Foundation Primitive Recognition Calculus Valid Comparison Examples Probability

The probability bridge

In probability theory, comparing two random events usually means comparing their probabilities. The Recognition Science framework formalizes this intuition through a ledger, a discrete record of events, and a bridge, a rule that translates between a native representation and a displayed value. The theorem probability_display_valid_iff states that for any finite number of possible outcomes N, two events E and F can be validly compared if and only if their rational counting probabilities are equal. In symbols: ValidComparison.IsValidComparison (probabilityDisplayBridge N) E F ↔ DeltaProbability.prob E = DeltaProbability.prob F. The bridge in question displays a native finite event as its rational counting probability, which is the fraction of equally likely outcomes that the event covers.

The theorem is a machine-checked result in the framework's library of formal theorems. It is a special case of a more general principle: a comparison is valid exactly when the native observable and the displayed observable agree. For probability, the native observable is the probability itself, and the display is the identity function on rationals. The proof is immediate from the general validity criterion, and the framework records it as a theorem with no unproved assumptions beyond the standard logical axioms. The headline result packages this with two companion bridges, one for real numbers and one for finite Hilbert spaces, showing the doctrine has concrete instances across different mathematical domains.

What the theorem does not claim is just as important as what it proves. It does not say that probability is the only way to compare events, nor that the rational counting probability is the only valid display. It establishes a conditional equivalence: if you use this particular bridge, then valid comparison collapses to equality of probabilities. The theorem also does not assert anything about infinite outcome spaces, continuous distributions, or the philosophical question of what probability means. It is a structural statement about a specific formal construction, not a claim about the nature of chance or randomness.

For the framework, the significance is that probability theory fits into the recognition calculus as a display layer. The rational counting probability is not an external add-on but a native observable that satisfies the validity criterion. This means the framework's comparison doctrine, which governs how recognition events are judged equal or different, extends naturally to probabilistic settings. The theorem provides a concrete anchor: when a reader encounters a probabilistic comparison in the framework, they can expect it to reduce to equality of rational probabilities, nothing more and nothing less.

theorem probability_display_valid_iff (N : ℕ) (E F : DeltaProbability.Event N) :
    ValidComparison.IsValidComparison (probabilityDisplayBridge N) E F
      ↔ DeltaProbability.prob E = DeltaProbability.prob F :=
  ValidComparison.validComparison_iff_native (probabilityDisplayBridge N) E F

What this page does not claim

The theorem does not claim that rational counting probability is the only valid display for events. It does not assert anything about infinite outcome spaces or continuous distributions. It does not address the philosophical interpretation of probability as chance or randomness.

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

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