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Dedicated to Bjarni Jónsson In this paper we consider distributive modal logic, a setting in which we may add modalities, such as classical types of modalities as well as weak forms of negation, to the fragment of classical propositional logic given by conjunction, disjunction, true, and false. For these logics we define both algebraic semantics, in the form of distributive modal algebras, and relational semantics, in the form of ordered Kripke structures. The main contributions of this paper lie in extending the notion of Sahlqvist axioms to our generalized setting and proving both a correspondence and a canonicity result for distributive modal logics axiomatized by Sahlqvist axioms. Our proof of the correspondence result relies on a reduction to the classical case, but our canonicity proof departs from the traditional style and uses the newly extended algebraic theory of canonical extensions.

The logic S5n is widely used as the logic of knowledge for ideal agents in a multi-agent system. Some extensions of S5n have been proposed for expressing knowledge sharing between the agents, but no systematic exploration of the possibilities has taken place. In this paper we present a spectrum of degrees of knowledge sharing by examining and classifying axioms expressing the sharing. We present completeness results and a diagram showing the relations between some of the principal extensions of S52 and discuss their usefulness. The paper considers the case of a group of two agents of knowledge.

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

A variety V of Boolean algebras with operators is singleton-persistent if it contains a complex algebra whenever it contains the subalgebra generated by the singletons. V is atom-canonical if it contains the complex algebra of the atom structure of any of the atomic members of V. This paper explores relationships between these “persistence ” properties and questions of whether V is generated by its complex algebras or its atomic members, or is closed under canonical embedding algebras or completions. It also develops a general theory of when operations involving complex algebras lead to the construction of elementary classes of

DAVID LEWIS proved in 1974 that all logics without iterative axioms are weakly complete. In this paper we extend LEWIS's ideas and provide a proof that such logics are canonical and so strongly complete. This paper also discusses the differences between relational and neighborhood frame semantics and poses a number of open questions about the latter. Canonicity for Intensional Logics Without Iterative Axioms Timothy J. Surendonk May 13, 1996 Keywords: Canonicity, strong completeness, non-normal modal logics, neighborhood frame semantics, non-iterative logics. Abstract DAVID LEWIS proved in 1974 that all logics without iterative axioms are weakly complete. In this paper we extend LEWIS's ideas and provide a proof that such logics are canonical and so strongly complete. This paper also discusses the differences between relational and neighborhood frame semantics and poses a number of open questions about the latter. 1 Introduction In 1974 DAVID LEWIS [13] showed that every intension...

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.

This paper looks at the concept of neighborhood canonicity introduced by BRIAN CHELLAS [2]. We follow the lead of the author's paper [8] where it was shown that every non-iterative logic is neighborhood canonical and here we will show that all logics whose axioms have a simple syntactic form---no intensional operator is in boolean combination with a propositional letter---and which have the finite model property are neighborhood canonical. One consequence of this is that KMcK, the McKinsey logic, is nieghborhood canonical, an interesting counterpoint to the results of ROBERT GOLDBLATT and XIAOPING WANG who showed, respectively, that KMcK is not relational canonical [7] and that KMcK is not relationally strongly complete [10]. Canonicity for Intensional Logics With Even Axioms Timothy J. Surendonk May 27, 1997 Keywords: Canonicity, strong completeness, non-normal modal logics, neighborhood frame semantics, uniform logics. Abstract This paper looks at the concept of neighborhood can...

This paper will remind the reader of neighborhood semantics for modal logics, compare them with relational semantics, and then look at some questions about neighborhood semantics that have been answered, and some which are still open. We will then introduce "ultrafilter semantics," a way of expressing all sets over a canonical frame in an `effable' way. This provides us with a conceptually easy way of dispatching some questions about intensional logics. In particular, we show that all non-iterative intensional logics are canonical and we go on to indicate how we can use ultrafilter semantics to demonstrate the canonicity of the Sahlqvist Logics. Neighborhoods, Ultrafilters, and Canonicity Timothy J. Surendonk September 20, 1996 Abstract This paper will remind the reader of neighborhood semantics for modal logics, compare them with relational semantics, and then look at some questions about neighborhood semantics that have been answered, and some which are still open. We will then i...

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...