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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. (Unpublished)

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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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Details

Item Type: University of Pittsburgh ETD
Status: Unpublished
Creators/Authors:
CreatorsEmailPitt UsernameORCID
Kishida, Koheikok6@pitt.edukok6@pitt.edu
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee CoChairBelnap, Nuel Dbelnap@pitt.edubelnap@pitt.edu
Committee CoChairAwodey, Steveawodey@cmu.eduawodey@cmu.edu
Committee MemberScott, Danadana.scott@cs.cmu.edudana.scott@cs.cmu.edu
Committee MemberManders, Kennethmandersk@pitt.edumandersk@pitt.edu
Committee MemberBrandom, Robertrbrandom@pitt.edurbrandom@pitt.edu
Date: 30 January 2011
Date Type: Completion
Defense Date: 17 November 2010
Approval Date: 30 January 2011
Submission Date: 8 December 2010
Access Restriction: 5 year -- Restrict access to University of Pittsburgh for a period of 5 years.
Institution: University of Pittsburgh
Schools and Programs: Dietrich School of Arts and Sciences > Philosophy
Degree: PhD - Doctor of Philosophy
Thesis Type: Doctoral Dissertation
Refereed: Yes
Uncontrolled Keywords: counterpart thoery; sheaf semantics; topological semantics; modal logic; neighborhood semantics
Other ID: http://etd.library.pitt.edu/ETD/available/etd-12082010-214917/, etd-12082010-214917
Date Deposited: 10 Nov 2011 20:09
Last Modified: 30 Jan 2016 06:15
URI: http://d-scholarship.pitt.edu/id/eprint/10224

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