Prospects for a Naive Theory of Classes

Notre Dame Journal of Formal Logic 58 (4):461-506 (2017)
  Copy   BIBTEX


The naive theory of properties states that for every condition there is a property instantiated by exactly the things which satisfy that condition. The naive theory of properties is inconsistent in classical logic, but there are many ways to obtain consistent naive theories of properties in nonclassical logics. The naive theory of classes adds to the naive theory of properties an extensionality rule or axiom, which states roughly that if two classes have exactly the same members, they are identical. In this paper we examine the prospects for obtaining a satisfactory naive theory of classes. We start from a result by Ross Brady, which demonstrates the consistency of something resembling a naive theory of classes. We generalize Brady’s result somewhat and extend it to a recent system developed by Andrew Bacon. All of the theories we prove consistent contain an extensionality rule or axiom. But we argue that given the background logics, the relevant extensionality principles are too weak. For example, in some of these theories, there are universal classes which are not declared coextensive. We elucidate some very modest demands on extensionality, designed to rule out this kind of pathology. But we close by proving that even these modest demands cannot be jointly satisfied. In light of this new impossibility result, the prospects for a naive theory of classes are bleak.

Similar books and articles

The Groundedness Approach to Class Theory.Jönne Kriener - 2014 - Inquiry: An Interdisciplinary Journal of Philosophy 57 (2):244-273.
Routes to triviality.Susan Rogerson & Greg Restall - 2004 - Journal of Philosophical Logic 33 (4):421-436.
Naïve set theory is innocent!A. Weir - 1998 - Mind 107 (428):763-798.
A model of tolerance.Elia Zardini - 2008 - Studia Logica 90 (3):337-368.
Plural quantification and classes.Gabriel Uzquiano - 2003 - Philosophia Mathematica 11 (1):67-81.
On the iterative explanation of the paradoxes.Christopher Menzel - 1986 - Philosophical Studies 49 (1):37 - 61.
Prädikative Klassen.Ralf-Dieter Schindler - 1993 - Erkenntnis 39 (2):209 - 241.
New Grounds for Naive Truth Theory.Stephen Yablo - 2004 - In J. C. Beall (ed.), Liars and Heaps: New Essays on Paradox. Clarendon Press. pp. 312-330.


Added to PP

584 (#18,679)

6 months
92 (#14,603)

Historical graph of downloads
How can I increase my downloads?

Author Profiles

Hartry Field
New York University
Tore Fjetland Øgaard
University of Bergen
Harvey Lederman
University of Texas at Austin

Citations of this work

Paths to Triviality.Tore Fjetland Øgaard - 2016 - Journal of Philosophical Logic 45 (3):237-276.
Naive set theory and nontransitive logic.David Ripley - 2015 - Review of Symbolic Logic 8 (3):553-571.
Model-theoretic semantics and revenge paradoxes.Lorenzo Rossi - 2019 - Philosophical Studies 176 (4):1035-1054.

View all 6 citations / Add more citations

References found in this work

Saving truth from paradox.Hartry H. Field - 2008 - New York: Oxford University Press.
Outline of a theory of truth.Saul Kripke - 1975 - Journal of Philosophy 72 (19):690-716.
Can there be vague objects?Gareth Evans - 1978 - Analysis 38 (4):208.
Universal Logic.Ross Brady - 2006 - CSLI Publications.
Can there be vague objects?Gareth Evans - 2004 - In Tim Crane & Katalin Farkas (eds.), Metaphysics: A Guide and Anthology. Oxford University Press.

View all 25 references / Add more references