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How much Time Does a Logical Inference Take?

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On Gödel and the Nonexistence of Time – Gödel und die Nichtexistenz der Zeit

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Zusammenfassung

The Kurt Gödel Award 2021, presented by the Kurt Gödel Circle of Friends Berlin, focuses on the question of what it “mean[s] for our world view if, according to Gödel, we also assume the non-existence of time”. In a physical world, the concept of time—has always been closely connected to the concept of causation. While Gödel himself regarded causation and time as two fundamental concepts in philosophy and metaphysics. In this paper, we now transfer the focus, which is usually seen in the connection between physical causation and time, into an epistemological setting, concentrating on the connection between logical inferences and the phenomenon of time.

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Notes

  1. 1.

    Kurt Gödel Papers, Box 11b, Folder 15, item accession 060168, on deposit with the Manuscripts Division, Department of Rare Books and Special Collections, Princeton University Library. Used with permission of the Institute for Advanced Study. Unpublished Copyright Institute for Advanced Study. All rights reserved. Transcriptions and translations by the author.

  2. 2.

    Physics II 3 and Metaphysics V 2

  3. 3.

    For detailed information about Aristotle’s Four Causes, see Falcon (2022).

  4. 4.

    Kurt Gödel Papers, Box 6b, Folder 69, item accession 030095. The book was written between November 1942 and March 1943.

  5. 5.

    Kurt Gödel Papers, Box 6b, Folder 67, item accession 030090. This book was written between May 1941 and April 1942.

  6. 6.

    In this case, the environment (i.e. the left-hand side of the judgement) is taken to be the empty set.

  7. 7.

    Kurt Gödel Papers, Box 6b, Folder 67, item accession 030091. The book was started in May 1942.

  8. 8.

    Gödel’s claim that intuitionistic logic is closely connected to the notion of knowledge is also strongly supported by the Gödel–McKinsey–Tarski translation (Gödel, 1933), which translates intuitionistic propositions into a (classical) modal logic S4, in which the usual provability operator may now be interpreted as a knowledge operator. Details about the modal language of knowledge can be found, for example, in van Benthem (2010, Chap. 12).

  9. 9.

    Kurt Gödel Papers, Box 6b, Folder 66, item accession 030089.

  10. 10.

    Here, H denotes some possible world. The “pieces of information” are represented by the propositional variables.

  11. 11.

    These problems seem to be supported by the fact that, in Łukasiewicz’s many-valued logic, the epistemological compatibility of conjunction and implication, expressed by the inequality \(p \wedge (p \rightarrow q) \le q\), is not valid. On the other hand, we have \(p \wedge q \le p \rightarrow q\). For details, see Lethen (2021a).

  12. 12.

    Kurt Gödel Papers, Box 12, Folder 43, item accession 060573.

  13. 13.

    The concept of branching time had been known to Gödel at least since 1935; see Lethen (2012b).

  14. 14.

    The reader may want to compare this quote to Fig.  12.3.

  15. 15.

    The first two items are marked as “analytisch” at the right margin.

  16. 16.

    We restrict ourselves to a very brief introduction here. For details, see J. Hindley and Seldin (2008) and Bimbo (2011).

  17. 17.

    As we do not consider the notion of strong reduction in this paper, we will simply speak of reduction.

  18. 18.

    The reader should note that the term \(\textsf{S}\textsf{K}\textsf{K}x\) reduces to x for any CL term x. It, thus, represents an identity operator with principal type \(a \rightarrow a\).

  19. 19.

    In the context of propositional logic, rule (pt) is often called rule D or condensed detachment.

  20. 20.

    Kurt Gödel Papers, Box 6b, Folder 70, item accession 030096. The book was written between March 1943 and January 1944.

  21. 21.

    Gödel adds the comment “ = ‘wenn’ ” above the second \(\supset \). (“wenn” means both “when” and “if.”)

  22. 22.

    Admissibility means that any proof of a proposition that uses this rule can be replaced by a proof without it.

  23. 23.

    For the difference between uni- and bi-directional unification, see Knight (1989).

  24. 24.

    Kurt Gödel Papers, Box 6a, Folder 59, item accession 030082.

  25. 25.

    Notebook Quantenmechanik II, Kurt Gödel Papers, Box 6b, Folder 78, item accession 030107.

  26. 26.

    For a critique, see, for example, Siedliñski (2017) and Ewert, Dembski, and Marks (2013). This is certainly not the place to give a comprehensive answer to this critique. Nevertheless, we rather prefer to have the question, whether the model is “inspiring either for computer scientists [\(\ldots \)] or for biologists” (Siedliñski, 2017, p. 143), be decided by the test of time.

  27. 27.

    Kurt Gödel Papers, Box 6b, Folder 77, item accession 030105.

  28. 28.

    Here, Chaitin follows the general idea that DNA may be regarded as software.

  29. 29.

    The instruction R moves a read/write head one position to the right. The instruction \(\lambda \) moves it one position to the left, after altering the symbol on the tape in a cyclic way. Parentheses indicate a loop that is executed as long as the current symbol on the tape is not the first one in the list of possible tape symbols.

  30. 30.

    A more elegant solution would consider the information content of an organism A, defined as the length of the shortest CL term, which reduces to A. As this measure is not computable, we stick to the more naive (but practical) solution.

  31. 31.

    The fact that Chaitin reduces his model to one single organism has frequently been criticised; see, for example, Siedliñski (2017). However, Bergson, (1911, Chap. 1) already argues: “Strictly speaking, there is nothing to prevent our imagining that the evolution of life might have taken place in one single individual by means of a series of transformations spread over thousands of ages. Or, instead of a single individual, any number might be supposed, succeeding each other in unilinear series”.

  32. 32.

    In epistemic logic, the veridicality property \(K_a\varphi \rightarrow \varphi \) states that what an agent a knows is always true, truth is included in the concept of knowledge.

  33. 33.

    In this connection, the reader may want to skip back to the closing remark of Sect. 12.4.

  34. 34.

    Gödel’s footnote: “erfordert Konzentration”

  35. 35.

    Gödel’s footnote: “requires concentration”

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Acknowledgements

The research for this article was conducted within the Gödeliana project led by Jan von Plato, Helsinki, Finland. The project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 787758) and from the Academy of Finland (Decision No. 318066).

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  1. Department of Philosophy, University of Helsinki, Helsinki, Finland

    Tim Lethen

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  1. Tim Lethen
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Editors and Affiliations

  1. Bergische Universität Wuppertal, Wuppertal, Deutschland

    Oliver Passon

  2. Universität Bamberg, Bamberg, Deutschland

    Christoph Benzmüller

  3. Technische Universität Dortmund, Dortmund, Deutschland

    Brigitte Falkenburg

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© 2023 Der/die Autor(en), exklusiv lizenziert an Springer-Verlag GmbH, DE, ein Teil von Springer Nature

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Lethen, T. (2023). How much Time Does a Logical Inference Take?. In: Passon, O., Benzmüller, C., Falkenburg, B. (eds) On Gödel and the Nonexistence of Time – Gödel und die Nichtexistenz der Zeit. Springer Spektrum, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-67045-3_12

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