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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
A Computational Interpretation of Modal Proofs
 Proof Theory of Modal Logics
, 1994
"... The usual (e.g. Prawitz's) treatment of natural deduction for modal logics involves a complicated rule for the introduction of the necessity, since the naive one does not allow normalization. We propose natural deduction systems for the positive fragments of the modal logics K, K4, KT, and S4, ..."
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Cited by 31 (2 self)
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The usual (e.g. Prawitz's) treatment of natural deduction for modal logics involves a complicated rule for the introduction of the necessity, since the naive one does not allow normalization. We propose natural deduction systems for the positive fragments of the modal logics K, K4, KT, and S4, extending previous work by Masini on a twodimensional generalization of Gentzen's sequents (2sequents). The modal rules closely match the standard rules for an universal quantifier and different logics are obtained with simple conditions on the elimination rule for 2. We give an explicit term calculus corresponding to proofs in these systems and, after defining a notion of reduction on terms, we prove its confluence and strong normalization. 1. Introduction Proof theory of modal logics, though largely studied since the fifties, has always been a delicate subject, the main reason being the apparent impossibility to obtain elegant, natural systems for intensional operators (with the excellent ex...
On an Intuitionistic Modal Logic
 Studia Logica
, 2001
"... . In this paper we consider an intuitionistic variant of the modal logic S4 (which we call IS4). The novelty of this paper is that we place particular importance on the natural deduction formulation of IS4our formulation has several important metatheoretic properties. In addition, we study models ..."
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Cited by 27 (6 self)
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. In this paper we consider an intuitionistic variant of the modal logic S4 (which we call IS4). The novelty of this paper is that we place particular importance on the natural deduction formulation of IS4our formulation has several important metatheoretic properties. In addition, we study models of IS4, not in the framework of Kripke semantics, but in the more general framework of category theory. This allows not only a more abstract definition of a whole class of models but also a means of modelling proofs as well as provability. 1. Introduction Modal logics are traditionally extensions of classical logic with new operators, or modalities, whose operation is intensional. Modal logics are most commonly justified by the provision of an intuitive semantics based upon `possible worlds', an idea originally due to Kripke. Kripke also provided a possible worlds semantics for intuitionistic logic, and so it is natural to consider intuitionistic logic extended with intensional modalities...
Intuitionistic Necessity Revisited
 PROCEEDINGS OF THE LOGIC AT WORK CONFERENCE
, 1996
"... In this paper we consider an intuitionistic modal logic, which we call IS42 . Our approach is different to others in that we favour the natural deduction and sequent calculus proof systems rather than the axiomatic, or Hilbertstyle, system. Our natural deduction formulation is simpler than other pr ..."
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Cited by 25 (7 self)
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In this paper we consider an intuitionistic modal logic, which we call IS42 . Our approach is different to others in that we favour the natural deduction and sequent calculus proof systems rather than the axiomatic, or Hilbertstyle, system. Our natural deduction formulation is simpler than other proposals. The traditional means of devising a modal logic is with reference to a model, and almost always, in terms of a Kripke model. Again our approach is different in that we favour categorical models. This facilitates not only a more abstract definition of a whole class of models but also a means of modelling proofs as well as provability.
Finitary Partial Inductive Definitions as a General Logic
, 1994
"... . We describe how the calculus of partial inductive definitions is used to represent logics. This calculus includes the powerful principle of definitional reflection. We describe two conceptually different approaches to representing a logic, both making essential use of definitional reflection. In t ..."
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Cited by 11 (1 self)
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. We describe how the calculus of partial inductive definitions is used to represent logics. This calculus includes the powerful principle of definitional reflection. We describe two conceptually different approaches to representing a logic, both making essential use of definitional reflection. In the deductive approach, the logic is defined by its inference rules. Only the succedent rules (in a sequent calculus setting  introduction rules in a natural deduction setting) need be given. The other rules are obtained implicitly using definitional reflection. In the semantic approach, the logic is defined using its valuation function. The latter approach often provides a more straightforward representation of logics with simple semantics but complicated proof systems. 1 Introduction: Finitary Partial Inductive Definitions We will describe how to use the calculus of partial inductive definitions as a general logic. That is, as a framework for representing various logics. Following common...
On the formalization of the modal µcalculus in the Calculus of Inductive Constructions
 Information and Computation
, 2000
"... This paper is part of an ongoing research programme at the Computer Science Department of the University of Udine on proof editors, started in 1992, based on HOAS encodings in dependent typed #calculus for program logics [15, 21, 16]. In this paper, we investigate the applicability of this approach ..."
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Cited by 6 (0 self)
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This paper is part of an ongoing research programme at the Computer Science Department of the University of Udine on proof editors, started in 1992, based on HOAS encodings in dependent typed #calculus for program logics [15, 21, 16]. In this paper, we investigate the applicability of this approach to the modal calculus. Due to its expressive power, we adopt the Calculus of Inductive Constructions (CIC), implemented in the system Coq. Beside its importance in the theory and verification of processes, the modal calculus is interesting also for its syntactic and proof theoretic peculiarities. These idiosyncrasies are mainly due to a) the negative arity of "" (i.e., the bound variable x ranges over the same syntactic class of x#); b) a contextsensitive grammar due the condition on x#; c) rules with complex side conditions (sequentstyle "proof " rules). These anomalies escape the "standard" representation paradigm of CIC; hence, we need to accommodate special techniques for enforcing these peculiarities. Moreover, since generated editors allow the user to reason "under assumptions", the designer of a proof editor for a given logic is urged to look for a Natural Deduction formulation of the system. Hence, we introduce a new proof system N # K in Natural Deduction style for K. This system should be more natural to use than traditional Hilbertstyle systems; moreover, it takes best advantage of the possibility of manipulating assumptions o#ered by CIC in order to implement the problematic substitution of formul for variables. In fact, substitutions are delayed as much as possible, and are kept in the derivation context by means of assumptions. This mechanism fits perfectly the stack discipline of assumptions of Natural Deduction, and it is neatly formalized in CIC. Bes...
Relational sheaves and predicate intuitionistic modal logic
, 1999
"... This paper generalises and adapts the theory of sheaves on a topological space to sheaves on a relational space: a topological space with a binary relation. The relational bundles on a relational space are defined as the continuous, relationpreserving functions into the space, and the relational se ..."
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This paper generalises and adapts the theory of sheaves on a topological space to sheaves on a relational space: a topological space with a binary relation. The relational bundles on a relational space are defined as the continuous, relationpreserving functions into the space, and the relational sections of a relational bundle are defined as the relationpreserving partial sections. This defines a functor to the category of presheaves on the space, which has a left adjoint. The presheaves which arise as the relational sections of a relational bundle are characterised by separation and patching conditions similar to those of a sheaf: we call them the relational sheaves. The relational bundles which arise from presheaves are characterised by local homeomorphism conditions: we call them the local relational homeomorphisms. The adjunction restricts to an equivalence between the categories of relational sheaves and local relational homeomorphisms. The paper goes on to investigate the structure of these equivalent categories. They are shown to be quasitoposes (thus modelling firstorder logic), and to have enough structure to model a certain firstorder modal logic described in a companion paper. 1
A Proof Search System for a Modal Substructural Logic Based on Labelled Deductive Systems
 Logic Programming and Automated Reasoning (LPAR'92), number 624 in LNAI (subseries of LNCS), St
, 1993
"... This paper describes a proof search system for a modal substructural (concatenation) logic based on Gabbay's Labelled Deductive System (LDS) as a case study. The logic combines resource (linear or Lambek Calculus) with modal features, and has applications in AI and natural language processing. ..."
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This paper describes a proof search system for a modal substructural (concatenation) logic based on Gabbay's Labelled Deductive System (LDS) as a case study. The logic combines resource (linear or Lambek Calculus) with modal features, and has applications in AI and natural language processing. We present axiomatic and LDS style proof theories and semantics for the logic, with soundness and completeness results. For nonclassical logic theorem proving, we show that LDS is flexible and generic, and can be mechanised directly. This partially verifies Gabbay's open claims that LDS is suitable to study combinations of logics. We believe our approach can be extended to any variant which combines substructurality and modality. 1 Introduction In recent years there has been considerable research on theorem proving in nonclassical logics (modal, intuitionistic, etc.). These logics have many useful applications in AI and Computer Science. Methodologies are mainly derived from the Gentzen Sequ...