We prove that modal logic formulated in a language with the cover modality is exponentially more succinct than the usual box-and-diamond version. In contrast with this, we show that adding the so-called public announcement operator to the latter results in a modal system that is exponentially more succinct than the one based on the cover modality.

Sahlqvist formulas are a syntactically specified class of modal formulas proposed by Hendrik Sahlqvist in 1975. They are important because of their first-order definability and canonicity, and hence axiomatize complete modal logics. The first-order properties definable by Sahlqvist formulas were syntactically characterized by Marcus Kracht in 1993. The present paper extends Kracht's theorem to the class of ‘generalized Sahlqvist formulas' introduced by Goranko and Vakarelov and describes an appropriate generalization of Kracht formulas.

We elaborate on semantically labelled syntax trees that provide a method of proving the non-existence of modal formulae satisfying certain syntactic properties and defining a given class of frames and use them to show that there are classes of Kripke frames that are definable by both non-Sahlqvist and Sahlqvist formulae, but the latter requires more propositional variables.

In the paper we present a technique for eliminating quantifiers of arbitrary order, in particular of first-order. Such a uniform treatment of the elimination problem has been problematic up to now, since techniques for eliminating first-order quantifiers do not scale up to higher-order contexts and those for eliminating higher-order quantifiers are usually based on a form of monotonicity w.r.t implication (set inclusion) and are not applicable to the first-order case. We make a shift to arbitrary relations “ordering” the underlying universe. (...) This allows us to incorporate background theories into higher-order quantifier elimination methods which, up to now, has not been achieved. The technique we propose subsumes many other results, including the Ackermann's lemma and various forms of fixpoint approaches when the “ordering” relations are interpreted as implication and reveals the common principle behind these approaches. (shrink)