Revisiting Peirce’s Rules of Transformation for Euler-Venn Diagrams

Charles S. Peirce introduced in 1903 a set a transformation rules for Euler-Venn diagrams. This innovation contrasted with earlier practices where logicians rather extracted the desired information by a simple ‘glance’ at their diagrams. Also, Peirce’s set of rules was the starting point of Sun-Joo Shin’s more recent systems which, in turn, inspired most subsequent modern diagrammatic systems. Despite their significance, these rules got little attention from both diagram and Peirce scholars. In this paper, we revisit Peirce’s rules of transformation and discuss the extent to which they ‘survived’ in modern diagrammatic systems. We will specifically consider their clarity and completeness to assess Peirce’s assumption that some of his rules may be simplified while others may have been overlooked.

1. 1.

   Unfortunately, manuscript MS 479 has still not been properly published. It has only been partially reproduced and poorly edited in Peirce’s Collected Papers [13]. This transcription, on which was based Shin’s account, should be used with extreme caution. The manuscript is also not reproduced in Ahti-Veikko Pietarinen’s edition of Peirce’s existential graphs [15], but additional text and variants are included [14]. Apparently, Peirce intended to include his manuscript as a chapter in a volume of Logical Tracts [14, p. 72]. The original manuscript MS 479 is freely accessible on the Peirce Archive repository (https://rs.cms.hu-berlin.de/peircearchive/pages/search.php). The page numbers we indicate for MS 479 are the file titles in the Peirce Archive.

2. 2.

   Lewis Carroll is a remarkable exception here. See [7, 8]. A comparison of Carroll’s rules with those of Peirce is found in [10].

3. 3.

   Peirce explained that he used rules “in the sense in which we speak of the “rules” of algebra; that is, as a permission under strictly defined condition” [12]. In his entry on ‘Symbolic Logic’, published a year earlier, Peirce defined a rule as “a permission under certain circumstances to make a certain transformation” [11, p. 450].

4. 4.

   Peirce’s mature Eulerian diagrams and Existential graphs were developed at the same time and share several features, including the formulation of transformation rules. But they differ significantly in their purpose: Eulerian diagrams served mainly for logical calculus while Existential graphs were designed for logical analysis. Roughly, calculus aims at carrying reasonings while analysis investigates them. On the opposition between calculus and analysis, see [3, 11, p. 450].

5. 5.

   Here cross and zero represents non-emptiness and emptiness of a region respectively. The connected lines between any of these symbols represent their disjunction

6. 6.

   This rule was written by Peirce on the margins of his manuscript, without further explanation. It seems to have been added later, as shown by the renumbering of the following rule.

7. 7.

   Shin also proposed to use rule 1 to get Fig. 12 and Fig. 14 from Fig. 11 and Fig. 13 respectively. She also proposed to use rule 2 to get Fig. 10 from Fig. 12.

8. 8.

   Additional difficulties may appear when the number of closed curves increases, if the diagrams are not simple or reducible. Peirce occasionally used Venn diagrams for more than 3 curves. Some examples are found in [15]. On the construction of diagrams for n number of curves, see [6].

We acknowledge with gratitude some fruitful suggestions given by Mihir Kumar Chakraborty and Ahti Pietarinen in the course of development of this research. The second author acknowledges support from TalTech internal grant SSGF21021.

Authors and Affiliations

1. School of Cognitive Science, Jadavpur University, Kolkata, India

   Reetu Bhattacharjee

2. Department of Mathematics, Mandsaur University, Mandsaur, India

   Reetu Bhattacharjee

3. Ragnar Nurkse Department of Innovation and Governance, Tallinn University of Technology, Tallinn, Estonia

   Amirouche Moktefi

Corresponding author

Correspondence to Reetu Bhattacharjee .

Editors and Affiliations

1. Jadavpur University, Kolkata, India

   Amrita Basu

2. University of Cambridge, Cambridge, UK

   Gem Stapleton

3. Lancaster University in Leipzig, Leipzig, Germany

   Sven Linker

4. Deakin University, Burwood, VIC, Australia

   Catherine Legg

5. Kyoto University, Kyoto, Japan

   Emmanuel Manalo

6. Universidade Federal Fluminense, Niterói, Brazil

   Petrucio Viana

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