Foundation Primitive Recognition Calculus Trace Logic Exists Intro

The existential rule

In ordinary mathematics, proving that something exists requires producing it. The declaration exists_intro in the Recognition Science framework's machine-checked library of formal theorems is the same rule, adapted to a setting where statements are not allowed to become false when new events arrive. The framework builds its logic on finite traces, which are discrete records of events, and a proposition is defined as a predicate on traces that persists under trace extension. That persistence is the key: if a statement holds on a short record, it must still hold on every longer record that extends it.

The existential rule then states that if a specific witness, call it a, is available and the property P holds for a on a trace T, then the existential statement "there exists something satisfying P" holds on T. The proof is immediate: the witness itself is the evidence. The framework's library formalizes this as a theorem, not a definition, because it derives the rule from the underlying structure of stable predicates rather than assuming it as an axiom.

What the rule does not claim is just as important. It does not say that existence is constructive in the sense of providing an algorithm to find the witness; it only requires that a witness be given. It does not establish that any particular object exists, only that the logical step from witness to existential claim is valid. And it says nothing about whether the witness is unique or how it might be discovered. The rule is a bridge from a specific example to a general statement, nothing more.

theorem exists_intro {α : Type} {P : α → TracePredicate} {T : Trace}
    (a : α) (h : (P a).holds T) :
    (exists_ P).holds T := by
  exact ⟨a, h⟩

What this page does not claim

The rule does not provide an algorithm for finding a witness. The rule does not establish that any particular object exists. The rule does not assert uniqueness of the witness.

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/PrimitiveRecognitionCalculus/TraceLogic.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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