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 meta-theory 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 self-justifying 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 first-order logic. It is also established that, in many cases, the natural deduction systems induce well-known intuitionistic modal logics, previously given by Hilbertstyle axiomatizations. The main benefit of the natural deduction systems over axiomatizations is their

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 two-dimensional generalization of Gentzen's sequents (2-sequents). 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...

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 Hilbert-style, 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.

. 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 IS4---our 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...

. 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...

We present !2 , a concise formulation of a proof term calculus for the intuitionistic modal logic S4 that is well-suited for practical applications. We show that, with respect to provability, it is equivalent to other formulations in the literature, sketch a simple type checking algorithm, and prove subject reduction and the existence of canonical forms for well-typed terms. Applications include a new formulation of natural deduction for intuitionistic linear logic, modal logical frameworks, and a logical analysis of staged computation and binding-time analysis for functional languages [6]. 1 Introduction Modal operators familiar from traditional logic have received renewed attention in computer science through their importance in linear logic. Typically, they are described axiomatically in the style of Hilbert or via sequent calculi. However, the Curry-Howard isomorphism between proofs and -terms is most poignant for natural deduction, so natural deduction formulations of modal and...

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 context-sensitive grammar due the condition on x#; c) rules with complex side conditions (sequent-style "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 Hilbert-style 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...

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, relation-preserving functions into the space, and the relational sections of a relational bundle are defined as the relation-preserving 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 quasi-toposes (thus modelling firstorder logic), and to have enough structure to model a certain firstorder modal logic described in a companion paper. 1

This work gives a natural deduction presentations to some of the sublogic of predicate logic [vB94] and also to some of Alechina and J. van Benthem's sublogics. These presentations are based on [BM92, Ben95, Gab94] and Gabbay's Labelled Deductive System LDS [Gab94]. It also proves some correspondence results for the sublogics presented in [vB94] and some completeness theorem to the classes of models presented in [Ale95]. 1 Introduction In J. van Benthem [vB94], predicate logic is treated as a many-modal dynamic logic, and it is well known that some of these logics are decidable. The motivation is to investigate sublogics of first order logic that are decidable and still have usual desirable meta-properties (Craig interpolation, Los-Tarski Preservation and etc). According to Tarki's semantics, the notion of satisfaction of an existential formula 9xff in a given model M for an assignment s is defined as: M; s j= 9xff iff for some d 2 jM j, M; s d x j= ff. Whilst in Dynamic Semantics i...

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 non--classical 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 non--classical logics (modal, intuitionistic, etc.). These logics have many useful applications in AI and Computer Science. Methodologies are mainly derived from the Gentzen Sequ...