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Absolutely Everything Winter Reading Group II - Absolute Generality

Intro

This is to be a space for the winter 2007 philosophy reading group. The text will be an anthology by Rayo and Uzquiano: Absolute Generality. If you are interested in joining, please email dsidi [*at*] u [*dot*] arizona [*dot*] edu.

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David Sidi's remarks are in indigo.

Links

Discussion

  • Motivations for studying absolute generality (ongoing additions to this section in the future)
    • philosophical claims seem to involve absolute generality
      • existence claims (mostly negative: no talking donkeys, no abstract objects, etc..., but also "what is there? ...Everything")
      • logical claims (everything is self-identical, identity is symmetric, Russell/Burali-Forti/Cantor/.../ Paradoxes - [ie does an absolutely unrestricted domain lead to paradox], can we consistently employ unrestrictedly unrestricted quantification)
      • mathematical claims (the empty set has no members, everything is countable)

Rayo and Uzquiano - Introduction

  • metaphysical question of absolute generality: is there an all-inclusive domain of discourse?
  • "availability" question of absolute generality: could an all-inclusive domain of discourse be available to us as a domain of inquiry?
    • domains of inquiry might be linguistic, epistemic, metaphysical?
    • if the answer to the latter is 'no', what grounds is there for accepting the former?
      • could it be we can indirectly show the answer to the metaphysical question is 'yes,' but not as a domain of inquiry (of any sort) in its own right?
    • if the answer to the former is 'no', we might still be able to quantify unrestrictedly -- so we could take an all-inclusive domain as a domain of (linguistic) inquiry, but since there is none, we can't.
    • it takes a 'yes' answer to both availability and metaphysical questions to accept absolutely general discourse
  • Is there not something wrong with Rayo and Uzquiano's claim that they are not assuming the existence of a 'domain' to not require the existence of a set or set-like object? Isn't there something wrong with taking 'domain' to require only the existence of objects (and since its all-inclusive, this includes sets), but no membership relation obtaining between the sets and the domain ? In what sense are they "in" the domain, then ?
    • this objection might be taken as another difficulty with the "all-in-one" principle of Richard Cartwright.
      All in One Principle The objects in a domain of discourse make up a set or some set-like object.
      • R and O mention three difficulties with All in One:
        1. Rejecting naive comprehension
        2. rejecting that the model-theoretic characterization of logical consequence requires (first order) quantification over all domains: to be materially adequate, you need only set sized domains.
        3. rejecting that first order quantification is required
  • Indefinite Extensibility (availability question)
    • an indefinitely extensible concept is one that lacks a definite extension, but are extendible according to principles which yield a hierarchy of increasingly inclusive extensions
    • explained using the schema of naive comprehension:
      $\exists y \forall x(x \in y \iff \phi (x))$ , where $\phi(x)$ is any formula not containing '$y$' free.
      • the point is that the substitution instance $\exists y \forall x(x \in r \iff x \notin x) $ shows that there is a set not in whatever extension you try to give.
        • It provides a principle for extending the extension, since we can always reapply the schema.
      • I don't understand how Rayo and Uzquiano explain the response to the person who claims that naive comprehension must be rejected: what is a "proper interpretation" of it which will deliver the existence of an r such that $ r \in r \iff r \notin r $?
  • Reconceptualization (metaphysical question)
    • ontology is relative to a language / conceptual scheme
      • an argument for that: semantic indeterminacy, from the Lowenheim-Skolem theorem. Use of a first order quantifier with all inclusive domain is also compatible with a not all inclusive domain, so first order quantifiers never determinately range over an all-inclusive domain.
        • Lewis' response is weird: he claims that certain collections of individuals are more natural than others (If I remember right, in an article on natural kinds, he leaves this naturalness as a brute fact). This breaks the indeterminacy in favor of the "natural" assignments of semantic values to an expression.
          • But isn't this question-begging? Its a brute fact of the all inclusive domain that its members are its members? How is this supposed to convince anyone? I must not be getting things right -- please help if you have some insight.-- DavidSidi? - 17 Jan 2007

Fine - Relatively Unrestricted Quantification

  • central concern is indefinite extensibility
    • quantification over all objects/sets can't happen, since reasoning that leads to Russell's paradox gets us the possibliity of quantifying over a larger domain
      • whats the simple version of this? from $\exists y(\forall x (x \in y \equiv \neg (x \in x)))$, you can derive $\exists y \forall x (x \neq y) $
      • while its true that set theoretical principles restrict our use of the quantifiers to sets that aren't too big, still we can find a new understanding of the quantifiers that ranges over sets that were (on the old understanding) too big.
        • Understanding quantification requires only a condition (any condition) to build a set whose members the quantifier ranges over.
          • The question of how many objects satisfy that condition is not required.
          • This is a bit like the distinction between Fregean extensions and Russellian classes for numbers
      • You can construct forms of Russell's paradox for non-sets, too: properties, singletons and mereological sums.-- DavidSidi? - 23 Jan 2007

Lavine - Something About Everything...

Linnebo - Sets, Properties, and Unrestricted Quantification

Rayo - Beyond Plurals

Shapiro and Wright - All Things Indefinitely Extensible

Williamson - Absolute Identity and Absolute Generality

-- DavidSidi? - 10 Dec 2006

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Topic revision: r20 - 25 Feb 2007 - 02:20:12 - ShaughanLavine

 
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