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Erschienen in:
2011 | OriginalPaper | Buchkapitel
A Modal Transcription of the Hausdorff Residue
verfasst von : Leo Esakia
Erschienen in: Logic, Language, and Computation
Verlag: Springer Berlin Heidelberg
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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.