Information-Complete Normative Divergence

Abstract

With distinct normative values, complete information permits disagreement; incomplete information permits consensus blind to a Boolean target.

Theorem 1.1 (Complete information permits normative divergence).

$$\begin{array}{c}(\forall X,I,U:\mathrm{Type},\\ \forall concept:\mathrm{Concept}\left(X,I\right),\\ \forall leftValue,rightValue:U,\\ (\mathrm{Nonempty}\left(X\right)\wedge leftValue\ne rightValue\wedge \mathrm{Injective}\left(concept\right))\Rightarrow \\ \exists leftNorm,rightNorm:\mathrm{Concept}\left(X,U\right),\\ \exists witness:X,leftNorm(witness)\ne rightNorm(witness))\wedge \\ (\forall X,I:\mathrm{Type},\\ \forall concept:\mathrm{Concept}\left(X,I\right),\\ \neg \mathrm{Injective}\left(concept\right)\Rightarrow \\ \exists leftNorm,rightNorm,target:\mathrm{Concept}\left(X,Bool\right),\\ \mathrm{NormativeConsensus}\left(leftNorm,rightNorm\right)\wedge \\ \neg (\exists answer:I\to Bool,target=answer\circ concept)).\end{array}$$

Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/Deliberation/InformationCompleteNormativeDivergence.complete_information_permits_normative_divergence (✓ std3). ∎

Source. Repository-derived.

Commentary.

For any nonempty state type and any two distinct normative values, an injective concept admits two normative functions that disagree at a state. The constant functions at the chosen values provide the disagreement, so informational completeness does not impose a unique norm.

For every noninjective concept, two equal Boolean normative functions witness consensus while a Boolean target remains impossible to recover from the concept. Consensus here is equality of the two normative functions, not a claim that all possible norms agree.

A collision in the concept fiber supplies the separating target: it distinguishes states that the concept identifies, and therefore cannot factor through any Boolean answer on the concept’s codomain. Agreement can thus coexist with blindness to a relevant target.

References

• Truth anchor: D5/S3/ConceptDynamics/Deliberation/InformationCompleteNormativeDivergence.complete_information_permits_normative_divergence
• Dependency: D5/S3/ConceptDynamics/Contracts/FutureObligationIncompleteness