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# Leśniewski, Stanisław

(b. Serpukhov, Russia, 18 March 1886; d. Warsaw, Poland, 13 May 1939),

philosophy of mathematics.

After studying philosophy at various German universities, Leśniewski received the Ph.D. under Kazimierz Twardowski at Lvov in 1912. From 1919 until his death he held the chair of philosophy of mathematics at the University of Warsaw and, with Jan Łukasiewicz, inspired and directed research at the Warsaw school of logic. As a student Leśniewski studied the works of John Stuart Mill and Edmund Husserl, but through the influence of Łukasiewicz he soon turned to mathematical logic and began to study the Principia and the writings of Gottlob Frege and Ernst Schröder. A thorough and painstaking analysis of Russell’s antinomy of the class of all those classes which are not members of themselves led Leśniewski to the construction of a system of logic and of the foundations of mathematics remarkable for its originality, elegance, and comprehensiveness. It consists of three theories, which he called protothetic, ontology, and mereology.

The standard system of protothetic, which is the most comprehensive logic of propositions, is based on a single axiom: and the functor of equivalence, “if and only if,” occurs in it as the only undefined term. The directives of protothetic include (1) three rules of inference: substitution, detachment, and the distribution of the universal quantifier; (2) the rule of protothetical definition; and (3) the rule of protothetical extensionality. Protothetic presupposes no more fundamental theory, whereas all other deductive theories which are not parts of protothetic must be based on it or on a part of it

Ontology is obtained by subjoining ontological axioms to protothetic, adapting the directives of protothetic to them, and allowing for a rule of ontological definition and a rule of ontological extensionality. The standard system of ontology is based on a singular ontological axiom, in which the functor of singular inclusion, the copula “is,” occurs as the only undefined term. Ontology comprises traditional logic and counterparts of the calculus of predicates, the calculus of classes, and the calculus of relations, including the theory of identity.

By subjoining mereological axioms to ontology and adapting the ontological directives to them, we obtain mereology, which is a theory of part-whole relations. The standard system of mereology, with the functor “proper or improper part of” as the only undefined term, can then be based on a single mereological axiom. No specifically mereological directives are involved. While ontology yields the foundations of arithmetic, mereology is the cornerstone of the foundations of geometry.

Leśniewski formulated the directives of his systems with unprecedented precision. In the art of formalizing deductive theories he has remained unsurpassed. Yet the theories that he developed never ceased for him to be interpreted theories, intended to embody a very general, and hence philosophically interesting, description of reality.

## BIBLIOGRAPHY

I. Original Works. Leśniewski’s writings include “O podstawach matematyki’ (“On the Foundation of Mathematics”), in Przeglad filozoficzny,30 (1927), 164-206; 31 (1928), 261; 32 (1929), 60-101; 33 (1930), 77-105, 142-170, with a discussion of the Russellian antinomy and an exposition of mereology; “Grundzüge eines neuen Systems der Grundlagen der Mathematik,” in Fundamenta mathematicae,14 (1929), 1-81, which gives an account of the origin and development of protothetic and contains the statement of protothetical directives; “Über die Grundlagen der Ontologie,” in Comptes rendus des seances de la Société des sciences et des letters de Varsovie, C1. III, 23 (1930), 11-132, containing the statement of the directives of ontology; “Über Definitionen in der sogenannten Theorie der Deduktion,” ibid.,24 (1931), 289-309, with the statement of the rules of inference and the rule of definition for a system of the classical calculus of propositions; and Einleitende Bemerkungen zur Fortsetzung meiner Mitteilung u.d. T. “Grundzüge eines neuen Systems der Grundlagen der Mathematik” (Warsaw, 1938), which includes a discussion of certain problems concerning protothetic. The last two works are available in an English trans. in Storrs McCall, ed., Polish Logic 1920-1939 (Oxford, 1967), pp. 116-169, 170-187. Grundzuge eines neuen Systems der Grundlagen der Mathematik § 12 (Warsaw, 1938) offers the deduction of an axiom system of the classical calculus of propositions from a single axiom of protothetic.

II. Secondary Literature. See T. Kotarbiński, La logique en Pologne(Rome, 1959); C. Lejewski, “A Contribution to Leśniewski’s Mereology,” in Polskie towarzystwo naukowe na obszyźnie. Rocznik,5 (1955), 43-50; “A New Axiom of Mereology,” ibid.,6 (1956),65-70; “On Leśniewski’s Ontology,” in Ratio,1 (1958), 150-176; “A Note on a Problem Concerning the Axiomatic Foundations of Mereology,” in Notre Dame Journal of Formal Logic,4 (1963), 135-139; “A Single Axiom for the Mereological Notion of Proper Part,” ibid.,8 (1967), 279-285; and “Consistency of Leśniewski’s Mereology,” in Journal of Symbolic Logic,34 (1969), 321-328; E. C. Luschei, The Logical Systems of Leśniewski (Amsterdam, 1962), a comprehensive and reliable presentation of the foundations of the systems constructed by Leśniewski; J. Slupecki, “St. Leśniewski’s Protothetics,” in Studia logica, 1 (1953), 44-112; “S. Leśniewski’s Calculus of Names,” ibid.,3 (1955), 7-76, which concerns ontology; “Towards a Generalised Mereology of Leśniewski,” ibid.,8 (1958), 131-163; B. Sobociński, “O kolejnych uproszczeniach aksjomatyki ‘ontologji’ Prof. St. Leśniewskiego” (“On Successive Simplifications of the Axiom System of Prof. S. Leśniewski’s ‘Ontology’”), in Fragmenty filozoficzne (Warsaw, 1934), pp. 144-160, available in English in Storrs McCall, ed., Polish Logic1920-1939 (Oxford, 1967), pp. 188-200; “L’analyse de l’antinomie Russellienne par Leśniewski,” in Methodos,1 (1949), 94-107, 220-228, 308-316; 2(1950), 237-257; “Studies in Leśiewski’s Mereology,” in Polskie towarzystwo naukowe na obcznie. Rocznik, 5 (1955), 34-43; “On Well-Constructed Axiom Systems,” ibid.,6 (1956), 54-65; “La génesis de la escuela polaca de lógica,” in Oriente Europeo,7 (1957), 83-95; and “On the Single Axioms of Protothetic,” in Notre Dame Journal of Formal Logic,1 (1960), 52-73; 2 (1961), 111-126, 129-148, in progress; and A. Tarski, “O wyrazie pierwotnym logistyki,” in Przeglad filozoficzny,26 (1923), 68-89, available in English in A. Tarski, Logic, Semantics, Metamathematics (Oxford, 1956), pp. 1-23, important for the study of protothetic.

CzesŁaw Lejewski