Abstract
Leśniewski’s systems deviate greatly from standard logic in some basic features. The deviant aspects are rather well known, and often cited among the reasons why Leśniewski’s work enjoys little recognition. This paper is an attempt to explain why those aspects should be there at all. Leśniewski built his systems inspired by a dream close to Leibniz’s characteristica universalis: a perfect system of deductive theories encoding our knowledge of the world, based on a perfect language. My main claim is that Leśniewski built his characteristica universalis following the conditions of de Jong and Betti’s Classical Model of Science (2008) to an astounding degree. While showing this I give an overview of the architecture of Leśniewski’s systems and of their fundamental characteristics. I suggest among others that the aesthetic constraints Leśniewski put on axioms and primitive terms have epistemological relevance.
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References
Cantù, P. (2008). Aristotle’s prohibition rule on kind-crossing and the definition of mathematics as a science of quantities. Synthese. doi:10.1007/s11229-008-9419-2.
Canty, J. (1967). Leśniewski’s ontology and Gödel’s incompleteness theorem. PhD Thesis, University of Notre Dame, Indiana.
Dascal M. (1987) Leibniz. Language, signs and thought. John Benjamins Publication Co., Amsterdam
de Jong, W. R. (2008). The analytic-synthetic distinction and the classical model of science: Kant, Bolzano and Frege. Synthese. doi:10.1007/s11229-008-9420-9.
de Jong W.R. (1996) Gottlob Frege and the analytic-synthetic distinction within the framework of the classical model of science. Kant-Studien 87: 290–324
de Jong, W. R., & Betti, A. (2008). The classical model of science: A millennia-old model of scientific rationality. Synthese.
Ditchen, R., Glibowski, E., & Kościk, S. (1963). O pewnym układzie pojȩć pierwotnych geometrii elementarnej. Acta Universitatis Wratislaviensis, 17, Matematyka Fizyka Astronomia IV, 5–11.
Frege, G. (1884). Die Grundlagen der Arithmetik. Eine logisch-mathematische Untersuchung über den Begriff der Zahl. Breslau: W. Koebner.
Husserl, E. (1900/1901). Logische Untersuchungen. Halle a.d.S.: Max Niemeyer = Husserliana XVIII, XIX/1, XIX/2. The Hague: M. Nijhoff. J. N. Findlay (Trans.). (1970). London: Routledge & Kegan Paul.
Korte, T. (2008). Frege’s Begriffsschrift as a lingua characteristica. Synthese. doi:10.1007/s11229-008-9422-9.
LeBlanc, A. O. V. (1983). A study of the axiomatic foundation of mereology. MA Thesis, University of Manchester.
LeBlanc, A. O. V. (1991). Leśniewski’s computative Protothetic. PhD Thesis, University of Manchester.
Leibniz, G. W. (1677). Leibniz an Jean Gallois. In Sämtliche Schriften und Briefe. Herausgegeben von der Berlin-Brandenburgischen Akademie der Wissenschaften und der Akademie der Wissenschaften in Göttingen, Zweite Reihe, Philosophischer Briefwechsel, Erster Band, Akademie Verlag (pp. 566–571). Eng. trans. Dascal (1987, p. 164, n. 2).
Leibniz, G. W. ( > 1684). Untitled. In C. I. Gerhardt (Ed.) (1875–1890), Die philosophischen Schriften von G. W. Leibniz, VII, repr. Hildesheim (1965, pp. 204–207). Trans. Thought, signs and the foundation of logic. In Dascal (1987, pp. 181–188).
Leśniewski, S. (1916). Podstawy ogólnej teorji mnogości. I (Czȩść. Ingredyens. Mnogość. Klasa. Element. Podmnogość. Niektóre ciekawe rodzaje klas). Prace Polskiego Koła Naukowego w Moskwie, Sekcja matematyczno-przyrodnicza. Eng. trans. Leśniewski (1991, pp. 129–173).
Leśniewski, S. (1929). Grundzüge eines neuen Systems der Grundlagen der Mathematik. Fundamenta Mathematicae, XIV, 1–81. Eng. trans. Leśniewski (1991, pp. 410–492).
Leśniewski, S. (1930). Über die Grundlagen der Ontologie. Sprawozdania z posiedzeń Towarzystwa Naukowego Warszawskiego – Comptes rendus des séances de la société des sciences et des lettres de Varsovie, Wydział III – nauk matematyczno-fizycznych, XXIII 1–3, 111–132. Eng. trans. Leśniewski (1991, pp. 606–628).
Leśniewski, S. (1931). Über Definitionen in der sogenannten Theorie der Deduktion. Sprawozdania z posiedzeń Towarzystwa Naukowego Warszawskiego – Comptes rendus des séances de la société des sciences et des lettres de Varsovie, Wydział III – nauk matematyczno-fizycznych XXIV, 289–309. Eng. trans. Leśniewski (1991, pp. 629–648).
Leśniewski, S. (1938). Einleitende Bemerkungen zur Fortsetzung meiner Mitteilung u. d. T. ‘Grundzüge eines neuen Systems der Grundlagen der Mathematik’. Collectanea Logica, 1, 1939, 1–60. Offprint dated 1938. Eng. trans. Leśniewski (1991, pp. 649–710).
Leśniewski, S. (1939). Grundzüge eines neuen Systems der Grundlagen der Mathematik, Sect. 12. Collectanea Logica, 1, 61–144. Eng. trans. Leśniewski (1991, pp. 492–605).
Leśniewski, S. (1991). In S. J. Surma, J. T. Szrednicki, D. I. Barnett, & V. F. Rickey (Eds.), Collected works. Dordrecht: Kluwer.
Łukasiewicz, J., & Tarski, A. (1930). Untersuchungen über den Aussagenkalkül. Sprawozdania z posiedzeń Towarzystwa Naukowego Warszawskiego - Comptes rendus des séances de la societé des sciences et des lettres de Varsovie XXIII, Cl. III, 30–50.
Luschei E.C. (1962) The logical systems of Leśniewski. North-Holland, Amsterdam
Scholz, H. (1930/1975). The ancient axiomatic theory. In J. Barnes, M. Schofield, & R. Sorabji (Eds.) (1975), Articles on Aristotle. I. Science (pp. 50–64). London: Duckworth. Trans. of: Die Axiomatik der Alten. (1930). Blätter für Deutsche Philosophie, IV, 259–278.
Sobociński B. (1955/1956) On well-constructed axiom systems. VI Rocznik Polskiego Towarzystwa Naukowego na Obczyźnie 6: 54–65
Srzednicki, J.T.J., Stachniak, Z. (eds) (1988) S. Leśniewski’s lecture notes in logic. Kluwer, Dordrecht
Sullivan T.H. (1971) Affine geometry having a solid as primitive. Notre-Dame Journal of Formal Logic 12: 1–61
Sullivan T.H. (1972) The name solid as primitive in projective geometry. Notre-Dame Journal of Formal Logic 13: 95–97
Tarski, A. (1929). Les fondements de la géométrie des corps (résumé). Ksiȩga Pamia̧tkowa Pierwszego Polskiego Zjazdu Matematycznego, Lwów, 7-10.IX.1927 – Supplement to Annales de la Société Polonaise de Mathématique, Krakow (1929, pp. 29–33). Eng. trans. Tarski, A. (1956). Logic, semantics, metamathematics. Papers from 1923 to 1938 (pp. 24–29). Oxford: Clarendon Press.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Betti, A. Leśniewski’s characteristica universalis . Synthese 174, 295–314 (2010). https://doi.org/10.1007/s11229-008-9423-6
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DOI: https://doi.org/10.1007/s11229-008-9423-6