, Volume 69, Issue 3, pp 363–376 | Cite as

Wittgensteinian Tableaux, Identity, and Co-Denotation

Original Article


Wittgensteinian predicate logic (W-logic) is characterized by the requirement that the objects mentioned within the scope of a quantifier be excluded from the range of the associated bound variable. I present a sound and complete tableaux calculus for this logic and discuss issues of translatability between Wittgensteinian and standard predicate logic in languages with and without individual constants. A metalinguistic co-denotation predicate, akin to Frege’s triple bar of the Begriffsschrift, is introduced and used to bestow the full expressive power of first-order logic with identity on W-logic in the presence of constants.


Predicate Symbol Sequent Calculus Individual Constant Open Branch Existential Sentence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



Thanks to Sam Hillier for various discussions concerning W-logic, and to Peter Schroeder-Heister for the suggestion to provide tableaux rules for W-logic. I also wish to acknowledge my gratitude to Ulrich Pardey for numerous illuminating discussions regarding identity. Finally, I am grateful to two anonymous referees for this journal for asking a number of technical and philosophical questions that helped improve this paper.


  1. Bergmann, M. (2005). Finite tree property for first-order logic with identity and functions. Notre Dame Journal of Formal Logic, 46(2), 173–180.CrossRefGoogle Scholar
  2. Bergmann, M., Moor, J., & Nelson, J. (2004). The logic book (4th ed.). New York: McGraw-Hill.Google Scholar
  3. Boolos, G. (1984). Trees and finite satisfiability: Proof of a conjecture of Burgess. Notre Dame Journal of Formal Logic, 25(3), 193–197.CrossRefGoogle Scholar
  4. Caton, C. E. (1976). The idea of sameness challenges reflection. In M. Schirn (Ed.), Studien zu Frege II–Studies on Frege II (pp. 167–180). Stuttgart-Bad Cannstatt: Frommann-Holzboog.Google Scholar
  5. Church, A. (1951). A formulation of the logic of sense and denotation. In P. Henle, H. M. Kallen, & S. K. Langer (Eds.), Structure, method, and meaning: Essays in honor of Henry M. Sheffer (pp. 3–24). New York: Liberal Arts Press.Google Scholar
  6. Frege, G. (1879). Begriffsschrift. Halle: Louis Nebert.Google Scholar
  7. Furth, M. (1967). Editor’s introduction to Gottlob Frege, The Basic Laws of Arithmetic: Exposition of the system (translated and edited, with an introduction, by Montgomery Furth). Berkeley and Los Angeles: University of California Press.Google Scholar
  8. Hintikka, J. (1956). Identity, variables, and impredicative definitions. Journal of Symbolic Logic, 21, 225–245.CrossRefGoogle Scholar
  9. Jeffrey, R. C. (1991). Formal logic: Its scope and limits (3rd ed.). New York: McGraw-Hill.Google Scholar
  10. Mendelsohn, R. L. (2005). The philosophy of Gottlob Frege. New York: Cambridge University Press.Google Scholar
  11. Pardey, U. (1994). Identität, Existenz und Reflexivität. Weinheim: Beltz Athenäum.Google Scholar
  12. Schütte, K. (1977). Proof theory. New York: Springer-Verlag.Google Scholar
  13. Shoenfield, J. (1967). Mathematical logic. Reading: Addison-Wesley.Google Scholar
  14. Smith, P. (1985). Review of [3], Mathematical Reviews MR744833 (85f:03005).Google Scholar
  15. Smullyan, R. M. (1968). First-order logic. New York: Springer-Verlag.Google Scholar
  16. Wehmeier, K. F. (2004). Wittgensteinian predicate logic. Notre Dame Journal of Formal Logic, 45(1), 1–11.CrossRefGoogle Scholar
  17. Williams, C. J. F. (1989). What is identity? Oxford: Clarendon Press.Google Scholar
  18. Wittgenstein, L. (1922). Tractatus Logico-Philosophicus (C. K. Ogden, Trans.). London: Routledge and Kegan Paul.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Logic & Philosophy of ScienceUniversity of CaliforniaIrvineUSA

Personalised recommendations