Actualism and Quantified Modal Logic

Dissertation, Princeton University (2002)

Authors
Reina Hayaki
University of Nebraska, Lincoln
Abstract
It has been alleged that actualism and quantified modal logic are incompatible. My aim in this dissertation is twofold: to defend thoroughgoing actualism with respect to possible objects, and to present a modified semantics for quantified modal logic that is compatible with such a position. The basic strategy is to draw on the parallels between fictions and possible worlds to develop a hierarchical system of worlds-within-worlds ;Actualists usually take first-order modal statements as being about the right objects, by stipulation. Any actualistically acceptable semantics must extend this approach to higher-order modal statements. "I could have had a brother who was a banker but could have been a pianist" means: there is a first-order world in which I have a banker brother, and there exists within that world a second-order world in which that very brother is a pianist. This eliminates the troubling need to identify non-actual objects across worlds of the same order. ;The concept of worlds-within-worlds is analogous to that of consistent fictions-within-fictions. Independent investigations of each concept reveal that both nesting structures are treelike, irreflexive and intransitive. Despite these restrictions, AA models are relatively tractable, being derivable from Kripke models for by unraveling. They are sound and complete for normal propositional modal logics, and for the quantified modal logic Q1R as long as constants are excluded from the language. ;In Chapter 1, I argue for actualism and against some proposed responses to the iterated modality problem. I then present the idea of nested worlds as a solution. Chapter 2 examines the analogous concept of nested fictions, to shed some light on consistent nesting structures. Chapter 3 returns to modality proper; AA models are defined and defended for both propositional and quantified modal logic. Chapter 4 contains the soundness and completeness results, and ends with a somewhat surprising philosophical argument motivated by AA semantics: unless it is metaphysically necessary that all objects have sufficient essences, the correct logic for metaphysical necessity must be weaker than B
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