Abstract
It is well known that Tarski proved a result which can be stated roughly as: no sufficiently rich, consistent, classical language can contain its own truth definition. Tarski's way around this problem is to deal with two languages at a time, an object language for which we are defining truth and a metalanguage in which the definition occurs. An obvious question then is: under what conditions can we construct a definition of truth for a given object language. Tarski claims that it is necessary and sufficient that the metalanguage be “essentially richer”. Our contention, put bluntly, is that this claim deserves more scrutiny from philosophers than it usually gets and in fact is false unless “essentially richer” means nothing else than “sufficient to contain a truth definition for the object language.”
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Bell, J. L. (1994): Fregean Extensions of First-order Logic, Mathematical Logic Quarterly 40: 27–30.
Bell, J. L. and Machover, M. (1977): A Course in Mathematical Logic, North-Holland, Amsterdam.
Carnap, R. (1937): The Logical Syntax of Language, transl. by A. Smeaton, Harcourt, Brace, New York.
Carnap, R. (1957): Introduction to Symbolic Logic and its Applications, transl. by W. H. Meyer and J. Wilkinson, Dover, New York.
DeVidi, D. and Solomon, G. (1995): Tolerance and Metalanguages in Carnap's Logical Syntax of Language, Synthese 103: 123–139.
Feferman, S. (1960): Arithmetization of Metamathematics in a General Setting, Fundament Mathematicae 49: 35–92.
Fine, K. and McCarthy, T. (1984): Truth Without Satisfaction, Journal of Philosophical Logic 13: 397–421.
Gupta, A. (1982): Truth and Paradox, Journal of Philosophical Logic 11: 1–60.
Hájek, P. and Pudlák, P. (1993): Metamathematics of First-Order Arithmetic, Springer-Verlag, Berlin.
Hazen, A. (1983): Predicative Logics, Handbook of Philosophical Logic, Vol. 1 (D. Gabbay and F. Guenther, eds.), Reidel, Dordrecht, pp. 331–407.
Kreisel, G. (1962): Review of Feferman (1960) Zentralblatt für Mathematik und ihre Grenzgebiete 95: 243–246.
Lawvere, W. and Schanuel, S. (1991): Conceptual Mathematics: A First Introduction to Category Theory (Preliminary Version).
McGee, V. (1990): Truth, Vagueness and Paradox, Hackett, Indianapolis.
McGee, V. (1992): Maximal Consistent Sets of Instances of Tarski's Schema (T), Journal of Philosophical Logic 21: 235–241.
Montague, R. and Vaught, R. L. (1959): Natural Models of Set Theories, Fundamenta Mathematicae 47: 219–242.
Mostowski, A. (1950): Some Impredicative Definitions in the Axiomatic Set Theory, Fundamenta Mathematicae 37, 111–124. Reprinted in Foundational Studies: Selected Works, vol. 1, (K. Kuratowski et al., eds.) North-Holland, Amsterdam, pp. 479–492.
Novak, I. (1950): A Construction for Models of Consistent Systems, Fundamenta Mathematicae 37: 87–100.
Parsons, C. (1974a): Informal Axiomatization, Formalization, and the Concept of Truth, Synthese 27: 22–47. Reprinted in C. Parsons Mathematics in Philosophy: Selected Essays, Cornell University Press, Ithaca, N.Y., 1983, pp. 71–91. Page references are to this reprint.
Parsons, C. (1974b): Sets and Classes, Nous 8: 1–12. Reprinted in C. Parsons, Mathematics in Philosophy, pp. 209–220.
Parsons, C. (1974c): The Liar Paradox, Journal of Philosophical Logic 3: 381–412. Reprinted with a Postscript in C. Parsons, Mathematics in Philosophy, pp. 221–267.
Rogers, R. (1963): A Survey of Formal Semantics, Synthese 15: 17–56.
Rogers, R. (1971): Mathematical Logic and Formalized Theories, North-Holland, Amsterdam.
Sheard, M. (1994): A Guide to Truth Predicates in the Modern Era, Journal of Symbolic Logic 59: 1032–1054.
Tarski, A. (1939): On Undecidable Statements in Enlarged Systems of Logic and the Concept of Truth, Journal of Symbolic Logic 4: 105–112.
Tarski, A. (1944): The Semantic Conception of Truth and the Foundations of Semantics, Philosophy and Phenomenological Research 4: 341–375. Reprinted in Readings in Philosophical Analysis (H. Feigl and W. Sellars, eds.), Appleton-Century-Crofts, New York, 1949, and in Basic Topics in the Philosophy of Language (R. M. Harnish, ed.), Prentice Hall, Englewood Cliffs, NJ, 1994.
Tarski, A. (1969): Truth and Proof, Scientific American 6: 63–77. Reprinted in A Philosophical Companion to First-Order Logic (R. I. G. Hughes, ed.), Hackett, Indianapolis, 1993.
Tarski, A., Mostowski, A. and Robinson, R. (1953): Undecidable Theories, North-Holland, Amsterdam.
Visser, A. (1989): Semantics and the Liar Paradox, in the Handbook of Philosophical Logic, vol. 4 (D. Gabbay and F. Guenther, eds.), Reidel, Dordrecht, pp. 617–706.
Wang, H. (1951): Arithmetic Translations of Axiom Systems, Transactions of the American Mathematical Society 71: 283–293. Reprinted in revised form inWang (1970).
Wang, H. (1952): Truth Definitions and Consistency Proofs, Transactions of the American Mathematical Society 73: 243–275. Reprinted in slightly revised form in Wang (1970).
Wang, H. (1970): Logic, Computers, and Sets, Chelsea Publishing, New York. First published as H. Wang, A Survey of Mathematical Logic, Science Press, Peking, 1962.
Wang, H. (1986): Beyond Analytic Philosophy: Doing Justice to What We Know, Bradford/ MIT Press, Cambridge, Mass.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
DeVidi, D., Solomon, G. Tarski on “essentially richer” metalanguages. Journal of Philosophical Logic 28, 1–28 (1999). https://doi.org/10.1023/A:1004294325183
Issue Date:
DOI: https://doi.org/10.1023/A:1004294325183