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Introduction

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Leśniewski's Systems of Logic and Foundations of Mathematics

Part of the book series: Trends in Logic ((TREN,volume 37))

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Abstract

I provide motivations for the research, describe the structure of the book, give a short biography of Leśniewski and provide details of primary sources.

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Notes

  1. 1.

    “In his zeal to demonstrate the outstanding merit of Leśniewski’s contributions to logic...Luschei sometimes devotes a disproportionate amount of space to denigrating the work of other logicians.”

  2. 2.

    “The book is expository. It does not contain proofs, and not many results are stated precisely. The author’s style makes reading difficult; many passages give the impression of being literal translations from some language unlike English. The book is also marred by being unduly polemical. It would have been relatively simple to present Leśniewski’s views and describe his practices, and then show how these differ from those of other logicians. Instead Luschei champions Leśniewski’s views, and mentions many other logicians only to show how far short they fall of a Leśniewskian ideal...Luschei’s explanations are less helpful, than they might be, for he uses a cumbersome, unfamiliar terminology, which sometimes makes his explanation as much trouble to work through as the T.E [Leśniewski’s Terminological Explanations]. Luschei also fails to provide helpful comparisons with more familiar systems.”

  3. 3.

    “From its title, one might be prepared to find in this book either a presentation and development of the systems in question, or a discussion and commentary about the systems (or both of these). In fact, Professor Luschei concerns himself almost entirely with the latter task, and indeed no proofs of theorems appear in the text at all, although some examples of ‘Leśniewski’s technique of “natural deduction” ’ are given in a note. Thus the book is primarily directed at readers who wish to learn something of the historical and philosophical background and foundations of Lesniewski’s systems of logic, rather than to those who wish to become familiar with specific theses of the logic or to acquire facility with the formal techniques involved.”

  4. 4.

    Habilitation is the highest academic qualification in some countries (like Poland and Germany). One usually has to obtain it to supervise Ph.D students and sometimes to teach certain courses.

  5. 5.

    For more details on Leśniewski’s biography see for example Betti (2005), Jadczak (1997), Woleński (1985) or Luschei (1962).

  6. 6.

    All page references to Leśniewski are to the English edition of Leśniewski’s Collected works published as Leśniewski (1991). The only exception are papers not included in those two volumes.

  7. 7.

    The story is a bit complicated here. First, Łukasiewicz published a book titled O zasadzie sprzeczności u Arystotelesa. Studium Krytyczne [On the principle of contradiction in Aristotle. A critical study, (Łukasiewicz 1910a)]. In the same year, he published a paper based on this book titled “Über den Satz des Widerspruchs bei Aristoteles” [On the principle of contradiction in Aristotle, (Łukasiewicz 1910c)]. The paper has been translated by Vernon Wedin as Łukasiewicz (1971). Since then, some people when they write about contradiction cite Łukasiewicz (1971) as the translation of Łukasiewicz (1910a, b, c) (see e.g. Horn 2009) , as if there was only one Łukasiewicz’s work from that year. As of now, the 1910 book hasn’t been translated into English. To make things even more confusing, there is a German translation of the 1910 book (Łukasiewicz 1993) under exactly the same title as Łukasiewicz’s 1910 paper. One of the authors who get the story right in their references is Betti (2004b).

  8. 8.

    The notion of equiformity played an important part. Instead of speaking of an expression type, say \(\phi \), he explicitly spoke of all tokens equiform to a token \(\phi \), where \(\phi \) was described in terms of its structure and equiformity of its constituents to certain other tokens. For example, instead of ‘\(p \wedge \lnot p\) is not provable in Protothetic’, he would say something more like ‘for every token whose first and fourth constituents are equiform to the following symbol: \(p\), whose second constituent is equiform to the following symbol: \(\wedge \), and whose third constituent is equiform to the following symbol: \(\lnot \), no system of Protothetic (that is, in no inscribed development state of this system) contains a proof (a sequence of inscription tokens satisfying appropriate conditions) which proves this token.’

  9. 9.

    In (Leśniewski 1927, 181) he wrote: “Steeped in the influence of John Stuart Mill in which I mainly grew up, and ‘conditioned’ by the problems of ‘universal–grammar’ and of logic-semantics in the style of Edward Husserl and by the exponents of the so–called Austrian School, I ineffectually attacked the foundations of ‘logistic’ from this point of view.”

  10. 10.

    A Polish journal which never came to being, mainly because of the Second World War.

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  1. Centre for Logic and Philosophy of Science, Ghent University, Blandijnberg 2, 9000, Ghent, Belgium

    Rafal Urbaniak

  2. Department of Philosophy, Sociology and Journalism, Gdańsk University, Gdańsk, Poland

    Rafal Urbaniak

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Urbaniak, R. (2014). Introduction. In: Leśniewski's Systems of Logic and Foundations of Mathematics. Trends in Logic, vol 37. Springer, Cham. https://doi.org/10.1007/978-3-319-00482-2_1

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