LEŚNIEWSKI, STANISŁAW
(1886–1939)

Leśniewski, Stanisław (1886–1939) was one of the founders of the Warsaw School of logic, which flourished from 1919 to 1939. He was the author of a highly original system for the foundations of mathematics, and one of the most innovative and unorthodox logicians of the twentieth century.

Life and Influence

Leśniewski was born in Serpukhov, Russia, and received his schooling in Irkutsk. After studying at German universities, including Leipzig and Munich, he moved in 1910 to Lwów where he studied philosophy with Kazimierz Twardowski and obtained his doctorate in 1912. Leśniewski published several papers before the First World War, which he spent in Moscow. His preoccupation with the logical antinomies, which began in 1911 when he read Jan Łukasiewicz's book On the Principle of Contradiction in Aristotle, shifted his interests permanently from philosophy of language to the logical foundations of mathematics. In 1919 he was appointed professor of the Philosophy of Mathematics at the University of Warsaw. From then until his early death from cancer he was at the center of developments in mathematical logic in Poland, first developing his systems, then from 1927 publishing his results. Leśniewski's notes, correspondence, and a monograph on the antinomies were destroyed in the 1944 Warsaw Uprising: After the war several of his surviving students worked to reconstruct the lost results.

Leśniewski's sole doctoral student Alfred Tarski—Leśniewski boasted proudly of having one hundred percent geniuses as doctoral students—inherited many of his teacher's attitudes, but Tarski's increasing willingness to embrace platonistic set theory for the sake of metamathematical results caused tensions between them. Other pupils such as Jerzy Słupecki, Bolesław Sobociński, Czesław Lejewski, and Henry Hiż remained closer to Leśniewski's views, but their influence was limited. Quine's concern with ontological commitment and the meaning of the quantifiers probably went back to discussions he had with Leśniewski in 1933 on the interpretation of higher-order quantification. Because of the inconvenience of his systems, his forbidding perfectionism, and the idiosyncrasy of his positions, Leśniewski's work remained outside the mainstream, but some aspects became widely influential outside Poland. These include: the object language/metalanguage distinction, exact canons of definition, the theory of semantic categories, and mereology.

Formal Systems

After learning about Russell's Paradox, Leśniewski set himself to produce an antinomy-free foundation for mathematics. Disconcerted by the inexactitudes of Russell's and Whitehead's Principia mathematica, he initially forswore logical symbolism and formulated his views in highly regimented Polish, but in 1920 Leon Chwistek persuaded him to formalize his work, which he did with unprecedented precision. The logical order of Leśniewski's three systems is the reverse of the chronological order of their discovery. Leśniewski diagnosed an ambiguity in the notion of class which he made responsible for Russell's Antinomy, and in 1916 developed the theory of concrete classes, later renamed mereology. Then he set about formalizing the underlying logic of names, predicates and higher-order functors, which he called ontology, axiomatizing it in 1920. Finally he formalized the theory of sentences, connectives, and quantification which underlay the other theories, calling the resulting system protothetic. The axiomatization of protothetic was assisted by Tarski's 1923 discovery that conjunction could be defined in terms of material equivalence and universal quantification. Leśniewski and others improved the results through the 1920s, and he published accounts of protothetic in a series of German papers, and mereology in a Polish series.

Mereology

Mereology (from Greek meros, part), a formal theory of the part-whole relation and cognate concepts, is Leśniewski's nominalistically acceptable partial substitute for set theory. It understands classes as concrete wholes literally composed of their members. Classes (now usually called mereological sums or fusions) are identical when they have the same parts, so the same sum may be determined by different pluralities of members—for example a chess board is the sum of its squares, but also of its ranks or its files. Sums exist if their members do, and a sum need not be spatiotemporally connected. There is no null or empty sum. Mereology can be axiomatized in many ways using many different undefined constants, its axiom(s) being added to ontology. The following perspicuous four-axiom system uses the primitive notion pt( ), meaning part of, and understood to include the case of identity or improper part; the variables in this system are all nominal (intended singular variables being capitalized), and the ontological notion presupposed is the copula "ε" of singular inclusion:
M1 ∀AB [ A ε pt(B ) → B ε B ]
M2 ∀ABC [ ((A ε pt(B ) & B ε pt(C )) → A ε pt(C )) ]
M3 ∀Ab [ A ε Sm(b ) ↔ (A ε A & ∀C [ C ε b → C ε pt(A ) ]
&
∀D [ D ε pt(A ) → ∃EF [ E ε b & F ε pt(E ) & F ε pt(D )] ] ) ]
M4 ∀Ab [ A ε b → Sm(b ) ε Sm(b ) ]

These axioms say: (M1) that whatever has a part is an individual; (M2) that parthood is transitive; (M3) define the sum of all the b s as that unique individual A which overlaps all and only b s; and state (M4) that if there is at least one b then the sum of all b s exists and is unique. Mereology is consistent relative to protothetic. It is independent of this system whether or not there are atoms—that is, objects without proper parts.

Mereology was the first system rigorously formulated by Leśniewski and remains the most thoroughly investigated. Its principles are much weaker than those of set theory, although some of its assumptions, especially the general sum principle M4, have been questioned on philosophical grounds. Especially when based on standard predicate logic rather than Leśniewski's ontology, mereology has come into standard use in ontology and cognitive science.

Ontology

Mereology presupposes ontology, so called because Leśniewski took it to formulate several meanings of be. He intended it as a modernized term logic of the sort formulated by Ernst Schröder, and in its admittance of empty and plural terms it is closer to traditional logic than to Frege-Russell predicate logic, whose terms are all singular. Like mereology, ontology can be based on many different primitives, but the most frequently used is the one chosen by Leśniewski, namely the singular inclusion functor "ε." The basic sentence-form "A ε b," readable as "A is a b " but best read perhaps as "A is one of the b s" captures the distributive rather than collective sense of "class": "A is a member of the class of the b s" just means "A is one of the b s" and no individual called "the class of the b s" is assumed.

Leśniewski's original (1920) axiom, though not the shortest, remains the most perspicuous:
O1∀Ab [ A ε b ↔ ( ∃C [ C ε A ] & ∀DE [ (D ε A & E ε A ) → D ε E ] &
∀F [ F ε A → F ε b ] ) ]

This says that A is a b if and only if (1) there is at least one A, and (2) there is at most one A, and (3) every A is a b. This axiomatic equivalence, which constitutes a sort of implicit self-definition of "ε" mirrors the analysis of singular definite descriptions by Russell, as can be seen by reading "A ε b " as "the A is a b." Existential import in ontology is located in the functor "ε." rather than the quantifiers: "A ε b " is only true if an A exists.

The axiom is not ontology's only source of logical power. Leśniewski allows new constants to be defined, and as these are introduced, new semantic categories of expression and thereby new categories of bindable variable become available. Each category of expression is subject to a principle of extensionality, and so the system grows in logical strength, ascending as required to higher types of variable. Thus although the axiom binds only nominal variables, later theses allow variables for predicates and other higher-order functors to be bound. Because Leśniewski allows plural names, his first-order calculus is equivalent in logical strength to standard monadic second-order predicate calculus. There is no axiom of choice in ontology, but a directive can be formulated allowing choice principles to be stated for each higher logical type (semantic category). Like mereology, ontology is consistent relative to protothetic. Despite its expressive power, ontology is ontologically neutral in that no thesis stating the existence of an individual can be derived. It is thus true of the empty universe as well as others.

Ontology is in many ways Leśniewski's most innovative system, combining features of traditional, Schröderian, and Fregean logic with a potential expressive power equivalent to that of the simple theory of types. Nevertheless, apart from some exploitation for purposes of historical comparison, and some development by Lejewski and others, it has found few supporters.

Protothetic

The basis of Leśniewski's logic is protothetic, a bivalent propositional calculus to which may be added propositional functors of any order, and incorporating the theory of quantification. It is equivalent in potential to a system of propositional types. Leśniewski, following Peirce, took quantification as embodying cardinally unconstrained conjunction and disjunction, and as part of the basis of logic rather than attaching primarily to nominal variables. The quantifiers ∀ and ∃ bind variables of any category. Leśniewski experimented from 1921 onwards with different axiomatic and combinatorial bases for protothetic. He chose material equivalence as sole undefined connective because he formulated definitions as object-language equivalences, and he developed the calculus of equivalent statements. But an intuitive axiomatization of prothetic using implication is:
P1     ∀pq [ p → (q → p ) ]
P2     ∀pqrf [ f (r p ) → (f (r ∀s [s ]) → f (r q )) ]

Quantifier apart, the first thesis is familiar from propositional calculus. The second exploits a variable f for functors taking two propositional arguments, with "∀s [s ]" a standard false sentence. Like ontology, protothetic derives much of its strength from the rules permitting the formulation of new definitions and extensionality principles for higher types. Each propositional type is finite in its extensions, starting from the basic types of sentences, which has just two extensions, the True and the False, so in principle the quantifiers can be replaced by computational principles running through the extensions for each type considered in a sentence. Leśniewski took great pains over protothetic but it remains the least discussed of his logical systems.

Though his published works covered mainly his own systems, with incidental but incisive criticism of such contemporaries as Russell and Whitehead, von Neumann and Zermelo, in his Warsaw lectures Leśniewski ranged more widely, finding single axioms for general and abelian group theory, developing Peano's axioms, investigating inductive definitions, comparing mereology with Whitehead's theory of events, and criticizing Łukasiewicz's many-valued logic.

Philosophical Metalogic

Though an unprecedentedly exact formalizer, Leśniewski deplored all formalism. Having come to logic through regimented ordinary language, he understood his logical systems throughout as interpreted with a determinate intended meaning, and intended his theses as general truths. From his first paper Leśniewski scrupulously distinguished use from mention of expressions, and literally failed to understand writings where this distinction was not observed, notably Principia. By contrast he admired and extolled the great rigor of Frege's formal systems, notwithstanding their inconsistency. Leśniewski's strictures on quotation were inherited and made influential by Tarski.

Leśniewski criticized Twardowski's platonism and strove to make his logical systems compatible with nominalism. This meant treating systems not as abstract entities but as concrete collections of physical inscriptions, growing in time by the addition of new inscriptions called theses. Because the systems as they develop allow new expressions to be introduced via definitions, and new types of variable to become available for quantification, the regulation of their growth had to be precise but schematic. Leśniewski achieved this by formulating for each system regulatory directives allowing new theses to be introduced. These directives are self-adjusting in that what they allow expands as the system grows. Leśniewski considered faulty definition to be responsible for the logical antinomies, and by bringing definitions within the system as object-language equivalences—rather than metalinguistic abbreviations—kept them under tight control. The highly complex directives for adding definitions in protothetic and ontology are Leśniewski's proudest achievement. To formulate them and the other directives governing substitution, quantifier distribution, modus ponens, and extensionality required a sequence of more than fifty metalogical definitions called terminological explanations. In his everyday logical working however, Leśniewski used an unofficial system of natural deduction from assumptions, understood as delivering an outline which could, if necessary be transformed into a proof according to the directives. This he never formalized. The complexities of formulating general terminological explanations and directives for variable-binding operators were beyond even Leśniewski, and he had to rest content in his official system with a sole syncategorematic operator, the universal quantifier.

The formulation of the directives employed Leśniewski's notion of semantic categories, a systematic logical grammar inspired by Frege's practice and Husserl's theory of Bedeutungskategorien, and intended as an ontologically parsimonious alternative to type theory. Though not codified by Leśniewski, the subsequent systematization by Ajdukiewicz and others has made this part of mainstream logic and linguistics. The basic category of protothetic was the sentence, to which ontology added the basic category of name. Mereology required no new categories or directives.

Leśniewski had definite ideas about the intellectual economy of logic. A system ought to have as few primitive notions, axioms, and directives as possible; the axioms ought to be as short as possible, logically independent, and organic—that is, not contain provable theses as subformulas. The search for ever shorter axioms was a general feature of the Warsaw School, which Leśniewski and his followers sometimes pursued at the expense of defending controversial aspects of the systems, such as their interpretation of quantification, their radical nominalism, and their thoroughgoing extensionalism.

Leśniewski's avowed metaphysical neutrality combined with his liberal use of quantifiers to bind non-nominal variables drew criticism from Quine. Leśniewski rejected Quine's accusation of platonism, and on reflection Quine came to regard Leśniewski's quantifiers as substitutional, committing not to corresponding entities, but to expressions to be substituted for variables bound by a quantifier. That Leśniewski cannot understand the quantifiers objectually is clear because a standard empty name "∧" can be substituted normally for bound variables: From the true "no ∧ exists" one may validly infer "for some a, no a exists," so "for some" (∃) cannot mean "there exists." In the light of its subsequent development, the substitutional interpretation fits Leśniewski no better than the objectual, because it would commit him to an infinity of platonic expression types. Comparison with standard accounts is complicated by Leśniewski's incriptional understanding of expressions and the import of his directives, which are conditional prescriptions rather than categorical descriptions. The directive "If A is the last thesis belonging to the system then a thesis B may be added if for some thesis C preceding A, B is a result of substitution from C into A " quantifies only over extant tokens. The question remains how expressions employed in a logical system (including the quantifiers) have their meanings. On this Leśniewski remains silent. How to theorize metalogically about meaning and truth within Leśniewski's strictures remains perhaps the biggest open question concerning his systems.

See also Logical Paradoxes; Łukasiewicz, Jan; Mereology; Syntactical and Semantic Categories; Tarski, Alfred; Twardowski, Kazimierz.

Bibliography

works by leśniewski

The only collected edition of Leśniewski's works is their English translation:

Leśniewski, Stanisław. Collected Works. 2 vols., edited by Stanisław J. Surma, Jan T. J. Srzednicki, and Dene I. Barnett, with an annotated bibliography by V. Frederick Rickey. Dordrecht, Netherlands: Kluwer, 1992. Rickey's bibliography lists the original Polish locations of these writings.

Srzednicki, Jan T. J., and Stachniak, Zbigniew, eds. S. Leśniewski's Lecture Notes in Logic. Dordrecht, Netherlands: Kluwer, 1988. Contains edited transcriptions of student notes from lecture courses Leśniewski gave in Warsaw.

works about leŚniewski

Luschei, Eugene C. The Logical Systems of Leśniewski. Amsterdam, Netherlands: North-Holland, 1962. The most comprehensive monograph on Leśniewski.

Miéville, Dénis. Un développement des systèmes logiques de Stanislaw Lesniewski: Protothétique–Ontologie–Méréologie. Berne, Switzerland: Lang, 1984. The nearest thing to an elementary introduction to Leśniewki.

Miéville, Dénis and Dénis Vernant, eds. Stanislaw Lesniewski Aujourd'hui. Grenoble, France: Université Pierre Mendès France, 1995. A collection of essays including an informative survey by Czesław Lejewski, "Remembering Stanislaw Lesniewski," pp. 25–66.

Srzednicki, Jan T. J., V. Frederick Rickey, and J. Czelakowski, eds. Leśniewski's Systems: Ontology and Mereology. Dordrecht, Netherlands: Kluwer 1984. Seminal commentaries and improvements by Lejewski, Słupecki, Robert E. Clay and others.

Srzednicki, Jan T. J., and Zbigniew Stachniak, eds. Leśniewski's Systems: Protothetic. Dordrecht, Netherlands: Kluwer, 1998. A similar compilation of important commentaries and improvements in protothetic.

Woleński, Jan. Logic and Philosophy in the Lvov-Warsaw School. Dordrecht, Netherlands: Kluwer, 1989. Definitive history, with a copious bibliography, situating Leśniewski in the context of the logical movement in Poland.

Peter Simons (2005)