Categorical Framework for Typed Extensional and Intensional Models in Formal Semantics

TL;DR

This paper introduces a category-theoretic framework for modeling intensional and extensional semantics, unifying their representations and clarifying their relationship in formal semantics.

Contribution

It constructs the category ModInt to represent intensional models and shows their equivalence to extensional models, providing a modular, order-independent approach.

Findings

Unified categorical representation of intensionality and extensionality

Equivalence of trivial intensional models to extensional models

Framework applicable to natural language semantics

Abstract

Intensional computation derives concrete outputs from abstract function definitions; extensional computation defines functions through explicit input-output pairs. In formal semantics: intensional computation interprets expressions as context-dependent functions; extensional computation evaluates expressions based on their denotations in an otherwise fixed context. This paper reformulates typed extensional and intensional models of formal semantics within a category-theoretic framework and demonstrates their natural representation therein. We construct $\textbf{ModInt}$, the category of intensional models, building on the categories $\textbf{Set}$ of sets, $\textbf{Rel}$ of relations, and $\textbf{Kr}$ and $\textbf{Kr}_\textbf{b}$ of Kripke frames with monotone maps and bounded morphisms, respectively. We prove that trivial intensional models are equivalent to extensional models, providing a unified categorical representation of intensionality and extensionality in formal semantics. This approach reinterprets the relationship between intensions and extensions in a categorical framework and offers a modular, order-independent method for processing intensions and recovering extensions; contextualizing the relationship between content and reference in category-theoretic terms. We discuss implications for natural language semantics and propose future directions for contextual integration and exploring $\textbf{ModInt}$'s algebraic properties.