Journal of Applied Non-Classical Logics 18 (2-3):175-211 (2008)

Valentin Goranko
Stockholm University
In a previous work we introduced the algorithm \SQEMA\ for computing first-order equivalents and proving canonicity of modal formulae, and thus established a very general correspondence and canonical completeness result. \SQEMA\ is based on transformation rules, the most important of which employs a modal version of a result by Ackermann that enables elimination of an existentially quantified predicate variable in a formula, provided a certain negative polarity condition on that variable is satisfied. In this paper we develop several extensions of \SQEMA\ where that syntactic condition is replaced by a semantic one, viz. downward monotonicity. For the first, and most general, extension \SSQEMA\ we prove correctness for a large class of modal formulae containing an extension of the Sahlqvist formulae, defined by replacing polarity with monotonicity. By employing a special modal version of Lyndon's monotonicity theorem and imposing additional requirements on the Ackermann rule we obtain restricted versions of \SSQEMA\ which guarantee canonicity, too.
Keywords modal logic  Algorithmic correspondence theory  canonicity  algorithm SQEMA
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DOI 10.3166/JANCL.18.175-211
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Modal Logic.Patrick Blackburn, Maarten de Rijke & Yde Venema - 2002 - Cambridge University Press.
Modal Logic.Alexander Chagrov - 1997 - Oxford University Press.
An Algebraic Theory of Normal Forms.Silvio Ghilardi - 1995 - Annals of Pure and Applied Logic 71 (3):189-245.

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