A Modal Transcription of the Hausdorff Residue

Abstract

The modal system S4.Grz is the system that results when the axiom (Grz) □(□(p → □p) → p) → □p is added to the modal system S4, i. e. S4.Grz = S4 + Grz. The aim of the present note is to prove in a direct way, avoiding duality theory, that the modal system S4.Grz admits the following alternative definition: S4.Grz = S4 + R-Grz, where R-Grz is an additional inference rule:

$$ (R-Grz)\;\;\;\;\;\;\;\;\;\; \frac{\vdash\Box(p \rightarrow \Box p) \rightarrow p}{\vdash p} $$

This rule is a modal counterpart of the following topological condition: If a subset A of a topological space X coincides with its Hausdorff residue ρ(A) then A is empty. In other words the empty set is a unique “fixed” point of the residue operator ρ(·).

We also present some consequences of this alternative axiomatic definition.