David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
Philosophical Perspectives 17 (1):467–490 (2003)
A number of authors have recently weighed in on the issue of whether it is coherent to have bound variables that range over absolutely everything. Prima facie, it is difficult, and perhaps impossible, to coherently state the “relativist” position without violating it. For example, the relativist might say, or try to say, that for any quantifier used in a proposition of English, there is something outside of its range. What is the range of this quantifier? Or suppose we ask the relativist if there are some things that cannot appear in the range of any bound variable. The likely response would be along these lines: “No. For each object o, it possible to include o in the range of quantifiers, but one cannot quantify over everything at once.” This sentence contains unrestricted quantifiers, or so it seems, pending some clever move from a relativist. On the other hand, in the context of set theory, the reasoning behind the Burali-Forti paradox strongly suggests that there are well-orderings strictly longer than the collection of all ordinals. And set theorists regularly do transfinite recursions and transfinite reductions along such well-orderings. The relativist simply points out that one can always define new ordinals, and thus expand the range of one’s bound variables. The purpose of this paper is to explore the iterative framework, proposed in Zermelo’s 1930 paper, “Über Grenzzahlen und Mengenbereiche” (“On boundary numbers and domains of sets”), in order to shed light on these issues, and see what is involved in resolving them.
|Keywords||No keywords specified (fix it)|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library|
References found in this work BETA
No references found.
Citations of this work BETA
Theodore Sider (2009). Williamson's Many Necessary Existents. Analysis 69 (2):250-258.
Similar books and articles
Robert Boyer, The Addition of Bounded Quantification and Partial Functions to a Computational Logic and its Theorem Prover.
John P. Burgess (1988). Sets and Point-Sets: Five Grades of Set-Theoretic Involvement in Geometry. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1988:456 - 463.
Jeremy Avigad (2002). An Ordinal Analysis of Admissible Set Theory Using Recursion on Ordinal Notations. Journal of Mathematical Logic 2 (01):91-112.
Øystein Linnebo (2006). Sets, Properties, and Unrestricted Quantification. In Gabriel Uzquiano & Agustin Rayo (eds.), Absolute Generality. Oxford University Press.
Eric Steinhart (2002). Why Numbers Are Sets. Synthese 133 (3):343 - 361.
H. Jerome Keisler (1998). Quantifier Elimination for Neocompact Sets. Journal of Symbolic Logic 63 (4):1442-1472.
James Beebe (2010). Moral Relativism in Context. Noûs 44 (4):691-724.
G. Hellman (2011). On the Significance of the Burali-Forti Paradox. Analysis 71 (4):631-637.
Added to index2009-01-28
Total downloads52 ( #34,846 of 1,140,113 )
Recent downloads (6 months)1 ( #147,976 of 1,140,113 )
How can I increase my downloads?