Foundation Ontology Predicates Rs True Classical Iff

The classical bridge

In classical logic, a proposition is true when it accurately describes the world. The Recognition Science declaration rs_true_classical_iff makes a precise statement about this: within the framework, the predicate it calls RSTrue_classical is logically equivalent to the ordinary truth of the proposition. In the machine-checked library of formal theorems, this is written as RSTrue_classical P ↔ P, and the proof is immediate, it is true by definition. This is not a claim about what makes something true in the world; it is a statement about how the framework's formal language connects to the standard one.

The declaration sits inside a larger project that defines existence and truth as selection outcomes, the results of a cost-minimization process. That project also proves, for example, that the only real number that exists in its sense is 1, and that the cost function has a specific form. But rs_true_classical_iff itself does none of that work. It is a bridge, not a discovery. It ensures that when the framework's proofs use the word true, a reader can translate that back into the ordinary meaning without loss or distortion. The statement is a consistency check, a guarantee that the framework's formal language has not drifted away from the classical one on this point.

The declaration does not claim that truth is a selection outcome in general. That is a separate, more ambitious idea, developed elsewhere in the framework's files. Here, the scope is narrow: for the specific predicate named RSTrue_classical, the equivalence holds. It also does not claim that the framework's other truth-like notions, such as stability under iteration, are the same as classical truth. Those are different concepts with their own definitions and theorems. The value of this declaration is that it fixes a reference point, a place where the framework and the classical tradition provably agree, so that more elaborate constructions can be checked against it.

theorem rs_true_classical_iff (P : Prop) : RSTrue_classical P ↔ P := Iff.rfl

What this page does not claim

This declaration does not claim that all truth in the framework is a selection outcome. It does not claim that the framework's other truth-like notions, such as stability under iteration, are equivalent to classical truth. It does not claim that the framework's notion of existence is the same as the classical one.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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