Foundation Ontology Predicates Rs True Neg Imp Neg Rs True

Truth and negation

In Recognition Science, the word "true" is not a primitive notion. The framework defines RSTrue (a stable configuration under recognition iteration) as a selection outcome: a proposition is true if it stabilizes under repeated recognition, rather than drifting. This makes truth a verifiable property of a process, not an assumed axiom. The declaration rs_true_neg_imp_neg_rs_true is the formal statement that if a proposition is true in this framework, then its negation is not also true in this framework. It is a consistency condition, a bridge between the framework's operational notion of truth and the classical logical law of non-contradiction.

The declaration does not claim that the framework's truth predicate is equivalent to classical truth in general. It only establishes a one-way implication: from framework-truth of a proposition to framework-falsity of its negation. The converse, that framework-falsity of a negation implies framework-truth of the proposition, is not part of this declaration. The declaration also does not claim that the framework's truth predicate is complete, meaning that every proposition is either true or false within the framework. It only rules out a specific inconsistency: a proposition and its negation cannot both be true.

In plain terms, the declaration is a guardrail. It ensures that the framework's notion of truth does not collapse into triviality, where every proposition and its opposite could both be accepted. This is a necessary property for any notion of truth that aims to be useful. The declaration is a formal theorem within the framework's machine-checked library of formal theorems, meaning it is a proved consequence of the framework's definitions, not an additional assumption. It is a small but load-bearing piece of the framework's logical foundation, ensuring that its operational ontology is coherent.

/-- Classical RSTrue: for pure propositions without dynamics context.
    Equivalent to the old placeholder `def RSTrue (P : Prop) : Prop := P`. -/
def RSTrue_classical (P : Prop) : Prop := P

/-- Classical RSTrue: for pure propositions without dynamics context.
    Equivalent to the old placeholder `def RSTrue (P : Prop) : Prop := P`. -/
def RSTrue_classical (P : Prop) : Prop := P

What this page does not claim

This declaration does not claim that the framework's truth predicate is equivalent to classical truth in general. This declaration does not claim that the framework's truth predicate is complete, meaning every proposition is either true or false. This declaration does not claim that the framework's truth predicate is sound with respect to any external notion of truth.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/OntologyPredicates.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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