## Generalized Topological Semantics for First-Order Modal Logic (2010)

### BibTeX

@MISC{Kishida10generalizedtopological,

author = {Kohei Kishida},

title = {Generalized Topological Semantics for First-Order Modal Logic },

year = {2010}

}

### OpenURL

### Abstract

This dissertation provides a new semantics for first-order modal logic. It is philosophically motivated 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, how the 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

### Citations

937 | Categories for the working mathematicians - Lane - 1971 |

192 |
Sheaves in Geometry and Logic: A First Introduction to Topos Theory, Universitext
- Lane, Moerdijk
- 1992
(Show Context)
Citation Context ...ullback diagram in Top below, if πD is open continuous and πE is a local homeomorphism, then pD is a local homeomorphism as well. D ×X E pE �� E pD D πD � X πE 8 Facts 55 and 56 are Exercise II.10 of =-=[12]-=-. 217�� � �� Draft of November 14, 2010 Corollary 9. LH/X has the same products as Top/X does. These structures of sheaves help us to achieve our desiderata (iii) and (iv) of Subsection VI.1.1; to se... |

94 |
Counterpart theory and quantified modal logic
- LEWIS
- 1968
(Show Context)
Citation Context ... = (i2) ∗ ◦ f ¯x. Thus the constant family of operations f is DoQ-restrictable with the binding of ¯y. □ 167sCHAPTER V Accessibility and Counterparts V.1. David Lewis’s Counterpart Theory David Lewis =-=[9]-=- puts forward an ontology of possible individuals that he calls counterpart theory, and that differs from the ontology of Kripke’s semantics, which we discussed in Chapter ??, in a crucial manner. 1 L... |

76 |
Advice on Modal Logic
- Scott
- 1970
(Show Context)
Citation Context ...ore modal logic E obtained by adding E to classical propositional logic is sound with respect to neighborhood semantics; and, indeed, it is also complete, in the following strong form: Theorem (Scott =-=[17]-=-, Montague [15], Segerberg [19]). For any consistent theory T containing E, there exists a neighborhood model (X, N, ⟦−⟧) that validates all and only theorems of T. I.1.2. Some Conditions on Neighborh... |

55 |
An essay in classical modal logic
- Segerberg
- 1971
(Show Context)
Citation Context ...dding E to classical propositional logic is sound with respect to neighborhood semantics; and, indeed, it is also complete, in the following strong form: Theorem (Scott [17], Montague [15], Segerberg =-=[19]-=-). For any consistent theory T containing E, there exists a neighborhood model (X, N, ⟦−⟧) that validates all and only theorems of T. I.1.2. Some Conditions on Neighborhood Frames. Though any set X ca... |

52 | The Nature of Necessity - Plantinga - 1974 |

35 |
Category theory
- Awodey
- 2006
(Show Context)
Citation Context ...∀r ◦ r ∗ = 1, because p∗ and r∗ are injective; therefore ∃p, ∀p, ∃r, ∀r are surjective. Next let us list some properties of the category-theoretic notion of pullbacks. While the reader should consult =-=[2]-=- for instance for a general definition of pullbacks, we can take the following fact as providing a definition for the particular case of the category Sets of sets. Fact 16. Given any maps f0 : X → Z a... |

28 | Counterpart-Theoretic Semantics for Modal Logic - Hazen - 1979 |

28 |
Semantical considerations on modal logic. Acta Philosophica Fennica 16:83–94
- Kripke
- 1963
(Show Context)
Citation Context ... out with Kripke semantics for S4, that is, a possibleworld semantics with a reflexive and transitive accessibility relation among possible worlds. As is 48Draft of November 14, 2010 shown in Kripke =-=[5]-=-, propositional S4 is sound and complete with respect to S4 Kripke semantics. This fact may appear to mean that the difference between topological and S4 Kripke semantics is not semantically or logica... |

20 |
Universal grammar”, Theoria 36
- Montague
- 1970
(Show Context)
Citation Context ... E obtained by adding E to classical propositional logic is sound with respect to neighborhood semantics; and, indeed, it is also complete, in the following strong form: Theorem (Scott [17], Montague =-=[15]-=-, Segerberg [19]). For any consistent theory T containing E, there exists a neighborhood model (X, N, ⟦−⟧) that validates all and only theorems of T. I.1.2. Some Conditions on Neighborhood Frames. Tho... |

15 | Reference and Modality - Linsky - 1971 |

4 | Semantics of Non-Classical First-Order Predicate Logics - Shehtman, Skvortsov - 1990 |

2 |
Topology and modality: The topological interpretation of first-order modal logic
- Awodey, Kishida
(Show Context)
Citation Context ...y extends Theorem 2 (Subsection 28Draft of November 14, 2010 I.1.2), the completeness S4 for propositional modal logic. This is one of the chief results of this dissertation. Theorem (Awodey-Kishida =-=[3]-=-). For any consistent theory T of first-order modal logic extending FOS4, there exists a topological-sheaf interpretation (π, ⟦−⟧) that validates all and only theorems of T. I.3.6. An Example of Inter... |

2 | Papers: Volume - Philosophical |

1 |
Notes on the Science of Logic, manuscript, version of 2009
- Belnap
(Show Context)
Citation Context ...nded beyond the classical one. First, we consider the notion of local determination. 7 An inference in L is a pair of a set of sentences of L and a sentence of L. 8 A respectable exception is Belnap, =-=[4]-=-, to whom I owe the notions (and their names) of local determination and semantics of substitution. 57Draft of November 14, 2010 Definition 7. Let (M, ⊨) be a classical-type satisfaction relation for... |

1 | The Blackwell Guide to Philosophical Logic - Lambert, Logics” - 2001 |

1 | Modal Modeling,” paper read at E - “Higher-Order - 2008 |