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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 23 (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.
Logical Modalities and MultiStage Programming
, 1999
"... . Multistage programming is a method for improving the performance of programs through the introduction of controlled program specialization. This paper makes a case for multistage programming with open code and closed values. We argue that a simple language exploiting interactions between two log ..."
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Cited by 22 (13 self)
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. Multistage programming is a method for improving the performance of programs through the introduction of controlled program specialization. This paper makes a case for multistage programming with open code and closed values. We argue that a simple language exploiting interactions between two logical modalities is well suited for multistage programming, and report the results from our study of categorical models for multistage languages. Keywords: Multistage programming, categorical models, semantics, type systems (multilevel typed calculi) , combination of logics (modal and temporal). 1 Introduction Multistage programming is a method for improving the performance of programs through the introduction of controlled program specialization [15, 13]. MetaML was the first language designed specifically to support this method. It provides a type constructor for "code" and staging annotations for building, combining, and executing code, thus allowing the programmer to have finer cont...
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 19 (4 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...
A General Semantics for Evaluation Logic
 Fundamenta Informaticae
, 1994
"... The semantics of Evaluation Logic proposed in [14] relies on additional properties of monads. This paper proposes an alternative semantics, which drops all additional requirement on monads, at the expense of stronger assumptions on the underlying category. These assumptions are satised by any topos, ..."
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Cited by 19 (3 self)
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The semantics of Evaluation Logic proposed in [14] relies on additional properties of monads. This paper proposes an alternative semantics, which drops all additional requirement on monads, at the expense of stronger assumptions on the underlying category. These assumptions are satised by any topos, but not by the category of cpos. However, in the setting of Synthetic Domain Theory (see [7, 23]) it is possible to reconcile the needs of Denotational Semantics with those of Logic. 1 Introduction Evaluation logic (EL T ) is a typed predicate logic originally proposed by [18], which is based on the metalanguage ML T for computational monads (see [12]) and permits statements about the evaluation of programs to values by the use of evaluation modalities: necessity and possibility. In particular, EL T might be used for axiomatizing computationrelated properties of a monad or devising computationally adequate theories (see [18]), and it appears useful when addressing the question of logic...
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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Cited by 1 (0 self)
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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 First Order Modal Logic and its Sheaf Models
"... Abstract: We present a new way of formulating first order modal logic which circumvents the usual difficulties associated with variables changing their reference on moving between states. This formulation allows a very general notion of model (sheaf models). The key idea is the introduction of synta ..."
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Abstract: We present a new way of formulating first order modal logic which circumvents the usual difficulties associated with variables changing their reference on moving between states. This formulation allows a very general notion of model (sheaf models). The key idea is the introduction of syntax for describing relations between individuals in related states. This adds an extra degree of expressiveness to the logic, and also appears to provide a means of describing the dynamic behaviour of computational systems in a way somewhat different from traditional program logics. 1
ComplementTopoi and Dual Intuitionistic Logic
, 2010
"... Abstract: Mortensen in [11] studies dual intuitionistic logic by dualizing topos internal logic, but he did not study a sequent calculus. In this paper I present a sequent calculus for complementtopos logic, which throws some light on the problem of giving a dualization for LJ. 1 introductory remar ..."
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Abstract: Mortensen in [11] studies dual intuitionistic logic by dualizing topos internal logic, but he did not study a sequent calculus. In this paper I present a sequent calculus for complementtopos logic, which throws some light on the problem of giving a dualization for LJ. 1 introductory remarks “Dual intuitionistic logic ” has been investigated to varying degrees of success through different perspectives. McKinsey and Tarski in [8] studied it from an algebraic point of view through the algebraic properties of “closure ” or Brouwerian algebras, the algebraic duals to Heyting algebras. 1 Curry in [2] presented what he called “Skolem lattices”, which comprise “absolute implicational lattices” and “absolute subtractive lattices”. Rauszer (cf. [14], [15]) used algebraic, Hilbertstyle and relational methods, but not Gentzen calculi, to investigate “intuitionistic logic with dual operators”, “pseudodifference ” being the dual to intuitionistic implication. From then the problem of dualizing intuitionistic logic has been carried