nLab dependent type theoretic methods in natural language semantics



Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type/path type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
completely presented setdiscrete object/0-truncated objecth-level 2-type/set/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels





A number of researchers are using dependent type theory as a formal tool to understand natural language. Some are using in particular homotopy type theory.


In the Preface to his book, Type-theoretic Grammar (Ranta 94), Aarne Ranta recounts how the idea of studying natural language in constructive type theory occurred to him in 1986:

In Stockholm, when I first discussed the project with Per Martin-Löf, he said that he had designed type theory for mathematics, and than natural language is something else. I said that similar work had been done within predicate calculus, which is just a part of type theory, to which he replied that he found it equally problematic. But his general attitude was far from discouraging: it was more that he was so serious about natural language and saw the problems of my enterprise more clearly than I, who had already assumed the point of view of logical semantics. His criticism was penetrating but patient, and he was generous in telling me about his own ideas. So we gradually developed a view that satisfied both of us, that formal grammar begins with what is well understood formally, and then tries to see how this formal structure is manifested in natural language, instead of starting with natural language in all it unlimitedness and trying to force it into some given formalism.

A noted early application of dependent type theory to natural language was Göran Sundholm's treatment (Sundholm 86) of the Donkey sentence

Any farmer who owns a donkey beats it,

which he rendered, using dependent product and dependent sum, as

z: x:Farmer( y:DonkeyOwns(x,y))Beats(p(z),p(q(z))). \prod_{z: \sum_{x: Farmer} (\sum_{y: Donkey}Owns(x, y))}Beats(p(z), p(q(z))).

This marks an improvement over first-order logic in which one has to rewrite the Donkey sentence as something like:

For all farmers and for all donkeys, if the farmer owns the donkey then he beats it.

xy(Farmer(x)Donkey(y)Own(x,y)Beat(x,y)). \forall x \forall y (Farmer(x) \wedge Donkey(y) \wedge Own(x, y) \to Beat(x, y)).


Papers using dependent type theory

  • René Ahn, Agents, Objects and Events: A computational approach to knowledge, observation and communication, Ph.D. dissertation, Eindhoven University, 2001.

  • Asher, N.: Lexical Meaning in Context: A Web of Words. Cambridge University Press (2011)

  • Asher, N., Luo, Z.: Formalisation of coercions in lexical semantics. In: Sinn und Bedeutung. vol. 17, pp. 63–80 (2012)

  • Daisuke Bekki: Representing anaphora with dependent types. In: Asher, N., Soloviev, S. (eds.) Logical Aspects of Computational Linguistics, Lecture Notes in Computer Science, vol.8535, pp. 14–29 (2014), Springer.

  • Daisuke Bekki, Asher, N.: Logical polysemy and subtyping. In: Motomura, Y., Butler, A., Bekki, D. (eds.) New Frontiers in Artificial Intelligence, Lecture Notes in Computer Science, vol. 7856, pp. 17–24. Springer (2013)

  • Daisuke Bekki, Satoh, M.: Calculating projections via type checking. In: Proceedings of TYTLES (to appear)

  • Robin Cooper, Austinian Truth, Attitudes and Type Theory, Research on Language and Computation. July 2005, Volume 3, Issue 2-3, pp 333-362 Springer

  • Chatzikyriakidis, S., Luo, Z.: Natural language inference in Coq. Journal of Logic, Language and Information 23(4), 441–480 (2014)

  • Zhaohui Luo: Type-theoretical semantics with coercive subtyping. In: Semantics and Linguistic Theory 20 (SALT 20). Vancouver (2010)

  • Zhaohui Luo: Contextual analysis of word meanings in type-theoretical semantics. In: Pogodalla, S., Prost, J.-P. (eds.) LACL 2011. LNCS, vol. 6736, pp. 159–174. Springer, Heidelberg (2011)

  • Zhaohui Luo: Common nouns as types. In: Béchet, D., Dikovsky, A. (eds.) LACL 2012. LNCS, vol. 7351, pp. 173–185. Springer, Heidelberg (2012), pdf.

  • Zhaohui Luo: Event Semantics with Dependent Types, pdf

  • Zhaohui Luo, Formal Semantics in Modern Type Theories: Is It Model-theoretic, Proof-theoretic, or Both?, pdf.

  • Koji Mineshima, A Presuppositional Analysis of Definite Descriptions in Proof Theory, New Frontiers in Artificial Intelligence Lecture Notes in Computer Science Volume 4914, 2008, pp 214-227 Springer

  • P. Piwek and E. Krahmer, ‘Presuppositions in Context: Constructing Bridges’, in Formal Aspects of Context, eds., P. Bonzon, M. Cavalcanti,and R. Nossum, volume 20 of Applied Logic Series, 85–106, Kluwer Academic Publishers, Dordrecht, (2000).

  • Aarne Ranta, Type-theoretical grammar. Oxford University Press (1994)

  • Göran Sundholm, Proof theory and meaning. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. 3, pp. 471–506. Springer (1986)

  • Tanaka, R., Mineshima, K., Bekki, D.: Resolving modal anaphora in Dependent Type Semantics. In: Proceedings of LENLS11. pp. 43–56 (2014), Springer.

Papers using homotopy type theory

  • Bahramian, H., Nematollahi, N., Sabry, A.: Copredication in Homotopy Type Theory, pdf
  • David Corfield, Expressing ‘the structure of’ in homotopy type theory, webpage

A worked example, homotopy type theory in the Grammatical Framework?


  • Daisuke Bekki, Anaphora and Presuppositions in Dependent Type Semantics (Theoretical side) slides
  • Daisuke Bekki, Anaphora and Presuppositions in Dependent Type Semantics (Empirical side) slides
  • Ribeka Tanaka, Generalized Quantifiers in Dependent Type Semantics slides
  • Yusuke Kubota and Robert Levine, Scope parallelism in coordination in Dependent Type Semantics slides

Last revised on December 9, 2020 at 09:19:40. See the history of this page for a list of all contributions to it.