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Categoricity by convention.

Julien Murzi1, Brett Topey1

  • 1Philosophy Department (KGW), University of Salzburg, Salzburg, Austria.

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|November 1, 2021
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Summary
This summary is machine-generated.

This study offers a naturalist solution to mathematical meaning, showing that following quantifier rules openly ensures intended interpretations for first-order and second-order logic, securing mathematical theories.

Keywords:
Carnap’s Categoricity ProblemCategoricityConventionalismHigher-order logicOpen-ended rulesPermutation invariancePutnam’s model-theoretic argument

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Area of Science:

  • Philosophy of Mathematics
  • Mathematical Logic
  • Naturalism

Background:

  • Mathematical terms' meanings are linked to language use and accepted principles.
  • First-order theories are non-categorical, posing challenges for meaning determination.
  • Second-order theories rely on full second-order logic, which itself faces categoricity issues.

Purpose of the Study:

  • To provide a naturalist-friendly solution to Carnap's Categoricity Problem for propositional and first-order logic.
  • To generalize this solution to secure the categoricity of second-order mathematical theories.
  • To explain how open-ended rule-following secures intended interpretations of quantifiers.

Main Methods:

  • Analyzing the implications of open-ended inferential dispositions for quantifier interpretation.
  • Applying Bonnay and Westerståhl's theorem on permutation invariance for first-order logic.
  • Generalizing this theorem to prove permutation invariance for second-order quantifiers.

Main Results:

  • The open-ended following of quantifier rules guarantees permutation-invariant interpretations.
  • This permutation invariance ensures the standard interpretation for first-order quantifiers.
  • The generalized theorem confirms permutation invariance for second-order quantifiers, securing full second-order logic.

Conclusions:

  • A naturalist account of mathematical meaning is possible by focusing on inferential dispositions.
  • The open-endedness of rule-following resolves the categoricity problems in logic.
  • This approach validates the intended interpretations of mathematical theories, including second-order ones.