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Logic KM: A Biography

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Leo Esakia on Duality in Modal and Intuitionistic Logics

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 4))

Abstract

This chapter is an attempt to collect under one roof all currently available facts related to logic \({\mathbf{KM }}\). Discovered as an equational class of the corresponding algebras, it has been developed as a natural intuitionistic counterpart of provability logic GL. We also outline the background, the work of thought, which had preceded and eventually had led to the birth of \({\mathbf{KM }}\). Where the results are new, the proofs are provided. Sometimes we derive conclusions, if they can be easily obtained from key results.

To the memory of my esteemed friend, Leo Esakia

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Notes

  1. 1.

    See the following comprehensive surveys [47, 71, 77, 80], as well as [82].

  2. 2.

    Although unrelated to the subject matter, which will be discussed below, the axiom \(A\rightarrow \square A\) had been used by Gödel in his ontological proof; see [32], pp. 431 and 435.

  3. 3.

    See a comprehensive account of provability logic in [4].

  4. 4.

    This was observed by Lindenbaum in 1934; see [75] for the proof of this property in a general setting. Church in his discussion [16], §29, on the history of substitution rule from Frege (1879) to von Neumann (1927), who proposed axiom schemata, does not mention this fact. However, see [16], §27, propositions \(^{*}270\) and \(^{*}271\).

  5. 5.

    Kripke’s result about this interconnection was reproduced in [13].

  6. 6.

    In the sequel, we will be omitting the sign \(\vdash \) while formulating a rule of inference.

  7. 7.

    See [74], Section 0.1, about the transformation of Hilbert-Bernays Derivability Conditions to Löb Derivability Conditions.

  8. 8.

    Kuznetsov was not formally a student at MGU and attended classes there as a freelance; see [61].

  9. 9.

    Unfortunately, Novikov’s approach along this line had been merely outlined in Kuztnetsov’s notes and was not included in the book at all.

  10. 10.

    All calculi in this chapter are assumed to define structural monotonic consequence operator.

  11. 11.

    Sambin proved that the identity \(\square x\le \square \square x\) is derivable from (7.1)–(7.3) (cf. [50]). Compare this with the elimination of \(\square p\rightarrow \square \square p\) as an axiom of GL in [12], Theorem 1.18.

  12. 12.

    Compare \(\mathbf D ^{\star }\) with \(D^{*}\) of [51].

  13. 13.

    Compare with Basic Working Hypothesis in [44, 45].

  14. 14.

    Indeed, one can prove that for any \({\fancyscript{L}_{\square \bigcirc }}\)-formula \(\alpha \), \( \mathbf{GL }\vdash s(\alpha ) \) if and only if \(\mathbf{GL }^{\bigcirc }\vdash \alpha \), where \( \mathbf{GL }^{\bigcirc }= \mathbf{GL }_{\square \bigcirc }\oplus \bigcirc p\leftrightarrow (p\wedge \square p)\).

  15. 15.

    This result was obtained in 1976 and first was announced in the abstract [43] with a subsequent publication in full detail in [44]. The preparation of the article [44] took eight months from October 1976 to June 1977, mainly due to Kuznetsov’s health instability and in part due to obtaining new results in the course of writing. Thus [44] had been submitted for publication in the summer of 1977; however, it took three more years for the editor of the collection to get it printed.

  16. 16.

    This system first had been considered by Sambin [65] in relation to effective fixed points in Magari algebras.

  17. 17.

    This notation was adapted by Kuznetsov from [6].

  18. 18.

    After having learned the story behind the footnote on p. 224 of [44], Leo Esakia began using the new name, \({\mathbf{KM }}\), instead of \(I^{\vartriangle }\), the first time in [26]. Thus Esakia can be regarded as the godfather of this logic.

  19. 19.

    A general setting for the separation property can be found in [17].

  20. 20.

    All these proofs were obtained independently and about the same time. The proof in [58] is algebraic, while the proof in [68, 70] uses syntactic means. Two last papers differ only in style.

  21. 21.

    Compare the fixed point property relative to all formulae with the Beth definability property for propositional logic (see e.g. [28]).

  22. 22.

    This is also a straightforward consequence of Lemma 2 in [60]: For any \({\mathbf{KM }}\)-algebra \(\mathfrak {A}\), there is a Magari algebra \(\mathfrak {B}\) such that \(\mathfrak {A}=\mathfrak {B}^{\square }\).

  23. 23.

    This property for the varieties of Heyting algebras has been known since the 1970s; see it implicitly (and with the use of Zorn’s lemma) in [83], Lemma 1, or explicitly (and without Zorn’s lemma) in [52], Lemma 1.

  24. 24.

    \(\text {d}(Y)\) is called the derived set of \(Y\).

  25. 25.

    This observation can be used in the proof of Corollaries 4.4.1 and 4.6.1.

  26. 26.

    The next proposition appeared as Corollary 2 in [41].

  27. 27.

    Indeed, one can prove that \(\mathbf{KM }\vdash A\) if and only if \(\mathbf{GL }^{\bigcirc }\vdash T(A) \).

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Acknowledgments

Over all period of the preparation of this text I have been in close contact with Alex Citkin. Discussions with him helped me understand better some key points and he provided me with useful references. Also, I am indebted to Grigori Mints for informing me that the last formula in Sect. 7.5 is not derivable in IntQ. It happened before I noticed Kuznetsov’s remark in [38]. Also, I am grateful to Srećko Kovač for drawing my attention to Gödel’s ontological proof (see Footnote 2).

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  1. Louisiana Scholars’ College, Northwestern State University, Natchitoches, LA, 71457, USA

    Alexei Muravitsky

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  1. New Mexico State University, Las Cruces, New Mexico, USA

    Guram Bezhanishvili

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Muravitsky, A. (2014). Logic KM: A Biography. In: Bezhanishvili, G. (eds) Leo Esakia on Duality in Modal and Intuitionistic Logics. Outstanding Contributions to Logic, vol 4. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-8860-1_7

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