Relation Between the Neutrality Properties

This module formalizes:

• se100.remark.PropertyRelation — Relation Between the Two Properties

Extension stability entails interpretive non-commitment.

Suppose that p is framework-variant. One admissible framework supports p, while another supports its object-language negation.

If the substrate committed to p, that commitment would remain available when the substrate was combined with the framework supporting ¬p. The combined theory would then entail both p and ¬p, and therefore entail contradiction.

Similarly, if the substrate committed to ¬p, combining it with the framework supporting p would entail contradiction.

Extension stability rules out both cases.

Only this direction is established. Interpretive non-commitment is not claimed to imply extension stability. The properties retain separate names because they identify distinct design failures: commitment to disputed content and failure of layerability.

theorem SE.NeutralSubstrate.Neutrality.interpretiveNonCommitment_of_extensionStability {L : Logic.Language.PropositionalLanguage} {R : Referent.ReferentCarriers} {C : Logic.ConsequenceSystem L} {S : Substrate.SubstrateSystem L R} {s : S.Carrier} {M : Framework.FrameworkSystem L.carrier} (hstability : ExtensionStability C S s M) : InterpretiveNonCommitment C S s M

Extension stability entails interpretive non-commitment.

A substrate that remains consistent with every admissible framework cannot commit to either side of a framework-variant proposition.