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Free Modal Algebras Revisited: The Step-by-Step Method

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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

We review the step-by-step method of constructing finitely generated free modal algebras. First we discuss the global step-by-step method, which works well for rank one modal logics. Next we refine the global step-by-step method to obtain the local step-by-step method, which is applicable beyond rank one modal logics. In particular, we show that it works well for constructing the finitely generated free algebras for such well-known modal systems as T, K4 and S4. This yields the notions of one-step algebras and of one-step frames, as well as of universal one-step extensions of one-step algebras and of one-step frames. We show that finitely generated free algebras for T, K4 and S4 and their dual spaces can be obtained by iterating the universal one-step extensions of one-step algebras and of one-step frames. In the final part of the chapter we compare our construction with recent literature, especially with [11] which undertakes a very similar approach.

In memory of Leo Esakia and Dito Pataraia

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Acknowledgments

The authors are very grateful to the referee for many useful suggestions that improved the presentation of the paper. They would also like to express special thanks to Guram Bezhanishvili for his invaluable help and encouragement in writing this paper. The first author would like to acknowledge the support of the Netherlands Organisation for Scientific Research under grant 639.032.918. The first and third authors would also like to acknowledge the support of the Rustaveli Science Foundation of Georgia under grant FR/489/5-105/11. The second author acknowledges the support of the PRIN 2010-2011 project "Logical Methods for Information Management" funded by the Italian Ministry of Education, University and Research (MIUR).

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Authors and Affiliations

  1. Institute for Logic, Language and Computation, University of Amsterdam, Amsterdam, The Netherlands

    Nick Bezhanishvili

  2. Department of Mathematics, Universita degli Studi di Milano, Milan, Italy

    Silvio Ghilardi

  3. Razmadze Mathematical Institute, Tbilisi State University, Tbilisi, Georgia

    Mamuka Jibladze

Authors
  1. Nick Bezhanishvili
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  2. Silvio Ghilardi
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  3. Mamuka Jibladze
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Correspondence to Nick Bezhanishvili .

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Editors and Affiliations

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

    Guram Bezhanishvili

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Bezhanishvili, N., Ghilardi, S., Jibladze, M. (2014). Free Modal Algebras Revisited: The Step-by-Step Method. 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_3

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