Documentation

Linglib.Theories.Sociolinguistics.EckertMontague

Eckert-Montague Lift (@cite{burnett-2019}, Definition 3.4) #

@cite{burnett-2019}

Burnett's set-theoretic reinterpretation of @cite{eckert-2008}'s indexical field. A variant's social meaning is formalized as the set of personae compatible with the properties it indexes.

The EM field #

Given a grounded indexical field F : Variant → Finset Property, the Eckert-Montague lift maps each variant to its compatible personae:

EM(F)(v) = { p ∈ Persona | F(v) ⊆ p.properties }

This is analogous to Montague's lift from entities to generalized quantifiers: a variant doesn't denote a single persona but a set of personae — those consistent with the social properties it indexes.

Bridge: fromIndexicalField #

The fromIndexicalField function converts sign-valued indexical fields (from Core.SocialMeaning) into grounded fields over the SCM property space. This bridges existing studies (BSB2022, B&S2024) to Burnett's formalism:

source
structure Sociolinguistics.EckertMontague.GroundedField (Variant : Type) (ps : PropertySpace) :
Type

A grounded indexical field: maps each variant to a consistent set of properties from a property space.

This is the set-theoretic version of Eckert's indexical field. Each variant indexes a set of social properties; the set must be internally consistent (no incompatible properties).

Instances For
    source
    def Sociolinguistics.EckertMontague.emField {Variant : Type} {ps : PropertySpace} (gf : GroundedField Variant ps) (v : Variant) :
    Finset (Finset ps.Property)

    The Eckert-Montague lift (Burnett Def. 3.4): the set of personae compatible with a variant's indexed properties.

    EM(F)(v) = { p ∈ allPersonaeSets | F(v) ⊆ p }

    More properties indexed → fewer compatible personae (antitonicity).

    Equations
    Instances For
      source
      theorem Sociolinguistics.EckertMontague.emField_antitone {Variant : Type} {ps : PropertySpace} (gf : GroundedField Variant ps) (v₁ v₂ : Variant) (h : gf.indexedProperties v₁ ⊆ gf.indexedProperties v₂) :
      emField gf v₂ ⊆ emField gf v₁

      The EM lift is antitone: more indexed properties → fewer compatible personae.

      source
      def Sociolinguistics.EckertMontague.scmPropertyIndexed {Variant : Type} (field : Core.SocialMeaning.IndexicalField Variant SCM.SocialDimension) (v : Variant) :
      SCM.SCMProperty → Prop

      Decidable predicate: whether an SCM property is indexed by a variant given a sign-valued field over social dimensions.

      Maps association signs to SCM poles:

      • association(v, d) > 0 → positive pole of dimension d
      • association(v, d) < 0 → negative pole of dimension d
      • association(v, d) = 0 → no property on dimension d
      Equations
      Instances For
        source
        instance Sociolinguistics.EckertMontague.instDecidablePredSCMPropertyScmPropertyIndexed {Variant : Type} (field : Core.SocialMeaning.IndexicalField Variant SCM.SocialDimension) (v : Variant) :
        DecidablePred (scmPropertyIndexed field v)
        Equations
        • One or more equations did not get rendered due to their size.
        source
        def Sociolinguistics.EckertMontague.scmPropertiesFromField {Variant : Type} (field : Core.SocialMeaning.IndexicalField Variant SCM.SocialDimension) (v : Variant) :
        Finset SCM.SCMProperty
        Equations
        Instances For
          source
          theorem Sociolinguistics.EckertMontague.scmPropertiesFromField_consistent {Variant : Type} (field : Core.SocialMeaning.IndexicalField Variant SCM.SocialDimension) (v : Variant) :
          SCM.scmSpace.isConsistent (scmPropertiesFromField field v) = true

          The SCM properties derived from a sign-valued field are always consistent: no variant can index both poles of the same dimension, because x > 0 and x < 0 cannot both hold.

          source
          def Sociolinguistics.EckertMontague.fromIndexicalField {Variant : Type} (field : Core.SocialMeaning.IndexicalField Variant SCM.SocialDimension) :
          GroundedField Variant SCM.scmSpace

          Convert a sign-valued IndexicalField to a GroundedField over the SCM property space.

          Equations
          Instances For
            source
            def Sociolinguistics.EckertMontague.emFieldMI {Variant : Type} {ps : PropertySpace} (gf : GroundedField Variant ps) (v : Variant) :
            Finset (Finset ps.Property)

            The Eckert–Montague lift with intersection semantics: a persona is compatible with variant v iff the persona shares at least one property with v's Eckert field.

            This is the Montagovian Individual interpretation from footnote 14 of @cite{burnett-2019}: EM({p₁, p₂}) = {π ∈ PERS | p₁ ∈ π ∨ p₂ ∈ π}.

            Compare emField (§2 above) which uses the subset semantics (all indexed properties must be in the persona). The intersection semantics gives a weaker / more inclusive meaning function, matching Burnett's Table 1.

            Equations
            Instances For
              source
              def Sociolinguistics.EckertMontague.emMeaningMI {Variant : Type} {ps : PropertySpace} (gf : GroundedField Variant ps) (v : Variant) (persona : Finset ps.Property) :
              Bool

              Meaning function from the intersection-based EM lift. Returns true iff persona shares at least one property with the Eckert field of v.

              Equations
              Instances For
                source
                theorem Sociolinguistics.EckertMontague.emFieldMI_monotone {Variant : Type} {ps : PropertySpace} (gf : GroundedField Variant ps) (v₁ v₂ : Variant) (h : gf.indexedProperties v₁ ⊆ gf.indexedProperties v₂) :
                emFieldMI gf v₁ ⊆ emFieldMI gf v₂

                The intersection-based EM lift is monotone: more indexed properties → more compatible personae (opposite of emField_antitone).