (2014-2020.4.05.19-0001) and (Pietarinen) the Basic Research Program of the HSE University and the TalTech grant SSGF21021.
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Notes
- 1.
On the history of residuation (adjunctions) and its relation to Galois connections, see [2].
- 2.
While Peirce emphasizes this ordering in his algebraic work, he says little about reading such an ordering off the graphs. The few such places on the ordering of the ‘hooks’ around the relation terms appear in the early drafts of EGs from late 1886, in which Peirce notices how the connections of the lines to the relations should be read “clockwise” or “counterclockwise” (their converses) “beginning at the left/right” of the relation term; see [18, pp.220,263,295,302,303]. A further advantage of the notation in [5] is that the ordering is always explicit.
- 3.
One such derivation is [21, p. 42] which we forego here due to space limitations. Schmidt speculates that equational reasoning using predicate logic results in derivations that are six times longer than the corresponding algebraic handling of relations [p. xi].
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Haydon, N., Pietarinen, AV. (2021). Residuation in Existential Graphs. In: Basu, A., Stapleton, G., Linker, S., Legg, C., Manalo, E., Viana, P. (eds) Diagrammatic Representation and Inference. Diagrams 2021. Lecture Notes in Computer Science(), vol 12909. Springer, Cham. https://doi.org/10.1007/978-3-030-86062-2_21
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