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The ProofTheory and Semantics of Intuitionistic Modal Logic
, 1994
"... Possible world semantics underlies many of the applications of modal logic in computer science and philosophy. The standard theory arises from interpreting the semantic definitions in the ordinary metatheory of informal classical mathematics. If, however, the same semantic definitions are interpret ..."
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Possible world semantics underlies many of the applications of modal logic in computer science and philosophy. The standard theory arises from interpreting the semantic definitions in the ordinary metatheory of informal classical mathematics. If, however, the same semantic definitions are interpreted in an intuitionistic metatheory then the induced modal logics no longer satisfy certain intuitionistically invalid principles. This thesis investigates the intuitionistic modal logics that arise in this way. Natural deduction systems for various intuitionistic modal logics are presented. From one point of view, these systems are selfjustifying in that a possible world interpretation of the modalities can be read off directly from the inference rules. A technical justification is given by the faithfulness of translations into intuitionistic firstorder logic. It is also established that, in many cases, the natural deduction systems induce wellknown intuitionistic modal logics, previously given by Hilbertstyle axiomatizations. The main benefit of the natural deduction systems over axiomatizations is their
On the relation between intuitionistic and classical modal logics. Algebra and Logic
, 1996
"... Intuitionistic propositional logic Int and its extensions, known as intermediate or superintuitionistic logics, in many respects can be regarded just as fragments of classical modal logics containing S4. Atthe syntactical level, the Godel translation t embeds every intermediate logic L = Int+ into m ..."
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Intuitionistic propositional logic Int and its extensions, known as intermediate or superintuitionistic logics, in many respects can be regarded just as fragments of classical modal logics containing S4. Atthe syntactical level, the Godel translation t embeds every intermediate logic L = Int+ into modal log1 ics in the interval L = [ L = S4 t (); L=Grz t ()]. Semantically this is re ected by the fact that Heyting algebras are precisely the algebras of open elements of topological Boolean algebras. From the latticetheoretic standpoint the map is a homomorphism of the lattice of logics containing S4 onto the lattice of intermediate logics, while, according to the Blok{Esakia theorem, is an isomorphism of the latter onto the lattice of extensions of the Grzegorczyk system Grz. Atthe philosophical level the Godel translation provides a classical interpretation of the intuitionistic connectives. And from the technical point of view this embedding is a powerful tool for transferring various kinds of results from intermediate logics to modal ones and back via preservation theorems.
A Uniform Tableau Method for Intuitionistic Modal Logics I
 STUDIA LOGICA
, 1993
"... We present tableau systems and sequent calculi for the intuitionistic analogues IK, ID, IT, IKB, IKDB, IB, IK4, IKD4, IS4, IKB4, IK5, IKD5, IK45, IKD45 and IS5 of the normal classical modal logics. We provide soundness and completeness theorems with respect to the models of intuitionistic logic e ..."
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We present tableau systems and sequent calculi for the intuitionistic analogues IK, ID, IT, IKB, IKDB, IB, IK4, IKD4, IS4, IKB4, IK5, IKD5, IK45, IKD45 and IS5 of the normal classical modal logics. We provide soundness and completeness theorems with respect to the models of intuitionistic logic enriched by a modal accessibility relation, as proposed by G. Fischer
On logics with coimplication
 Journal of Philosophical Logic
, 1998
"... This paper investigates (modal) extensions of HeytingBrouwer logic, i.e., the logic which results when the dual of implication (alias coimplication) is added to the language of intuitionistic logic. We rst develop matrix as well as Kripke style semantics for those logics. Then, by extending the God ..."
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This paper investigates (modal) extensions of HeytingBrouwer logic, i.e., the logic which results when the dual of implication (alias coimplication) is added to the language of intuitionistic logic. We rst develop matrix as well as Kripke style semantics for those logics. Then, by extending the Godelembedding of intuitionistic logic into S4, itisshown that all (modal) extensions of HeytingBrouwer logic can be embedded into tense logics (with additional modal operators). An extension of the BlokEsakiaTheorem is proved for this embedding. 1
A Hybrid Intuitionistic Logic: Semantics and Decidability
 Journal of Logic and Computation
, 2005
"... An intuitionistic, hybrid modal logic suitable for reasoning about distribution of resources was introduced in [17, 18]. The modalities of the logic allow validation of properties in a particular place, in some place and in all places. We give a sound and complete Kripke semantics for the logic exte ..."
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An intuitionistic, hybrid modal logic suitable for reasoning about distribution of resources was introduced in [17, 18]. The modalities of the logic allow validation of properties in a particular place, in some place and in all places. We give a sound and complete Kripke semantics for the logic extended with disjunctive connectives. The extended logic can be seen as an instance of Hybrid IS5. We also give a sound and complete birelational semantics, and show that it enjoys the finite model property: if a judgement is not valid in the logic, then there is a finite birelational countermodel. Hence, we prove that the logic is decidable.
Knowledge on Treelike Spaces
 Studia Logica
, 1997
"... This paper presents a bimodal logic for reasoning about knowledge during knowledge acquisition. One of the modalities represents (effort during) nondeterministic time and the other represents knowledge. The semantics of this logic are treelike spaces which are a generalization of semantics used fo ..."
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This paper presents a bimodal logic for reasoning about knowledge during knowledge acquisition. One of the modalities represents (effort during) nondeterministic time and the other represents knowledge. The semantics of this logic are treelike spaces which are a generalization of semantics used for modeling branching time and historical necessity. A finite system of axiom schemes is shown to be canonically complete for the formentioned spaces. A characterization of the satisfaction relation implies the small model property and decidability for this system. 1
Topological Semantics and Bisimulations for Intuitionistic Modal Logics and Their Classical Companion Logics ⋆
"... Abstract. We take the wellknown intuitionistic modal logic of Fischer Servi with semantics in birelational Kripke frames, and give the natural extension to topological Kripke frames. Fischer Servi’s two interaction conditions relating the intuitionistic preorder (or partialorder) with the modal ..."
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Abstract. We take the wellknown intuitionistic modal logic of Fischer Servi with semantics in birelational Kripke frames, and give the natural extension to topological Kripke frames. Fischer Servi’s two interaction conditions relating the intuitionistic preorder (or partialorder) with the modal accessibility relation generalise to the requirement that the relation and its inverse be lower semicontinuous with respect to the topology. We then investigate the notion of topological bisimulation relations between topological Kripke frames, as introduced by Aiello and van Benthem, and show that their topologypreserving conditions are equivalent to the properties that the inverserelation and the relation are lower semicontinuous with respect to the topologies on the two models. Our first main result is that this notion of topological bisimulation yields semantic preservation w.r.t. topological Kripke models for both intuitionistic tense logics, and for their classical companion multimodal logics in the setting of the Gödel translation. After giving canonical topological Kripke models for the Hilbertstyle axiomatizations of the Fischer Servi logic and its classical multimodal companion logic, we show that the syntactic Gödel translation induces a natural semantic map from the intuitionistic canonical model into the canonical model of the classical companion logic, and this map is itself a topological bisimulation. 1
Products Of `transitive' Modal Logics Without The (abstract) Finite Model Property
"... It is well known that many twodimensional products of modal logics with at least one `transitive' (but not `symmetric') component lack the product finite model property. Here we show that products of two `transitive' logics (such as, e.g., K4 K4, S4 S4, GrzGrz and GLGL) do not have the (abstr ..."
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It is well known that many twodimensional products of modal logics with at least one `transitive' (but not `symmetric') component lack the product finite model property. Here we show that products of two `transitive' logics (such as, e.g., K4 K4, S4 S4, GrzGrz and GLGL) do not have the (abstract) finite model property either. These are the first known examples of 2D modal product logics without the finite model property where both components are natural unimodal logics having the finite model property.
Dualities for Some Intuitionistic Modal Logics
"... We present a duality for the intuitionistic modal logic IK introduced by Fischer Servi in [8, 9]. Unlike other dualities for IK reported in the literature (see for example [13]), the dual structures of the duality presented here are ordered topological spaces endowed with just one extra relation, wh ..."
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We present a duality for the intuitionistic modal logic IK introduced by Fischer Servi in [8, 9]. Unlike other dualities for IK reported in the literature (see for example [13]), the dual structures of the duality presented here are ordered topological spaces endowed with just one extra relation, which is used to define the settheoretic representation of both ✷ and ✸. Also, this duality naturally extends the definitions and techniques used by Fischer Servi in the proof of completeness for IK via canonical model construction [10]. We also give a parallel presentation of dualities for the intuitionistic modal logics IntK ✷ and IntK✸. Finally, we turn to the intuitionistic modal logic MIPC, which is an axiomatic extension of IK, and we give a very natural characterization of the dual spaces for MIPC introduced in [2] as a subcategory of the category of the dual spaces for IK introduced here.