A dependently-typed calculus of event telicity and culminativity

We present a dependently-typed cross-linguistic framework for analyzing the telicity and culminativity of events, accompanied by examples of using our framework to model English sentences. Our framework consists of two parts. In the nominal domain, we model the boundedness of noun phrases and its relationship to subtyping, delimited quantities, and adjectival modification. In the verbal domain we define a dependent event calculus, modeling telic events as those whose undergoer is bounded, culminating events as telic events that achieve their inherent endpoint, and consider adverbial modification. In both domains we pay particular attention to associated entailments. Our framework is defined as an extension of intensional Martin-Löf dependent type theory, and the rules and examples in this paper have been formalized in the Agda proof assistant.

💡 Research Summary

The paper introduces a novel, dependently‑typed framework for modeling the linguistic notions of telicity (the presence of an inherent endpoint in an event) and culminativity (the actual attainment of that endpoint). Built as an extension of intensional Martin‑Löf dependent type theory, the system is formalized and mechanically verified in the Agda proof assistant.

The authors split the analysis into two domains. In the nominal domain, they define a primitive notion of boundedness for noun phrases. A noun phrase is bounded if it can supply an inherent endpoint for an event (e.g., “three apples”, “five liters of water”), and unbounded otherwise (e.g., “apples”, “water”). Boundedness is captured by a subtyping relation (Bounded ⊂ NP) and is linked to the classic in‑adverbial test: only sentences with bounded undergoers can combine with temporal in‑adverbials (“in ten minutes”). This definition is deliberately cross‑linguistic; it does not rely on the presence of determiners, allowing the same analysis to apply to languages without articles (e.g., Russian) by focusing on the ability to provide an endpoint rather than syntactic form.

In the verbal domain, events are introduced as a type Event. A dependent predicate Telic : Event → Set classifies those events whose undergoer is of bounded type; thus telicity is directly tied to the boundedness of the argument. Culminativity is then defined as a further dependent type Culminates : Telic e → (Entails ResultState e). In other words, a telic event culminates precisely when it entails its associated intended resulting state (the “endpoint”). This captures the classic entailment pattern: “John ate the soup” entails “There is no soup left”, while “John ate soup” (with an unbounded object) does not entail any specific resulting state.

The paper provides extensive linguistic examples to illustrate the calculus. English sentences such as “John ate the soup” and “John repaired my computer” are shown to be telic and, under the usual reading, also culminating because the verb’s lexical semantics entails the resulting state. The authors also discuss borderline cases, such as “John wiped the table”, where a result‑oriented reading yields a telic event whose culmination is not logically forced by the verb alone; this is reflected in the type system by allowing a telic event without an automatic Culminates proof. Mandarin examples demonstrate that the perfective particle –le marks event completion without guaranteeing culmination, thereby providing a genuine telic but non‑culminating construction that the framework can model.

A key contribution is the focus on entailment relations rather than merely on the origins of telicity. By encoding the entailment of the resulting state as a type‑level condition, the framework makes explicit which inferences are semantically valid and which are not, distinguishing semantic entailment from pragmatic implicature. This aligns with the view that entailment is central to formal semantics.

The authors compare their approach to prior work, noting that most existing formal analyses adopt either a mereological (parts‑of‑events) or lexico‑semantic (predicate decomposition) perspective. Their dependent‑type calculus, by contrast, leverages type‑level dependencies to capture the interaction between noun‑phrase boundedness, verb‑level telicity, and event‑level culminativity in a single, compositional system. The implementation in Agda verifies all typing rules, definitions, and example derivations, providing a machine‑checked guarantee of consistency.

In summary, the paper delivers a rigorously formalized, cross‑linguistically motivated calculus that connects nominal boundedness to verbal telicity and further to culminativity via entailment. It demonstrates how dependent types can serve as a powerful tool for modeling fine‑grained aspectual phenomena, opening avenues for extending the framework to other aspects of meaning such as outer aspect, tense, and more complex argument structures.