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Logic Journal of IGPL Advance Access originally published online on September 9, 2007
Logic Journal of IGPL 2008 16(1):43-73; doi:10.1093/jigpal/jzm018
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© The Author, 2007. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Term Definable Classes of Boolean Functions and Frame Definability in Modal Logic

Miguel Couceiro 1

Department of Mathematics, Statistics and Philosophy, University of Tampere, Kalevantie 4, 33014 Tampere, Finland. E-mail: miguel.couceiro{at}uta.fi

Lauri Hella 2

Department of Mathematics, Statistics and Philosophy, University of Tampere, Kalevantie 4, 33014 Tampere, Finland. E-mail: lauri.hella{at}uta.fi

Jari Kivelä 3

Department of Mathematics, Statistics and Philosophy, University of Tampere, Kalevantie 4, 33014 Tampere, Finland. E-mail: jari.j.kivela{at}uta.fi


  Abstract

We establish a connection between term definability of Boolean functions and definability of finite modal frames. We introduce a bijective translation between functional terms and uniform degree-1 formulas and show that a class of Boolean functions is defined by functional terms if and only if the corresponding class of Scott-Montague frames is defined by the translations of these functional terms, and vice versa. As a special case, we get that the clone {Lambda}1 of all conjunctions corresponds to the class of all Kripke frames. We also characterize some classes of Scott-Montague frames corresponding to subclones of {Lambda}1 by restricting the class of Kripke frames in a natural way. Furthermore, by modifying Kripke semantics, we extend our results to correspondences between linear clones and classes of Kripke frames equipped with modified semantics.

Key Words: Boolean functions • functional terms • term definable classes • clones • propositional logic • modal logic • modal axioms • Kripke frames • Scott-Montague frames

Received for publication 17 January 2006. Revision received 27 February 2006.

1The work of the first author was partially supported by the Graduate School in Mathematical Logic MALJA and by the grant #28139 from the Academy of Finland.

2The work of the second author was partially supported by the grant #28139 from the Academy of Finland.

3The work of the third author was supported by the grant #28139 from the Academy of Finland.


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