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The Temporal Logic of Coalgebras via Galois Algebras
, 1999
"... This paper introduces a temporal logic for coalgebras. Nexttime and lasttime operators are dened for a coalgebra, acting on predicates on the state space. They give rise to what is called a Galois algebra. Galois algebras form models of temporal logics like Linear Temporal Logic (LTL) and Computatio ..."
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Cited by 35 (7 self)
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This paper introduces a temporal logic for coalgebras. Nexttime and lasttime operators are dened for a coalgebra, acting on predicates on the state space. They give rise to what is called a Galois algebra. Galois algebras form models of temporal logics like Linear Temporal Logic (LTL) and Computation Tree Logic (CTL). The mapping from coalgebras to Galois algebras turns out to be functorial, yielding indexed categorical structures. This gives many examples, for coalgebras of polynomial functors on sets. Additionally, it will be shown how \fuzzy" predicates on metric spaces, and predicates on presheaves, yield indexed Galois algebras, in basically the same coalgebraic manner. Keywords: Temporal logic, coalgebra, Galois connection, fuzzy predicate, presheaf Classication: 68Q60, 03G05, 03G25, 03G30 (AMS'91); D.2.4, F.3.1, F.4.1 (CR'98). 1 Introduction This paper combines the areas of coalgebra and of temporal logic. Coalgebras are simple mathematical structures (similar, but dual, to...
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.
Developing Theories of Types and Computability via Realizability
, 2000
"... We investigate the development of theories of types and computability via realizability. ..."
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Cited by 23 (6 self)
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We investigate the development of theories of types and computability via realizability.
Counterpart Semantics  A Foundational Study on Quantified Modal Logics
, 2002
"... Counterpart semantics is proposed as the appropriate semantical framework for a foundational investigation of quantified modal logics. It turns out to be a limit case of the categorical semantics of relational universes introduced by Ghilardi and Meloni in 1988. The main result is a deeper understan ..."
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Cited by 3 (0 self)
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Counterpart semantics is proposed as the appropriate semantical framework for a foundational investigation of quantified modal logics. It turns out to be a limit case of the categorical semantics of relational universes introduced by Ghilardi and Meloni in 1988. The main result is a deeper understanding of the interplay between substitution, quantification and identity wherever modalities are present. Languages with types and explicit substitutions are the tools used to clarify such an interplay and to disintangle classical problems related to modalities in firstorder languages. It is shown that controversial modal principles are neither valid nor provable. Quine's worries are dispelled.
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
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
Topological Completeness of FirstOrder Modal Logic
"... As McKinsey and Tarski [20] showed, the Stone representation theorem for Boolean algebras extends to algebras with operators to give topological semantics for (classical) propositional modal logic, in which the “necessity ” operation is modeled by taking the interior of an arbitrary subset of a topo ..."
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As McKinsey and Tarski [20] showed, the Stone representation theorem for Boolean algebras extends to algebras with operators to give topological semantics for (classical) propositional modal logic, in which the “necessity ” operation is modeled by taking the interior of an arbitrary subset of a topological space. This topological interpretation was recently extended in a natural way to arbitrary theories of full firstorder logic by Awodey and Kishida [3], using topological sheaves to interpret domains of quantification. This paper proves the system of full firstorder S4 modal logic to be deductively complete with respect to such extended topological semantics. The techniques employed are related to recent work in topos theory, but are new to systems of modal logic. They are general enough to also apply to other modal systems. Keywords: Firstorder modal logic, topological semantics, completeness.
Part II Local Realizability Toposes and a Modal Logic for
"... 5.1 Definition and Examples 5.1.1 Definition and Definability Results A tripos is a weak tripos with disjunction which has a (weak) generic object. Explicitly we define: ..."
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5.1 Definition and Examples 5.1.1 Definition and Definability Results A tripos is a weak tripos with disjunction which has a (weak) generic object. Explicitly we define: