Bibliography on the Logical Work of Stanisław Leśniewski (Sin-Z)

Sinisi, Vito. 1961. "Nominalism and common names." The Philosophical Review no. 71:230-235.

"Edwin Allaire, Gustav Bergmann and Reinhardt Grossmann have objected to the nominalistic analysis of "this is red and that is red" which treats "red" as a common name. Such an analysis, they argued, must assimilate the copula in this sentence to the "is" of identity. sinisi claims that this objection is mistaken. Using a logical system developed by Stanisław Leśniewski, he shows that it is possible to construe "red" as a common name without taking the copula as the "is" of identity."

———. 1965. "Discussion: ``∈ '' and Common Names." Philosophy of Science no. 32:281-286.

"In [6] I tried to show how an objection to "the nominalist's" analysis of (a) "This is red" and (b) "That is red" on the basis of "the doctrine of common names" might be overcome.(1) The objection is that "the nominalist," attempting to analyze (a) and (b) by construing the pronouns in these sentences as two different proper names and "red" as a common name, is forced thereby to construe the copula in both sentences as the "is" of identity, and hence (it is claimed) this and that are identical, i.e., that there is only one red spot and not two. I attempted to show that by using Leśniewski's original axiom of ontology "the nominalist" could construe the pronouns in (a) and (b) as proper names, and "red" as a common name without taking the copula to express identity; he would not be forced to identify this with that.(2) The cogency of my explanation has been recently challenged by Mr. Grossmann [1]. In the first part of this note I shall answer some of his criticisms of my paper, and in the second I shall answer two questions he asks." (p. 281)

(1) The objectors are Edwin B. Allaire, Gustav Bergmann, and Reinhardt Grossmann. references see [6].

(2) Familiarity with [6] is assumed.

References

[1] Grossmann, Reinhardt, "Common Names", ed., Edwin B. Allaire et al., Iowa Publications in Philosophy, vol. 1, Iowa City, The Hague, 1963, 64-7.

[6] Sinisi, Vito F., "Nominalism and Common Names", The Philosophical Review, LXXI (1962), 230-235.

———. 1966. "Leśniewski's analysis of Whitehead's theory of events." Notre Dame Journal of Formal Logic no. 7:323-327.

"Stanisław Leśniewski (1886-1939) was a leading member of the famous Warsaw school of logicians which flourished between the two Wars. The works of Lejewski and Sobocinski have made many readers of this journal familiar with Leśniewski's three systems of logic: protothetίc, ontology, and mereology. What does not seem to be generally known is that in the course of setting forth mereology [1]: Leśniewski proved that A. N. Whitehead's axiomatic basis for the concept of event is an inadequate foundation for Whitehead's theory of events.

The purpose of this note is to recapitulate Leśniewski's analysis (available only in Polish) of Whitehead's theory of events. Perhaps a knowledge of this analysis will be of value not only to those interested in Leśniewski's work but to that growing number of philosophers concerned with Whitehead's metaphysics and philosophy of science." (p. 323)

[1] Leśniewski, Stanisiaw, "O podstawach matematyki," (On the Foundations of Mathematics), section 4, Przegląd Filozoficzny (Philosophical Review), XXXI (1923), 261-291.

———. 1969. "Leśniewski and Frege on collective classes." Notre Dame Journal of Formal Logic no. 10:239-246.

"Between 1927 and 1931 Leśniewski published a series of articles on the foundations of mathematics in the Polish journal Przegląd Filozoficzny.65% of the work is devoted to various axiomatizations of Leśniewski's mereology (a theory of collective classes) while the remainder takes up various related issues. In the third part of this series Leśniewski informally sets forth his notion of a collective class, criticizes certain descriptions of distributive classes, and argues that there is no justification in Frege's statement that the conception of a class as consisting of individuals, so that the individual thing coincides with the unit class, cannot in any case be supported.

Leśniewski's refutation of Frege's statement appears to be unknown to western logicians and philosophers. None of the recent books on Frege (e.g., Angelelli, Egidi, Sternfeld, Thiel, Walker) mentions it. Luschei, in his The Logical Systems of Leśniewski, mentions it but does not present it.

My purpose here is to state and explain Leśniewski's refutation in the hope that it will help stimulate interest in his work." (p. 239)

———. 1976. "Leśniewski's Analysis of Russell's Antinomy." Notre Dame Journal of Formal Logic no. 17:19-34.

"As mentioned above, Leśniewski's third analysis of Russell's antinomy has been published by Sobocinski in [49-50]. Luschei in [62] summarized Leśniewski's second analysis, which was published in [27]. However, the historically important first analysis of 1914, the analysis which determined the character of Leśniewski's later logical theories, is not accessible to those who do not read Polish.(5) This analysis should be of interest not only to those concerned with Leśniewski's Ontology and Mereology but also to those concerned with the antinomy. My purpose here is to state and explain the main points of Leśniewski's 1914 paper "Czy klasa klas, niepodporzadkowanych sobie, jest podporzadkowana sobie?" in order to reveal some of the factors which determined the form of Mereology, and to

help stimulate interest in his work." (pp. 19-20)

(5) Luschei [62], p. 20, asserts that Leśniewski repudiated his paper "Czy klasa klas . . . , " and in his bibliography of Leśniewski's works, p. 321, he lists this paper under the heading "Early writings, later repudiated." Luschei also lists Leśniewski's [16] under this heading. Unfortunately, I have not found any textual evidence to support the claim that Leśniewski repudiated these two works. In [27], pp. 182-183, Leśniewski listed four articles, two published in 1911, and two published in 1913, which he solemnly repudiated, but his list does not contain either the paper "Czy klasa klas . . . " or his [16]. Leśniewski said that he mentioned these four works " . . . because I wish to indicate that I am very distressed that they were published at all, and I herewith solemnly 'repudiate' these works, which I have already done from a university lectern, and assert the bankruptcy of the 'philosophico'-grammatical enterprises of the first period of my research." Furthermore, Luschei [62], p. 67, summarizes Leśniewski's second analysis of the antinomy, which appeared for the first time in Leśniewski's [27], pp. 182-189, but incorrectly he attributes the analysis to Leśniewski's "Czy klasa klas . . . " of 1914. The analyses of [27] and of "Czy klasa klas . . . " are distinct. "Czy klasa klas . . . " is not, as Luschei says, recapitulated in [27].

[The paper of 1914 "Czy klasa klas . . . , " is translated as "Is the Class of Classes not Subordinated to Themselves, Subordinated to Itself?" in the first volume of Lesniwski, Collected Works, pp. 115-128]

References

[16] Leśniewski, Stanisław, Podstawy ogólnej teoryi mnogości. I (Foundations of general set theory. I), Prace Polskiego Kola Naukowego w Moskwie, Sekcya matematyczno-przyrodnicza, No. 2, Moscow (1916).

[27] Leśniewski, Stanisław, "O podstawach matematyki" (On the foundations of mathematics), Przegląd Filozoficzny, vol. XXX (1927), pp. 164-206.

[49-50] Sobocinski, Bolesiaw, "L'analyse de l'antinomie russellienne par Leśniewski," Methodos, vol. I (1949), pp. 94-107, 220-228, 308-316, and vol. II (1950), pp. 237-257.

[62] Luschei, Eugene C., The Logical Systems of Leέniewski, North-Holland Publishing Company, Amsterdam (1962).

———. 1983. "The Development of Ontology." Topoi no. 2:53-62.

"Leśniewski published only two works devoted exclusively to Ontology: an article, 'Über die Grundlagen der Ontologie', in 1930, and, a year later, Chapter XI, 'O zdaniach "jednostkowych" typu "A e b" ' ['Singular' sentences of the type 'A e b'], in his series of articles on the foundations of mathematics(1) The first is essentially a highly formalized presentation of the terminological explanations for Ontology, breathtaking in its brevity and rigor; it concludes with a laconic summary of some results obtained by Leśniewski, Sobocifnski, and Tarski during 1921-1929. The second differs radically in style and content; written in colloquial Polish, almost a memoir, it narrates the development of Ontology. My purpose here is to give an analytical review of this narrative which includes many informative and illuminating discussions of the conceptual basis of Leśniewski's Ontology (or 'calculus of names', as it is sometimes called), and which may be helpful to those who are interested in Leśniewski's work but who do not read Polish." (p. 53)

[Chapter XI. "On 'singular' propositions of the type Aεb" is translated in Collected Works, pp. 364-382]

———. 1983. "Leśniewski's Foundations of Mathematics." Topoi no. 3:3-52.

"During 1927-1931 Leśniewski published a series of articles (169 pages) entitled 'O podstawach matematyki' [On the Foundations of Mathematics] in the journal Przegląd Filozoficzny [Philosophical Review], and an abridged English translation of this series is presented here. With the exception of this work, all of Leśniewski's publications appearing after the first World War were written in German, and hence accessible to scholars and logicians in the West.

This work, however, since written in Polish, has heretofore not been accessible to most Western readers, and it is hoped that this translation will encourage both the study of Leśniewski's works as well as the further development of his theories.

Leśniewski's foundations of mathematics consists of three theories: Protothetic, which, according to Leśniewski, roughly corresponds "to what is known in the discipline as the 'calculus of equivalent statements', 'Aussagenkalkul', 'theory of deduction' joined with the 'theory of apparent variables' "; Ontology, "which is a kind of modernized 'traditional logic', and with respect to its content and 'strength' most closely approaches Schröder's 'Klassenkatkul' considered as including a theory of 'individuals' "; and Mereology, an axiomatization of the part-whole relation, which Leśniewski initially called "general set theory".

Mereology presupposed Ontology, which in turn is grounded in Protothetic. When Leśniewski began the publication of his 'O podstawach matematyki' he intended to present all three theories, but was able to present only Mereology.

There is an informal discussion of Ontology in Chapter XI of this series; and the terminological explanations, as his rules were called, for Ontology were published in another journal in 1930. Protothetic was first presented in 1929 in Fundamenta Mathematicae, and in 1938 his last work on Protothetic appeared. A year later, he died at the age of 53. During the Warsaw Insurrection of August, 1944 all of his unpublished manuscripts and lecture notes were destroyed." (p. 3, notes omitted)

"The essay "On the Foundations of Mathematics" is translated in Collected Works, vol. 1, pp. 174-382]

Slupecki, Jerzy. 1953. "St Leśniewski's Protothetics." Studia Logica no. 1:44-111.

Reprinted in Jan Srzednicki, Zbigniews Stachniak (eds.), S. Leśniewski's Systems: Protothetic, Dordrecht: Kluwer 1998, pp. 85-152.

"This paper does not bear the character of a historical study on Professor Leśniewski's system as the notes on which it is based were not written by himself. Moreover, the exposition of protothetics given here deviates in many points from that contained in those notes. This seemed to be indicated, in the first place, for didactic reasons, my aim being to make this paper comprehensible also to non-specialists,(2) although it has, of course, been taken for granted that the reader is familiar with the fundamental branches of mathematical logic.

I have taken account of all the results contained in the notes with the exception of a few which I consider either unessential or indeed obsolete.

In order to make the exposition of protothetics both concise and complete I was bound to amplify considerably the material found in the notes. However, all the results given here and not contained in the notes were undoubtedly known to Professor Leśniewski. The paper includes no results of my own but a considerable part of the proofs has been entirely worked out by myself. I mention this so that I alone should be held responsible for any inaccuracies and errors which may have occurred.

This paper is not written with that exactitude which Professor Leśniewski always observed in his lectures and publications. This reservation, however, refers only to the meta-logical considerations whereas in the proofs of theorems of the system I have strictly followed the proofs found in the notes.

The lectures "On Certain Problems of Protothetics", the notes of which form the basis of this paper, were delivered by Professor Leśniewski at the Warsaw University during the academic year 1932-1933. The subject-matter of this paper is discussed in a more detailed way in Section 2." (p. 86 of the reprint)

(2) With the exception of Section 13 (proof of completeness of protothetics).

———. 1955. "S. Leśniewski's Calculus of Names." Studia Logica no. 3:7-70.

Reprinted in Jan T. J. Srzednicki, V, F, Rickey (eds.), Leśniewski's Systems: Ontology and Mereology, The Hague: Martinus Nijhoff 1984, pp. 59-122.

"The only primitive term in Leśniewski's system of the Calculus of Names is the verb 'is' for which the participle 'being' corresponds to the Greek 'ον' (gen. 'ὄντως'). This was by no means the only reason for Leśniewski's use for his system a name indicating one ofthe main branches of philosophy. Thus in Leśniewski's article "On the Foundations of Mathematics"(2) we read:

... I used the term 'outology' for the theory I developed, as this was not opposed to my 'linguistic intuition', just in view of the fact that I formulated in that theory a sort of 'general principles of being'.

In the title of this paper I nevertheless thought it better to use "Calculus of Names", as 'ontology' might cause some misunderstanding, but I shall be using throughout the text 'ontology' for Leśniewski's system." (p. 59 of the reprint)

(...)

"I have subdivided this paper into four sections. Section I discusses the intuitive and formal foundations of ontology. Section II deals with theorems of that part of Leśniewski's system to which I shall refer as elementary ontology(7) and which contains the simplest theorems of the whole system and those nearest to intuition. It is in this section too that I discuss the relation between ontology and traditional logic as well as the algebra of sets. Section III deals with the remaining part of the system, i.e. the non-elementary ontology. However, I shall not give a systematic presentation of non-elementary ontology, as this could only be done in a separate and large treatise. I shall only adduce the theorems and definitions I consider most characteristic of non-elementary ontology. Further, I shall present all those theorems and definitions contained in the notes from Leśniewski's lectures which, though exceeding the limits of elementary ontology, are nevertheless closely related to its theorems and notions. Section IV contains brief methodological considerations on the system."

(2) 2 Leśniewski [1927-1931, Ch. XI, p. 163]. (Editorial Note: English translation Leśniewski [1983].)

(7) The definition of elementary ontology is given in II, § 1.

References

Leśniewski, Stanisław. [1927-1931] O podstawach matematyki (On the Foundations of Mathematics) Przegląd Filozoficzny, Vol. XXX (1927), 164-206; Vol. XXXI (1928), 261-291;

Yol. XXXII (1929), 60-101; Yol. XXXIII (1930), 77-105; Vol. XXXIV (1931), 142-170. (Polish). (English translation - Leśniewski [1983].)

_________ [1983] On the Foundations of Mathematics, Topoi, Vol. II, No.1, 7-52. (This is the abridged English translation by Vito F. Sinisi of Leśniewski [1927-1931].)

———. 1958. "Towards a generalized mereology of Leśniewski." Studia Logica no. 8:131-154.

"The antinomies of the set theory have made it imperative for logicians and mathematicians to investigate its basic assumptions. As a result, consistent systtems were formulated, but at the same time the intuitive interpretations of the "naive" set theory were lost. In all those systems, except the mereology of S. Leśniewski, the set is interpreted so that even sets of perceivable objects are not perceivable objects; thus, e. g., libraries are not sets of books, and constellations are not sets of stars. Mereology, however, is essentially "poorer" than other systems of the set theory. It is, for instance, impossible to build in mereology the arithmetic of natural numbers. In this paper Leśniewski's system is enriched so as to be suitable for laying the foundations of mathematics in the same degree as is the case of other systems of the set theory. This is achieved by including in mereology certain new definitions and by using much stronger logical means than it was done by Leśniewski. The extended system of mereology, however, retains the basic intuitive assumptions of the original system." (p. 131)

Smirnov, Vladimir Aleksandrovich. 1983. "Embedding the Elementary Ontology of Stanisław Leśniewski into the Monadic Second-Order Calculus of Predicates." Studia Logica no. 42:197-207.

"The elementary ontology of Leśniewski and the standard calculus of predicates are based on different categorial systems. Categories of name and sentence are fundamental syntactical categories of Le?niewski's ontology, categories of proper name and sentence are fundamental categories of the calculus of predicates. Is it possible to compare logical systems built on different systems of categories? We give an affirmative answer to this question in the case of the elementary ontology and the standard calculus of predicates." (p. 197)

———. 1983. "A Correction to 'Embedding the Elementary Ontology of Stanisław Leśniewski into the Monadic Second-Order Calculus of Predicates'." Studia Logica no. 45:231.

Sobocinski, Boleslaw. 1949. "An Investigations of Protothetic." Cahiers de l’Institut d’Etudes Polonaises en Belgique no. 5:1-39.

Reprinted in Storrs McCall (ed.), Polish Logic 1920-1939, Oxford: Clarendon Press 1967, pp. 201-206.

"This paper was intended to appear, under the title 'Z badan natl prototetyką', in vol. 1 of the periodical Collectanea Logica (Warsaw, 1939), pp. 171-6. (...) An English translation of the paper, made by Dr. Sobocinski, appeared as no. 5 of the Cahiers de l'lnstitut d'Etudes polonaises en Belgique (Brussels, 1949). This version is translated anew from the Polish by Z. Jordan." (p. 201 of the reprint).

Reprinted with a new Introduction (pp. 69-75) in Jan Srzednicki, Zbigniews Stachniak (eds.), S. Leśniewski's Systems: Protothetic, Dordrecht: Kluwer 1998, pp. 75-83.

"Protothetic is a deductive theory constructed by the late Stanisław Leśniewski. As we know, he based the whole system of contemporary mathematics on three deductive theories, protothetic, ontology, and mereology, which he conceived and constructed. [ ... ]

I will neither describe the characteristic features of these theories nor comment on the theoretical basis of St. Leśniewski's system. I will only state that:

1. The precision and the conciseness of the formalization, symbolism, and the formulation of the rules of procedure for the above theories are unparalleled among the known deductive systems.

2. The principles on which these theories are based differ in many respects from those that were usually accepted before the discovery of the Russell Antinomy. They allow, nevertheless, to formulate and prove all the theorems of classical logic.

3. There are straightforward proofs that these theories and the entire system are consistent, and, consequently, that no known logical antinomy can be reconstructed in them." (p. 72)

(...)

"While investigating various problems of protothetic, I observed that a number of theorems which, as far as I could ascertain, remained unknown at that time (December 1935), were theses of protothetic. 26 As this finding is closely associated with the theorem of Dr. Alfred Tarski concerning the definability, in protothetic, of conjunction in terms of equivalence, I have decided to publish the theses discovered by myself." (p. 75, notes omitted)

———. 1954-1955. "Studies in Leśniewski’s mereology." Rocznik Polskiego Towarzystwa Naukowego na Obczyźnie no. 5:34-43.

Reprinted in Jan T. J. Srzednicki, V, F, Rickey (eds.), Leśniewski's Systems: Ontology and Mereology, The Hague: Martinus Nijhoff 1984, pp. 217-227.

———. 1960. "On the single axioms of protothetic." Notre Dame Journal of Formal Logic no. 1:52-73.

Reprinted in Jan Srzednicki, Zbigniews Stachniak (eds.), S. Leśniewski's Systems: Protothetic, Dordrecht: Kluwer 1998, pp. 153-171.

"In this paper I should like to present the results of my unpublished investigations concerning axiom-systems of protothetic. Strictly speaking, only the system of protothetic called S [Gothic] 5 will be considered here. It seems to me that this investigation may interest students of propositional calculus and the related subjects, since the deductions which will be used, sometimes unexpected and rather difficult, not only explain to some degree the structure of protothetic, but can also throw light upon several problems concerning various systems of propositional calculus. Because, generally, protothetic is still a little known theory, at the beginning I have to give several, possibly short, explanations concerning it. Without them the subject of this paper and the proofs presented below would hardly be understandable for the reader.

Thus, in the first section a short description of protothetic and the necessary information about the rules of procedure of the system S [Gothic] 5 will be given.

There will also be added some history of the researches concerning the single axioms of protothetic and related problems. Especially, I shall discuss briefly the metatheorems L (of Leśniewski) and the stronger S (mine). In the second section I shall present a combined proof: (1) that my axiom A_n can serve as a single (and probably the shortest) axiom of the system S [Gothic] 5 of protothetic, and (2) that the above mentioned metatheorem S is sufficient to check the completeness of any axiom system of protothetic. In the third and the last section it will be shown, in the shortest possible way, how the classical propositional calculus and the quantification theory for protothetical formulas can be obtained in the field of the system S [Gothic] 5 **.

Instead of the authentic symbolism of Lesniewski2 introduced by him mostly in order to formulate the rules of procedure in the most precise way, I shall use here a more convenient Peano-Russelian symbolism modified in a manner which will satisfy the requirements of protothetic. Anyone who is familiar with [elementary] logic will understand these modifications without difficulty." (p. 153 of the reprint, a note omitted)

** [Ed. Note: The third section was never published by Sobocinski; its intended contents is discussed by Rickey in paper II of this volume.]

———. 1961. "On the single axioms of protothetic. II." Notre Dame Journal of Formal Logic no. 2:111-126.

Reprinted in Jan Srzednicki, Zbigniews Stachniak (eds.), S. Leśniewski's Systems: Protothetic, Dordrecht: Kluwer 1998, pp. 171-188.

———. 1961. "On the single axioms of protothetic. III." Notre Dame Journal of Formal Logic no. 2:129-148.

Reprinted in Jan Srzednicki, Zbigniews Stachniak (eds.), S. Leśniewski's Systems: Protothetic, Dordrecht: Kluwer 1998, pp. 188-216.

———. 1967. "Successive simplifications of the axiom-system of Leśniewski's Ontology." In Polish Logic 1920-1939, edited by McCall, Storrs, 188-200. Oxford: Clarendon Press.

"The aim of this paper is to provide a detailed account of the successive steps by which the axiom of Ontology was simplified.(1)

I originally intended to present only the results that I obtained myself. However, in my research I took advantage of various contributions, published and unpublished, made by others, namely by Professor Stanisław Leśniewski and Dr. Alfred Tarski.

If no detailed account of their results were given, the reader could have difficulties in following my own proofs. Moreover, this omission could obscure the development of all these investigations, and, contrary to my wishes, belittle the share of some contributors or their contributions. To avoid such misunderstandings and to present a complete picture of the results obtained in this field of research, I decided to give an account of all of them, for which the persons concerned granted me their kind permission." (p. 188)

(1) For the sake of conciseness, in this paper the term 'Ontology' is always used instead of the expression 'Ontology of Stanisław Leśniewski'.

[This paper appeared originally under the title 'O kolejnych uproszczeniach aksjomatyki "ontologji" prof. St. Lesniewskiego' in Fragmenty Filozoficzne, a volume in commemoration of fifteen years' teaching in the University of Warsaw by Professor T. Kotarbiński, Warsaw, 1934, pp. 143-60. Translated by Z. Jordan.]

———. 1984. "Leśniewski's Analysis of Russell's Paradox (1949) (*)." In S. Leśniewski's Systems: Ontology and Mereology, edited by Srzednicki, Jan, Rickey, Frederick V. and Czelakowski, Janusz, 11-44. The Hague: Martinus Nijhoff.

Published in French in four parts: Methodos, I - II - III: vol. 1. (1949) pp. 94-107; 220-228; 308-316; IV: vol. 2 (1950) pp. 237-257.

"The purpose of this article is to give, with a minimum of symbolism, a simple, accessible, unpolemical exposition of Leśniewski's analysis of Russell's Paradox.(1)

This analysis merits attention for several reasons. Mainly, because it was the point of departure for the construction of Leśniewski's system for the foundation of mathematics. His method of overcoming the paradox in question is very different from those employed by others. From the start it forced Leśniewski to take a path on which he had to overcome great difficulties related to the problem of the paradoxes; it determined the character of the theories which constitute his system. This system, which differs in many ways from contemporary systems, is non-contradictory (which is easy to prove), and is an adequate base for the construction of contemporary mathematics. However, it is not very easy to get the feel of the system, nor is it easy to penetrate the psychology from which it arose-what precisely were Leśniewski's thoughts about Russell's Paradox.(2)" (p. 11)

* Editorial Note: Translated from the French by Robert E. Clay.

(1) Stanisław Leśniewski (born March 18, 1886, died May 13, 1939) was professor of philosophy of mathematics at the University of Warsaw from 1919 until his death.

(2) Leśniewski presented the essentials of his views on Russell's paradox - Leśniewski [1927-1931, Chs. II-III] (Editorial Note: English translation - Leśniewski [1983].)

However, the formal reasonings given here have never been published.

References

Leśniewski, Stanisław. [1927-1931] O podstawach matematyki (On the Foundations of Mathematics) Przegląd Filozoficzny, Vol. XXX (1927), 164-206; Vol. XXXI (1928), 261-291;

Yol. XXXII (1929), 60-101; Yol. XXXIII (1930), 77-105; Vol. XXXIV (1931), 142-170. (Polish). (English translation - Leśniewski [1983].)

_________ [1983] On the Foundations of Mathematics, Topoi, Vol. II, No.1, 7-52. (This is the abridged English translation by Vito F. Sinisi of Leśniewski [1927-1931].)

Srzednicki, Jan, Rickey, Frederick V., and Czelakowski, Janusz, eds. 1984. S. Leśniewski's Systems: Ontology and Mereology. The Hague: Martinus Nijhoff.

Contents: Editorial Note 7; 1. Z. Kruszewski: Ontology without Axioms (1925) 9; 2. B. Sobocinski: Leśniewski's Analysis of Russell's Paradox (1949) 11; 3. C. Lejewski: Logic and Existence (1954) 45; 4. J. Slupecki: S. Leśniewski's Calculus of Names (1955) 59; 5. C. Lejewski: On Leśniewski's Ontology (1958) 123; 6. J. Canty: Ontology: Leśniewski's Logical Language (1969) 149; 7. B. Iwanus: On Leśniewski's Elementary Ontology (1973) 165; 8. B. Sobocinski: Studies in Leśniewski's Mereology (1954) 217; 9. E. Clay: On the Definition of Mereological Class (1966) 229; 10. C. Lejewski: Consistency of Leśniewski's Mereology (1969) 231; 11. E. Clay: The Dependence of a Mereological Axiom (1970) 239; 12. E. Clay: Relation of Leśniewski's Mereology to Boolean Algebra (1974) 241; Bibliography 253; Index of Names 261-262.

Srzednicki, Jan, and Stachniak, Zbigniew, eds. 1998. S. Leśniewski's Systems: Protothetic. Dordrecht: Kluwer.

"This edition of papers concerning Leśniewski's logical system Protothetic completes the four volume project - The Leśniewski Collection - a collected edition of Leśniewski's papers and major contributions to Leśniewski's system of the foundations of mathematics. The three volumes published so far are:

(1) Leśniewski's Systems. Ontology and Mereology Edited by J.T.J. Srzednicki, Y.F. Rickey, and J. Czelakowski. Nijhoff International Philosophy Series, 13 (1984).

(2) S. Leśniewski's Lecture Notes in Logic Edited by J.T.J. Srzednicki and Z. Stachniak. Nijhoff International Philosophy Series, 24 (1988).

(3) Stanisław Leniewski: Collected Works Edited by J.T.J. Srzednicki, S.J. Surma, and D. Barnett with an Annotated Bibliography by Y.F. Rickey. Nijhoff International Philosophy Series, 44 (1992)." (Editor's Foreword, p. VII)

Contents: Editor's Foreword VII; 1. Peter M. Simons: Nominalism in Poland (1983) 1; 2. V. Frederick Rickey: A survey of Leśniewski's logic (1977) 23; 3. Alfred Tajtelbaum-Tarski: On the primitive term of logistic (1923) 43; 4. Boleslaw Sobocinski: An investigation in Protothetics (1949) 69; 5. Jerzy Slupecki: St. Leśniewski's Protothetics (1953) 85; 6. Boleslaw Sobocinski: On the single axiom of Protothetic (1960) 153; 7. V. Frederick Rickey: Axiomatic inscriptional syntax. Part II. The syntax of Protothetic (1973) 217; VIII. Audoënus Le Blanc: Investigations in Protothetic (1985) 289; Protothetic bibliography 299; Author Index 309.

Stachniak, Zbigniew. 1981. Introduction to Model Theory for Leśniewski's Ontology. Wroclaw: Wydawnictwo Uniwersytetu Wroclaskiego.

This monograph presents a formal theory of models for a certain extension of Leśniewski's Ontology.

Contents: Chapter 1. Ontology L_DF

1.1 Categories; 1.2 Basic language L; 1.3 Ontological definitions; 1.4 The language of ontology L_DF; 1.5 The system of ontology L_DF; 1.6 Non-creativity of ontological definitions;

Chapter 2. Model Theory

2.1 Ontological atomic Boolean systems; 2.2 Basic atomic Boolean models; 2.3 Generalized atomic Boolean models; 2.4 Completeness and compactness;

Chapter 3. Omitting types theorem and the fundamental theorem of ultraproducts;

3.1 Omitting types theorem; 3.2 The fundamental theorem of ultraproducts.

Sundholm, Göran. 2003. "Tarski and Leśniewski on Languages with Meaning versus Languages without Use. A 60th Birthday Provocation for Jan Wolenski." In Philosophy and Logic in Search of the Polish Tradition: Essays in Honour of Jan Wolenski on the Occasion of His 60th Birthday, edited by Kijania-Placek, Katarzyna, 109-128. Dordrecht: Kluwer.

"Conclusion

Around 1930 Alfred Tarski, a mathematician by inclination, training, and ability, very much like other contemporary researchers, attempted to apply the techniques of mathematics to problems in logic. Out of necessity this demanded that the formal languages of logic had to be converted into objects of study, from having been major tools for research. For him personally this entailed a conflict between the foundational stance that he had taken over from his teacher Leśniewski and the metamathematical laisser faire towards which he, as a mathematician, was inclined. He resolved this dilemma between 1933 and 1935 and his unequivocal choice was in favour of metamathematics. I have suggested that contributing factors in this decision were, possibly among others, (1) the impact of the achievements of of metamathematics; (2) Tarski's own experience of metamathematical work; (3) the availability of an attractive alternative foundation, namely, Zermelo's axiomatic set theory in relation to the cumulative hierarchy; and (4) unfortunate personal conflicts among his teachers and collaborators." (p. 123)

Surma, Stanisław. 1977. "On the work and influence of Stanisław Leśniewski." In Logic Colloquium 76, edited by Gandy, Robin and Hyland, John Martin, 191-220. Amsterdam: North-Holland.

"Concluding remarks

Stanisław Leśniewski's efforts to solve the problem of antinomies resulted in the construction of what he called the New System of the foundations of mathematics, distinguished by originality, comprehensiveness and elegance, a pioneering achievement of the 20-ties.

Leśniewski played a considerable role during the period of elaborarating modern tendencies of mathematical logic and the foundations of mathematics. He was the forerunner and originator of many ideas now incorporated into logical and foundational text-books.

But Leśniewski's writings are done in a highly condensed and difficult style, most cumbersome in practice, his famous terminological explanations are hardly intelligible. He invented a special symbolism, the so called wheel and spoke notation the use of which was an additional factor determining his isolation on the international scene.

This is why Leśniewski's systems have not been so popular as they deserved. Another reason is that general trends of logical research had meanwhile drifted away from "system building" to metalogical investigations mostly of first order languages. All this is quite unfortunate but the fact remains that Leśniewski·systems are not generally accepted as a tool in the foundational practice, and they" are not the systems a mathematician in the street makes use of. But on the other hand, Leśniewski's work has greatly influenced the very phflosophy of logic and of the foundational studies. In this field Leśniewski had worked out an original point of view he had called the "intuitionistic formalism" which he characterized by these sentences:

"I might take this opportunity to point out that. for many months I have devoted a considerable expenditure of systematic work towards the formalization of these systems / ••• / through a clear formulation of their directives using a number of the auxiliary terms whose meaning I fixed in the terminological explanations / •.. /. Having no predilection for various "mathematical games" that consist in writing out according to one or another conventional rule various more or less picturesque formulae which need not be meaningful, or even - as some of the "mathematical gamers" might prefer - which should necessarily be meaningless, I would not have taken the trouble to systematize and to often check quite scrupulously the directives of my system, had I not imputed to its theorems a certain specific and completely determined sense, in virtue of which its axioms, definitions, and final directives / .•. / have for me an irresistably intuitive validity" and further "I know no method more effective for acquainting the reader with my logical intuitions tban the method of formalizing any deductive theory to be set forth./Compare for this Leśniewski /1929/, p.78/." (pp. 212-213)

References

Leśniewski, S. 1929. Grundzuege eines neuen Systems der Grundlagen der Mathematik. §1-§11. Fundamenta Mathematicae,14/1929/,l-81.

Świetorzecka, Kordula, and Marek, Porwolik. 2018. "Bolesław Sobociński on Universals. Leśniewski’s Nominalism and Sobociński’s Metaconceptualism." In The Lvov-Warsaw School. Past and Present, edited by Garrido, Ángel and Wybraniec-Skardowska, Urszula, 615-632. Cham (Switzerland): Birkhäuser.

Abstract: "The present paper proposes a comparative analysis of two standpoints on the existence and nature of universals hold by Stanisław Leśniewski and Bolesław Sobocinski. We consider first the nominalistic argumentation of Leśniewski formalized by Sobocinski and described in the correspondence with J. M. Bochenski in 1956. Sobocinski’s formalization revealed a fundamental pragmatic weakness of the reconstructed argumentation which was also mentioned by Sobocinski. He himself was aware of the difficulties connected with an adequate interpretation of the crucial axiom, whose

acceptance Leśniewski imputed to supporters of all theories of universals. Finally, the problem of the existence and nature of universals was elaborated by Sobocinski also in a separate typescript “Uwagi w sprawie powszechników” (Remarks on universals).

The view formulated by Sobocinski comes from a combination of the methodology of deductive systems and the conceptualist standpoint. From the philosophical perspective Sobocinski’s idea is both interesting and original, but it remained unknown to philosophers and logicians in general. For these reason we describe it and compare it with Lesniewski’s approach. We use in this description epistemological notions of R. Suszko. Our analysis enables to speak about universals in sense of Leśniewski, which are described by some universal in sense of Sobocinski."

Tajtelbaum-Tarski, Alfred. 1998. "On the primitive term of logistic. Doctoral Dissertation [1923]." In S. Leśniewski's Systems: Protothetic, edited by Srzednicki, Jan and Stachniak, Zbigniew. Dordrecht: Kluwer.

"This paper appeared in print in Polish under the title 'O wyrazie pierwotnym logistyki', Przegląd Filozojiczny XXVI (1923), 68-89, by permission of Jan Tarski. Translated by Z. Stachniak." (p. 43)

"The considerations carried out in the present work belong to the area of logistic; the sentences on which I based these considerations are generally accepted among researchers working in this field of knowledge. I do not, however, carry out my considerations on the basis of any specific system of logistic; in particular, I do not make my reasonings dependent on the best known theory of logical types by Russell. Although it is not possible, as I see it, to develop a consistent system of logistic without this or that theory of types, among all the theories of types which could be constructed(2) there unquestionably exist those according to which my arguments, in their present general form, are faultless. One such theory was developed by S. Leśniewski during his lectures on the foundations of arithmetic at the University of Warsaw (in 1920-1921).(3)

The main objective of the present work is to settle the following problem: is it possible to construct a system of logistic with the sign of equivalence as the sole primitive term (in addition, of course, to the quantifiers(4) )?" (p. 43)

(2) The possibility of constructing different theories of logical types was already anticipatedby the inventor of the first of them - Russell. Cf. A.N. Whitehead and B. Russell, Principia Mathematica, Cambridge 1910, Vol. I, p. vii.

(3) One way in which Leśniewski's theory of types affected the layout of this work is that for functions, whose arguments are not sentences, I am using distinct parentheses. Cf. Def. 4 in Section 2 and Def. 6 in Section 3.

(4) I am using the term 'quantifier' in the sense of Peirce ([cf.] 'On the Algebra of Logic', American Journal of Mathematics VII, 1885, p. 197), who denotes with this term the symbols 'Π' (universal quantifier) and 'Σ' (particular quantifier), representing abbreviations of the expressions: 'for every signification of terms ... ' and 'for some signification of terms ... '.

Takano, Mitio. 1985. "A Semantical Investigation into Leśniewski's Axiom of His Ontology." Studia Logica no. 44:71-77.

———. 1987. "Embeddings between the Elementary Ontology with an Atom and the Monadic Second-Order Predicate Logic." Studia Logica no. 46:247-253.

Tanaja, Shôtarô. 1969. "Lesniewski’s Protothetics S1, S2. I." Procedings of Japan Academy no. 45:97-101.

"The systems S1 and S2 are defined originally by S. Leśniewski [1].

The definitions, theorems and some relations between S1 and S2 are also shown by K. Iski [2]. The equivalences of some laws in S1 are proved by K. Chikawa [3].

In this paper we shall prove that every theorem of S2 is a theorem of S1."

References

(1) S. Leśniewski: Grundzfige eines neuen Systems der Grundlagen der Mathematik. Fundamenta Mathematicae, 65, 1-81 Warszawa (1929).

(2) K. Iski. Symbolic Logic Propositional Calculi (in Japanese.), Vol. 1, Maki publisher (1968).

(3) K. Chikawa: On equivalences of laws in elementary protothetics. I. Proc. Japan Acad., 43, 743-747 (1967).

———. 1969. "Lesniewski’s Protothetics S1, S2. II." Procedings of Japan Academy no. 45:259-262.

———. 1969. "Lesniewski’s Protothetics S1, S2. III." Procedings of Japan Academy no. 45:263-266.

Thom, Paul. 1986. "A Lesniewskian Reading of Ancient Ontology: Parmenides to Democritus." History and Philosophy of Logic no. 7:155-166.

Abstract: "Parmenides formulated a formal ontology, to which various additions and alternatives were proposed by Melissus, Gorgias, Leucippus and Democritus. These systems are here interpreted as modifications of a minimal Lesniewskian Ontology."

"There is a tradition of ontological theorising which commences with Parmenides and whose central arguments can can be given a purely formal interpretation. This, of course, is not their only possible interpretation. It is, nonetheless, worthy of consideration, as a means of articulating the continuities and discontinuities within that tradition, and of investigating the prehistory of logic.

The main thesis of this paper is that such a purely formal interpretation of Parmenides, his followers and critics, is best expressed in the language (or, if you wish, in some of the languages) of Leśniewski's Ontology." (p. 155)

Trentman, John. 1966. "Leśniewski's Ontology and Some Medieval Logicians." Notre Dame Journal of Formal Logic no. 7:361-364.

"In a recent issue of this journal (Oct., 1964) Professor Desmond Paul Henry [*] has shown that, although it may be the case that Ockham's descensus in his supposition theory cannot be adequately rendered in the lower functional calculus (Cf. [7]), it can be adequately rendered in the Ontology of S. Leśniewski. Professor Henry, furthermore, suggests that Ontology would be an appropriate tool for analyzing other medieval logical theories, claiming, "It is not difficult to multiply examples of the facility and directness with which Ontology can furnish formal analyses of medieval logical theories, including those cases which are despaired of in the histories." (P. 292)

In this note I wish to suggest an important limitation upon this claim.

For a very fundamental reason Ontology is not an appropriate tool for analyzing a certain class of fourteenth-century logical theories. One can best make this point, however, by emphasizing its usefulness for explicating Ockham's doctrines. Not only will it allow one to express the descensus; it also provides a very close and illuminating explication of Ockham's doctrine of predication, and this is the matter that most concerns me in this note." (p. 361)

[*] Ockham, suppositio, and modern logic, pp. 290-292.

References

[7] Gareth B. Matthews, "Ockham's Supposition Theory and Modern Logic", The Philosophical Review, vol. LXXIII, pp. 91-99.

———. 1976. "On Interpretation, Leśniewski's Ontology, and the Study of Medieval Logic." Journal of the History of Philosophy no. 14:217-222.

"The most characteristic thing about D. P. Henry's interesting studies of medieval logic is his persistent use of and appeal to the logical system called Leśniewski's Ontology.

Whether this characteristic of his books and essays is useful or repellent to the reader is a matter of controversy, and the use of this particular interpretative tool has, in fact, been challenged on the ground that if Leśniewski's Ontology is taken as an interpreted system in the sense in which medieval logics must be seen to be interpreted, it must be regarded as committed to a two-name doctrine of predication and, hence, is good Oekhamism but a dubious device for the expression of anti-Ockhamist logics.(1)

Henry has responded that this objection is mistaken.(2) There is no reason to believe that Ontology is thus limited. Nevertheless, Ontology is an interpreted system in the strong sense intended in the criticism. Indeed, he says about my characterization of the difference between logic and an uninterpreted calculus, "In fact the view of logic propounded is exactly the one adopted by Leśniewski. ''(3)

The aim of this note is to attempt some clarification of this controversy about the use of Leśniewski's Ontology as an analytical and historiographic tool. Henry's understanding of what the medieval logicians were trying to do is totally unexceptionable, but I shall suggest it is far from easy to get, either from him or from Leśniewski's other interpreters, a coherent and consistent understanding of the philosophical point of view of Ontology, upon which one might base a judgement about its historiographic usefulness.

The key to the problem is to be found in understanding what it means or can mean to talk about the interpretation of logic; it is here that an attempt at clarifying these issues must begin." (p. 217)

(1) 1 John Trentman, "Leśniewski's Ontology and some Medieval Logicians," Nolre Dame Journal oJ Formal Logic, VII (1966), 361-364.

2 His response first appeared in "Leśniewski's Ontology and some Medieval Logicians," NDJFL, X (1969), 324-326; the substance of his arguments is repeated in his Medieval Logic and Metaphysics (London, 1972). In this note I shall concentrate on Henry's use of Leśniewski and his defence of that use in this book (hereafter cited as MLM).

(3) MLM, p. 54.

Urbaniak, Rafal. 2006. "On Ontological functors of Leśniewski's Ontology." Reports on Mathematical Logic no. 40:15-43.

Abstract: "We present an algorithm which allows to define

any possible sentence-formative functor of Leśniewski's Elementary Ontology (LEO), arguments of which belong to the category of names. Other results are: a recursive method of listing possible functors, a method of indicating the number of possible n-place

ontological functors, and a sketch of a proof that LEO is functionally complete with respect to { ∧, ¬ ∀, ∊ }."

———. 2006. "On Representing Sentential Connectives of Leśniewski's Elementary Protothetic." Journal of Logic and Computation no. 16:451-460.

Abstract: "After a brief presentation of Leśniewski's notation for 1- and 2-place sentential connectives of protothetic, the article discusses a method of extending this method to n ≥ 3-place sentential connectives. Such a method has been hinted at b Luschei, but in fact, no general effective method of defining such functors has been clearly and explicitly given. The purpose of this article is to provide such a method."

———. 2006. "Some non-standard interpretations of the axiomatic basis of Lesniewski’s Ontology." Australasian Journal of Logic no. 3:13-46.

"Intuitively, in a slogan, when we give axioms for a given axiomatic system one of our purposes is to characterize constants occurring in these axioms. Following this idea, axioms of Lesniewski’s Ontology aim to characterize ‘univocally’ the primitive constants of this system. Usually, there is only one such a constant specific to Ontology; it is " (sometimes, there are other constants: see

Lejewski [9]). Hence, Lejewski writes:

In the original system of Ontology . . . the meaning of the copula ‘is’ (‘ε"axiomatizations. In order to proceed, we shall (i) introduce the language we will be talking about (ii) say what axioms and rules of inference were accepted in Ontology in some axiomatizations, (iii) present some possible interpretations of quantifiers in Ontology, (iv) explain what is meant by ’semantic interpretation of a given functor’, and, when it will be done, (v) obtain the answer for the main problem.’ in symbols) is determined axiomatically . . . [10, p. 323]

Our purpose will be to investigate, whether in fact axiomatizations of Ontology determine a unique semantic interpretation of the primitive constant(s) of this axiomatizations. In order to proceed, we shall (i) introduce the language we will be talking about (ii) say what axioms and rules of inference were accepted in Ontology in some axiomatizations, (iii) present some possible interpretations

of quantifiers in Ontology, (iv) explain what is meant by ’semantic interpretation of a given functor’, and, when it will be done, (v) obtain the answer for the main problem." (pp. 13-14)

References

[9] Lejewski, C., On Lesniewski’s Ontology, Lesniewski’s Systems. Ontology and Mereology, Editors: Jan T. J. Srzednicki, V. F. Rickey, J. Czelakowski, Polish Academy of Sciences, Institute of Philosophy and Sociology, Martinus Nijhoff Publishers, The Hague, 1984, pp. 123-149

[10] Lejewski, C., Systems of Lesniewski’s Ontology with the Functor of Weak Inclusion as the only Primitive Term, Studia Logica, 1977, XXXVI, 4, pp. 323–349

———. 2008. "Leśniewski and Russell's Paradox: Some Problems." History and Philosophy of Logic no. 29:115-146.

Abstract: "Sobocinski in his paper on Leśniewski's solution to Russell's paradox (L'analyse de l'antinomie russellienne par Leśniewski, 1949) argued that Leśniewski has succeeded in explaining it away. The general strategy of this alleged explanation is presented. The key element of this attempt is the distinction between the collective (mereological) and the distributive (set-theoretic) understanding of the set. The mereological part of the solution, although correct, is likely to fall short of providing foundations of mathematics. I argue that the remaining part of the solution which suggests a specific reading of the distributive interpretation is unacceptable. It follows from it that every individual is an element of every individual. Finally, another Lesniewskian-style approach which uses so-called higher-order epsilon connectives is used and its weakness is indicated."

———. 2010. "Response to a critic (definability and ontology)." Reports on Mathematical Logic no. 10:255-259.

Reply to Leslaw Borowski (2010).

"I would like to thank Mr Borowski for his comments, I appreciate his time and effort. It is always uplifting to learn that a topic which one has considered quite hermetic can stir up such emotions. I’ll just briefly respond in a rather relaxed manner to what I think the main points raised by Mr Borowski are." (p. 255)

(..)

"Once again, I would like to thank to Mr Borowski for his criticism – he pointed out a mistake in one of the definitions, and the review brought up to my attention the need for extreme clarity which I, over-relying on the reader’s common-sense, might have neglected." (p. 259)

———. 2013. Leśniewski's Systems of Logic and Foundations of Mathematics. Dordrecht: Springer.

"Leśniewski’s work is interesting for a few reasons.

• If one is interested in history of logic in general, it is hard to deny that Leśniewski was one of the key figures in one of the most important schools of logic in twentieth century. He devoted his research to developing an alternative to the system of Principia Mathematica and this attempt is worth studying in his own right.

• If one is interested in the development of Tarski’s thought it might be useful to learn what his Ph.D. supervisor’s views were and how Leśniewski’s work and Tarski’s ideas are (or are not) related.

• Philosophical discussions in which Leśniewski participated pertained to issues which are discussed quite lively even today. His approach to semantical and set-theoretic paradoxes and his views on the validity of the principle of excluded middle and of the principle of contradiction are philosophically interesting.

• Leśniewski was a nominalist and his systems were a nominalistic attempt to provide a system of foundations of mathematics. It is a major attempt of this sort and as such it is worth an examination.

• His metalogic is quite specific. Nominalist as he was, he wanted to develop a purely inscriptional syntactic description of his systems in a way that did not make any reference to expression types. It is interesting to see how he proceeded.

• His systems have some interesting properties. For instance, in all of them definitions can be creative (and this is not considered to be a problem). The generality of Prothetic admits interesting extensions (intuitionistic (see López-Escobar and Miraglia 2002) or modal (see the works of Suszko and in general, see Sect. 3.7 for references). The language of Ontology (which, in a way, can be viewed as one of the first free formal logics) is, arguably, more suitable for capturing certain aspects of predication and abstract noun phrases as they work in natural language.

This book is devoted to a presentation of Leśniewski’s achievements and their critical evaluation. I discuss his philosophical views, describe his systems, and evaluate the role they can play in the foundations of mathematics. It was my purpose to focus on primary sources and present Leśniewski’s own views and results rather than those present in secondary literature. For this reason, later developments are not treated in detail but rather either mentioned in passing, or described in sections devoted to secondary literature included in some chapters." (Preface, pp. VII-VIII)

References

López-Escobar, E., & Miraglia, F. (2002). Definitions: The primitive concept of logics or the Leśniewski-Tarski Legacy, Dissertationes Mathematicae (Vol. 401). Warszawa: Polska Akademia Nauk.

———. 2015. "Stanisław Leśniewski: Rethinking the Philosophy of Mathematics." European Review no. 23:125-138.

Abstract: "Near the end of the XIXth century part of mathematical research was focused on unification: the goal was to find ”one sort of thing” that mathematics is (or could be taken to be) about. Quite quickly sets became the main candidate for this position. While the enterpize hit a rough patch with Frege’s failure and set-theoretic paradoxes, by the 1920s mathematicians (roughly speaking) settled on a promising axiomatization of set theory and considered it foundational. Quite parallel to this development was the work of Stanisław Leśniewski (1886-1939), a Polish logician who did not accept the existence of abstract (aspatial, atemporal and acausal) objects such as sets. Leśniewski attempted to find a nominalistically acceptable replacement for set theory in the foundations of mathematics. His candidate was Mereology — a theory which instead of sets and elements spoke of wholes and parts. The goal of my talk will be to present Mereology in this context, to evaluate the feasibility of Lesniewski’s project and to briefly comment on its contemporary relevance."

———. 2016. "Potential Infinity, Abstraction Principles and Arithmetic (Leśniewski Style)." Axioms no. 5:1-20.

Abstract: "The paper starts with an explanation of how the logistic project can be approached within the framework of Lesniewski’s systems. One nice feature of the system is that Hume’s Principle is derivable in it from an explicit definition of natural numbers. I generalize this result to show that all predicative abstraction principles corresponding to any second-level relation which is provably an equivalence relation are provable. However, the system fails, despite being much neater than the construction of Principia Mathematica. One of the key reasons is that just as in the case of the system of PM, without the assumption that infinitely many objects exist, (renderings of) most of 8 the standard axioms of Peano Arithmetic are not derivable in the system. I prove that introducing modal quantifiers meant to capture the intuitions behind potential infinity results in the (renderings of) axioms of PA being valid in all relational models of the extended language. The second, historical part of the paper contains a user-friendly description of Lesniewski’s own arithmetic and a brief investigation into its properties."

Urbaniak, Rafal, and Severi Hämäri, K. 2012. "Busting a Myth about Leśniewski and Definitions." History and Philosophy of Logic no. 33:159-189.

Abstract: "A theory of definitions which places the eliminability and conservativeness requirements on definitions is usually called the standard theory. We examine a persistent myth which credits this theory to Leśniewski, a Polish logician. After a brief survey of its origins, we show that the myth is highly dubious. First, no place in Leśniewski's published or unpublished work is known where the standard conditions are discussed. Second, Leśniewski's own logical theories allow for creative definitions. Third, Leśniewski's celebrated ‘rules of definition’ lay merely syntactical restrictions on the form of definitions: they do not provide definitions with such meta-theoretical requirements as eliminability or conservativeness. On the positive side, we point out that among the Polish logicians, in the 1920s and 1930s, a study of these meta-theoretical conditions is more readily found in the works of Łukasiewicz and Ajdukiewicz."

Vanderveken, Daniel R. 1976. "The Leśniewski-Curry Theory of Syntactical Categories and the Categorially Open Functors." Studia Logica no. 32:191-201.

"The present paper is concerned with the problems which are posed for the Leśniewski-Curry theory of syntactical categories by the categorially open functors.

A categorially open functor of a language L is any functor of L whose syntactical category in L ranges at each occurrence over a set of several different syntactical categories admitted in L, and is determined in each case effectively in function of the categories of one or several of the expressions which it then takes as arguments.

For example, the quantifiers and the identity sign of a logical language with several types of objects are categorially open functors in this language.

The deficiencies of the Leśniewski-Curry theory of categories with respect to the categorially open functors have often been mentioned in the literature, for example, in A. Tarski 1936, § 4, 11th note and in A. N. Prior, 1971, ch. 3, § 7.

Our fundamental purpose in this paper is to define a consistent extension of this theory adequate for the characterisation of these functors.

Since the usual categorial base components of transformational grammars are constructed on the model of Leśniewski-Curry's theory, this new theory of syntactical categories will naturally have interesting consequences for those components." (p. 191)

References

[10] A. N. Proir, Object of thought, edited by P. T. Geach and A. J. P. Kenny, Oxford, Clarendon Press, 1971.

[11] A. Tarski, Der Wahrheitsbegriff in den formalisierten Sprachen, Studia Philosophica 1, 1936.

Vasiukov, Vladimir L. 1998. "Non-elementary Exegesis of Twardowski's Theory of Presentation." In The Lvov-Warsaw School and Contemporary Philosophy, edited by Kijania-Placek, Katarzyna and Wolenski, Jan, 153-167. Dodrecht: Kluwer.

"In spite of the historical proximity of S. Leśniewski to K. Twardowski, an attempt to look at Twardowski's heritage through Leśniewski's eyes leads to striking results. Firstly, it results in a wider framework than Leśniewski's Elementary Ontology and secondly, it involves a transition from Formal Ontology to Formal Phenomenology. In this paper an extension of Leśniewski's Non-Elementary Ontology is presented which is suitable for investigating Twardowski's Theory of Presentation." (p. 153)

Vasyukov, Vladimir. 1993. "A Lesniewskian Guide to Husserl's and Meinong's Jungles." Axiomathes no. 4:59-74.

"The borderline between modem and traditional logics can hardly be drawn in the case of the Lvov-Warsaw Philosophical School. If chronologically we regard 1879 (when Frege's Begriffsschrift was published) as the beginning of the revolution in logic, then the date of the beginning of the activity of the Lvov-Warsaw school (1895) allows us to consider it prima-facie as wholly belonging to modem logic. And undoubtedly many results and ideas of such members of the school as Łukasiewicz, Tarski, Ajdukiewicz etc. belong to the great development of 'modem' twentieth century logic.

(...)

Taking into account that Leśniewski's Ontology is also the theory of objects, it will not be such a surprising endeavour to analyse Husserl's and Meinong's views from the standpoint of Ontology, in search of the common basic features.

The problem is that perhaps the language of Ontology is too poor to describe some aspects of these other theories. And this is quite natural: Leśniewski's task was an inquiry into the deepest and thus simplest intuitive concepts of objects as such.

Our proposals in this case probably would not meet with the approval of Leśniewski himself, but extending logical theories is common enough, so this is the approach that is undertaken here. We shall consider two ways of extending Leśniewski's Ontology which allow us to interpret some aspects of Husserl's and Meinong's theories of objects. I do not think that this will be precisely the remedy for overcoming the "horrors of Meinong's (Husserl's) jungle" but rather an attempt to yield an ontologically oriented language as background for further expeditions and penetrations deep into the heart of these "magical territories." (pp. 59-60)

Waragai, Toshiharu. 1979. "Ontological Burden of Grammatical Categories." Annals of the Japan Association for Philosophy of Science no. 5:185-205.

"Now the problem about the predicate-quantification disappears, for the category convention of predicates is not that which the category of names has.

They are related to the realm of entities regarding their meanings, or thier extensions.

Having an extention is not the same as naming a set We must introduce some special category, if we want to speak about sets. For some philosophical reason, I shall name the interpretation which I proposed the subjectivistic interpretation of quantification. My philosophical intuition for it is more or less the following.

Our world consists of subjects, which I understand in the sense of traditional ontology; they have their own inner structures within the framework of which they can appear in the world, related to each other again within this framework.

Their classification according as what they are gives us the categories, or predicates3. If we replace a word designating some so-called substantia prima, e.g. Socrates, in a sentence containing this word, say 'Socrates is wise', with 'something' (aliquid, or better aliqua res), then we may be said to be committed to some entity by the use of the sentence 'something is wise', but as to 'wise', the resulting

sentence which we get by replacing this word with 'something' does not make us commit ourselves to any kind of entity. The sentence 'Socrates is something' does not force us to accept any new kind of entity like idea. It only says that Socrates is in some mode of being. Only quantification of the word for the substantial prima forces us to commit ourselves to entities. Hence I call my interpretation

subjectivistic." (p. 199, notes omitted)

"Now let me summarize what I have discussed until now in this chapter. The language I considered has as to noun expressions only one category, and they are in two ways related to the reality by the category conventions C(U) and C(ob).

In general, names are related by C(U) to the reality as to their extensions, but those names which can be the subject of the sentence 'x€y' are related to the reality as entity names. Hence, it is clear that the quantification in this language is not merely substitutional, but rather should be regarded as subjectivistic.(2)

I may stress this fact by saying that existence is not what quantification expresses but what the grammar of a regimented language does." (pp. 201-202)

———. 1980. "Leśniewski on General Objects." Journal of Gakugei no. 29:19-22.

———. 1981. "Leśniewski's Refutation of General Object on the Basis of Ontology." Journal of Gakugei no. 30:49-54.

———. 1981. "The Ontological Law of Contradiction and Its Logical Structure." Annals of the Japan Association for Philosophy of Science no. 6:43-58.

"§ 7. System Lo and Leśniewski's Ontology

Though the logico-semantical analysis of Aristotle's argument of descending chain of predicates [*] which is essentially ontological, and at the same time through the analysis of the everyday usage of names and negations, I made clear the logico-semantical content of the ontological law of identity and that of the ontological law of contradiction by constructing a first-order language Lo which is strong enough to perform the logico-semantical analysis of the two ontological dicta.

Historically speaking, this system is a proper part of a more comprehensive logical system constructed by Stanisław Leśniewski (1886-1939), which he named Ontology.(1)

He stated:

I used the name Ontology for the system I constructed, since, when I consider the circumstances that I formulated in the system a kind of "general principle of being", the name did not hurt my "feeling of language"(2).

But what we regret is that he does not seem to have mentioned any philosophical relation between his system Ontology and traditional Ontology.

In this paper, we obtained logical system through a philosophical and logical analysis of Aristotle's argument of descending chain of predicates, and it became clear that the analysis leads us to a logical analysis which is of its essential nature Lesniewskian. This fact helps understand the philosophical relation between Leśniewski's Ontology and traditional Ontology." (p. 58)

[*] described in § 2, with reference to Atistotle, Analytica Posteriora, A, XIX-XXII.

(1) The axiom of Ontology was found during the summer semester in 1919/1920, and officially announced in 1921. On this point; cf. Leśniewski, S.: 'O Podstawach Matematyki', Rozdzial XI ('On the Foundations of Mathematics', Chapter XI). Przegląd Filozoficzny, 34, 1931. (...)

(2) p. 163 of Leśniewski's work mentioned in (1).

———. 1998. "On Some Essential Subsystems of Leśniewski's Ontology and the Equivalence between the Singular Barbara and the Law of Leibniz in Ontology." In The Lvov-Warsaw School and Contemporary Philosophy, edited by Kijania-Placek, Katarzyna and Wolenski, Jan, 169-180. Dodrecht: Kluwer.

"The main aim of this paper is to show that in Leśniewski's Ontology, the law of Leibniz and a special case of Barbara which we refer to as the singular Barbara are equivalent to each other. To show this, we choose a group of theses of Ontology which are sufficient to establish the intended equivalence. From the result it follows that the problematic characters concerning the law of Leibniz reduce either to the validity of the singular Barbara or that of the theses used in establishing the intended equivalence.

We suggest that the most dubious thesis is the one which correctly expresses the operation of comprehension. According to this result, we claim 1) that the dubiousness of the law of Leibniz reduces to that of the notion of comprehension and 2) that not every property is convertible to a name. With respect to these results, the doctrine of limitation of size in set theories will be criticized.

Since I presented and used an extended version of Ontology to show the main results at the conference held in Lvov, I will give a sketch of the system I made use of at the conference. The logical relation between the sole axiom of Leśniewski's Ontology and Russell's theory of description will be made clear, too." (p. 169)

Wolenski, Jan. 1986. "Reism and Leśniewski's Ontology." History and Philosophy of Logic no. 7:167-176.

Abstract: "This paper examines relations between Reism, the metaphysical theory invented by Tadeusz Kotarbiński, and Leśniewski's calculus of names. It is shown that Kotarbinski's interpretation of common nouns as genuine names, i.e., names of things is essentially based on Leśniewski's logical ideas. It is pointed out that Lesniewskian semantics offers better prospects for Nominalism than does semantics of the standard first-order predicate calculus."

———. 1989. Logic and Philosophy in the Lvov-Warsaw School. Dordrecht: Kluwer.

Chapter VII: Leśniewski's Systems, pp. 141-161.

———. 1995. "Leśniewski's Logic and the concept of Being." In Stanisław Leśniewski aujourd'hui, edited by Miéville, Denis and Vernant, Denis, 93-101. Grenoble: Recherches sur la Philosophie et le Langage.

Abstract: "This paper applies Leśniewski's logical ideas to an analysis of the concept of being. The analysis follows the classical ontology which is based on a distinction of two concepts of being : being in the distributive sense and being in the collective sense. Now it is argued that Leśniewski's ontology (calculus of names) is a much better device for analysizing being in the distributive sense than the standard first-order predicate logic. Moreover, basic intuition connected with the being in the collective sense are nicely captured by mereology."

———. 1995. "Mathematical logic in Poland 1900-1939. People, circles, institutions, ideas." Modern Logic no. 5:363-405.

"Assume that someone would try to predict the development of mathematical logic circa 1900. Probably, he would point out Germany, England, and perhaps France as the central countries. Certainly, this person would not mention Poland, and not only because there was no such country at that time. Thirty year later, Heinrich Scholz, the first modern historian of logic, called Warsaw one of the capitals of mathematical logic. How did a country without special traditions in logic so quickly arrive at the top of this field? What happened that permitted Fraenkel and Bar-Hillel to write: "There is probably no country which has contributed, relative to the size of its population, so much to mathematical logic and set theory as Poland"? This paper tries to explain the phenomenon called "Polish logic" by pointing out the wider context in which logic in Poland was done." (p. 363)

———. 2004. "Polish Logic." Journal of Logic and Computation no. 12:399-428.

Abstract: "This paper outlines the history of logic in Poland in the years 1918-1939 (which some addictions concerning the period before 1918 and after 1945). The disciplinary and social history of logical investigations in Poland is widely described. The author stresses topics characteristic for Polish logic, namely prositional calculus, many-valued logic, Leśniewski's systems, Chwistek's systems and the works in the history of logic."

Section K. Leśniewski's systems, pp. 418-422.

———. 2012. "Truth is Eternal if and only if it is Sempiternal." In Studies in the Philosophy of Herbert Hochberg, edited by Tegtmeier, Erwin, 223-230. Frankfurt: Ontos Verlag.

"The problem addressed in this paper goes back to Aristotle and his considerations about tomorrow’s sea battle. In a famous passage in De Interpretatione (19a 25-30; after The Works of Aristotle, vol. 1: Categoriae and De Intepretatione, tr. by E. M. Edghill, Oxford University Press, Oxford 1928), the Stagirite says:

Everything must be either be or not be, whether in the present or in the future, but it is not always possible to distinguish and state determinately which of these alternatives must necessarily come about.

Let me illustrate. A sea-fight must take place tomorrow or not, but it is not necessary that it either should not take place tomorrow, neither it is necessary that it should not take place, yet it is necessary that it either should or should not take place tomorrow. Since propositions correspond with facts, it is evident that when in future events there is a real alternative, and a potentiality in contrary directions, the corresponding affirmation and denial have the same character.

These words initiated a considerable discussion about the relation between truth and time. Is truth relative and dependent on temporal coordinates or absolute and timeless? The debate concerns several problems, in particular, the validity of some logical principles, fatalism, God’s omniscience, free-will and determinism (see Bernstein 1992, Cahn 1967, Gaskin 1995, Hintikka 1977, Lucas 1989, Prior 1953, Vuillemin 1996. This paper concentrates almost entirely on logical issues." (p. 223)

References

Bernstein, M. H., 1992, Fatalism, University of Nebraska Press, Lincolm

Cahn, S. M. 1967, Fate, Logic and Time, Yale University Press, New Haven.

Gaskin, D. 1995, The Sea Battle Argument and the Master Argument. Aristotle and Diodorus Cronus, de Gruyter, Berlin 1995.

Hintikka, J. (in collaboration with U. Remes and S. Knuuttila) 1977, Aristotle on Modality and Determinism, North-Holland, Amsterdam.

Lucas, J. R 1989, The Future. An Essay on God, Temporality and Truth, Basil Blackwell, Oxford.

Prior, A. N. 1953, ‘Three-Valued Logic and Future Contingents’, The Philosophical Quarterly, 317-326.

Vuillemin, J. 1996, Necessity or Contingency. The Master Argument, CSLI Publications, Stanford.

———. 2016. "Truth-Theories in the Lvov-Warsaw School." In Tradition of the Lvov-Warsaw School: Ideas and Continuations, edited by Brożek, Anna, Chybińska, Alicja, Jadacki, Jacek and Woleński, Jan, 73-91. Leiden: Brill Rodopi.

"Final Remarks

There is an esplicit continuity of aletheiology in LWS [Lwow-Warsaw School] from Twardowski to Tarski. Most Polish authors followed Aristotle’s ideas, eventually in a Brentanist shape, accepted the weak correspondence and considered truth as absolute. (SDT) offers a very sophisticated account of these ideas. Although Łukasiewicz was an exception, he defended the eternality of truth, which can be accepted as a weakened absoluteness. Aletheiology in LWS was developed parallel to the growth of mathematical logic. Twardowski, Kotarbiński, early Leśniewski and early Łukasiewicz explained their ideas informally and with quite old-fashioned formal equipment. On the other hand, later works of Leśniewski, Łukasiewicz and. particularly Tarski, involved strong formal logical devices. And this last point is perhaps the most important Polish contribution to contemporary aletheiology." (p. 88)

SDT = A sentence A is true if and only if it is satisfied by every infinite sequence of objects (equivalently; by the empty sequence; by some sequence).

Wong, Sen. 2021. "On Reading Leśniewski." History and Philosophy of Logic no. 42:160-179.

"It is well-known that Lesniewski’s logical works are not reader-friendly, which certainly hinters any attempt to approach and not to mention understand his ideas for first-timers.

Peter M. Simons’s On Understanding Leśniewski of this journal (Simons 1982) presents a picture of how to interpret names and the constant ‘ε’ of Le´sniewski’s second system Ontology. As Ontology assumes Protothetic, it seems a good idea to deal with Protothetic first as it is Lesniewski’s first system from the perspective of logical precedence." (p. 160)

(...)

"What follows in this paper is a rough presentation of Protothetic as a template for constructing a nominalist propositional calculus. Some special techniques will be discussed and a method of graphic exposition will be used to describe some of the original TE

[terminological explanations] formulae." (p. 161)

Żełaniec, Wojciech. 1998. "Is "Being" Predicated in Only One Sense, after All?" Logic amd Logical Philosophy no. 6:241-258.

"This piece is going to be on a particular difficulty with translating ordinary language (English, in this case) sentences into logical idiom. Difficulties in this area are quite abundant, as is known to everyone who has ever taught logic, but there is one specific family of problems that appears to be of crucial importance. The chief member of this family is this problem: How is the “is” of ordinary English to be translated into logical idiom?

Traditionally, we distinguish between the “is” of predication — the copula —, the “is” of existence and the “is” of identity. This is not an exclusive classification, however, because the “is” of identity is, syntactically speaking, a special case of the “is” of predication: in sentences of form “A is identical with B” the predicate “. . . is identical with B” can be discerned, alongside two others. As regards the “is” of existence, a lot of ink has been spilled on “proving” that existence is or is not a “genuine predicate” — which is itself a piece of evidence that things are not at all clear here. Indeed, apart from quite singular sentences such as “God is” or “He’s the power that was” (said of a politician) we usually make our existence statements in sentences that do not look much different from “ordinary predications”, such as, for instance, “Soldiers are there”, “This technology is available” and the like.

If there is any difference from “ordinary predications” here, it is that of the “is” of localisation (being there, being at some definite place or within some definite domain) and all the other kinds of predication — a difference on which Professor Perzanowski has taught us a lot in a number of articles.

I shall be concerned here with just this: How to distinguish between the “is” of identity and the “is” of other kinds of predication. More precisely, I shall be concerned with the question of which kind of logic allows us to make this distinction with more accuracy. From among all possible kinds of logic as competitors, I shall concentrate on just these two: first-order predicate calculus, as, in the words of Hodges ([8], p. 2), “the simplest, the most powerful and the most applicable branch of modern logic” and Leśniewski’s Ontology, as a modern version of the calculus of names." (p. 241)

References

[8] Hodges, W., “Elementary predicate logic”, in D. M. Gabbay and F. Guenthner, Handbook of Philosophical Logic, vol. I, chapter I.1, p. 2–131, D. Reidel, Dordrecht 1983.

Stanisław Leśniewski: bibliography in English (Sin - Z)

Contents of this Section

Studies on Leśniewski in English