, Note that (15) is ill-typed for two reasons now: the first x is a variable not assumed anywhere and the term bark(x, e) is ill-typed as well. Employing dependent event types in the Montagovian semantics (i.e., in C e as described in Sect. 4.1), would still leave a small possibility of some formally legal but incorrect semantics. For instance, one might consider the following semantics for (8): (16) (#) ?e : Event. ¬?x : e. dog(x) & bark(e) Note that, although (16) is incorrect, it is still well-typed because e is just an event, not an event with x as agent. 9 This, however, 14) ¬?x : Dog. ?e : Evt A (x). bark(x, e) (15) (#) ?e : Evt A (x). ¬?x : Dog. bark(x, e)

N. Asher and Z. Luo, Formalisation of coercions in lexical semantics, Sinn und Bedeutung, vol.17, 2012.

L. Champollion, The interaction of compositional semantics and event semantics, Linguist. Philos, vol.38, pp.31-66, 2015.

S. Chatzikyriakidis and Z. Luo, Modern Perspectives in Type-Theoretical Semantics, 2017.

S. Chatzikyriakidis and Z. Luo, Formal Semantics in Modern Type Theories, 2018.

A. Church, A formulation of the simple theory of types, J. Symb. Log, vol.5, issue.1, pp.56-68, 1940.

D. Davidson, The logical form of action sentences, 1967.

P. De-groote and Y. Winter, A type-logical account of quantification in event semantics, Logic and Engineering of Natural Language Semantics, vol.11, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01102261

H. Goguen, A typed operational semantics for type theory, 1994.

F. Landman, The Handbook of Contemporary Semantic Theory, 1996.

Z. Luo, A problem of adequacy: conservativity of calculus of constructions over higher-order logic, LFCS report series ECS-LFCS-90-121, 1990.

Z. Luo, Computation and Reasoning: A Type Theory for Computer Science, 1994.

Z. Luo, Coercive subtyping in type theory, CSL 1996, vol.1258, pp.275-296, 1997.

Z. Luo, Formal semantics in modern type theories with coercive subtyping, Linguist. Philos, vol.35, issue.6, pp.491-513, 2012.

Z. Luo, S. Soloviev, and T. Xue, Coercive subtyping: theory and implementation, Inf. Comput, vol.223, pp.18-42, 2012.
URL : https://hal.archives-ouvertes.fr/hal-01130574

P. Martin-löf, Intuitionistic Type Theory. Bibliopolis, 1984.

R. Montague, Collected papers Ed, 1974.

T. Parsons, Events in the Semantics of English, 1990.

A. Ranta, Type-Theoretical Grammar, 1994.

A. Williams, Arguments in Syntax and Semantics, 2015.

Y. Winter and J. Zwarts, Event semantics and abstract categorial grammar, MOL 2011. LNCS (LNAI), vol.6878, pp.174-191, 2011.

T. Xue, Theory and implementation of coercive subtyping, 2013.