Extensional First-Order Structure

Definition

Let $\mathcal{L}$ be a first-order logical language with vocabulary $V$ and $S=(D,\mathbf{R})$ an extensional relational structure.

An extensional first-order structure (also called model for $\mathcal{L}$) is a tuple:

$M=(S,I)$

where $I$ (called extensional interpretation function) is a total function:

$I:V\to D\cup \mathbf{R}$

that maps each vocabulary symbol in $V$ to either an element of $D$ or an extensional relation belonging to the set $\mathbf{R}$.

Key Characteristics

• Tarskian semantics: This is the standard model-theoretic semantics for first-order logic
• Extensional interpretation: Assigns denotations to non-logical symbols based on current world state
• No modal content: Cannot capture intended meaning across possible worlds
• Snapshot semantics: Provides interpretation only for a single state of affairs

Limitations

As Guarino emphasizes, extensional interpretation is insufficient for capturing intended meaning. Two key problems arise:

1. Conceptual ambiguity: The same linguistic theory could be satisfied by radically different conceptualizations
2. Modal inadequacy: Cannot distinguish essential from accidental properties

Without intensional anchoring, two agents might agree on all extensional facts while harboring fundamentally incompatible concepts—precisely because concepts are individuated intensionally, not extensionally.

Relationship to Intensional Structures

The extensional first-order structure serves as the foundation for the more sophisticated intensional first-order structure:

• Extensional: $M=(S,I)$ where $S=(D,\mathbf{R})$
• Intensional: $K=(\mathcal{C},\mathcal{I})$ where $\mathcal{C}=(D,W,\mathfrak{R})$

The transition from extensional to intensional captures the move from:

• Single-world snapshots → Multi-world conceptualizations
• Extensional relations → Intensional relations (functions over worlds)
• Surface agreement → Deep conceptual commitment

Examples

Taxonomic Classification

For a biological taxonomy:

• Vocabulary $V$: {mammal, dog, is-a, has-property}
• Domain $D$: {Felix, Fido, Rover, …}
• Relations $\mathbf{R}$: {is-a $\subseteq D\times D$, has-property $\subseteq D\times D$}

The extensional interpretation $I$ maps:

• dog → the set {Fido, Rover, …} ⊆ $D$
• is-a → the subset relation in $\mathbf{R}$

But this doesn’t tell us whether “is-a” means taxonomic classification, mereological parthood, or instance-of relationship.

Enterprise Domain

For an organizational model:

• Vocabulary $V$: {employee, department, works-for}
• Domain $D$: {Alice, Bob, Engineering, Sales, …}
• Relations $\mathbf{R}$: {works-for $\subseteq D\times D$}

An extensional structure captures who currently works for which department, but not:

• What it means to work for a department
• Whether employment is essential or contingent
• How the relation would behave under counterfactual organizational changes

Philosophical Context

This is standard Tarskian semantics: an interpretation assigns denotations to non-logical symbols. The limitation is not in the mathematics, but in what can be expressed: extensional structures lack the modal resources to capture conceptual content that transcends particular states.

Key References

On Truth and Models

Alfred Tarski

The foundational work establishing model-theoretic semantics for first-order logic.

On What Ontology Ought to Be

Jonathan Schaffer (2008)

Critiques truthmaker theory and argues for grounding-based ontological commitment, which aligns with the intensional approach.

Related Concepts

• extensional-relational-structure - The underlying relational structure $(D,\mathbf{R})$
• ontological-commitment - The intensional analogue that maps to conceptual relations
• intensional-relation - What extensional interpretations cannot capture
• conceptualization - The intensional structure that overcomes extensional limitations

Bibliography Keys

• Schaffer2008