this paper for a discussion on the merits or otherwise of Kripke semantics and its "sufficiency" extension. Just as Kripke frames are dual to a class of Boolean algebras with modal operators [18, 24], one can build a duality for frames and Boolean algebras with sufficiency operators. Mixed structures occur when modal and sufficiency operators arise from the same accessibility relation. In this paper we introduce the classes of sufficiency algebras and that of mixed algebras which include both a modal and a sufficiency operator, and study representation and duality theory for these classes of algebras. We also give examples for classes of first-order definable frames, where such operators are required for a "modal-style" axiomatisation. 2 Why sufficiency and mixed algebras?

Monotonic modal logics form a generalization of normal modal logics...

We propose a generalization of Sahlqvist formulae to polyadic modal languages by representing modal polyadic languages in a combinatorial style and thus, in particular, developing what we believe to be the right approach to Sahlqvist formulae at all. The class of polyadic Sahlqvist formulae PSF defined here expands essentially the so far known one. We prove first-order definability and canonicity for the class PSF.

There exist modal logics that are validated by their canonical frames but are not sound and complete for any elementary class of frames. Continuum many such bimodal logics are exhibited, including one of each degree of unsolvability, and all with the finite model property. Monomodal examples are also constructed that extend K4 and are related to the proof of non-canonicity of the McKinsey axiom. We dedicate this paper to Max Cresswell, a pioneer in the study of canonicity, on the occasion of his 65th birthday. 1

This paper addresses the question for which varieties of boolean algebras with operators membership of an atomic algebra A is determined by its atom structure At A. We prove a positive answer for conjugated Sahlqvist varieties; we also show that the conjugation condition is necessary. As a corollary to the positive result and a recent result by I. Hodkinson, we prove that the variety RRA of representable relation algebras, although canonical, cannot be axiomatised by Sahlqvist equations. 1 1

This work explores the interconnections between a number of different perspectives on the formalisation of space. We begin with an informal discussion of the intuitions that motivate these formal representations.

Abstract. SCAN is an algorithm for reducing monadic existential second-order logic formulae to equivalent simpler formulae, often first-order logic formulae. It is provably impossible for such a reduction to first-order logic to be always successful, even if there is an equivalent first-order formula for a second-order logic formula. In this paper we show that SCAN successfully computes the first-order equivalents of all Sahlqvist formulae in the classical (multi-)modal language. 1

this paper we continue the development of algebraic counterparts of logics arising from information systems which we have begun in [5] and extend some results to reasoning about relative relations

This is a review of those aspects of the theory of varieties of Boolean algebras with operators (BAO's) that emphasise connections with modal logic and structural properties that are related to natural properties of logical systems.

Generalizing an example from Fine [1] and inspired by a theorem in J' onsson [4], we prove that any modal formula of the form ß(p q) $ ß(p) ß(q) (with ß(p) a positive formula) is canonical. We also prove that any such formula is strongly sound and complete with respect to an elementary class of frames, definable by a first order formula which can be read off from ß. 1 Introduction For quite a while now, modal logicians have been interested in the relation between first order logic and canonical modal formulas; recall that the latter are formulas that are valid on the underlying frame of the canonical model. Some very interesting connections have been discovered, but there are also some intriguing open problems. Examples of important results are Fine's Theorem (cf. [1]) that the modal logic of an elementary class of frames is canonical, and Sahlqvist's Theorem (cf. [6]) identifying a class of modal formulas each of which is canonical and corresponds to a first order formula which can...