Compositionality & Typing in Semantics

• Compositionality and typing are central in formal semantics, ensuring that complex expressions are interpreted via the meanings of their parts and strict type constraints.
• Fine-grained type systems, ranging from commonsense ontologies to higher-order logic, enable the systematic resolution of polysemy, copredication, and selectional restrictions.
• Techniques such as coercive subtyping and category-theoretic models simplify semantic derivations while maintaining robust, scalable interpretations of natural language.

Compositionality and typing are central constructs in formal theories of semantic interpretation for natural language. The principle of compositionality requires that the meaning of any complex expression is uniquely determined by the meanings of its constituent parts and the rules used to combine them. In modern formal semantics and computational linguistics, precise type systems—ranging from commonsense-grounded ontological hierarchies to higher-order type theories—are leveraged to enforce, constrain, and operationalize compositional assembly of meaning. Recent frameworks systematically integrate fine-grained typing to capture selectional restrictions, polysemy, copredication, context adaptation, and metonymic reference, rendering many traditionally complex phenomena nearly trivial within appropriately typed systems.

1. Strongly Typed Ontologies and Commonsense-Based Semantic Composition

Type systems in compositional semantics are frequently grounded in hierarchies that mirror commonsense ontological distinctions. Canonical type lattices contain sorts such as $$\mathsf{human} \sqsubseteq \mathsf{animal} \sqsubseteq \mathsf{entity}$$ and $$\mathsf{artifact} \sqsubseteq \mathsf{physical} \sqsubseteq \mathsf{entity}$$, with further sortal refinements like $\mathsf{substance}$, $\mathsf{event}$, $\mathsf{content}$, and $\mathsf{activity}$ (0704.3886, 0708.2303). Each predicate or relation encodes precise argument type signatures, such as $$\mathsf{Hungry} : \mathsf{animal} \to \mathsf{Prop}$$ or $$\mathsf{Painted} : \mathsf{human} \times \mathsf{painting} \to \mathsf{Prop}$$. Crucially, the system distinguishes between actual and merely conceptual (abstract) existence at the level of types, where $t^a$ (e.g., $x : \mathsf{event}^a$) denotes entities that need not exist in actuality.

Semantic interpretation proceeds by mapping syntactic categories to variables annotated with the most specific compatible type, merging type information via intersection or greatest lower bound (meet) operators. For example, in the phrase "Sheba is hungry," the unification resolves $x : \mathsf{entity}$ (from the name) with $x : \mathsf{animal}$ (from the property), yielding $x : \mathsf{animal}$ (0704.3886, 0708.2303). Type mismatch (resulting in $\bot$) signals a commonsense violation, halting composition.

2. Type-Driven Lexicon and Logical Forms

The typed lexicon assigns each lexical item a denotation of suitable type. Quantifiers are type-annotated, e.g., $$(\exists x : t)\, \varphi(x),\ (\forall x : t)\, \varphi(x)$$. Logical forms are constructed in a manner that every application (function-argument combination or conjunction) is mediated by type unification. For noun compounds, semantic composition may introduce relational templates according to the head-modifier pairing—e.g., "book review" introduces $\mathsf{ReviewOf}(y,x)$ with $y : \mathsf{review}, x : \mathsf{book}$ (0708.2303).

A robust unification mechanism obviates the need for ad hoc solutions such as dot-types or intensional logic for handling intensional verbs, metonymic shifts, and the conceptual-actual existence shift. This reduction to function application, quantifier closure, and intersection within a monotonic type lattice forms the core of "nearly trivial" compositional semantics (0704.3886).

3. Polymorphism, Higher-Order Types, and Adjectival Modification

Polymorphic typing enables the uniform treatment of adjectival modification and the construction of compositional theories for [Adj N] compounds. Adjectives are modeled as higher-order polymorphic functions, e.g., $$\mathsf{BEAUTIFUL}: \forall\alpha \leq \mathsf{entity}. \alpha \to t$$, where each occurrence is instantiated at the most specific type the modified noun licenses (0801.4746). The composition rule for [Adj N] enforces application only when the noun type $s$ satisfies $s \sqsubseteq T_0$ (adjective's domain), with the meet operation resolving multiple type assignments.

This architecture derives not only the semantics of individual [Adj N] compounds but also adjective ordering restrictions: orderings that would otherwise introduce incompatible or non-unifiable types render the sequence ungrammatical (0801.4746).

4. Second-Order Type Systems and Coercive Subtyping

The Montagovian Generative Lexicon operationalizes compositionality via Girard’s System F, which allows universal quantification over types (second-order lambda calculus) (Retoré, 2013, Retoré, 2013). Base types include a fine-grained $e_{i}$-system reflecting grammatical sorts, with predicates and argument slots typed accordingly.

Coercive subtyping handles ontological inclusions: a global, acyclic coercion relation $A_0 <: A$ is maintained, along with explicit coercion terms $c_{A_0,A}: A_0 \to A$. Function application automatically inserts these coercions as needed, enabling, for instance, "A chair barks" to fail type-checking unless a chair→Dog coercion is present (Retoré, 2013). Copredication and sense adaptation are implemented via polymorphic conjunction terms (e.g., $$\text{PolAnd} : \forall\alpha.\forall\beta.(\alpha\to t)\to(\beta\to t)\to\forall\xi.\xi\to(\xi\to\alpha)\to(\xi\to\beta)\to t$$).

This approach provides a uniform mechanism to treat determiners, quantifiers, meaning transfers, copredication, and nuanced selectional restrictions in a fully compositional, lexicalized, and type-safe fashion (Retoré, 2013, Retoré, 2013).

5. Category-Theoretic Typing in Quantitative Compositional Semantics

The DisCoCat (Distributional Compositional Categorical) model situates syntactic and semantic typing within category theory, employing a compact closed monoidal category (often a pregroup grammar) for syntax and finite-dimensional real vector spaces ($\mathbf{FdVect}_{\mathbb{R}}$) for semantics (Grefenstette, 2013). A strong monoidal functor $F$ maps syntactic types and reductions to their semantic counterparts, commuting with all structural morphisms.

The compact closed categorical axioms enforce strict typing and compositionality: only linear maps prescribed by the grammatical reduction (via contractions in the tensor category) produce valid semantic vectors. Thus, every grammatical parse yields a unique compositional recipe, guaranteeing bracketing independence and type safety in the vector space model.

Extensions to richer grammatical frameworks or semantic representations (e.g., neural embeddings, co-occurrence vectors, SVD/NMF) are allowed as long as the target remains a monoidal category and the functoriality is preserved (Grefenstette, 2013).

6. Illustrative Derivations and Operational Simplicity

Within these frameworks, classic semantic puzzles such as intensional verbs, nominal compounds, and dot-type phenomena are resolved via type unification and coercive subtyping, without recourse to specialized intensional operators or duplication of logical structures (0704.3886, 0708.2303, Retoré, 2013). For example, the semantic derivation for "John painted a large elephant" introduces both a painting and a conceptually abstract elephant, relating them via $\mathsf{PaintingOf}(p, e)$, and maintains proper type constraints through unification (0704.3886).

Similarly, the coordinated copredication in "Liverpool is large and defeated Chelsea" is assembled through polymorphic "and" and the appropriate coercions, ensuring the subject is interpreted both as a location (for “large”) and as a football club (for “defeated Chelsea”) without incoherence (Retoré, 2013). The strong typing disciplines obviate the need for higher-order intensional or dot-type mechanisms present in older systems.

7. Implications and Extensions

The adoption of rich type systems and compositional architectures has resulted in substantial simplification and unification of semantic composition mechanisms. Nearly all complexity is relegated to the structure of the ontology and the accompanying subtyping or coercive relations, with compositional rules collapsing to function application, type unification, and relation template instantiation. Scalability is maintained: new lexical or grammatical phenomena are accommodated by expanding the ontological base rather than altering the assembly machinery (0708.2303, Retoré, 2013).

Different theoretical frameworks—commonsense-grounded ontologies, polymorphic System F type theories, and categorical distributional models—have converged on the conclusion that fine-grained typing, maintained throughout composition, is both necessary and sufficient for capturing the central principle of compositionality in natural language semantics (0704.3886, Grefenstette, 2013).