Quantifier Variance and Indefinite Extensibility

Philosophical Review 126 (1):81-122 (2017)
Abstract
This essay clarifies quantifier variance and uses it to provide a theory of indefinite extensibility that I call the variance theory of indefinite extensibility. The indefinite extensibility response to the set-theoretic paradoxes sees each argument for paradox as a demonstration that we have come to a different and more expansive understanding of ‘all sets’. But indefinite extensibility is philosophically puzzling: extant accounts are either metasemantically suspect in requiring mysterious mechanisms of domain expansion, or metaphysically suspect in requiring nonstandard assumptions about mathematical objects. Happily, the view of quantifier meanings that underwrites quantifier variance can be used to provide an account of indefinite extensibility that is both metasemantically and metaphysically satisfying. Section 1 introduces the puzzle of indefinite extensibility; section 2 develops and clarifies the metasemantics of quantifier variance; section 3 solves section 1's puzzle of indefinite extensibility by applying section 2's account of quantifier meanings; and section 4 compares the theory developed in section 3 to several other theories in the literature.
Keywords quantifier variance  indefinite extensibility  set-theoretic paradoxes  metaontology
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DOI 10.1215/00318108-3683622
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Quantifier Variance and the Demand for a Semantics.Eli Hirsch & Jared Warren - forthcoming - Philosophy and Phenomenological Research.

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