Notre Dame Journal of Formal Logic

Tarski on Logical Consequence

Mario Gómez-Torrente

Abstract

This paper examines from a historical perspective Tarski's 1936 essay, "On the concept of logical consequence." I focus on two main aims. The primary aim is to show how Tarski's definition of logical consequence satisfies two desiderata he himself sets forth for it: (1) it must declare logically correct certain formalizations of the $\omega$-rule and (2) it must allow for variation of the individual domain in the test for logical consequence. My arguments provide a refutation of some interpreters of Tarski, and notably John Etchemendy, who have claimed that his definition does not satisfy those desiderata. A secondary aim of the paper is to offer some basic elements for an understanding of Tarski's definition in the historical logico-philosophical context in which it was proposed. Such historical understanding provides useful insights on Tarski's informal ideas on logical consequence and their internal cohesion.

Article information

Source
Notre Dame J. Formal Logic, Volume 37, Number 1 (1996), 125-151.

Dates
First available in Project Euclid: 16 December 2002

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1040067321

Digital Object Identifier
doi:10.1305/ndjfl/1040067321

Mathematical Reviews number (MathSciNet)
MR1379554

Zentralblatt MATH identifier
1186.03010

Subjects
Primary: 03-03: Historical (must also be assigned at least one classification number from Section 01)
Secondary: 01A60: 20th century 03A05: Philosophical and critical {For philosophy of mathematics, see also 00A30}

Citation

Gómez-Torrente, Mario. Tarski on Logical Consequence. Notre Dame J. Formal Logic 37 (1996), no. 1, 125--151. doi:10.1305/ndjfl/1040067321. https://projecteuclid.org/euclid.ndjfl/1040067321


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References

  • Carnap, R., Logische Syntax der Sprache, Springer, Vienna, 1934. Zbl 0182.00301 MR 38:24
  • Carnap, R., The Logical Syntax of Language, Routledge & Kegan Paul, London, 1937.
  • Carnap, R., “Intellectual autobiography,” pp. 1–84 in The Philosophy of Rudolf Carnap, edited by P. A. Schilpp, Open Court, La Salle, 1963.
  • Coffa, J. A., The Semantic Tradition from Kant to Carnap, Cambridge University Press, Cambridge, 1991.
  • Corcoran, J., “Conceptual structure of classical logic,” Philosophy and Phenomenological Research, vol. 33 (1972), pp. 25–47. Zbl 0657.03005
  • Corcoran, J., “Meanings of implication,” Diálogos, vol. 4 (1973), pp. 59–76. MR 1260674
  • Etchemendy, J., “The doctrine of logic as form,” Linguistics and Philosophy, vol. 6 (1983), pp. 319–334.
  • Etchemendy, J., “Tarski on truth and logical consequence,” The Journal of Symbolic Logic, vol. 53 (1988), pp. 51–79. Zbl 0645.03002 MR 89g:01051
  • Etchemendy, J., “Models, semantics and logical truth,” Linguistics and Philosophy, vol. 11 (1988), pp. 91–106. Zbl 0653.03004
  • Etchemendy, J., The Concept of Logical Consequence, Harvard University Press, Cambridge, 1990. Zbl 0743.03002 MR 92f:03003
  • Gödel, K., “Die Vollständigkeit der Axiome des logischen Funktionenkalküls,” Monatshefte für Mathematik und Physik, vol. 37 (1930), pp. 349–360. Translated as “The completeness of the axioms of the functional calculus of logic,” pp. 583–591 in From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, edited by J. van Heijenoort, Harvard University Press, Cambridge, 1967.
  • Gödel, K., “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I,” Monatshefte für Mathematik und Physik, vol. 38 (1931), pp. 173–198. Translated as “On formally undecidable propositions of Principia Mathematica and Related Systems I,” pp. 596–616 in From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, edited by J. van Heijenoort, Harvard University Press, Cambridge, 1967.
  • Gómez-Torrente, M., “On a fallacy attributed to Tarski,” forthcoming in History and Philosophy of Logic. Zbl 01678958 MR 2000f:03006
  • Hempel, C. G., “On the nature of mathematical truth,” pp. 377–393 in The Philosophy of Mathematics, second edition, edited by P. Benacerraf and H. Putnam, Cambridge University Press, Cambridge, 1983. Zbl 0060.01909 MR 7,193b
  • Hilbert, D., and W. Ackermann, Grundzüge der theoretischen Logik, Springer, Berlin, 1928.
  • Hilbert, D., and W. Ackermann, Grundzüge der theoretischen Logik, second edition, Springer, Berlin, 1938. Zbl 0018.19301
  • Hilbert, D., and W. Ackermann, Principles of Mathematical Logic, Chelsea, New York, 1950. Zbl 0040.00402
  • Hodges, W., “Truth in a structure,” Proceedings of the Aristotelian Society, new series, vol. 86 (1985–86), pp. 135–151.
  • Moore, G. H., “The emergence of first-order logic,” pp. 95–135 in History and Philosophy of Modern Mathematics, edited by W. Aspray and P. Kitcher, Minnesota Studies in the Philosophy of Science, vol. XI, University of Minnesota Press, Minneapolis, 1988. MR 89i:03001
  • Quine, W. V., “Quine to Carnap,” pp. 294–300 in Dear Carnap, Dear Van. The Quine-Carnap Correspondence and Related Work, University of California Press, Berkeley, 1990.
  • Ramsey, F. P., “The foundations of mathematics,” pp. 1–61 in The Foundations of Mathematics and other Logical Essays, Kegan Paul, London, 1931.
  • Sher, G., The Bounds of Logic, M.I.T. Press, Cambridge, 1991. MR 94g:03014
  • Tarski, A., “On definable sets of real numbers,” pp. 110–142 in Logic, Semantics, Metamathematics, second edition, Hackett, Indianapolis, 1983.
  • Tarski, A., “Some observations on the concepts of $\omega$-consistency and $\omega$-completeness,” pp. 279–295 in Logic, Semantics, Metamathematics, second edition, Hackett, Indianapolis, 1983.
  • Tarski, A., “Bemerkung der Redaktion,” Fundamenta Mathematicæ, vol. 23 (1934), p. 161.
  • Tarski, A., and A. Lindenbaum, “On the limitations of the means of expression of deductive theories,” pp. 384–392 in Logic, Semantics, Metamathematics, second edition, Hackett, Indianapolis, 1983.
  • Tarski, A., “The concept of truth in formalized languages,” pp. 152–278 in Logic, Semantics, Metamathematics, second edition, Hackett, Indianapolis, 1983.
  • Tarski, A., “On the foundations of Boolean algebra,” pp. 320–341 in Logic, Semantics, Metamathematics, second edition, Hackett, Indianapolis, 1983.
  • Tarski, A., “Foundations of the calculus of systems,” pp. 343–383 in Logic, Semantics, Metamathematics, second edition, Hackett, Indianapolis, 1983.
  • Tarski, A., “On the concept of logical consequence,” pp. 409–420 in Logic, Semantics, Metamathematics, second edition, Hackett, Indianapolis, 1983.
  • Tarski, A., Einführung in die mathematische Logik und in die Methodologie der Mathematik, Springer, Vienna, 1937. Zbl 0018.00101
  • Tarski, A., Introduction to Logic and to the Methodology of Deductive Sciences, Oxford University Press, New York, 1941. Zbl 0025.00401 MR 2,209a
  • Tarski, A., “A philosophical letter of Alfred Tarski,” Journal of Philosophy, vol. 84 (1987), pp. 28–32. MR 88d:03008
  • Tarski, A., “What are logical notions?,” History and Philosophy of Logic, vol. 7 (1986), pp. 143–154. Zbl 0622.03004 MR 88b:03010
  • Tarski, A., “A general method in proofs of undecidability,” pp. 1–35 in Undecidable Theories, by A. Tarski, A. Mostowski, and R. Robinson, North-Holland, Amsterdam, 1953.
  • Vaught, R. L., “Model theory before 1945,” pp. 153–172 in Proceedings of the Tarski Symposium, edited by L. Henkin et al., Proceedings of Symposia in Pure Mathematics, vol. XXV, American Mathematical Society, Providence, 1974. Zbl 0306.02005 MR 57:12157
  • Vaught, R. L., “Tarski's work in model theory,” The Journal of Symbolic Logic, vol. 51 (1986), pp. 869–882. Zbl 0619.03007 MR 88g:03008a