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
- Syntactic and semantic analyses:
- initial examples
- the PTQ fragment [Montague 1973]
- discontinuity [Morrill et al. 2011]
- Focusing [Andreoli 1992]
- Cut-elimination [Lambek 1958]
- Count-invariance [van Benthem 1991]
- J. M. Andreoli.
Logic programming with focusing in linear logic. Journal of Logic and Computation, 2(3):297–347, 1992.
- Kosta Dosen and Peter Schroder–Heister, editors.Substructural Logics. Number 2 in Studies in Logic and Computation. Clarendon Press, 1993.
- 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.
- Jean–Yves Girard. Linear logic. Theoretical Computer Science, 50:1–102, 1987.
- 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.)
- 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.
- Glyn Morrill, Oriol Valentín, and Mario Fadda. The Displacement Calculus. Journal of Logic, Language and Information, 20(1):1–48, 2011.
- 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 text: Morrill and Valentín (in preparation)Categorial Logic for Syntax and Semantics
- Software: CatLog2 Prolog parser/theorem-prover