Numerical Abstraction via the Frege Quantifier

Notre Dame Journal of Formal Logic 51 (2):161-179 (2010)
  Copy   BIBTEX

Abstract

This paper presents a formalization of first-order arithmetic characterizing the natural numbers as abstracta of the equinumerosity relation. The formalization turns on the interaction of a nonstandard cardinality quantifier with an abstraction operator assigning objects to predicates. The project draws its philosophical motivation from a nonreductionist conception of logicism, a deflationary view of abstraction, and an approach to formal arithmetic that emphasizes the cardinal properties of the natural numbers over the structural ones.

Categories

Keywords

Reprint years

DOI

10.1215/00294527-2010-010

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 92,227

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

My notes

Similar books and articles

Logicism, quantifiers, and abstraction.Aldo Antonelli - manuscript
Frege meets dedekind: A neologicist treatment of real analysis.Stewart Shapiro - 2000 - Notre Dame Journal of Formal Logic 41 (4):335--364.
Abstraction Relations Need Not Be Reflexive.Jonathan Payne - 2013 - Thought: A Journal of Philosophy 2 (2):137-147.
Which abstraction principles are acceptable? Some limitative results.Øystein Linnebo & Gabriel Uzquiano - 2009 - British Journal for the Philosophy of Science 60 (2):239-252.
Fregean abstraction, referential indeterminacy and the logical foundations of arithmetic.Matthias Schirn - 2003 - Erkenntnis 59 (2):203 - 232.
Frege’s Attack on “Abstraction” and his Defense of the “Applicability” of Arithmetic.Daniël F. M. Strauss - 2003 - South African Journal of Philosophy 22 (1):63-80.
Nominalist Neologicism.Rafal Urbaniak - manuscript
Bad company tamed.Øystein Linnebo - 2009 - Synthese 170 (3):371 - 391.
Predicate abstraction, the limits of quantification, and the modality of existence.Philip Percival - 2011 - Philosophical Studies 156 (3):389-416.
Frege’s Logicism and the Neo-Fregean Project.Matthias Schirn - 2014 - Axiomathes 24 (2):207-243.
Hume’s Big Brother: counting concepts and the bad company objection.Roy T. Cook - 2009 - Synthese 170 (3):349 - 369.
Notions of Invariance for Abstraction Principles.G. A. Antonelli - 2010 - Philosophia Mathematica 18 (3):276-292.

Analytics

Added to PP
2010-08-13

Downloads
133 (#139,067)

6 months
9 (#317,143)

Historical graph of downloads
How can I increase my downloads?

Author's Profile

G. Aldo Antonelli
University of California, Davis

Citations of this work

On Number-Set Identity: A Study.Sean C. Ebels-Duggan - 2022 - Philosophia Mathematica 30 (2):223-244.
Notions of Invariance for Abstraction Principles.G. A. Antonelli - 2010 - Philosophia Mathematica 18 (3):276-292.
Relative categoricity and abstraction principles.Sean Walsh & Sean Ebels-Duggan - 2015 - Review of Symbolic Logic 8 (3):572-606.

View all 9 citations / Add more citations

References found in this work

What are logical notions?Alfred Tarski - 1986 - History and Philosophy of Logic 7 (2):143-154.
Completeness in the theory of types.Leon Henkin - 1950 - Journal of Symbolic Logic 15 (2):81-91.
On a generalization of quantifiers.Andrzej Mostowski - 1957 - Fundamenta Mathematicae 44 (2):12--36.
Cardinality, Counting, and Equinumerosity.Richard G. Heck - 2000 - Notre Dame Journal of Formal Logic 41 (3):187-209.
Neo-Fregeanism: An Embarrassment of Riches.Alan Weir - 2003 - Notre Dame Journal of Formal Logic 44 (1):13-48.

View all 13 references / Add more references