Abstract
Gareth Evans proved that if two objects are indeterminately equal then they are different in reality. He insisted that this contradicts the assumption that there can be vague objects. However we show the consistency between Evans's proof and the existence of vague objects within classical logic. We formalize Evans's proof in a set theory without the axiom of extensionality, and we define a set to be vague if it violates extensionality with respect to some other set. There exist models of set theory where the axiom of extensionality does not hold, so this shows that there can be vague objects.
Similar content being viewed by others
References
Akiba, K. (2000): Vagueness as a modality, Philos. Q. 50, 359–370.
Boolos, G. (1971): The iterative conception of set, J. Philos. 68, 215–232.
Copeland, B. J. (1995): On vague objects, fuzzy logic and fractal boundaries, South. J. Philos. 33, 83–95.
Dummett, M. (1975): Wang's paradox, Synthese 30, 301–324, Reprinted in [12, 99–118].
Evans, G. (1978): Can there be vague objects?, Analysis 38, 208, Reprinted in [12, 317].
Friedman, H. (1973): The consistency of classical set theory relative to a set theory with intuitionistic logic, J. Symb. Log. 38, 315–319.
Garrett, B.(1991): Vague identity and vague object, Noûs 25, 341–351.
Grisn, V. N. (1982): Predicate and set-theoretic caliculi based on logic without contractions, Math. USSR Izv. 18, 41–59.
Hajek, P. and Hanikova, Z. (2003): A development of set theory in fuzzy logic, Theory and applications of multiple-valued logic, pp. 273–285.
Keefe, R. (1995): Contingent identity and vague identity, Analysis 55, 183–190.
Keefe, R. and Smith, P. (1997): Introduction: Theories of Vagueness, in [12, 1–57].
Keefe, R. and Smith, P. (eds.) (1997): Vagueness: A Reader, MIT, Cambridge, Massachusetts.
Krajicek, J. (1987): A possible modal formulation of comprehension scheme, Z. Math. Log. Grundl. Math. 33(5) 461–480.
Krajicek, J. (1988): Some results and problems in the modal set theory MST, Z. Math. Log. Grundl. Math. 34(2) 123–134.
Noonan, H. W. (1990): Vague identity yet again, Analysis 50, 157–162.
Noonan, H. W. (2004): Are there vague objects?, Analysis 64, 131–134.
Sainsbury, R. M. (1990): Concepts without Boundaries, Vagueness: A Reader, MIT Press, Cambridge, Massachusetts, pp. 251–264.
Tye, M. (1994): Sorites paradoxes and the semantics of vagueness, Philosophical Perspectives, 8: Logic and language, J. E. Thomberlin (ed.), pp. 189–206, Reprinted in [12, 281–293].
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Yatabe, S., Inaoka, H. On Evans's Vague Object from Set Theoretic Viewpoint. J Philos Logic 35, 423–434 (2006). https://doi.org/10.1007/s10992-005-9022-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10992-005-9022-7