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A Sahlqvist theorem for distributive modal logic
 Annals of Pure and Applied Logic 131, Issues
, 2002
"... 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 ..."
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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.
How completeness and correspondence theory got married
 Diamonds and Defaults, Synthese
, 1993
"... It has been said that modal logic consists of three main disciplines: duality theory, completeness theory and correspondence theory; and that they are the pillars on which this edifice called modal logic rests. This seems to be true if one looks at the history of modal logic, for all three discipli ..."
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Cited by 29 (5 self)
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It has been said that modal logic consists of three main disciplines: duality theory, completeness theory and correspondence theory; and that they are the pillars on which this edifice called modal logic rests. This seems to be true if one looks at the history of modal logic, for all three disciplines have been explicitly defined around the same time, namely
Y.: Sahlqvist’s theorem for Boolean algebras with operators with an application to cylindric algebras. Studia Logica 54
, 1995
"... with an Application ..."
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A spectrum of modes of knowledge sharing between agents
 Intelligent Agents VI: Agent Theories, Architectures, and Languages, Lecture Notes in Artificial Intelligence 1757
, 2000
"... Abstract. The logic S5 is widely used as the logic of knowledge for ideal agents in a multiagent system. Some extensions of S5 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 spec ..."
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Abstract. The logic S5 is widely used as the logic of knowledge for ideal agents in a multiagent system. Some extensions of S5 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 S5 and discuss their usefulness. The paper considers the case of a group of two agents of knowledge. 1
On canonical modal logics that are not elementarily determined. Logique et Analyse
 181:77— 101, 2003. Published October 2004. 20 Robert Goldblatt, Ian Hodkinson, and Yde
, 2004
"... 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 als ..."
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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 noncanonicity of the McKinsey axiom. 1
Canonicity for Intensional Logics without Iterative Axioms
 JOURNAL OF PHILOSOPHICAL LOGIC
, 1996
"... 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 semanti ..."
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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.
Persistence and atomic generation for varieties of Boolean algebras with operators
 STUDIA LOGICA
, 2001
"... A variety V of Boolean algebras with operators is singletonpersistent if it contains a complex algebra whenever it contains the subalgebra generated by the singletons. V is atomcanonical if it contains the complex algebra of the atom structure of any of the atomic members of V. This paper explores ..."
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A variety V of Boolean algebras with operators is singletonpersistent if it contains a complex algebra whenever it contains the subalgebra generated by the singletons. V is atomcanonical 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
Canonical PseudoCorrespondence
 Advances in Modal Logic Volume 2
, 1998
"... 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 ..."
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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...
BINARY SUBTREES WITH FEW LABELED PATHS
"... We prove several quantitative Ramseyan results involving ternary complete trees with {0, 1}labeled edges where we attempt to nd a complete binary subtree with as few labels as possible along its paths. One of these is used to answer a question of Simpson's in computability theory; we show tha ..."
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We prove several quantitative Ramseyan results involving ternary complete trees with {0, 1}labeled edges where we attempt to nd a complete binary subtree with as few labels as possible along its paths. One of these is used to answer a question of Simpson's in computability theory; we show that there is a bounded Π 0 1 class of positive measure which is not strongly (Medvedev) reducible to DNR3; in fact, the class of 1random reals is not strongly reducible to DNR3.