rs_true_classical_iff
The equivalence states that the classical recognition truth predicate applied to any proposition P holds exactly when P is true. Researchers formalizing the operational ontology of Recognition Science cite this when moving from bare propositions to cost-based stability conditions. The proof is a direct reflexivity step on the definition of the classical predicate.
claimFor any proposition $P$, the classical recognition truth predicate satisfies $RSTrue_{classical}(P) ↔ P$.
background
The module OntologyPredicates supplies the operational ontology of Recognition Science, where existence and truth arise as outcomes of cost minimization under the J function rather than as primitives. RSTrue_classical is introduced as the classical placeholder that simply returns the input proposition, in contrast to the full RSTrue that requires stabilization under recognition iteration. Upstream results include the definition of RSExists as the conjunction of positivity and zero defect from CostFirstExistence, together with the Configuration structure from InitialCondition that supplies the discrete ledger entries on which J-cost is evaluated.
proof idea
The proof is a one-line term that applies Iff.rfl directly to the defining equation of RSTrue_classical.
why it matters in Recognition Science
This theorem supplies the classical bridge inside the ontology predicates, permitting direct importation of ordinary propositions before the J-cost and defect conditions of the full framework are imposed. It supports the module's selection rule that a proposition is true precisely when it stabilizes under recognition iteration. The declaration sits upstream of any later theorems that equate RSTrue with RSExists or RSReal, though no downstream uses are recorded in the current graph.
scope and limits
- Does not impose J-cost stability or defect collapse.
- Does not reference the phi-ladder or discrete configuration space.
- Does not connect to RSExists or RSReal predicates.
- Does not address the eight-tick octave or spatial dimension forcing.
formal statement (Lean)
213theorem rs_true_classical_iff (P : Prop) : RSTrue_classical P ↔ P := Iff.rfl
proof body
Term-mode proof.
214
215/-! ## RSReal: Existence in the Discrete Configuration Space -/
216
217/-- **RSReal**: A value x is "real" in the RS sense if:
218 1. RSExists x (stable under J)
219 2. x is in the discrete configuration space (quantized)
220
221 For now, we model discreteness as being algebraic in φ. -/