Frege and Russell as Inspirers of Leśniewski's Formalisation of the Concept of Collective Set

Andrzej Pietruszczak

doi:10.18778/0138-0680.2026.27

Authors

Andrzej Pietruszczak Nicolaus Copernicus University in Toruń, Poland Author https://orcid.org/0000-0001-9133-5081

DOI:

https://doi.org/10.18778/0138-0680.2026.27

Keywords:

collective class, being of a part of, being of ingrediens of, Leśniewski's mereology, Russell's paradox, Frege, Leśniewski, Russell

Abstract

In this paper, we show that Frege's approach to classes and the relation of being an element was influential in Leśniewski's formalisation of these notions in a collective version. This was further influenced by Leśniewski's attempt to solve Russell's paradox. This happened even though Frege and Russell wrote about distributive classes, while Leśniewski recognised collective classes. We will present this in terms of Frege's approach to classes and the relation of being an element of. It allows us to discuss Russell's paradox in its form as Leśniewski saw it.

References

G. Frege, Grundgesetze der Arithmetik: begriffsschriftlich abgeleitet, vol. 1, Hermann Pohle, Jena (1893).

G. Frege, Grundgesetze der Arithmetik: begriffsschriftlich abgeleitet, vol. 2, Hermann Pohle, Jena (1903).

E. Husserl, Aufsätze und Rezensionen (1890–1910), vol. 22 of Husserliana: Edmund Husserl – Gesammelte Werke, Martinus Nijhoff Publishers, The Hague (1979).

S. Leśniewski, O podstawach matematyki, Przegląd Filozoficzny, vol. 30(2–3) (1927), pp. 164–206, part I: introduction and chapters I–III.

S. Leśniewski, O podstawach matematyki, Przegląd Filozoficzny, vol. 31(3) (1928), pp. 261–291, part II: chapter IV.

S. Leśniewski, O podstawach matematyki, Przegląd Filozoficzny, vol. 34(2–3) (1931), pp. 142–170, part V: chapters X–XI.

S. Leśniewski, Collected Works, vol. 1 of Nijhoff International Philosophy Series, Kluwer Academic Publishers, Dordrecht (1992), series volume 44; annotated bibliography by V. Frederick Rickey; co-published with PWN–Polish Scientific Publishers, Warsaw. The publisher lists a release date of 31 December 1991 but a 1992 copyright/imprint date.

A. Pietruszczak, Metamereology, Nicolaus Copernicus University Scientific Publishing House, Toruń (2018), DOI: https://doi.org/10.12775/3961-4, 328 pages. The listed ISBN and DOI identify the softcover; hardcover ISBN 978-83-231-3975-1 has a separate DOI, 10.12775/9751.

A. Pietruszczak, Foundations of the Theory of Parthood: A Study of Mereology, vol. 54 of Trends in Logic, Springer International Publishing, Cham (2020), DOI: https://doi.org/10.1007/978-3-030-36533-2, e-book ISBN shown; hardcover ISBN 978-3-030-36532-5.

W. V. O. Quine, From a Logical Point of View: Nine Logico-Philosophical Essays, Harvard University Press, Cambridge, MA (1953), first edition; do not confuse with the second revised edition (1961, reprinted 1980), which has separate digital identifiers.

W. V. O. Quine, Philosophy of Logic, Foundations of Philosophy Series, Prentice-Hall, Englewood Cliffs, NJ (1970), first edition. The Harvard University Press second edition (1986) is a different edition and has DOI 10.4159/9780674042445.

W. V. O. Quine, Mathematical Logic, revised ed., Harvard University Press, Cambridge, MA (1981), DOI: https://doi.org/10.4159/9780674042469, 1981 printing of the revised edition; DOI identifies the later electronic publication of this text.

W. V. O. Quine, Quiddities: An Intermittently Philosophical Dictionary, Harvard University Press, Cambridge, MA (1987), DOI: https://doi.org/10.4159/9780674042438, first published in 1987; DOI identifies a later Harvard University Press electronic edition.

D. E. Rowe, On the Origins of Cantor’s Paradox: What Hilbert Left Unsaid at the 1900 ICM in Paris, The Mathematical Intelligencer, vol. 46(2) (2024), pp. 102–116, DOI: https://doi.org/10.1007/s00283-022-10259-x, published online 28 March 2023; issue publication in June 2024.

B. Russell, The Principles of Mathematics, Cambridge University Press, Cambridge (1903), URL: https://fair-use.org/bertrand-russell/the-principles-of-mathematics/, original title page identifies this as Volume I; a planned second volume did not appear.

H. Wang, What Is Logic?, The Monist, vol. 77(3) (1994), pp. 261–277, DOI: https://doi.org/10.5840/monist199477312.

J. Woleński, Tajemnica warszawskiej szkoły logicznej, Wiadomości Matematyczne, vol. 26(2) (1985), pp. 133–153, URL: https://wydawnictwa.ptm.org.pl/index.php/wiadomosci-matematyczne/article/view/4056.

E. Zermelo, Neuer Beweis für die Möglichkeit einer Wohlordnung, Mathematische Annalen, vol. 65(1) (1908), pp. 107–128, DOI: https://doi.org/10.1007/BF01450054, the original journal volume is dated 1908; Springer online metadata inconsistently gives March 1907.