## Notre Dame Journal of Formal Logic

### Numerical Abstraction via the Frege Quantifier

G. Aldo Antonelli

#### 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 (but still first-order) 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.

#### Article information

Source
Notre Dame J. Formal Logic Volume 51, Number 2 (2010), 161-179.

Dates
First available in Project Euclid: 11 June 2010

http://projecteuclid.org/euclid.ndjfl/1276284780

Digital Object Identifier
doi:10.1215/00294527-2010-010

Mathematical Reviews number (MathSciNet)
MR2667904

Zentralblatt MATH identifier
05758435

#### Citation

Antonelli, G. Aldo. Numerical Abstraction via the Frege Quantifier. Notre Dame J. Formal Logic 51 (2010), no. 2, 161--179. doi:10.1215/00294527-2010-010. http://projecteuclid.org/euclid.ndjfl/1276284780.

#### References

• [1] Antonelli, A., and R. May, "Frege's other program", Notre Dame Journal of Formal Logic, vol. 46 (2005), pp. 1--17.
• [2] Antonelli, G. A., "Free quantification and logical invariance", pp. 61--73 in Il Significato Eluso. Saggi in Onore di Diego Marconi, edited by M. Andronico, A. Paternoster, and A. Voltolini, vol. 33, Rosenberg & Sellier, Torino, 2007. Special issue.
• [3] Antonelli, G. A., "The nature and purpose of numbers", forthcoming in Journal of Philosophy.
• [4] Antonelli, G. A., "Notions of invariance for abstraction principles", forthcoming in Philosophia Mathematica.
• [5] Frege, G., Die Grundlagen der Arithmetik, eine der arithmetischen nachgebildete Formelsparche des reines Denkens, Nebert, Breslau, 1884. English translation by J. L. Austin as [frege1950?].
• [6] Frege, G., Grundgesetze der Arithmetik, Hermann Pohle, Jena, 1903. English translation by M. Furth as [frege1967?].
• [7] Frege, G., The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number, Blackwell, Oxford, 1950.
• [8] Frege, G., The Basic Laws of Arithmetic, University of California Press, Berkeley, 1967. Translated from the German, edited, and with an introduction by M. Furth.
• [9] Hale, B., and C. Wright, The Reason's Proper Study. Essays towards a Neo-Fregean Philosophy of Mathematics, The Clarendon Press, Oxford, 2001.
• [10] Härtig, K., "Über einen Quantifikator mit zwei Wirkungsbereichen", pp. 31--36 in Colloquium on the Foundations of Mathematics, Mathematical Machines and Their Applications (Tihany, 1962), edited by L. Kalmár, Akadémiai Kiadó, Budapest, 1965.
• [11] Heck, R. G., Jr., "On the consistency of second-order contextual definitions", Noûs, vol. 26 (1992), pp. 491--94.
• [12] Heck, R. G., Jr., "Cardinality, counting, and equinumerosity", Notre Dame Journal of Formal Logic, vol. 41 (2000), pp. 187--209. Logicism and the Paradoxes: A Reappraisal, I (Notre Dame IN, 2001).
• [13] Henkin, L., "Completeness in the theory of types", The Journal of Symbolic Logic, vol. 15 (1950), pp. 81--91.
• [14] Herre, H., M. Krynicki, A. Pinus, and J. Väänänen, "The Härtig quantifier: A survey", The Journal of Symbolic Logic, vol. 56 (1991), pp. 1153--83.
• [15] Lambert, K., "On the philosophical foundations of free logic", Analysis, vol. 24 (1981), pp. 147--203.
• [16] MacFarlane, J., Logical constants,'' in Stanford Encyclopedia of Philosophy, edited by E. N. Zalta, http://plato.stan% ford.edu/entries/abstract-objects/, 2005. Spring 2006Edition.
• [17] Montague, R., English as a formal language,'' in Formal Philosophy, edited by R. H. Thomason, Yale University Press, 1974. Originally published 1969.
• [18] Mostowski, A., "On a generalization of quantifiers", Fundamenta Mathematicae, vol. 44 (1957), pp. 12--36.
• [19] Peters, S., and D. Westerståhl, Quantifiers in Language and Logic, Oxford University Press, Oxford and New York, 2006.
• [20] Rescher, N., "Plurality quantification", The Journal of Symbolic Logic, vol. 27 (1962), pp. 373--74.
• [21] Rosen, G., Abstract objects,'' in Stanford Encyclopedia of Philosophy, edited by E. N. Zalta, http://plato.stan% ford.edu/entries/abstract-objects/, 2006. Spring 2006Edition.
• [22] Tarski, A., What are logical notions?'' History and Philosophy of Logic, vol. 7 (1986), pp. 143--54.
• [23] Weir, A., "Neo-Fregeanism: An Embarrassment of Riches", Notre Dame Journal of Formal Logic, vol. 44 (2003), pp. 13--48.