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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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Cited by 102 (0 self)
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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
Intuitionistic modal logics as fragments of classical bimodal logics
, 1998
"... Gödel's translation of intuitionistic formulas into modal ones provides the wellknown embedding of intermediate logics into extensions of Lewis' system S4, which reflects and sometimes preserves such properties as decidability, Kripke completeness, the finite model property. In this paper we establ ..."
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Cited by 19 (5 self)
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Gödel's translation of intuitionistic formulas into modal ones provides the wellknown embedding of intermediate logics into extensions of Lewis' system S4, which reflects and sometimes preserves such properties as decidability, Kripke completeness, the finite model property. In this paper we establish a similar relationship between intuitionistic modal logics and classical bimodal logics. We also obtain some general results on the finite model property of intuitionistic modal logics first by proving them for bimodal logics and then using the preservation theorem.
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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Cited by 16 (4 self)
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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
Almost duplicationfree tableau calculi for propositional Lax logics
 In TABLEAUX'96
, 1996
"... In this paper we provide tableau calculi for the intuitionistic modal logics PLL and PLL 1 , where the calculus for PLL 1 is duplicationfree while among the rules for PLL there is just one rule that allows duplication of formulas. These logics have been investigated by Fairtlough and Mendler in re ..."
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Cited by 3 (0 self)
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In this paper we provide tableau calculi for the intuitionistic modal logics PLL and PLL 1 , where the calculus for PLL 1 is duplicationfree while among the rules for PLL there is just one rule that allows duplication of formulas. These logics have been investigated by Fairtlough and Mendler in relation to the problem of Formal Hardware Verification. In order to develop these calculi we extend to the modal case some ideas presented by Miglioli, Moscato and Ornaghi for intuitionistic logic. Namely, we enlarge the language containing the usual sings T and F with the new sign F c . PLL and PLL 1 logics are characterized by a Kripkesemantics which is a "weak" version of the semantics for ordinary intuitionistic modal logics. In this paper we establish the soundness and completeness theorems for these calculi.
2003, ‘On decidability of intuitionistic modal logics
 In: Third Workshop on Methods for Modalities
"... We prove a general decidability result for a class of intuitionistic modal logics. The proof is a slight modification of the Ganzinger, Meyer and Veanes [6] result on decidability of the two variable monadic guarded fragment of first order logic with constraints on the guard relations expressible in ..."
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We prove a general decidability result for a class of intuitionistic modal logics. The proof is a slight modification of the Ganzinger, Meyer and Veanes [6] result on decidability of the two variable monadic guarded fragment of first order logic with constraints on the guard relations expressible in monadic second order logic. 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
On the Finite Model Property of Intuitionistic Modal Logics over MIPC
, 1998
"... It is shown that every normal intuitionistic modal logic L over MIPC has the finite model property if there exists a universal firstorder sentence 8 such that (1) L is characterized by the class of Kripke frames satisfying 8 and (2) every Kripke frame that validates L satisfies 8. Here, MIPC is ..."
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It is shown that every normal intuitionistic modal logic L over MIPC has the finite model property if there exists a universal firstorder sentence 8 such that (1) L is characterized by the class of Kripke frames satisfying 8 and (2) every Kripke frame that validates L satisfies 8. Here, MIPC is a wellknown intuitionistic modal logic introduced by Prior (1957).
Information Frames, Implication Systems and Modalities
, 1996
"... We investigate the logical systems which result from introducing the modalities 2 and 3 into the family of substructural implication logics (including relevant, linear and intuitionistic implication). Our results lead to the formulation of a uniform labelled refutation system for these logics. 1 Int ..."
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We investigate the logical systems which result from introducing the modalities 2 and 3 into the family of substructural implication logics (including relevant, linear and intuitionistic implication). Our results lead to the formulation of a uniform labelled refutation system for these logics. 1 Introduction The classical theory of modalities is based on the notion of possible world. Possible worlds can be seen as maximally consistent sets (or innite conjunctions) of propositions. However, the idea of something being possible relative to or accessible from something else is by no means restricted to such a context. For instance, we may sensibly speak of a theory T 2 being \possible" relative to another theory T 1 , meaning that T 2 is obtained from T 1 by modifying some peripheral assumption which is not part of the \hardcore" of the theory. In this context, neither T 1 nor T 2 need to be maximally consistent sets of propositions: indeed, they are certainly not maximal and, perhaps...
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.