A Note on Identity and Higher Order Quantification

  • Rafal Urbaniak Centre for Logic and Philosophy of Science, Ghent University / Chair of Logic, Methodology and Philosophy of Science, Gdańsk University

Abstract

It is a commonplace remark that the identity relation, even though not expressible in a first-order language without identity with classical set-theoretic semantics, can be defined in a language without identity, as soon as we admit second-order, set-theoretically interpreted quantifiers binding predicate variables that range over all subsets of the domain. However, there are fairly simple and intuitive higher-order languages with set-theoretic semantics (where the variables range over all subsets of the domain) in which the identity relation is not definable. The point is that the definability of identity in higher-order languages not only depends on what variables range over, but also is sensitive to how predication is construed. This paper is a follow-up to (Urbaniak 2006), where it has been proven that no actual axiomatization of Leśniewski’s Ontology determines the standard semantics for the epsilon connective.

Author Biography

Rafal Urbaniak, Centre for Logic and Philosophy of Science, Ghent University / Chair of Logic, Methodology and Philosophy of Science, Gdańsk University
Published
2009-06-11
How to Cite
URBANIAK, Rafal. A Note on Identity and Higher Order Quantification. The Australasian Journal of Logic, [S.l.], v. 7, june 2009. ISSN 1448-5052. Available at: <https://ojs.victoria.ac.nz/ajl/article/view/1807>. Date accessed: 18 july 2019. doi: https://doi.org/10.26686/ajl.v7i0.1807.