Against pluralism

Hazen, Allen

doi:10.1080/00048409312345142

Cited by 32 publications

(10 citation statements)

References 13 publications

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“…The purported advantage is that by using the logical apparatus of quantification to talk about a plurality of critics, we are not ontologically committed to the existence of a collection of critics only to the existence of the critics from that plurality. This claim is itself not without its critics, although this is beyond the scope of this paper (Hazen 1993;Resnik 1988).…”

Section: They're Not Sets They're Pluralitiesmentioning

confidence: 92%

Sets and supersets

Meadows

2015

Synthese

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It is a commonplace of set theory to say that there is no set of all wellorderings nor a set of all sets. We are implored to accept this due to the threat of paradox and the ensuing descent into unintelligibility. In the absence of promising alternatives, we tend to take up a conservative stance and tow the line: there is no universe (Halmos, in: Naive set theory, 1960). In this paper, I am going to challenge this claim by taking seriously the idea that we can talk about the collection of all the sets and many more collections beyond that. A method of articulating this idea is offered through an indefinitely extending hierarchy of set theories. It is argued that this approach provides a natural extension to ordinary set theory and leaves ordinary mathematical practice untouched.Keywords Philosophy of mathematics · Philosophy of set theory · Formal theories of truth · Absolute generality · Indefinite extensibility You're gonna need a bigger boat.(Jaws)The idea that there ought to be a set containing all of the sets is the focus of this paper. It is not a new idea. Moreover, I suspect that my way of addressing this issue will have occurred to many readers, although usually they they will have set it aside. I do not want to suggest that the techniques explored in this paper are of themselves particularly original; we shall see that related approaches have been explored from a technical angle for many years. Rather, what I am aiming to do is give this group of ideas a more philosophical twist by drawing out their underlying motivations and B Toby Meadows

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Section: They're Not Sets They're Pluralitiesmentioning

confidence: 92%

Sets and supersets

Meadows

2015

Synthese

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“…See also Cartwright (1993) and Uzquiano (2003). For examples of critical responses, see Resnik (1988), Hazen (1993), Linnebo (2003), and Jané (2005). Lavine (1994) offers an alternative to our COM(ω): namely, the scheme "The F numbers form a set" where it is understood that we can replace the place-holder F with any predicate first-order definable in some expansion of our mathematical vocabulary.…”

Section: Sets Of Natural Numbersmentioning

confidence: 97%

Mathematical determinacy and the transferability of aboutness

Pollard

2006

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“…Moreover, we articulate a generalized notion of ontological commitment according to which plural logic is not, after all, innocent. This provides, for the first time, a precise development of some ideas adumbrated by Parsons (: section ), Hazen (), Shapiro (), and Linnebo (). Our focus is on plural logic, though much of what we say would apply, mutatis mutandis , to second‐ and higher‐order logics which quantify into predicate position.…”

Section: Introductionmentioning

confidence: 93%

“…Of course, Plural Innocence is not uncontroversial (see Resnik , Parsons , Hazen , and Linnebo ); we too take issue with it below. But if the thesis is false, so is an essential premise of the argument we wish to reject, and we are done.…”

Section: Does Ontological Innocence Ensure Determinacy?mentioning

confidence: 99%

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On the Innocence and Determinacy of Plural Quantification

Florio

Linnebo

2015

Nous

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Plural logic is widely assumed to have two important virtues: ontological innocence and determinacy. It is claimed to be innocent in the sense that it incurs no ontological commitments beyond those already incurred by the first‐order quantifiers. It is claimed to be determinate in the sense that it is immune to the threat of non‐standard (Henkin) interpretations that confronts higher‐order logics on their more traditional, set‐based semantics. We challenge both claims. Our challenge is based on a Henkin‐style semantics for plural logic that does not resort to sets or set‐like objects to interpret plural variables, but adopts the view that a plural variable has many objects as its values. Using this semantics, we also articulate a generalized notion of ontological commitment which enables us to develop some ideas of earlier critics of the alleged ontological innocence of plural logic.

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Against pluralism

Cited by 32 publications

References 13 publications

Sets and supersets

Sets and supersets

Mathematical determinacy and the transferability of aboutness

On the Innocence and Determinacy of Plural Quantification

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