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Generalized Topological Semantics for First-Order Modal Logic

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
    ETD Committee:
    ETD Committee TypeCommittee MemberEmailORCID
    Committee CoChairBelnap, Nuel Dbelnap@pitt.edu
    Committee CoChairAwodey, Steveawodey@cmu.edu
    Committee MemberScott, Danadana.scott@cs.cmu.edu
    Committee MemberManders, Kennethmandersk@pitt.edu
    Committee MemberBrandom, Robertrbrandom@pitt.edu
    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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