Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-20T01:13:20.287Z Has data issue: false hasContentIssue false
  • English
  • Français

THE SEMANTICS OF VALUE-RANGE NAMES AND FREGE’S PROOF OF REFERENTIALITY

Published online by Cambridge University Press:  19 April 2018

MATTHIAS SCHIRN
MATTHIAS SCHIRN*
Affiliation:
Munich Center for Mathematical Philosophy, University of Munich
*
*MUNICH CENTER FOR MATHEMATICAL PHILOSOPHY UNIVERSITY OF MUNICH LUDWIGSTRASSE 31 80539 MUNICH, GERMANY E-mail:[email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this article, I try to shed some new light on Grundgesetze §10, §29–§31 with special emphasis on Frege’s criteria and proof of referentiality and his treatment of the semantics of canonical value-range names. I begin by arguing against the claim, recently defended by several Frege scholars, that the first-order domain in Grundgesetze is restricted to value-ranges (including the truth-values), but conclude that there is an irresolvable tension in Frege’s view. The tension has a direct impact on the semantics of the concept-script, not least on the semantics of value-range names. I further argue that despite first appearances truth-value names (sentences) play a distinguished role as semantic “target names” for “test names” in the criteria of referentiality (§29) and do not figure themselves as “test names” regarding referentiality. Accordingly, I show in detail that Frege’s attempt to demonstrate that by virtue of his stipulations “regular” value-range names have indeed been endowed with a unique reference, can plausibly be regarded as a direct application of the context principle. In a subsequent section, I turn to some special issues involved in §10. §10 is closely intertwined with §31 and in my and Richard Heck’s view would have been better positioned between §30 and §31. In a first step, I discuss the piecemeal strategy which Frege applies when he attempts to bestow a unique reference on value-range names in §3, §10–§12. In a second step, I critically analyze his tentative, but predictably unsuccessful proposal (in a long footnote to §10) to identify all objects whatsoever, including those already clad in the garb of value-ranges, with their unit classes. In conclusion, I present two arguments for my claim that Frege’s identification of the True and the False with their unit classes in §10 is illicit even if both the permutation argument and the identifiability thesis that he states in §10 are regarded as formally sound. The first argument is set out from the point of view of the syntax of his formal language. It suggests though that a reorganization of the exposition of the concept-script would have solved at least one of the problems to which the twin stipulations in §10 give rise. The second argument rests on semantic considerations. If it is sound, it may call into question, if not undermine the legitimacy of the twin stipulations.

Keywords

03-03all-encompassing domaincriteria of referentialitycontext principleidentifiability thesispiecemeal strategyreferential indeterminacyregular value-range namesunit classes
Type
Research Article
Information
The Review of Symbolic Logic , Volume 11 , Issue 2 , June 2018 , pp. 224 - 278
Copyright
Copyright © Association for Symbolic Logic 2018 

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.)

References

BIBLIOGRAPHY

Blanchette, P. (2012a). Frege on shared belief and total functions. Journal of Philosophy, 109, 9–39.CrossRefGoogle Scholar
Blanchette, P. (2012b). Frege’s Conception of Logic. New York: Oxford University Press.CrossRefGoogle Scholar
Blanchette, P. (2015). Reply to Cook, Rossberg and Wehmeier. Journal of the History of Analytic Philosophy, 3, 1–13.Google Scholar
Blanchette, P. (2016). Frege on mathematical progress. In Costreie (2016). pp. 3–19.Google Scholar
Cook, R. (2013). How to read Grundgesetze. In Frege, G. Basic Laws of Arithmetic. Derived Using Concept-Script, Vol. I & II, translated and edited by Ebert, P. A. and Rossberg, M., with Wright, C.. Oxford: Oxford University Press, pp. A1–A42.Google Scholar
Cook, R. T. (2014). Review of Blanchette (2012b). Philosophia Mathematica, 22, 108–120.CrossRefGoogle Scholar
Cook, R. T. & Ebert, P. A. (2016). Frege’s recipe. The Journal of Philosophy, 113, 309–345.CrossRefGoogle Scholar
Costreie, V. S. (editor) (2016). Early Analytic Philosophy. New Perspectives on the Tradition. New York, London: Springer.CrossRefGoogle Scholar
Dummett, M. (1973). Frege. Philosophy of Language. London: Duckworth.Google Scholar
Dummett, M. (1981). The Interpretation of Frege’s Philosophy. London: Duckworth.Google Scholar
Dummett, M. (1991). Frege. Philosophy of Mathematics. London: Duckworth.Google Scholar
Frege, G. (1884). Die Grundlagen der Arithmetik. Eine logisch mathematische Untersuchung über den Begriff der Zahl. Breslau: W. Koebner.Google Scholar
Frege, G. (1892). Über Begriff und Gegenstand. Vierteljahrsschrift für wissenschaftliche Philosophie, 16, 192–205. In Frege (1967), pp. 167–178.Google Scholar
Frege, G. (1893). Grundgesetze der Arithmetik. Begriffsschriftlich abgeleitet, Vol. I. Jena: H. Pohle.Google Scholar
Frege, G. (1902). Letter to Russell of 3.8.1902. In Frege (1976), pp. 225–226.Google Scholar
Frege, G. (1903). Grundgesetze der Arithmetik. Begriffsschriftlich abgeleitet, Vol. II. Jena: H. Pohle.Google Scholar
Frege, G. (1967). Kleine Schriften, edited by Angelelli, I.. Hildesheim: Georg Olms.Google Scholar
Frege, G. (1976). Wissenschaftlicher Briefwechsel, edited by Gabriel, G., Hermes, H., Kambartel, F., Thiel, C., and Veraart, A.. Hamburg: Felix Meiner.Google Scholar
Heck, R. G. (1997). Grundgesetze der Arithmetik I §§29–32. Notre Dame Journal of Formal Logic, 38, 437–474.Google Scholar
Heck, R. G. (1999). Grundgesetze der Arithmetik I §10. Philosophia Mathematica, 7, 258–292.CrossRefGoogle Scholar
Heck, R. G. (2005). Julius Caesar and Basic Law V. Dialectica, 59, 161–178.CrossRefGoogle Scholar
Heck, R. G. (2011). Frege’s Theorem. Oxford: Oxford University Press.Google Scholar
Heck, R. G. (2012). Reading Frege’s Grundgesetze. Oxford: Oxford University Press.Google Scholar
Linnebo, Ø. (2004). Frege’s proof of referentiality. Notre Dame Journal of Formal Logic, 45, 73–98.CrossRefGoogle Scholar
Resnik, M. (1986). Frege’s proof of referentiality. In Haaparanta, L. & Hintikka, J., editors. Frege Synthesized. Essays on the Philosophical and Foundational Work of Gottlob Frege. Dordrecht, Boston: D. Reidel, pp. 177–195.Google Scholar
Ruffino, M. (2002). Logical objects in Frege’s Grundgesetze, section 10. In Reck, E., editor. From Frege to Wittgenstein: Perspectives on Early Analytical Philosophy. Oxford: Oxford University Press, pp. 125–148.CrossRefGoogle Scholar
Schirn, M. (2003). Fregean abstraction, referential indeterminacy and the logical foundations of arithmetic. Erkenntnis, 59, 203–232.CrossRefGoogle Scholar
Schirn, M. (2006a). Concepts, extensions, and Frege’s logicist project. Mind, 115, 983–1005.CrossRefGoogle Scholar
Schirn, M. (2006b). Hume’s principle and axiom V reconsidered: Critical reflections on Frege and his interpreters. Synthese, 148, 171–227.CrossRefGoogle Scholar
Schirn, M. (2010). On translating Frege’s Die Grundlagen der Arithmetik. History and Philosophy of Logic, 31, 47–72.CrossRefGoogle Scholar
Schirn, M. (2013). Frege’s approach to the foundations of analysis (1873–1903). History and Philosophy of Logic, 34, 266–292.CrossRefGoogle Scholar
Schirn, M. (2014a). Frege on quantities and real numbers in consideration of the theories of Cantor, Russell and others. In Link, G., editor. Formalism and Beyond. On the Nature of Mathematical Discourse. Boston and Berlin: Walter de Gruyter, pp. 25–95.Google Scholar
Schirn, M. (2014b). Frege’s logicism and the neo-Fregean project. Axiomathes, 24, 207–243.CrossRefGoogle Scholar
Schirn, M. (2016). On the nature, status, and proof of Hume’s Principle in Frege’s logicist project. In Costreie (2016), pp. 49–96.Google Scholar
Schirn, M. (2018a). Frege on the Foundations of Mathematics, Synthese Library, Studies in Epistemology, Logic, Methodology, and Philosophy of Science. New York, London: Springer.Google Scholar
Schirn, M. (2018b). Second-order abstraction before and after Russell’s paradox. In Ebert, P. and Rossberg, M., editors. Essays on Frege’s Basic Laws of Arithmetic. Oxford: Oxford University Press, pp. 433–491.Google Scholar
Schroeder-Heister, P. (1987). A model-theoretic reconstruction of Frege’s permutation argument. Notre Dame Journal of Formal Logic, 28, 69–79.CrossRefGoogle Scholar
Thiel, C. (1975). Zur Inkonsistenz der Fregeschen Mengenlehre. In Thiel, C., editor. Frege und die Moderne Grundlagenforschung. Meisenheim am Glan: Anton Hain, pp. 134–159.Google Scholar
Wehmeier, K. F. (1999). Consistent fragments of Grundgesetze and the existence of non-logical objects. Synthese, 121, 309–328.CrossRefGoogle Scholar
Wehmeier, K. F. (2015). Critical remarks on Frege’s conception of logic by Patricia Blanchette. Journal of the History of Analytic Philosophy, 3, 1–9.Google Scholar
Wehmeier, K. F. & Schroeder-Heister, P. (2005). Frege’s permutation argument revisited. Synthese, 147, 43–61.CrossRefGoogle Scholar