Hostname: page-component-848d4c4894-pftt2 Total loading time: 0 Render date: 2024-05-18T01:54:47.826Z Has data issue: false hasContentIssue false


Published online by Cambridge University Press:  03 February 2015

Department of Philosophy, University of Pittsburgh


We present an extension of the basic revision theory of circular definitions with a unary operator, □. We present a Fitch-style proof system that is sound and complete with respect to the extended semantics. The logic of the box gives rise to a simple modal logic, and we relate provability in the extended proof system to this modal logic via a completeness theorem, using interpretations over circular definitions, analogous to Solovay’s completeness theorem for GL using arithmetical interpretations. We adapt our proof to a special class of circular definitions as well as to the first-order case.

Research Article
Copyright © Association for Symbolic Logic 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)



Antonelli, A. (1994). The complexity of revision. Notre Dame Journal of Formal Logic, 35(1), 6772.Google Scholar
Asmus, C. M. (2013). Vagueness and revision sequences. Synthese, 190(6), 953974.Google Scholar
Belnap, N. (1982). Gupta’s rule of revision theory of truth. Journal of Philosophical Logic, 11(1), 103116.Google Scholar
Blackburn, P., de Rijke, M., & Venema, Y. (2002). Modal Logic. Cambridge, UK: Cambridge University Press.Google Scholar
Boolos, G. (1993). The Logic of Provability. New York City, NY: Cambridge University Press.Google Scholar
Bruni, R. (2013). Analytic calculi for circular concepts by finite revision. Studia Logica, 101(5), 915932.Google Scholar
Chapuis, A. (1996). Alternative revision theories of truth. Journal of Philosophical Logic, 25(4), 399423.Google Scholar
Gupta, A. (1982). Truth and paradox. Journal of Philosophical Logic, 11(1). A revised version, with a brief postscript, is reprinted in Martin (1984), pp. 160.Google Scholar
Gupta, A. (1988–89). Remarks on definitions and the concept of truth. Proceedings of the Aristotelian Society, 89, 227246. Reprinted in Gupta (2011).Google Scholar
Gupta, A. (2006). Finite circular definitions. In Bolander, T., Hendricks, V. F., and Andersen, S. A., editors. Self-Reference. Stanford, CA: CSLI Publications, pp. 7993.Google Scholar
Gupta, A. (2011). Truth, Meaning, Experience. New York City, NY: Oxford University Press.Google Scholar
Gupta, A., & Belnap, N. (1993). The Revision Theory of Truth. Cambridge, MA: MIT Press.Google Scholar
Gupta, A., & Standefer, S. (2014). Conditionals in theories of truth. Manuscript.Google Scholar
Herzberger, H. G. (1982). Notes on naive semantics. Journal of Philosophical Logic, 11(1), 61102.Google Scholar
Horsten, L., Leigh, G., Leitgeb, H., & Welch, P. D. (2012). Revision revisited. Journal of Philosophical Logic, 5(4), 642664.Google Scholar
Hughes, G. E., & Cresswell, M. J. (1996). A New Introduction to Modal Logic. New York City, NY: Routledge.Google Scholar
Kremer, P. (1993). The Gupta-Belnap systems S # and S* are not axiomatisable. Notre Dame Journal of Formal Logic, 34(4), 583596.Google Scholar
Kühnberger, K.-U., Löwe, B., Möllerfeld, M., & Welch, P. (2005). Comparing inductive and circular definitions: Parameters, complexity and games. Studia Logica, 81(1), 7998.Google Scholar
Löwe, B., & Welch, P. D. (2001). Set-theoretic absoluteness and the revision theory of truth. Studia Logica, 68(1), 2141.Google Scholar
Martin, R. L., editor. (1984). Recent Essays on Truth and the Liar Paradox. New York City, NY: Oxford University Press.Google Scholar
Martinez, M. (2001). Some closure properties of finite definitions. Studia Logica, 68(1), 4368.Google Scholar
Orilia, F. (2000). Property theory and the revision theory of definitions. Journal of Symbolic Logic, 65(1), 212246.Google Scholar
Shapiro, L. (2006). The rationale behind revision-rule semantics. Philosophical Studies, 129(3), 477515.Google Scholar
Solovay, R. M. (1976). Provability interpretations of modal logic. Israel Journal of Mathematics, 25(3–4), 287304.Google Scholar
Standefer, S. (2013). Truth, semantic closure, and conditionals. PhD Thesis, University of Pittsburgh.Google Scholar
Welch, P. D. (2001). On Gupta-Belnap revision theories of truth, Kripkean fixed points, and the next stable set. Bulletin of Symbolic Logic, 7(3), 345360.Google Scholar
Yaqūb, A. M. (1993). The Liar Speaks the Truth: A Defense of the Revision Theory of Truth. New York City, NY: Oxford University Press.CrossRefGoogle Scholar