Advertisement

Are (Linguists’) Propositions (Topos) Propositions?

  • Carl Pollard
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6736)

Abstract

Lambek([22]) proposed a categorial achitecture for natural language grammars, whereby syntax and semantics are modelled by a biclosed monoidal category (bmc) and a cartesian closed category (ccc) respectively, and semantic interpretation by a functor from syntax to semantics that preserves the biclosed monoidal structure; essentially this same architecture underlies the framework of abstract categorial grammar (ACG, de Groote [12]), except that the bmc is now symmetric, in keeping with the collapsing of Lambek’s directional implications / and \ into the linear implication \(\multimap\). At the same time, Lambek proposed that the semantic ccc bears the additional structure of a topos, and that the meanings of declarative sentences—linguist’s propositions—can be identified with propositions in the sense of topos theory, i.e. morphisms from the terminal object 1 to the subobject classifier Ω. Here we show (1) that this proposal as it stands is untenable, and (2) that a serviceable framework results if a preboolean algebra object distinct from Ω is employed instead. Additionally we show that the resulting categorial structure provides ‘for free’, via Stone duality, an account of the relationship between fine-grained ‘hyperintensional’ semantics ([6],[33],[27],[28]) and the familiar coarse-grained intensional semantics of Carnap ([2]) and Montague ([26]).

Keywords

proposition hyperintension intension topos Stone duality preboolean algebra 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adams, R.: Theories of actuality. Noûs 8, 211–231 (1974)CrossRefGoogle Scholar
  2. 2.
    Carnap, R.: Meaning and Necessity. University of Chicago Press, Chicago (1947)zbMATHGoogle Scholar
  3. 3.
    Ajdukiewicz, K.: Die syntaktische Konnexita ̈t. Studia Philosophica 1, 1–27 (1935); English translation in McCall, S. (ed.) Polish Logic, 1920-1939, 207–231. Oxford University Press, Oxford zbMATHGoogle Scholar
  4. 4.
    Bar-Hillel, Y.: A quasi-arithmetical notation for syntactic description. Language 29, 47–58 (1953)CrossRefzbMATHGoogle Scholar
  5. 5.
    Church, A.: A formulation of the simple theory of types. Journal of Symbolic Logic 5, 56–68 (1940)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cresswell, M.J.: Hyperintensional logic. Studia Logica 34, 25–48 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Diaconescu, R.: Axiom of choice and complementation. Proc. Amer. Math. Soc. 51, 176–178 (1975)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Fox, C., Lappin, S., Pollard, C.: A higher-order fine-grained logic for intensional semantics. In: Proceedings of the Seventh International Symposium on Logic and Language, Pécs, Hungary, pp. 37–46 (2002)Google Scholar
  9. 9.
    Frege, G.: On sense and reference. In: Geach, P., Black, M. (eds.) Translations from the Philosophical Writings of Gottlob Frege, 3rd edn., pp. 56–78. Basil Blackwell, Oxford (1980)Google Scholar
  10. 10.
    Gallin, D.: Intensional and Higher Order Modal Logic. North-Holland, Amsterdam (1975)zbMATHGoogle Scholar
  11. 11.
    Goldblatt, R.: Topoi: the categorial analysis of logic. North-Holland, Amsterdam (1983)zbMATHGoogle Scholar
  12. 12.
    de Groote, P.: Toward abstract categorial grammars. In: Proceedings of the 39th Annual Meeting and 10th Conference of the European Chapter of the Association for Computational Linguistics, pp. 148–155 (2001)Google Scholar
  13. 13.
    Henkin, L.: Completeness in the theory of types. Journal of Symbolic Logic 15, 81–91 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Johnstone, P.: Stone Spaces. Cambridge University Press, Cambridge (1982)zbMATHGoogle Scholar
  15. 15.
    Jónsson, B., Tarski, A.: Boolean algebras with operators, part 1. American Journal of Mathematics 73(4), 891–939 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kripke, S.: A completeness theorem in modal logic. Journal of Symbolic Logic 24, 1–14 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kripke, S.: Semantic analysis of modal logic I: normal modal propositional calculi. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 9, 67–96 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lambek, J.: The Mathematics of sentence structure. American Mathematical Monthly 65, 154–170 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lambek, J.: Deductive systems and categories I: Syntactic calculus and residuated categories. Mathematical Systems Theory 2, 287–318 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lambek, J.: Deductive systems and categories II: Standard constructions and closed categories. In: Dold, A., Eckmann, B. (eds.) Category Theory, Homology Theory, and their Applications. Springer Lecture Notes in Mathematics, vol. 86, pp. 76–122 (1969)Google Scholar
  21. 21.
    Lambek, J.: From λ-calculus to cartesian closed categories. In: Hindley, J., Seldin, J. (eds.) To H.B. Curry: Essays on Combinatorial Logic, Lambda Calculus, and Formalism, pp. 375–402. Academic Press, New York (1980)Google Scholar
  22. 22.
    Lambek, J.: Categorial and categorical grammars. In: Oehrle, R., Bach, E., Wheeler, D. (eds.) Categorial Grammars and Natural Language Structures. Reidel, Dordrecht (1988)Google Scholar
  23. 23.
    Lambek, J., Scott, P.J.: Introduction to Higher Order Categorical Logic. Cambridge University Press, Cambridge (1986)zbMATHGoogle Scholar
  24. 24.
    Martin, S., Pollard, C.: Dynamic hyperintensional semantics: enriching contexts for type-theoretic discourse analysis. In: 15th International Conference on Formal Grammar (FG 2010), Copenhagen. Springer Lecture Notes in Artificial Intelligence, (August 2010) (in press)Google Scholar
  25. 25.
    Martin, S., Pollard, C.: Under revision. A higher-order theory of presupposition. To appear in Special Issue of Studia Logica on Logic and Natural languageGoogle Scholar
  26. 26.
    Montague, R.: The proper treatment of quantification in English. In: Thomason, R. (ed.) Formal Philosophy: Selected Papers of Richard Montague, pp. 247–270. Yale University Press, New Haven (1974)Google Scholar
  27. 27.
    Muskens, R.: Sense and the computation of reference. Linguistics and Philosophy 28(4), 473–504 (2005)CrossRefGoogle Scholar
  28. 28.
    Pollard, C.: Hyperintensions. Journal of Logic and Computation 18(2), 257–282 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Pollard, C.: Remarks on categorical semantics of natural language. Invited talk presented at the Workshop on Logic, Categories, and Semantics, Bordeaux (November 2010), http://www.ling.ohio-state.edu/~pollard/lcs/remarks.pdf
  30. 30.
    Stalnaker, R.: Inquiry. Bradford Books/MIT Press, Cambridge, MA (1984)Google Scholar
  31. 31.
    Stone, M.: The theory of representation for boolean algebras. Transactions of the American Mathematical Society 40, 37–111 (1936)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Stone, M.: Topological representation of distributive lattices and Brouwerian logics. Časopis pešt. mat. fys. 67, 1–25 (1937)zbMATHGoogle Scholar
  33. 33.
    Thomason, R.: A model theory for propositional attitudes. Linguistics and Philosophy 4, 47–70 (1980)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Carl Pollard
    • 1
  1. 1.Department of LinguisticsThe Ohio State UniversityColumbusUSA

Personalised recommendations