On Universality and Formality in 19th Century Symbolic Logic: The Case of Schröder’s “Absolute Algebra”

  • Javier LegrisEmail author
Part of the Studies in Universal Logic book series (SUL)


This paper deals with conceptions of formality underlying 19th century symbolic logic, where notations and manipulation of signs played an important role. It is devoted specifically to the case of Ernst Schröder’s “formal algebra”, which extended with the algebra of relatives (as developed by C.S. Peirce) constituted the basis for a Pasigraphy as a universal notation system. The discussion will begin with the well-known distinction devised by Gottlob Frege between two sorts of formal theories. In the paper, both conceptions of formality will be connected with the corresponding attempts of constructing universal scientific notations (Schröder’s Pasigraphy and Frege’s Begriffsschrift). It will be shown that the Pasigraphy was an interpretation of that formal algebra. As a further conclusion, it will be suggested that each of the two conceptions of formality places logic in different levels and determines different conceptions of universality.


History of logic Algebra of logic Formality Ernst Schröder 

Mathematics Subject Classification

01A55 0303 03A05 



It is a truism that the idea of universality is essential to Jean-Yves Beziau’s work. In some respect, Jean-Yves devotes his life to this idea. The following historical comments can be interesting in connection with it. This paper is a result of a research project supported by the Consejo Nacional de Investigaciones Científicas y Técnicas from Argentina (PIP 11220080101334, CONICET). It also benefited from the additional financial support of the Alexander von Humboldt Foundation. I wish to thank Volker Peckhaus for helpful and valuable comments on an earlier draft of this paper.


  1. 1.
    Angelelli, I.: Studies on Gottlob Frege and Traditional Philosophy. Dordrecht, Reidel (1967) CrossRefzbMATHGoogle Scholar
  2. 2.
    Corcoran, J.: Schemata: the concept of schema in the history of logic. Bull. Symb. Log. 12, 219–240 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Detlefsen, M.: Hilbert’s Program. An Essay on Mathematical Instrumentalism. Dordrecht, Reidel (1986) zbMATHGoogle Scholar
  4. 4.
    Dutilh Novaes, C.: The different ways in which logic is (said to be) formal. Hist. Philos. Logic 32, 303–332 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Frege, G.: Über formale Theorien der Arithmetik. Sitz.ber. Jeaneischen Ges. Med. Nat.wiss. 12, 94–104 (1885/1886). Reprinted in Frege G.: Kleine Schriften, 2nd edn. (1990) Edited by Ignacio Angelelli. Olms, Hildesheim–Zürich–New York, Olms, pp. 103–111 Google Scholar
  6. 6.
    Frege, G.: Grundgesetze der Arithmetik, vol II. Jena (1903). Reprinted Hildesheim, Olms, 1966 Google Scholar
  7. 7.
    Frege, G.: Über die Grundlagen der Geometrie. Jahresber. Dtsch. Math.-Ver. 15, 293–309 (1906). Reprinted in Frege 1990, pp. 281–323 zbMATHGoogle Scholar
  8. 8.
    Hodges, W.: A formality. CD Festschrift for the 50th birthday of Johan van Benthem, Vossiuspers AUP (1999).
  9. 9.
    Kant, I.: KrV. Kritik der reinen Vernunft, 1st edn. 1781, 2nd edn. 1787. In: Weischedel, W. (ed.) Werke in zwölf Bänden, vol. III. Suhrkamp, Frankfurt (1968) Google Scholar
  10. 10.
    Legris, J.: Deux approches des relations logique-mathématiques Frege et Schröder. In: En Justifier en Mathématiques, comp. por Dominique Flament y Philippe Nabonnand, pp. 215–254. Editions de la Maison des sciences de l´homme, Paris (2011) Google Scholar
  11. 11.
    Legris, J.: Universale Sprache und Grundlagen der Mathematik bei Ernst Schröder. In: Mathematik—Logik—Philosophie. Ideen und ihre historischen Wechselwirkungen, comp. por Günther Löffladt. pp. 255–269. Harri Deutsch, Frankfurt am Main (2012) Google Scholar
  12. 12.
    MacFarlane, J.: What does it mean to say that logic is formal? PhD Dissertation, University of Pittsburgh (2000) Google Scholar
  13. 13.
    Peckhaus, V.: Schröder’s logic. In: Gabbay, D.M., Woods, J. (eds.) Handbook of the History of Logic. Vol. 3. The Rise of Modern Logic: From Leibniz to Frege, pp. 557–609. Elsevier-North Holland, Amsterdam (2004) Google Scholar
  14. 14.
    Schröder, E.: Lehrbuch der Arithmetik und Algebra. Teubner, Lepizig (1873) zbMATHGoogle Scholar
  15. 15.
    Schröder, E.: Review of Begriffsschrift by Gottlob Frege. Z. Math. Phys. 25, 81–94 (1880) Google Scholar
  16. 16.
    Schröder, E.: Über Pasigraphie, ihren gegenwärtige Stand und die pasigraphische Bewegung in Italien. In: Rudio, F. (ed.) Verhandlungen der Ersten Internationales Mathematiker-Kongresses in Zürich vom 9. bis 11. August 1897. The Monist, vol. 9, pp. 147–162. Teubner, Leipzig (1898). Reprint Nendeln, Kraus, 1967, English translation: on Pasigraphy. Its Present State and The Pasigraphic Movement in Italy en The monist 9, 1898, pp. 44–62 (corrigenda, p. 320) Google Scholar
  17. 17.
    van Heijenoort: Logic as calculus and logic as language. Synthese 24, 324–330 (1967) CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.IIEP-BAIRES, Facultad de Ciencias EconómicasUniversidad de Buenos AiresCiudad Autónoma de Buenos AiresArgentina

Personalised recommendations