Neo-Davidsonian Event Semantics

• Neo-Davidsonian event semantics is a formal framework that models natural language using explicit event variables and binary thematic-role predicates.
• It enables fine-grained analysis of phenomena like verb phrase anaphora, adverbial modification, and discourse structure through type-theoretic and dynamic implementations.
• The approach integrates proof-theoretic and continuation-based methods to achieve modularity, context sensitivity, and systematic handling of ambiguities.

Neo-Davidsonian event semantics is a foundational paradigm in formal semantics which models natural language verb phrases and sentences using explicit event variables and associated thematic-role predicates. This approach supports fine-grained compositional analysis of event-related phenomena, including verb phrase anaphora, adverbial modification, and discourse structure. Recent developments integrate Neo-Davidsonian ideas within modern proof-theoretic and dynamic semantic frameworks, enabling uniform treatments of anaphora, ellipsis, and discourse-level phenomena in type-theoretic and continuation-based systems (Itegulov et al., 2018, Qian et al., 2011).

1. Core Principles of Neo-Davidsonian Event Semantics

The Neo-Davidsonian approach distinguishes itself by introducing an explicit atomic event type, commonly denoted as $\mathit{event}$ or $v$, which serves as the semantic anchor for action predicates. Every verb introduces an implicit event variable, and thematic roles (e.g., agent, patient, theme) and modifiers (e.g., adverbials, locatives) are realized as first-class (typically binary) predicates over events and individuals, such as $\mathit{agent}(e, x)$ and $\mathit{patient}(e, y)$. This allows event properties to be stackable and independently composable.

In dependent type systems, an atomic event type is postulated: $\mathit{event} : \mathit{type}$ Dependent-pair ($\Sigma$) and dependent-function ($\Pi$) types can then be constructed over events, mirroring the treatment of individuals.

Neo-Davidsonian event representations generalize to:

• Simple event:

$\Sigma\,e:\mathit{event}.\,\mathit{agent}(e, a)$

• Fully specified event (e.g., eating the cake quietly at midnight by John):

$$\Sigma\,e:\mathit{event}.\;(\mathit{ate}(e)\times\mathit{agent}(e, j)\times\mathit{patient}(e, \mathit{cake})\times\mathit{quietly}(e)\times\mathit{at}(e, \mathit{midnight}))$$

Each “$\times$” signifies a dependent pair, yielding an extensible record of event properties (Itegulov et al., 2018).

2. Verb Phrase Anaphora and Event Reconstruction

A major application of this framework is the uniform modeling of verb phrase anaphora (VPA), such as "Mary did too" in response to "John left." Handling such phenomena requires not simply referencing old events but constructing new events that retain certain properties of prior ones while swapping a specific participant (agent/patient).

This is achieved by introducing an extended $\mathit{@}_i$-operator: $$@_i : \gamma_i \to (x:\mathit{entity}) \to \Sigma\,e:\mathit{event}.\;\mathit{agent}(e, x)$$ Here,

• $\gamma_i$ encodes left context (accessible discourse referents),
• The operator slots in a new agent $x$.

Event reconstruction is accomplished by “replace” functions in the global context, such as: $$\mathit{replaceA}: (p:\mathit{entity} \to (\mathit{event} \to \mathit{type})) \to \mathit{entity} \to \mathit{entity} \to (\Sigma\,e':\mathit{event}.p\,\mathit{orig}\,e') \to (\Sigma\,e'':\mathit{event}.p\,\mathit{new}\,e'')$$ This replaces the agent while leaving other event properties specified by $p$ intact.

Example derivations illustrate this mechanism:

• "John left. Mary did too." yields reconstruction of a new "leaving" event for Mary, inheriting non-agent properties (Itegulov et al., 2018).

3. Type-Theoretic and Proof-Theoretic Implementation

This approach situates all event arguments as internal dependent-pair components, aligning with the Curry–Howard correspondence between types and proofs. Adverbial and other modifiers become additional $\Sigma$ components, enabling fine-grained compositionality and extensibility:

• Adverbial modifiers (e.g., “quietly,” “at midnight”) are encoded alongside roles via further dependent pairs.

Strict versus sloppy readings in anaphoric constructions (such as "John likes his hat. Fred does too.") are modeled as admissible variations in the definition of the $\mathit{@}$-operator, without requiring additional mechanisms. The proof-theoretic setting eliminates the need for a distinct model theory for events; all evidence for event properties is constructed within the type system itself (Itegulov et al., 2018).

4. Subtyping, Event Property Refinement, and Continuation-Based Semantics

A subtyping relation for event types enables abstract operators to flexibly match various event structures: $$\Sigma\,e:\mathit{event}.(\mathit{left}(e) \times \mathit{agent}(e, a)) <: \Sigma\,e:\mathit{event}.\mathit{agent}(e, a) <: \Sigma\,e:\mathit{event}.\top = \mathit{event}$$ This guarantees that a term supplying only an agent proof is valid where finer event details (e.g., “left,” “agent”) are expected, provided supporting proofs are reconstructible.

In continuation-based compositional dynamic semantics, as in (Qian et al., 2011), events are embedded in lambdas of the form: $v \to \alpha \to (\alpha \to t) \to t$ Here, the left context $\alpha$ is a list of accessible events. Existential closure occurs only at boundaries via an EOS-like operator. Discourse composition is handled via two sets of functions distinguishing subordinating and coordinating rhetorical relations:

• Subordinating: $\llbracket Sub_{Adv}\rrbracket$ introduces dependencies linked by $\mathit{Rel}$,
• Coordinating: $\llbracket Coor_{Adv}\rrbracket$ introduces abstract coordination super-events and manages context accessibility through operators like $\mathit{Del}$ and $\mathit{Sel}$.

A table contrasts key constructs:

Construct Type	DTS/Proof-Theoretic	Continuation-Based (Dynamic)
Event Typing	$\mathit{event} : \mathit{type}$	$v$ (atomic type)
Thematic Role	Predicate (e.g., $\mathit{agent}(e,x)$)	Predicate (e.g., $Agent(e, x)$)
Discourse Context	$\gamma_i$ (left context)	$\alpha$ (left context, event list)
Anaphora/Ellipsis	$\mathit{@}_i$ replaces roles	Context update via dynamic binding

5. Illustrative Examples and Derivational Workflows

Both dependent-type and dynamic frameworks provide stepwise derivations for key phenomena:

• "John left. Mary did too." is derived by reconstructing a new event with inherited properties using $\textit{replaceA}$ or the corresponding continuation semantics.
• "John quietly ate the cake last night. Mary did too." involves passing a predicate $p$ describing relevant event features to $\textit{replaceA}$.
• Discourse-level examples, such as composing "John kisses Mary in the plaza. She smiles," are handled by sequential context extension and existential closure, with accessibility managed according to the rhetorical structure.

Strict/sloppy ambiguities arise naturally by selecting different variants of the $\mathit{@}_i$ operator, with type checking accommodating multiple readings (Itegulov et al., 2018).

6. Advantages, Limitations, and Open Issues

Neo-Davidsonian event semantics in proof-theoretic and dynamic frameworks enables modular encoding of argument structure, modifiers, and anaphora, supporting:

• Uniform treatment of VPA and ellipsis by reconstructing new events via replace-functions,
• Systematic modeling of strict/sloppy ambiguities as alternative admissible terms,
• Internalization of event properties as dependent types, obviating the need for external model-theoretic constructs,
• Explicit handling of discourse structuring and accessibility through compositions of continuations and event-lists (Itegulov et al., 2018, Qian et al., 2011).

Noted limitations include:

• The necessity to define a growing family of replace-functions for all role combinations requiring replacement,
• Open challenges in extending the approach to "do so" anaphora, modal ellipsis, adjectival anaphora, and in managing world accessibility and quantifier-scope interactions within proof-theoretic semantics,
• In the dynamic setting, careful maintenance of event context lists and rhetorical relation tracking to ensure correct anaphoric resolution and Right-Frontier Constraint enforcement.

A plausible implication is that continued refinement of these frameworks may bridge the gap between logical formalisms and computational pragmatics, facilitating implementations for parsing and semantic interpretation in computational linguistics.

7. Integration with Dynamic and Discourse Semantics

The paradigmatic merger of Neo-Davidsonian event semantics with continuation-based and type-theoretic frameworks has produced comprehensive, compositional pipelines from surface syntax to proof-theoretic or dynamic discourse representations. This integration preserves compositionality, supports context-sensitive anaphoric binding, and enables uniform existential event quantification at natural discourse boundaries. By separating the processes of event quantification and context update, the frameworks precisely model the interplay of event reference, accessibility, and discourse relations as found in natural language (Itegulov et al., 2018, Qian et al., 2011).