Abstract
Charles Peirce’s important manuscript on Euler diagrams (Ms 479, 1903) was partially printed in the Collected Papers in 1933 (CP 4.347-371). That transcription omitted many paragraphs, figures and important variants of the main text, and diagrams were reproduced misleadingly or imprecisely. Another important and wholly unpublished paper of his (Ms 481, 1896-7) presents a novel extension of Euler’s diagrams for negative terms. Third, among the discarded pages of a published article (Ms 1147, 1901) we find a variant on logical graphs suggesting similar extensions. Ms 855 (1911) is yet another unpublished note in which Peirce deals with existentials and shading. The present paper restores Peirce’s original drawings from these four manuscripts and explains their main innovations. As Euler diagrams were not designed to reason about relative terms, Peirce’s interest was not in their mathematical application or problem solving but in showing what the basic elements of syllogistic reasoning are.
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Notes
- 1.
These five shortcomings of the theory of Euler diagrams are (i) its inadequacy of dealing with every syllogistic form; (ii) that it “cannot affirm the existence of any description of an object”; (iii) that it fails to represent disjunctions in the general case; (iv) that it “affords no means of expressing any other than dichotomous” information; and (v) that these diagrams fail to exhibit “relational or abstractional” kind of reasoning, meaning that the system “has no vital power of growth beyond the point to which it has been carried” (Ms 491; Ms 479).
- 2.
For reasons of space, we omit reproducing Figs. 1–24, which mostly are examples of ordinary Euler diagrams and Venn’s modifications (see CP 4.350-8). Figures 25–36, 54–66 in Sect. 1 and in Variants 1 and 2 are photographic images from Peirce’s original manuscripts deposited at the Harvard Houghton Library, slightly enhanced digitally for readability.
- 3.
Other unfortunate typographical inaccuracies in the CP were using the letter x for crosses and the number 0 for circles, as well as using capital letters as labels placed outside the circles. Peirce wrote the letters on the circles. He might have adopted this convention from Johann Christian Lange’s 1712 Nucleus Logicae Weisianae, the work which Peirce, just as Venn, thought “anybody familiar with such literature the title proclaims it to be a work by [Christian] Weise probably with a running commentary or copious notes by Lange” (Ms 479: 16). Weise’s Nucleus Logicae was indeed originally published, Peirce remarks, as a small booklet in 1691 and edited and expanded into an over 800-page volume published 1712 by Weise’s student Lange.
- 4.
Generically, count the number of connected systems of lines separated by such circles.
- 5.
This figure, numbered Fig. 57 in CP 4.364 was published there with a different but equivalent orientation.
- 6.
A reviewer suggests a general result may be that representing n individuals requires n + 1 contours. Peirce’s simplification that follows here suggests that n contours suffice.
- 7.
This is a common convention that was reintroduced, in a different sense, in [10], in order to strike out all constituents of a certain region and not to demolish a particular individual.
- 8.
I thank a reviewer for a highly relevant reference [17] on projections of Euler diagrams, similar to this anticipation of Peirce’s. The iterated diagram was omitted from the publication of Ms 479 in CP, as indeed was another large diagram that applies iteration to a complex example taken from Ladd-Franklin (SiL: 58–61). Peirce explains: “In order to illustrate the method I will apply it (without any preconsideration at all) to the following problem by Mrs. Franklin […]” (Ms 479: 59). A three-page explanation of this example then follows. Appendix reproduces the image of this large diagram.
- 9.
Another reviewer assumes these concave curves lack the kind of iconicity or well-matchedness that Peirce might have desired them to have. One way of viewing them could be denoting ‘cuts’ in a hyperbolic space, which might then restore their presumed iconic character.
- 10.
We surmise that all disjunctions can be expressed by these boundary dots, if the shading of zones to deny existence is also used (see Sect. 4).
- 11.
There are variations concerning the 12 schemas of syllogisms in Ms 481 and the 16 schemas in Ms 1147: Figs. 10, 11, 14 and 15 from Ms 481 are not represented in this latter table.
- 12.
See [1]. However, a few draft pages exist in which Peirce uses up to five different types of shadings (e.g. Ms L 76, written on the back of a letter, not microfilmed). Their semantics remains veiled. See Appendix, second figure.
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Acknowledgments
Supported by projects PUT267 (Estonian Research Council) and 127335 (Academy of Finland): Diagrammatic Mind: Logical and Cognitive Aspects of Iconicity; 2014–15 Foreign Experts Program of State Administration of Foreign Experts Affairs, China; and the 2012 Harvard Houghton Library Fellowship (Principal Investigator A.-V. Pietarinen).
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A Appendix
A Appendix
The images below are from Ms 479 (top/right) and Ms L 76 (bottom/left).
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Pietarinen, AV. (2016). Extensions of Euler Diagrams in Peirce’s Four Manuscripts on Logical Graphs. In: Jamnik, M., Uesaka, Y., Elzer Schwartz, S. (eds) Diagrammatic Representation and Inference. Diagrams 2016. Lecture Notes in Computer Science(), vol 9781. Springer, Cham. https://doi.org/10.1007/978-3-319-42333-3_11
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