Foundation Universal Forcing Ethics Realization Ethics Cost Symm

The symmetry of ethical cost

In Recognition Science, the framework's ledger, a discrete record of events, extends to ethics through a deliberately thin model. The carrier of ethical value is simply the count of morally meaningful improvement steps, represented as a natural number. The cost function ethicsCost assigns a cost of 0 when two states are identical and 1 when they differ. This is a definitional choice, not a claim about the nature of goodness.

The theorem ethicsCost_symm proves that this cost is symmetric: for any two states a and b, the cost from a to b equals the cost from b to a. In plain language, moving from one moral state to another costs exactly the same as moving back. The proof is a simple case analysis: if the states are equal, both sides are 0; if they differ, both sides are 1. This symmetry is a formal property of the chosen cost function, verified in the machine-checked library of formal theorems.

The theorem does not claim that moral improvement is reversible in practice, nor that all steps are equally valuable. It only states a structural property of the cost function itself. The framework deliberately does not rebuild a full domain theory of ethics; it formalizes only the identity and step comparison structure needed for Universal Forcing. The symmetry result is a building block, not a moral philosophy.

theorem ethicsCost_symm (a b : MoralImprovementStep) : ethicsCost a b = ethicsCost b a := by
  by_cases h : a = b
  · subst h; simp [ethicsCost]
  · have h' : b ≠ a := by intro hb; exact h hb.symm
    simp [ethicsCost, h, h']

def ethicsCost (a b : MoralImprovementStep) : Nat :=
  if a = b then 0 else 1

What this page does not claim

The theorem does not claim that moral improvement is reversible or that all improvement steps are equally valuable. The framework does not claim to have a complete theory of ethics, only a thin structural model. The symmetry result does not imply that the cost of moral failure equals the cost of moral success.

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

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