Plural Identity Theory Judgments

Abstract

Distinct identity concepts can issue opposite judgments under distinct relations.

Theorem 1.1 (Identity theories can disagree on distinct propositions).

$$\begin{gathered} \exists C1,C2:Bool\to Bool, \\ C1\ne C2\wedge \\ \mathrm{ConceptIdentity}\left(C1\right)\ne \mathrm{ConceptIdentity}\left(C2\right)\wedge \\ \mathrm{ConceptIdentity}\left(C1,false,true\right)\wedge \\ \neg \mathrm{ConceptIdentity}\left(C2,false,true\right). \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/Identity/PluralIdentityTheoryJudgments.identity_theories_can_disagree_on_distinct_propositions (✓ std3). ∎

Source. Repository-derived.

Commentary.

The first Boolean concept is constant, while the second is the identity readout. They are different concepts and induce different concept-relative compatibility relations.

The constant concept identifies false with true, whereas the identity concept distinguishes them. Because each judgment names its own compatibility relation, the disagreement is not a proposition and its negation inside one theory.

References

• Truth anchor: D5/S3/ConceptDynamics/Identity/PluralIdentityTheoryJudgments.identity_theories_can_disagree_on_distinct_propositions
• Dependency: D5/S3/ConceptDynamics/Identity/ConceptRelativeIdentity