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The paper also records the indexed representation layer motivating the terminology. For a finite closure system, an indexed truth space selects closed theories as contexts and represents each element by the region of selected closed theories containing it. Closure consequence is always sound for region inclusion, and the full indexed space of all closed theories recovers the original closure consequence exactly; reduced indexed spaces can therefore create spurious region consequences by deleting separating closed theories. A formal opposite gives a four-region membership decomposition - only one, only the other, both, and neither - unless additional separation assumptions are imposed. Finally, overlap-compatible local closed theories glue by canonical union under the atlas-generated closure. The framework is finite, structural, and closure-theoretic; the logical terminology is used only as an interpretation of the underlying closure data.
From: Jaehwan Kim [view email]
[v1]
Fri, 19 Jun 2026 01:27:18 UTC (23 KB)
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