Kishida, Kohei (2011) *Generalized Topological Semantics for First-Order Modal Logic.* Doctoral Dissertation, University of Pittsburgh.

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## Abstract

This dissertation provides a new semantics for first-order modal logic. It is philosophicallymotivated by the epistemic reading of modal operators and, in particular, three desiderata in the analysis of epistemic modalities.(i) The semantic modelling of epistemic modalities, in particular verifiability and falsifiability, cannot be properly achieved by Kripke's relational notion of accessibility. It requires instead a more general, topological notion of accessibility.(ii) Also, the epistemic reading of modal operators seems to require that we combine modal logic with fully classical first-order logic. For this purpose, however, Kripke's semantics for quantified modal logic is inadequate; its logic is free logic as opposed to classical logic.(iii) More importantly, Kripke's semantics comes with a restriction that is too strong to let us semantically express, for instance, that the identity of Hesperus and Phosphorus, even if metaphysically necessary, can still be a matter of epistemic discovery.To provide a semantics that accommodates the three desiderata, I show, on the one hand, howthe desideratum (i) can be achieved with topological semantics, and more generally neighborhood semantics, for propositional modal logic. On the other hand, to achieve (ii) and (iii), it turns out that David Lewis's counterpart theory is helpful at least technically. Even though Lewis's ownformulation is too liberal---in contrast to Kripke's being too restrictive---to achieve our goals, this dissertation provides a unification of the two frameworks, Kripke's and Lewis's. Through a series of both formal and conceptual comparisons of their ontologies and semantic ideas, it is shown that structures called sheaves are needed to unify the ideas and achieve the desiderata (ii) and (iii). In the end, I define a category of sheaves over a neighborhood frame with certain properties, and show that it provides a semantics that naturally unifies neighborhood semantics for propositional modal logic, on the one hand, and semantics for first-order logic on the other. Completeness theorems are proved.

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Item Type: | University of Pittsburgh ETD |
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ETD Committee: | |

Title: | Generalized Topological Semantics for First-Order Modal Logic |

Status: | Unpublished |

Abstract: | This dissertation provides a new semantics for first-order modal logic. It is philosophicallymotivated by the epistemic reading of modal operators and, in particular, three desiderata in the analysis of epistemic modalities.(i) The semantic modelling of epistemic modalities, in particular verifiability and falsifiability, cannot be properly achieved by Kripke's relational notion of accessibility. It requires instead a more general, topological notion of accessibility.(ii) Also, the epistemic reading of modal operators seems to require that we combine modal logic with fully classical first-order logic. For this purpose, however, Kripke's semantics for quantified modal logic is inadequate; its logic is free logic as opposed to classical logic.(iii) More importantly, Kripke's semantics comes with a restriction that is too strong to let us semantically express, for instance, that the identity of Hesperus and Phosphorus, even if metaphysically necessary, can still be a matter of epistemic discovery.To provide a semantics that accommodates the three desiderata, I show, on the one hand, howthe desideratum (i) can be achieved with topological semantics, and more generally neighborhood semantics, for propositional modal logic. On the other hand, to achieve (ii) and (iii), it turns out that David Lewis's counterpart theory is helpful at least technically. Even though Lewis's ownformulation is too liberal---in contrast to Kripke's being too restrictive---to achieve our goals, this dissertation provides a unification of the two frameworks, Kripke's and Lewis's. Through a series of both formal and conceptual comparisons of their ontologies and semantic ideas, it is shown that structures called sheaves are needed to unify the ideas and achieve the desiderata (ii) and (iii). In the end, I define a category of sheaves over a neighborhood frame with certain properties, and show that it provides a semantics that naturally unifies neighborhood semantics for propositional modal logic, on the one hand, and semantics for first-order logic on the other. Completeness theorems are proved. |

Date: | 30 January 2011 |

Date Type: | Completion |

Defense Date: | 17 November 2010 |

Approval Date: | 30 January 2011 |

Submission Date: | 08 December 2010 |

Access Restriction: | 5 year -- Restrict access to University of Pittsburgh for a period of 5 years. |

Patent pending: | No |

Institution: | University of Pittsburgh |

Thesis Type: | Doctoral Dissertation |

Refereed: | Yes |

Degree: | PhD - Doctor of Philosophy |

URN: | etd-12082010-214917 |

Uncontrolled Keywords: | counterpart thoery; sheaf semantics; topological semantics; modal logic; neighborhood semantics |

Schools and Programs: | Dietrich School of Arts and Sciences > Philosophy |

Date Deposited: | 10 Nov 2011 15:09 |

Last Modified: | 21 May 2012 14:12 |

Other ID: | http://etd.library.pitt.edu/ETD/available/etd-12082010-214917/, etd-12082010-214917 |

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