On the referential indeterminacy of logical and mathematical concepts

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Abstract

Hartry Field has recently examined the question whether our logical and mathematical concepts are referentially indeterminate. In his view, (1) certain logical notions, such as second-order quantification, are indeterminate, but (2) important mathematical notions, such as the notion of finiteness, are not (they are determinate). In this paper, I assess Field's analysis, and argue that claims (1) and (2) turn out to be inconsistent. After all, given that the notion of finiteness can only be adequately characterized in pure second-order logic, if Field is right in claiming that second-order quantification is indeterminate (see (1)), it follows that finiteness is also indeterminate (contrary to (2)). After arguing that Field is committed to these claims, I provide a diagnosis of why this inconsistency emerged, and I suggest an alternative, consistent picture of the relationship between logical and mathematical indeterminacy.

Original languageEnglish (US)
Pages (from-to)65-79
Number of pages15
JournalJournal of Philosophical Logic
Volume34
Issue number1
DOIs
StatePublished - Dec 1 2005
Externally publishedYes

Keywords

  • Finiteness
  • Henkin semantics
  • Indeterminacy
  • Second-order logic

ASJC Scopus subject areas

  • Philosophy

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