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


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.


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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: 02 july 2020. doi: https://doi.org/10.26686/ajl.v7i0.1807.