Algorithmic correspondence and completeness in modal logic. IV. Semantic extensions of SQEMA
Journal of Applied Non-Classical Logics 18 (2-3):175-211 (2008)
Abstract
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.
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