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Leo Esakia on Duality in Modal and Intuitionistic Logics Outstanding Contributions to Logic

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Leo Esakia on Duality in Modal and Intuitionistic Logics Outstanding Contributions to Logic Outstanding Contributions to Logic 4 Guram Bezhanishvili Editor Leo Esakia on Duality in Modal and Intuitionistic Logics Outstanding Contributions to Logic Volume 4 Editor-in-Chief Sven Ove Hansson, Royal Institute of Technology, Stockholm, Sweden Editorial Board Marcus Kracht, Universität Bielefeld Lawrence Moss, Indiana University Sonja Smets, Universiteit van Amsterdam Heinrich Wansing, Ruhr-Universität Bochum For further volumes: http://www.springer.com/series/10033 Guram Bezhanishvili Editor Leo Esakia on Duality in Modal and Intuitionistic Logics 123 Editor Guram Bezhanishvili New Mexico State University Las Cruces, NM USA ISSN 2211-2758 ISSN 2211-2766 (electronic) ISBN 978-94-017-8859-5 ISBN 978-94-017-8860-1 (eBook) DOI 10.1007/978-94-017-8860-1 Springer Dordrecht Heidelberg New York London Library of Congress Control Number: 2014936452 Ó Springer Science+Business Media Dordrecht 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface This volume is dedicated to Leo Esakia’s contributions to the theory of modal and intuitionistic systems. Leo Esakia was one of the pioneers in developing duality theory for modal and intuitionistic logics, and masterfully utilizing it to obtain some major results in the area. The volume consists of 10 chapters, written by leading experts, that discuss Leo’s original contributions and consequent devel- opments that have shaped the current state of the field. I would like to express sincere gratitude to the authors as well as to the referees without whose outstanding job the volume would not have been possible. It is my belief that the volume will serve as an excellent tribute to Leo Esakia’s pioneering achievements in developing algebraic and topological semantics of modal and intuitionistic logics, which have paved the way for the next generations of researchers interested in this area. Guram Bezhanishvili v Contents 1 Esakia’s Biography and Bibliography ..................... 1 2 Canonical Extensions, Esakia Spaces, and Universal Models .... 9 Mai Gehrke 3 Free Modal Algebras Revisited: The Step-by-Step Method...... 43 Nick Bezhanishvili, Silvio Ghilardi and Mamuka Jibladze 4 Easkia Duality and Its Extensions ........................ 63 Sergio A. Celani and Ramon Jansana 5 On the Blok-Esakia Theorem ........................... 99 Frank Wolter and Michael Zakharyaschev 6 Modal Logic and the Vietoris Functor..................... 119 Yde Venema and Jacob Vosmaer 7 Logic KM: A Biography ............................... 155 Alexei Muravitsky 8 Constructive Modalities with Provability Smack ............. 187 Tadeusz Litak 9 Cantor-Bendixson Properties of the Assembly of a Frame ...... 217 Harold Simmons 10 Topological Interpretations of Provability Logic ............. 257 Lev Beklemishev and David Gabelaia 11 Derivational Modal Logics with the Difference Modality ....... 291 Andrey Kudinov and Valentin Shehtman vii Introduction Leo Esakia’s lifelong passion for modal and intuitionistic logics started to develop in the 1960s. Soon after it became apparent that Kripke semantics [28], although very attractive and intuitive, was not adequate for handling large classes of modal logics (the phenomenon of Kripke incompleteness). It was already understood that Kripke frames provide a nice representation for modal algebras, but a modal algebra can in general be realized only as a subalgebra of the modal algebra arising from a Kripke frame. Leo’s main interest at the time was Gödel’s translation [23] of the intuitionistic propositional logic Int into Lewis’ modal system S4, and the corresponding classes of Heyting algebras and S4-algebras. Influenced by the work of Stone [38, 39], Tarski (and his collaborators McKinsey and Jónsson) [26, 27, 30–32], and Halmos [25], Leo realized that the missing link between the algebraic and relational semantics of these systems is topology. This yielded the notion of what we now call (quasi-ordered) Esakia spaces (namely quasi-ordered Stone spaces with additional properties) and the representation of S4-algebras as the algebras of clopen subsets of Esakia spaces. This representation extends to full duality between the categories of S4-algebras and Esakia spaces. In his discussions with Sikorski, Leo also realized an apparent need for duality for Heyting algebras. He was able to obtain such a duality as a particular case of his duality for S4-algebras, thus obtaining a powerful machinery to study modal logics over S4 and superintuitionistic logics (extensions of Int). These ground-breaking results were published in Esakia’s 1974 paper [10], which remains one of the most cited papers by Leo. Around the same time (mid 1970s), Goldblatt and Thomason came to the same realization, and developed what later became known as the descriptive frame semantics for modal logic. These findings were published in Goldblatt [21, 22]. Note that although Esakia worked with quasi-ordered Stone Spaces, replacing a quasi-order with an arbitrary binary relation in an Esakia space yields the descriptive frame semantics of Goldblatt and Thomason. The machinery Leo developed was powerful in many respects. In particular, what we now call the Esakia lemma was a consequence of his duality (in fact, Leo developed the lemma to obtain the morphism correspondence of his duality). As ix x Introduction was shown by Sambin and Vaccaro [35] it plays a crucial role in developing the Sahlqvist completeness and correspondence in modal logic. Subsequently, many generalizations of Sahlqvist’s theorem have been obtained that utilize Esakia’s lemma. The volume opens with the chapter by Mai Gehrke which discusses Esakia duality for S4-algebras, and how to derive Esakia duality for Heyting algebras from it. Gehrke provides a more general setting for this approach, which also yields the celebrated Priestley duality for bounded distributive lattices [33, 34]. All this is done utilizing the theory of canonical extensions, a very active field of research of today. Gehrke also discusses Esakia’s lemma and gives a modern account of how to construct free finitely generated Heyting algebras and their Esakia duals. The dual description of free finitely generated Heyting algebras and S4-algebras was initiated by Esakia and his student Grigolia in the mid 1970s. They developed the so-called coloring technique [19, 20] which became very useful in describing ‘‘upper-parts’’ of the dual spaces of the free finitely generated Heyting and modal algebras. This important topic was further developed in the 1980s by Shehtman, Rybakov, Grigolia, and Belissima. In the 1990s, Ghilardi published a series of papers which gave a novel perspective on the topic. This paved the way for the follow-on papers by N. Bezhanishvili, A. Kurz, M. Gehrke, D. Coumans, S. van Gool, and others. An up-to-date survey of this topic is given in the chapter by Nick Bezhanishvili, Silvio Ghilardi, and Mamuka Jibladze. Over the years, several generalizations of Esakia duality have been developed. To name a few, Leo himself generalized his duality to the setting of bi-Heyting algebras and temporal algebras [11, 13] (see also F. Wolter [40]), G. Bezhanishvili generalized Esakia duality to monadic Heyting algebras [1], S. Celani and R. Jansana generalized it to weak Heyting algebras [9], and G. Bezhanishvili and R. Jansana to implicative semilattices [3]. The chapter by Sergio Celani and Ramon Jansana discusses Esakia duality for Heyting algebras and its generaliza- tions to weak Heyting algebras and implicative semilattices. It also discusses how to obtain the duals of maps between Heyting algebras that only preserve part of the Heyting algebra structure. These turn out to be partial Esakia morphisms that play a crucial role in developing Zakharyaschev’s canonical formulas, which provide an axiomatization of superintuitionistic
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