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V.A. Yankov on Non-Classical Logics, History and Philosophy of Mathematics pp 71–124Cite as

Jankov Formulas and Axiomatization Techniques for Intermediate Logics

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Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 24))

Abstract

We discuss some of Jankov’s contributions to the study of intermediate logics, including the development of what have become known as Jankov formulas and a proof that there are continuum many intermediate logics. We also discuss how to generalize Jankov’s technique to develop axiomatization methods for all intermediate logics. These considerations result in what we term subframe and stable canonical formulas. Subframe canonical formulas are obtained by working with the disjunction-free reduct of Heyting algebras and are the algebraic counterpart of Zakharyaschev’s canonical formulas. On the other hand, stable canonical formulas are obtained by working with the implication-free reduct of Heyting algebras and are an alternative to subframe canonical formulas. We explain how to develop the standard and selective filtration methods algebraically to axiomatize intermediate logics by means of these formulas. Special cases of these formulas give rise to the classes of subframe and stable intermediate logics, and the algebraic account of filtration techniques can be used to prove that they all posses the finite model property (fmp). The fmp results about subframe and cofinal subframe logics yield algebraic proofs of the results of Fine and Zakharyaschev. We conclude by discussing the operations of subframization and stabilization of intermediate logics that this approach gives rise to.

In the English literature there are two competing spellings of Jankov’s name: Jankov and Yankov. We follow the former because this is more common usage in the mathematical literature. The latter usage is more common in the philosophical literature, and is commonly used in this volume.

We are very happy to be able to contribute to this volume dedicated to V.  A. Jankov. His work has been very influential for many generations of logicians, initially in the former Soviet Union, but eventually also abroad. In particular, it had a profound impact on our own research. While we have never met Professor Jankov in person, we have heard lots of interesting stories about him from our advisor Leo Esakia. Jankov is not only an outstanding logician, but also a role model citizen, who stood up against the Soviet regime. Because of this, he ended up in the Soviet political camps. A well-known Georgian dissident and human rights activist Levan Berdzenishvili spent several years there with Jankov. We refer to his memoirs (Berdzenishvili 2019) about the Soviet political camps of 1980s in general and about Jankov in particular. One chapter of the book “Vadim” (pp. 127–141) is dedicated to Jankov, in which he is characterized as follows: “I can say with certainty that in our political prison, Vadim Yankov, omnipotent and always ready to help, embodied in the pre-Internet era the combined capabilities of Google, Yahoo, and Wikipedia”.

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Notes

  1. 1.

    The results of Bezhanishvili and Bezhanishvili (2017) were obtained earlier than those in Bezhanishvili et al. (2016a). However, the latter appeared in print earlier than the former.

  2. 2.

    Again, the results of Bezhanishvili and Bezhanishvili (2017) were obtained earlier but appeared later than those in Bezhanishvili et al. (2016b).

  3. 3.

    While the assumption that A is finite is not essential, it suffices for our purposes.

  4. 4.

    This lemma is closely related to Chang and Keisler (1990, Proposition 2.1.8).

  5. 5.

    In Bezhanishvili and Bezhanishvili (2009, Definition 3.30) these morphisms were called well partial Esakia morphisms.

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Authors and Affiliations

  1. New Mexico State University, New Mexico, USA

    Guram Bezhanishvili

  2. University of Amsterdam, Amsterdam, Netherlands

    Nick Bezhanishvili

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  1. Guram Bezhanishvili
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Correspondence to Nick Bezhanishvili .

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Editors and Affiliations

  1. Metropolitan Telecommunications, New York, NY, USA

    Alex Citkin

  2. Hellenic Open University, Athens, Greece

    Ioannis M. Vandoulakis

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Bezhanishvili, G., Bezhanishvili, N. (2022). Jankov Formulas and Axiomatization Techniques for Intermediate Logics. In: Citkin, A., Vandoulakis, I.M. (eds) V.A. Yankov on Non-Classical Logics, History and Philosophy of Mathematics. Outstanding Contributions to Logic, vol 24. Springer, Cham. https://doi.org/10.1007/978-3-031-06843-0_4

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