Property Spaces and Personae (@cite{burnett-2019}, Definitions 3.1–3.3)

@cite{burnett-2019}

Burnett's formalization of @cite{eckert-2008}'s indexical field as a property space — a set of social properties with incompatibility constraints — from which personae emerge as maximal consistent subsets.

Core concepts

Property space (Def. 3.1): A finite set of properties plus a symmetric, irreflexive incompatibility relation. Two properties are incompatible if they cannot co-occur in any coherent persona (e.g., "competent" and "incompetent").

Consistency (Def. 3.2): A set of properties is consistent if no two distinct members are incompatible.

Persona (Def. 3.3): A maximal consistent subset of the property space — a persona that takes a stance on every dimension (every bipolar pair has exactly one pole selected).

structure Sociolinguistics.PropertySpace : Type 1

A property space (Burnett Def. 3.1): a finite set of social properties with a symmetric, irreflexive incompatibility relation.

Incompatible properties cannot co-occur in any coherent persona. For example, "competent" and "incompetent" are incompatible: a coherent persona selects one pole of each bipolar dimension.

• Property : Type

  The type of social properties.

• incompatible : self.Property → self.Property → Bool

  Two properties are incompatible (cannot co-occur in a persona).

• incomp_symm (p q : self.Property) : self.incompatible p q = true → self.incompatible q p = true

  Incompatibility is symmetric.

• incomp_irrefl (p : self.Property) : self.incompatible p p = false

  No property is incompatible with itself.

• propFintype : Fintype self.Property

  Properties form a finite type.

• propDecEq : DecidableEq self.Property

  Properties have decidable equality.

def Sociolinguistics.PropertySpace.isConsistent (ps : PropertySpace) (S : Finset ps.Property) : Bool

A set of properties is consistent if no two distinct members are incompatible (Burnett Def. 3.2).

Equations

• ps.isConsistent S = decide (∀ p ∈ S, ∀ q ∈ S, p ≠ q → ps.incompatible p q = false)

structure Sociolinguistics.Persona (ps : PropertySpace) : Type

A persona (Burnett Def. 3.3): a maximal consistent subset of the property space. Every dimension is decided — a persona takes a stance on each bipolar opposition.

The maximality condition ensures that a persona is as specific as possible: you can't add any property without creating an incompatibility.

• properties : Finset ps.Property

  The properties that characterize this persona.

• consistent : ps.isConsistent self.properties = true

  The property set is consistent (no incompatible pairs).

• maximal (p : ps.Property) : p ∉ self.properties → ∃ q ∈ self.properties, ps.incompatible p q = true

  The property set is maximal: adding any property breaks consistency.

def Sociolinguistics.PropertySpace.allPersonaeSets (ps : PropertySpace) : Finset (Finset ps.Property)

Enumerate all personae by filtering the powerset for maximal consistent subsets.

For small property spaces (like the SCM with 6 properties), this is tractable. The result is a Finset (Finset ps.Property) containing exactly the property sets of all personae.

Equations

• ps.allPersonaeSets = {S ∈ Finset.univ.powerset | (ps.isConsistent S && decide (∀ (p : ps.Property), p ∈ S ∨ ∃ q ∈ S, ps.incompatible p q = true)) = true}