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A Note on the Historical Development of Logic Diagrams: Leibniz, Euler and Venn

Published online by Cambridge University Press:  03 November 2016

Margaret E. Baron
Margaret E. Baron*
Affiliation:
Stockwell College, Bromley, Kent
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Extract

Lessons on sets have become commonplace in schools today. Venn diagrams proliferate and, even in primary schools, children can be seen sorting and classifying objects by size, colour and shape and placing them in spaces marked out on the floor by chalk outlines or wooden hoops. Older children learn that such diagrams are named after the English logician, John Venn, and that through them we can represent the relations of membership and inclusion and the operations of union, intersection and complementation. A rectangle is drawn to represent the universe U: subsets of U are represented by the interiors of circles, or other closed curves within U, i.e. subspaces of the rectangle. The elements of U are represented by points within the rectangle, the elements of a subset A by points within the corresponding subspace of the rectangle and the elements of A′ by points within the rectangle but outside the region representing A.

Type
Research Article
Information
The Mathematical Gazette , Volume 53 , Issue 384 , May 1969 , pp. 113 - 125
Copyright
Copyright © Mathematical Association 1969

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References

page 113 note * Pitt, E. B.: Mathematics in the Infant School: Mathematics Teaching, XXXIV (1966): pp. 45-53.

Mathematics Begins: Nuffield Mathematics Teaching Project: London, 1967: pp. 10-26.

page 113 note † Brissendon, T. H. F.: Modern Maths for Sec Moderns⌍: Mathematics Teaching, XXXIV (1966): pp. 19-21. Brissendon says “We don’t really teach anything, just suggest some standard terminology and, generally, foist Venn diagrams on the class.”

page 115 note * See Lange, F. A.: Logische Studien: Iserlohn, 1877: p. 10.

page 115 note † See Kneale, W. and M.: The Development of Logic: Oxford, 1962: pp. 71-72.

page 115 note ‡ See Gardner, M.: Logic Machines and Diagrams: New York, 1958.

page 115 note § Lull, R.: Opera ea quae ad adinventam …”. Argintinae, 1617. Ramon Lull (c. 1235-1315), Catalan author, mystic and missionary, sought to establish Christian doctrines as necessary truths by mechanical processes which he claimed to be completely general.

page 115 note ‖ The meaning of this diagram is that nothing exists which does not possess unity, truth and goodness.

page 115 note See Lange, J. C.: Inventum novum quadrati logici universalis: Giessen, 1711. Many of these ideas are attributed to C. Weise (d. 1711). Most of Lange’s diagrams are shown as mounted on pedestals which suggests that they were intended as teaching aids.

page 116 note * Alsted, J. R.: Elementale mathematicum: Frankfort, 1611.

page 116 note † Couturat, L.: Opuscules et fragments inédits de Leibniz: Paris, 1903.

page 116 note ‡ Couturat, L.: La Logique de Leibniz: Paris, 1901: p. 1.

page 117 note * Leibniz uses an ellipse wherever it appears more convenient.

page 118 note * Couturat, L.: Opuscules et fragments: op. oit.: pp. 301–312, et passim.

page 118 note † Ploucquet, G.: Sammlung der Schriften welche den logischen Oalcul betreffen: Tübingen, 1773.

page 118 note ‡ Lambert, J. H.: Neues Organon: Leipzig, 1764.

page 119 note * Ibid.: pp. 110-111, pp. 121-122.

page 119 note † Euler, L.: Lettres à une Princesse d’Allemagne: St. Petersburg, 1768. (Trans. Hunter, H.: Letters to a German Princess: London, 1795, see pp. 453-454.)

page 120 note * Gergonne, J. D.: Essai de dialectique rationelle: Annales de mathématiques, VII (1816-1817): pp. 189-228.

page 120 note † Kneale, W. and Kneale, M.: The Development of Logic: op. cit.: pp 350-352.

page 120 note ‡ Faris, J. A.: The Gergonne Relations: The Journal of Symbolic Logic, XX (1955); pp. 207-231.

page 121 note * Venn notes that, of 60 logical treatises published during the century, 34 appealed to the aid of diagrams, nearly all making use of the Eulerian scheme.

page 121 note † Hamilton, W.: Lectures: Edinburgh, 1860: IV, p. 465.

page 121 note ‡ Venn, J.: Symbolic Logic: London, 1881. Chap. XX, Historic Notes.

page 121 note § Boole, G.: The Mathematical analysis of logic: Cambridge, 1847

page 121 note ‖ Ibid.: p. 5.

page 121 note Venn notes the publication of a system virtually identical with his own in the same year: Seheffler, H.: Die Naturgesetze, III (1880), XXX.

page 121 note ** Symbolic Logic: op. cit.: p. 22.

page 122 note * Ibid.: p. 103.

page 123 note * Ibid.: pp. 107-108.

page 123 note † Ibid.: p. 110.

page 124 note * The null class was not recognised as a class.

page 124 note † Ibid.: p. 261.

page 125 note * Ibid.: pp. 186-187.

page 125 note † Carroll, L.: Symbolic Logic: London, 1896. Given from Dover Publications, reprint, New York, 1955, p. 176.