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Metalogic, Schopenhauer and Universal Logic

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Language, Logic, and Mathematics in Schopenhauer

Part of the book series: Studies in Universal Logic ((SUL))

Abstract

Schopenhauer used the word “metalogical” since his first work, On the Fourfold Root of the Principle of Sufficient Reason (1813), being the first to give it a precise meaning and a proper place within a philosophical system. One century later the word “Metalogic” started to be used and promoted in modern logic by the Russian logician Nicolai Vasiliev and the Polish School (Łukasiewicz, Tarski, Wajsberg). The aim of this paper is to examine the relations between the different uses of this word and doing that to try to have a better understanding of what Metalogic is and also logic tout court.

In a first section we examine and clarify the meaning of Metalogic in modern logic, comparing Metalogic to Metamathematics and Universal Logic. We make in particular a distinction between two trends in Metalogic that can be crystallized through metatheorem vs. meta-axiom.

In a second section we present Schopenhauer’s use of the word, which is essentially through the notion of metalogical truths. We describe their locations within Schopenhauer’s framework, standing side by side with other kinds of truths (metaphysical truths, logical truths, empirical truths), constituting altogether the Principle of Sufficient Reason (PSR) of Knowledge, one of the four roots of the PSR. We explain why Schopenhauer thinks that mathematical truths do not need to have a logical ground and present his view according to which metalogical truths are fundamental laws of thought that cannot be changed. We discuss the feminine nature he attributes to them and establish a parallel with Aristotle’s vision of logic.

In a third section we examine how modern logic arose from a double challenge of the fundamental laws of logic: their reformulation and relocation, their relativization and rejection. We emphasize that this dynamic evolution was performed on the basis of some semiotical and conceptual changes at the heart of logic and Metalogic.

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Notes

  1. 1.

    The same can be said about a central terminology and a central character of modern logic: “truth-value” and Gottlob Frege (1848–1925), see [20].

  2. 2.

    This is also expressed as an opposition between semantics and syntax. “Proof Theory” as coined by Hilbert concentrates on mathematical proofs from a syntactical point of view, according to which mathematics, Hilbert says, “becomes a stock of formulae” [62]. “Model Theory” was coined by Tarski [114] and deals with the interpretation of the syntax, the models of the theories.

  3. 3.

    The two books by Hilbert on logic are entitled Grundzüge der theoretischen Logik (1928), co-authored with Wilhelm Ackermann, 1896–1962 [63] (in English: Principles of Mathematical Logic) and Grundlagen der Mathematik (in English: Foundations of Mathematics), (Volume 1, 1934 - Volume 2, 1939), co-authored with Paul Bernays, 1888–1977 [64].

  4. 4.

    Kleene later on (1967) published another textbook entitled Mathematical Logic [69] full of model theory, giving up Metamathematics both syntactically and semantically. In particular he uses in this book the expression “Model Theory” for propositional logic, which is up to now unfortunately not so common.

  5. 5.

    Gentzen also worked on “natural deduction”, developing formal systems supposed to catch in a more natural way reasoning. For a discussion about that see [103], which makes a connection with Schopenhauer who was already concerned by this point.

  6. 6.

    To fix the ideas we have symbolically put here as dates of birth and death of this school, respectively, the coming of Łukasiewicz to Warsaw University and his departure from this university. Of course one can argue that this school started before 1915 and did not stop in 1944, that it is still alive, see the recent book The Lvov-Warsaw School, Past and Present [50].

  7. 7.

    See also two papers by Vasiliev of the same period: [118] and [119]. For a general presentation of Vasiliev and his work, see [2, 3].

  8. 8.

    This figure is extracted from our previous paper “Is Modern Logic Non-Aristotelian?” [22] related to a lecture presented at a conference in honor of Vasiliev, October 24–25, 2012 at Lomonosov Moscow State University. And it was published in a book with other papers presented at this conference.

  9. 9.

    “Sentential Logic” and “Propositional Logic” are both used. The first expression is used by people who want to emphasize, not to say to force, a syntactic or/and linguistic interpretation.

  10. 10.

    Translation from Polish courtesy of Robert Purdy—checked and revised by Zygmunt.

  11. 11.

    For a presentation of the different kinds of proof-theoretical systems, the relation between them and their metalogical features, see [12].

  12. 12.

    Moreover Tarski was not considering a relation but a function, an “operator” acting on theories. It seems that for developing his theory he was influenced by the topological work of Kuratowski, with whom he collaborated at some point. The three properties presented here are not the same as, but are equivalent to, the ones of Tarski’s consequence operator which look like those of a topological space, see [107].

  13. 13.

    Abraham Robinson (1918–1974) however talked about “The metamathematics of algebra” [91].

  14. 14.

    Compare with what S.Haack says in the section Logic, philosophy of logic, metalogic of her 1978 book.

  15. 15.

    Maybe Schopenhauer is too harsh with the philosopher known for claiming that we are living in the best of all possible worlds, by contrast to Schopenhauer’s idea, according to which we maybe are in the worst of all possible worlds. For a more neutral assessment of Leibniz on the Principle of Reason see [85].

  16. 16.

    English translation courtesy of Jens Lemanski. No English translation of this Handwritten Manuscript has yet been published.

  17. 17.

    About the distinction between form and matter in syllogistic, see [32,33,34].

  18. 18.

    PA’ is here the conjunction of the propositions of a finite subtheory of PA, from which IP is a consequence.

  19. 19.

    Roy Cook says: “Metalogic can be captured, loosely, by the slogan reasoning about reasoning”, we agree with him but we do not reduce reasoning about reasoning to metalogic as he describes it, i.e., the “mathematical study of formal systems that are intended to capture correct reasoning.” [36, p. 188].

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  1. UFRJ - University of Brazil, Rio de Janeiro, Brazil

    Jean-Yves Beziau

  2. CNPq - Brazilian Research Council, Brasilia, Brazil

    Jean-Yves Beziau

  3. ABF - Brazilian Academy of Philosophy, Rio de Janeiro, Brazil

    Jean-Yves Beziau

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    Jens Lemanski

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Beziau, JY. (2020). Metalogic, Schopenhauer and Universal Logic. In: Lemanski, J. (eds) Language, Logic, and Mathematics in Schopenhauer. Studies in Universal Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-33090-3_13

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