Mathematical logic and linguistics

Glyn Morril (UPC) and Oriol Valentín (UPC)
 Mon-Wed 17h-19h, from Oct 5, 2015
Room 102, Facultat de Matemàtiques i Estadística, UPC

One of the new directions taken by mathematical logic in the late 20th century was the turn towards substructural or resource-conscious inference [Girard 1987].
The non-commutative variety of this had already been anticipated in linguistics thirty years earlier [Lambek 1958]. Recently this convergence of mathematical logic and linguistics has consolidated in a particular categorial logic for syntax and semantics, which it is the objective of this course to present.

  • Syntactic types; grammar as an intuitionistic sublinear logic. Tree-based hypersequent calculus; absorbing structural rules
  • Operations on sets; semantic types. Semantic representation language; higher-order logic as a simply typed lambda-calculus with logical constants
  • Rules of grammatical inference; linguistic applications of connectives
  • Algebraic and frame semantics [Dosen and Schroder-Heister 1993]; [Galatos et al. 2007]
  • Soundness and completeness
  • Lexicon
  • Syntactic and semantic analyses:
    • initial examples
    • the PTQ fragment [Montague 1973]
    • discontinuity [Morrill et al. 2011]
    • relativization
    • coordination
  • Focusing [Andreoli 1992]
  • Cut-elimination [Lambek 1958]
  • Count-invariance [van Benthem 1991]
  1. J. M. Andreoli.
    Logic programming with focusing in linear logic. Journal of Logic and Computation, 2(3):297–347, 1992.
  2. Kosta Dosen and Peter Schroder–Heister, editors.Substructural Logics. Number 2 in Studies in Logic and Computation. Clarendon Press, 1993.
  3. Nikolaos Galatos, Peter Jipsen, Tomasz Kowalski, and Hiroakira Ono, editors. Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Studies in Logic and the Foundations of Mathematics. Elsevier, 2007.
  4. Jean–Yves Girard. Linear logicTheoretical Computer Science, 50:1–102, 1987.
  5. Joachim Lambek. The mathematics of sentence structure. American Mathematical Monthly, 65:154–170, 1958 (Reprinted in Buszkowski, Wojciech, Wojciech Marciszewski, and Johan van Benthem, editors, 1988,Categorial Grammar}, Linguistic & Literary Studies in Eastern Europe volume 25, John Benjamins, Amsterdam, 153–172.)
  6. Richard Montague. The Proper Treatment of Quantification in Ordinary English. In J. Hintikka, J.M.E. Moravcsik, and P. Suppes, editors, Approaches to Natural Language: Proceedings of the 1970 Stanford Workshop on Grammar and Semantics, pages 189–224. D. Reidel, Dordrecht, 1973. Reprinted in R.H.~Thomason, editor, 1974, Formal Philosophy: Selected Papers of Richard Montague, Yale University Press, New Haven, 247–270.
  7. Glyn Morrill, Oriol Valentín, and Mario Fadda. The Displacement Calculus. Journal of Logic, Language and Information, 20(1):1–48, 2011.
    Doi 10.1007/s10849-010-9129-2.
  8. J. van Benthem. Language in Action: Categories, Lambdas, and Dynamic Logic. Number 130 in Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1991. Revised student edition printed in 1995 by the MIT Press.
Course Materials
  • Course text: Morrill and Valentín (in preparation)Categorial Logic for Syntax and Semantics
  • Software: CatLog2 Prolog parser/theorem-prover