Homogeneity

A trivalent proposition (Trivalent.Prop3) is homogeneous when its extension gap is nonempty ([Kri16]): true when the predicate holds throughout its specification points, false when it fails throughout, and undefined in between. Instantiations differ in the specification-point sort — atoms of a plurality (Homogeneity.Plural), closest antecedent worlds (Homogeneity.Conditional), overlapping pluralities (Homogeneity.Collective), best modal worlds (Studies/AghaJeretic2022). Homogeneity removers (all, necessarily, completely) denote the Beaver-Krahmer assertion operator Prop3.metaAssert, which collapses the gap into the negative extension; the pragmatics of the gap lives in Homogeneity.Usable.

References

• M. Križ, Homogeneity, Non-Maximality, and All
• M. Križ, Aspects of Homogeneity in the Semantics of Natural Language

def Semantics.Homogeneity.isHomogeneous {W : Type u_1} (p : Trivalent.Prop3 W) : Prop

A proposition is homogeneous if its extension gap is nonempty. The gap is what enables non-maximal readings.

Equations

• Semantics.Homogeneity.isHomogeneous p = p.gapExt.Nonempty

theorem Semantics.Homogeneity.isHomogeneous_of_gap {W : Type u_1} (p : Trivalent.Prop3 W) (w : W) (h : p w = Trivalent.indet) : isHomogeneous p

A single gap-world witnesses homogeneity.

theorem Semantics.Homogeneity.isBivalent_iff_not_isHomogeneous {W : Type u_1} (p : Trivalent.Prop3 W) : p.isBivalent ↔ ¬isHomogeneous p

Bivalence and homogeneity are complementary.

theorem Semantics.Homogeneity.not_isHomogeneous_metaAssert {W : Type u_1} (p : Trivalent.Prop3 W) : ¬isHomogeneous p.metaAssert

A meta-asserted proposition is never homogeneous: gap removers yield bivalence.