## Relational sheaves and predicate intuitionistic modal logic (1999)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Hilken99relationalsheaves,

author = {Barnaby P. Hilken},

title = {Relational sheaves and predicate intuitionistic modal logic},

year = {1999}

}

### OpenURL

### Abstract

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

### Citations

156 |
A semantical analysis of modal logic i: Normal modal propositional calculi. Zeitschrift fur Mathematische Logic und Grundlagen der Mathematik 9:67–97
- Kripke
- 1963
(Show Context)
Citation Context ...hus modelling firstorder logic), and to have enough structure to model a certain firstorder modal logic described in a companion paper. 1 Introduction The Kripke “many world” semantics of modal logic =-=[11]-=- models the modal connectives ✷ and ✸ in terms of a set S of “possible worlds” (or “states”) 1sand a binary relation ↠ of “accessibility” between worlds (or “transition” between states). The truth of ... |

98 |
Quantification in modal logic
- Garson
- 1977
(Show Context)
Citation Context ... which world each variable should be interpreted in, and which set Dt each quantifier should range over. In order to avoid ambiguity and make the semantics coherent, most approaches (see, for example =-=[3, 5]-=-) restrict the relations ↠ts to subset inclusions Dt ⊆ Ds. Our approach to this problem is to change the syntax (see [8] for details), thus allowing ourselves a more general class of models. The struc... |

22 |
Topology and duality in modal logic
- Sambin, Vaccaro
- 1988
(Show Context)
Citation Context ...d in this paper, and we answer it by developing a notion of “relational sheaf.” Various other approaches to predicate intuitionistic modal logic have been proposed in the literature (see, for example =-=[12, 13, 14, 1, 4, 6]-=-), often of a topological or category-theoretic nature. However, they are usually either 2slimited to special cases (such as S4 modality or constant domains) or else have a very ad-hoc feel: the struc... |

20 |
Intuitionistic tense and modal logic
- Ewald
- 1986
(Show Context)
Citation Context ...d in this paper, and we answer it by developing a notion of “relational sheaf.” Various other approaches to predicate intuitionistic modal logic have been proposed in the literature (see, for example =-=[12, 13, 14, 1, 4, 6]-=-), often of a topological or category-theoretic nature. However, they are usually either 2slimited to special cases (such as S4 modality or constant domains) or else have a very ad-hoc feel: the struc... |

16 |
Handbook of Categorical Algebra 3: Categories of Sheaves. Encyclopedia of Mathematics and its Applications 52
- Borceux
- 1994
(Show Context)
Citation Context ...st. That they have a rich mathematical theory (similar to that of sheaves) is one of the conclusions of this paper. Our development of the theory follows that of traditional sheaf theory (see [15] or =-=[2]-=-). In Section 2 we describe bundles and presheaves over a relational space. We show that the “relational sections” of a bundle form a presheaf, that the fibres of a presheaf form a bundle, and that th... |

15 |
Completeness results for intuitionistic and modal logic in a categorical setting, Annals od Pure and Applied Logic, 72
- Makkai, Reyes
- 1995
(Show Context)
Citation Context ...d in this paper, and we answer it by developing a notion of “relational sheaf.” Various other approaches to predicate intuitionistic modal logic have been proposed in the literature (see, for example =-=[12, 13, 14, 1, 4, 6]-=-), often of a topological or category-theoretic nature. However, they are usually either 2slimited to special cases (such as S4 modality or constant domains) or else have a very ad-hoc feel: the struc... |

13 |
Modal and Tense Predicate Logic: Models in Presheaves and Categorical Conceptualization
- Ghilardi, Meloni
- 1987
(Show Context)
Citation Context |

11 |
A constructive presentation for the modal connective of necessity
- Benevides, Maibaum
- 1992
(Show Context)
Citation Context |

10 | Topological duality for intuitionistic modal algebras
- Hilken
(Show Context)
Citation Context ...ion ↠ ⊆ S × S; for each t ∈ S a set Dt; for each related pair t ↠ s a relation ↠ts ⊆ Dt × Ds. The aim of this paper is to adapt this structure to the intuitionistic case. In the author’s recent paper =-=[7]-=-, the Kripke semantics of propositional modal logic is extended to intuitionistic modal logic by considering a topology on the set of worlds, and interpreting propositions as open sets in the topology... |

10 |
Topos-Theoretic Approaches to Modality
- Reyes, Zolfaghari
- 1990
(Show Context)
Citation Context |

5 |
Semantical aspects of quantified modal logic
- Corsi, Ghilardi
- 1992
(Show Context)
Citation Context ... which world each variable should be interpreted in, and which set Dt each quantifier should range over. In order to avoid ambiguity and make the semantics coherent, most approaches (see, for example =-=[3, 5]-=-) restrict the relations ↠ts to subset inclusions Dt ⊆ Ds. Our approach to this problem is to change the syntax (see [8] for details), thus allowing ourselves a more general class of models. The struc... |

5 | Topos Theory. Number 10 in L.M.S. Monographs - Johnstone - 1977 |

1 |
A first-order modal logic and its sheaf models. Draft paper in preparation
- Hilken, Rydeheard
(Show Context)
Citation Context ...ambiguity and make the semantics coherent, most approaches (see, for example [3, 5]) restrict the relations ↠ts to subset inclusions Dt ⊆ Ds. Our approach to this problem is to change the syntax (see =-=[8]-=- for details), thus allowing ourselves a more general class of models. The structure used to model first-order modal logic therefore consists of: a set S; a relation ↠ ⊆ S × S; for each t ∈ S a set Dt... |

1 |
Sheaf Theory. Number 20
- Tennison
- 1975
(Show Context)
Citation Context ...c interest. That they have a rich mathematical theory (similar to that of sheaves) is one of the conclusions of this paper. Our development of the theory follows that of traditional sheaf theory (see =-=[15]-=- or [2]). In Section 2 we describe bundles and presheaves over a relational space. We show that the “relational sections” of a bundle form a presheaf, that the fibres of a presheaf form a bundle, and ... |