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# PRECURSORS OF MODERN LOGIC: EULER

The noted mathematician Leonhard Euler (17071783) is remembered in logic chiefly for his geometrical illustrations of syllogistic, "Euler's diagrams" or "Euler's circles." Similar devices were used by J. C. Sturm (1661), Leibniz (see Bocheński, History of Formal Logic, plate facing p. 260), Joachim Lange (1712), and Gottfried Ploucquet (1759), and in a very general way the idea of spatial illustration goes back at least to Juan Luis Vives, who used triangles to illustrate the Barbara syllogism ("De Censura Veri," in Opera, Basel, 1555). But because of Euler's fame as a mathematician and the popularity of his charming Lettres à une princesse d'Allemagne (the relevant letters are CII ff., dated 1761) such diagrams are traditionally named for him.

Euler used proper inclusion for the universal affirmative proposition, exclusion for the universal negative, and intersection for both the particulars. If his interpretation is followed systematically, it correctly decides the validity or invalidity of all three-term syllogisms with all terms distinct but fails for the laws of identity and contradiction and for degenerate syllogisms depending on them. Apparently nobody developed full syllogistic along these lines until J. D. Gergonne (1816), whose five relations give a complete system and can indeed be defined by three of them (see Ivo Thomas, "Eulerian Syllogistic," and references supplied there), but not by Euler's three. The extensional approach evidenced by Euler's interpretation of the universal affirmative was a healthy influence.

Euler also lent his authority to the doctrine that singular propositions are equivalent to universal ones (Lettres, CVII), a thesis propounded by John Wallis (from 1638; see Appendix to his Institutio Logica, Oxford, 1687). Bertrand Russell severely criticized this doctrine as confusing class membership with inclusion, but of course we can get an inclusive proposition equivalent to a membership proposition by taking the unit class of the singular subject.

## Bibliography

Faris, J. A. "The Gergonne Relations." Journal of Symbolic Logic 20 (1955): 207231.

Gardner, Martin. Logic Machines and Diagrams. New York: McGraw-Hill, 1958.

Hamilton, William. Lectures on Logic. London, 1860.

Hocking, W. E. "Two Extensions of the Use of Graphs in Elementary Logic." University of California Publications in Philosophy 2 (2) (1909): 3144.

More, Trenchard. "On the Construction of Venn Diagrams." Journal of Symbolic Logic 24 (1959): 303304.

Thomas, Ivo. "Eulerian Syllogistic." Journal of Symbolic Logic 22 (1957): 1516.

Thomas, Ivo. "Independence of Faris-Rejection-Axioms." Notre Dame Journal of Formal Logic 1 (1959): 4851.

Venn, John. Symbolic Logic. 2nd ed. London: Macmillan, 1894.

Ivo Thomas (1967)