Spatial reasoning is essential for many AI applications. In most existing systems the representation is primarily numerical, so the information that can be handled is limited to precise quantitative data. However, for many purposes the ability to manipulate high-level qualitative spatial information in a flexible way would be extremely useful. Such capabilities can be proveded by logical calculi; and indeed 1st-order theories of certain spatial relations have been given [20]. But computing inferences in 1st-order logic is generally intractable unless special (domain dependent) methods are known. 0-order modal logics provide an alternative representation which is more expressive than classical 0-order logic and yet often more amenable to automated deduction than 1st-order formalisms. These calculi are usually interpreted as propositional logics: non-logical constants are taken as denoting propositions. However, they can also be given a nominal interpretation in which the constants stand...

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The main goal of this paper is to explain the link between the algebraic and the Kripke-style models for certain classes of propositional logics. We start by presenting a Priestley-type duality for distributive lattices endowed with a general class of well-behaved operators. We then show that finitely-generated varieties of distributive lattices with operators are closed under canonical embedding algebras. The results are used in the second part of the paper to construct topological and non-topological Kripke-style models for logics that are sound and complete with respect to varieties of distributive lattices with operators in the above-mentioned classes. Introduction In the study of non-classical propositional logics (and especially of modal logics) there are two main ways of defining interpretations or models. One possibility is to use algebras -- usually lattices with operators -- as models. Propositional variables are interpreted over elements of these algebraic models, an...

This study examines BDI logics in the context of Gabbay's fibring semantics. We show that dovetailing (a special form of fibring) can be adopted as a semantic methodology to combine BDI logics. We develop a set of interaction axioms that can capture static as well as dynamic aspects of the mental states in BDI systems, using Catach's incestual schema $G^{a, b, c, d}$. Further we exemplify the constraints required on fibring function to capture the semantics of interactions among modalities. The advantages of having a fibred approach is discussed in the final section.

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

The main goal of this paper is to show that the Priestley duality for SHn-algebras can help to establish a link between the algebraic and Kripke-style semantics for SHn- logics. We present a Priestley duality theorem for SHn-algebras, and note that the dual space of an SHn-algebra satis es in particular the properties of a Kripke model for SHn-logics. We then show that Priestley duality can help in proving the soundness and completeness of SHn-logics with respect to the class of SHn-frames in a direct way, by using only soundness and completeness of SHn-logics with respect to the variety of SHn-algebras.

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.

The variety MBA of monadic bounded algebras consists of Boolean algebras with a distinguished element E, thought of as an existence predicate, and an operator ∃ reflecting the properties of the existential quantifier in free logic. This variety is generated by a certain class FMBA of algebras isomorphic to ones whose elements are propositional functions. We show that FMBA is characterised by the disjunction of the equations ∃E = 1 and ∃E = 0. We also define a weaker notion of “relatively functional ” algebra, and show that every member of MBA is isomorphic to a relatively functional one. In [1], an equationally defined class MBA of monadic bounded algebras was introduced. Each of these algebras comprises a Boolean algebra B with a distinguished element E, thought of as an existence predicate, and an operator ∃ on B reflecting the properties of the existential quantifier in logic without existence assumptions. MBA was shown to be generated by a certain proper

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Algebraic Logic. In preparation. Manuscript.