Skip to main content Accessibility help
×
×
Home

What is Tarski's Common Concept of Consequence?

  • Ignacio Jané (a1)

Abstract

In 1936 Tarski sketched a rigorous definition of the concept of logical consequence which, he claimed, agreed quite well with common usage—or, as he also said, with the common concept of consequence. Commentators of Tarski's paper have usually been elusive as to what this common concept is. However, being clear on this issue is important to decide whether Tarski's definition failed (as Etchemendy has contended) or succeeded (as most commentators maintain). I argue that the common concept of consequence that Tarski tried to characterize is not some general, all-purpose notion of consequence, but a rather precise one, namely the concept of consequence at play in axiomatics. I identify this concept and show that Tarski's definition is fully adequate to it.

Copyright

References

Hide All
[1] Awodey, Steve and Carus, André W., Carnap, completeness, and categoricity: The Gabelbarkeitssatz of 1928, Erkenntnis, vol. 54 (2001), pp. 145172.
[2] Awodey, Steve and Reck, Erich H., Completeness and categoricity, part I: Nineteenth-century axiomatics to twentieth-century metalogic, History and Philosophy of Logic, vol. 23 (2002), pp. 130.
[3] Bays, Timothy, On Tarski on models, The Journal of Symbolic Logic, vol. 66 (2001), pp. 17011726.
[4] Carnap, Rudolf, Bericht über Untersuchungen zur allgemeinen Axiomatik, Erkenntnis, vol. 1 (1930), pp. 302307.
[5] Carnap, Rudolf, The logical syntax of language, Routledge and Kegan Paul, London, 1937.
[6] Etchemendy, John, The concept of logical consequence, Harvard University Press, Cambridge, Massachusetts, 1990.
[7] Ewald, William (editor), From Kant to Hilbert: A source book in the foundations of mathematics, vol. II, Clarendon Press, Oxford, 1996.
[8] Ferreirós, José, Labyrinth of thought, Birkhäuser, Basel, 1999.
[9] Fraenkel, Adolf, Einleitung in die Mengenlehre, third ed., Julius Springer, Amsterdam, 1928.
[10] Gödel, Kurt, Collected works, vol. I: Publications 1929–1936. Feferman, S. et al. (eds.), Oxford University Press, New York, 1986.
[11] Gómez-Torrente, Mario, Tarski on logical consequence, Notre Dame Journal of Formal Logic, vol. 37 (1996), pp. 125151.
[12] Gómez-Torrente, Mario, On a fallacy attributed to Tarski, History and Philosophy of Logic, vol. 19 (1998), pp. 227234.
[13] Hilbert, David, Über den Zahlbegriff, Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 8 (1900), pp. 180184, Translated into English as “On the concept of number”, in [7], pp. 1092–1095, from which references are made.
[14] Hilbert, David and Ackermann, Wilhelm, Grundzüge der theoretischen Logik, Springer, Berlin, 1928.
[15] Hilbert, David and Bernays, Paul, Grundlagen der Mathematik, vol. 1, Springer, Berlin, 1934.
[16] Huntington, Edward V., Sets of independent postulates for the algebra of logic, Transactions of the American Mathematical Society, vol. 5 (1904), pp. 288309.
[17] Huntington, Edward V., The fundamental propositions of algebra, Monographs on topics of modern mathematics (Young, John Wesley, editor), Longmans, Green and Co., New York, 1911, pp. 149207.
[18] Huntington, Edward V., A set of postulates for abstract geometry expressed in terms of the simple relation of inclusion, Mathematische Annalen, vol. 73 (1913), pp. 522559.
[19] Kleene, Stephen C., Introduction to metamathematics, Wolters-Noordhoff, Groningen, 1971.
[20] Kline, Morris, Mathematical thought from ancient to modern times, Oxford University Press, New York, 1972.
[21] Langford, C. H., Some theorems on deducibility, Annals of Mathematics, vol. 28 (1926), pp. 1640.
[22] Langford, C. H., Theorems on deducibility, Annals of Mathematics, vol. 28 (1927), pp. 459471.
[23] MacLane, Saunders, Review of Tarski: Einführung in die Mathematische Logik und in die Methodologie der Mathematik, The Journal of Symbolic Logic, vol. 3 (1938), pp. 5152.
[24] Mancosu, Paolo, Tarski onmodels and logical consequence, The architecture of modern mathematics (Ferreirós, J. and Gray, J., editors), Oxford University Press, Forthcoming.
[25] Padoa, Alessandro, Essai d'une théorie algébrique des nombres entiers, précédé d'une introduction logique à une théorie déductive quelconque, Bibliothèque du Congrès international de philosophie, Paris, 1900, vol. 3 (1900), pp. 309365, Partial English translation in [64], pp. 118–123, to which references are made.
[26] Pasch, Moritz, Vorlesungen über neuere Geometrie, second ed., Springer, Berlin, 1926, The first edition appeared in 1882.
[27] Peano, Giuseppe, Studii di logica matematica, Atti della Reale Accademia delle Scienze di Torino, vol. 32 (1897), pp. 565583, reprinted in [28], pp. 201–217.
[28] Peano, Giuseppe, Opere scelte, Unione Matematica Italiana, Edizioni Cremonese, Roma, 1958.
[29] Peano, Giuseppe, The principles of arithmetic, presented by a new method, In van Heijenoort, [64], pp. 8397.
[30] Pieri, Mario, Sui principii che reggono la geometria di posizione. Nota I, Atti della Reale Accademia delle Scienze di Torino, vol. 30 (1895), pp. 607641, reprinted in [35], pp. 13–48.
[31] Pieri, Mario, I principii della geometria di posizione composti in sistema logico deduttivo, Memorie della Reale Accademia delle Scienze di Torino, vol. 48 (1898), pp. 162, reprinted in [35], pp. 101–162.
[32] Pieri, Mario, Sur la géometrie envisagée comme un système purement logique, Bibliothèque du Congrès international de philosophie, Paris, 1900, vol. 3 (1900), pp. 367404, reprinted in [35], pp. 235–272.
[33] Pieri, Mario, Sur la compatibilité des axiomes de l'arithmétique, Revue de Métaphysique et de Morale, vol. 14 (1906), pp. 196207, reprinted in [35], pp. 377–388.
[34] Pieri, Mario, La geometria elementare istituita sulle nozione di “punto” e “sfera”, Memorie della Società Italiana delle Scienze, vol. 15 (1908), pp. 345450, reprinted in [35], pp. 455–560.
[35] Pieri, Mario, Opere sui fondamenti della matematica, Unione Matematica Italiana, Edizioni Cremonese, Bologna, 1980.
[36] Ray, Greg, Logical consequence: A defense of Tarski, Journal of Philosophical Logic, vol. 25 (1996), pp. 617677.
[37] Sagüillo, José Miguel, Logical consequence revisited, this Bulletin, vol. 3 (1997), pp. 216241.
[38] Scanlan, Michael J., Who were the American postulate theorists?, The Journal of Symbolic Logic, vol. 56 (1991), pp. 9811002.
[39] Scanlan, Michael J., American postulate theorists and Alfred Tarski, History and Philosophy of Logic, vol. 24 (2003), pp. 307325.
[40] Sher, Gila, The bounds of logic, The MIT Press, Cambridge, Massachusetts, 1991.
[41] Sher, Gila, Did Tarski commit “Tarski's fallacy”?, The Journal of Symbolic Logic, vol. 61 (1996), pp. 653686.
[42] Szczerba, L.W., Tarski and geometry, The Journal of Symbolic Logic, vol. 51 (1986), pp. 907912.
[43] Tarski, Alfred, Einige Betrachtungen über die Begriffe der ω-Widerspruchsfreiheit und der ω-Vollständigkeit, Monatshefte für Mathematik und Physik, vol. 40 (1933), pp. 97112, reprinted in [59], pp. 621–636.
[44] Tarski, Alfred, Über den Begriff der logischen Folgerung, Actes du Congrès International de Philosophie Scientifique, vol. 7 (1936), pp. 111, reprinted in [60], pp. 271–281.
[45] Tarski, Alfred, Einführung in die Mathematische Logik und in die Methodologie der Mathematik, Julius Springer, Wien, 1937.
[46] Tarski, Alfred, Sur la méthode deductive, Travaux du IXe Congrés International de Philosophie, vol. 6 (1937), pp. 95103, in [60], pp. 325–333.
[47] Tarski, Alfred, What is elementary geometry?, The axiomatic method, with special reference to geometry and physics (Henkin, L., Suppes, P., and Tarski, A., editors), North Holland, Amsterdam, 1959, pp. 1629.
[48] Tarski, Alfred, The concept of truth in formalized languages, In Logic, Semantics, Metamathematics [52], pp. 152278.
[49] Tarski, Alfred, Foundations of the calculus of systems, In Logic, Semantics, Metamathematics [52], pp. 342383.
[50] Tarski, Alfred, Foundations of the geometry of solids, In Logic, Semantics, Metamathematics [52], pp. 2429.
[51] Tarski, Alfred, Fundamental concepts of the methodology of the deductive sciences, In Logic, Semantics, Metamathematics [52], pp. 60109.
[52] Tarski, Alfred, Logic, semantics, metamathematics, translated by Woodger, J. H., edited and introduced by Corcoran, J., second ed., Hackett, Indianapolis, Indiana, 1983.
[53] Tarski, Alfred, On definable sets of real numbers, In Logic, Semantics, Metamathematics [52], pp. 110142.
[54] Tarski, Alfred, On some fundamental concepts of metamathematics, In Logic, Semantics, Metamathematics [52], pp. 3037.
[55] Tarski, Alfred, On the concept of logical consequence, In Logic, Semantics, Metamathematics [52], pp. 409420.
[56] Tarski, Alfred, On the foundations of boolean algebra, In Logic, Semantics, Metamathematics [52], pp. 320341.
[57] Tarski, Alfred, Some methodological investigations on the definability of concepts, In Logic, Semantics, Metamathematics [52], pp. 296319.
[58] Tarski, Alfred, Some observations on the concepts of ω-consistency and ω-completeness, In Logic, Semantics, Metamathematics [52], pp. 279295.
[59] Tarski, Alfred, Collected papers. vol 1: 1921–1934, Birkhäuser, Basel, 1986, edited by Givant, Steven R. and McKenzie, Ralph N..
[60] Tarski, Alfred, Collected papers. vol 2: 1935–1944, Birkhäuser, Basel, 1986, edited by Givant, Steven R. and McKenzie, Ralph N..
[61] Tarski, Alfred, On the concept of following logically. Translated from the Polish by Magda Stroińska and David Hitchcock, History and Philosophy of Logic, vol. 23 (2002), pp. 155196.
[62] Tarski, Alfred and Givant, Steven, Tarski's system of geometry, this Bulletin, vol. 5 (1999), pp. 175214.
[63] Tarski, Alfred and Lindenbaum, Adolf, On the limitations of themeans of expression of deductive theories, In Logic, Semantics, Metamathematics [52], pp. 384392.
[64] van Heijenoort, Jan (editor), From Frege to Gödel: A source book in mathematical logic, Harvard University Press, Cambridge, Massachusetts, 1967.
[65] Vaught, Robert L., Alfred Tarski's work in model theory, The Journal of Symbolic Logic, vol. 51 (1986), pp. 869882.
[66] Veblen, Oswald, A system of axioms for geometry, Transactions of the American Mathematical Society, vol. 5 (1904), pp. 343384.
[67] Veblen, Oswald and Young, John Wesley, Projective geometry, vol. I, Ginn and Company, Boston, 1910.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Bulletin of Symbolic Logic
  • ISSN: 1079-8986
  • EISSN: 1943-5894
  • URL: /core/journals/bulletin-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed