The third rung · what ought to be

Deontic Arithmetic

The invariant layer says what cannot be. The heuristic layer says what is better. Neither says what ought to be — obligations, permissions, prohibitions, and what you owe once an obligation is broken. That is the deontic layer: a small lattice of normative status, a join that accrues norms, priority that resolves their conflicts, and a contrary-to-duty mechanism that escalates rather than fails silently.

IThe algebra

A norm assigns an action one of four deontic statuses. They form a diamond lattice ordered by how much they constrain the agent: OPTIONAL (permitted, no constraint) at the bottom; OBLIGATORY and FORBIDDEN as incomparable middles; CONFLICT (both at once — over-constrained) at the top.

Permission, obligation, and prohibition are the classical deontic square, interdefinable through negation: P(p) ≡ ¬O(¬p), F(p) ≡ O(¬p) ≡ ¬P(p). A consistent norm set obeys the SDL axiom D: O(p) → P(p) — you cannot be obligated to do the impermissible. Exactly where that fails is the CONFLICT node.

Norms and operations

A Norm is a conditional rule of one modality, with a priority and an optional contrary-to-duty repair: { modality, condition, priority, ctd }. Six operations work over norms and the statuses they induce.

Precedence is principled. The three modalities compose in a fixed order: alethic (can't be violated, ever) over deontic (ought — violable, but a breach has consequences) over axiological (preference). So a forbidden option is excluded but recorded as overridable; an obligatory feasible option is forced over anything that merely scores higher; and an obligation whose only satisfier is alethically infeasible triggers escalation — never a silent fall-back.

IIThe laws

Twelve laws, the same discipline as the other rungs. Nine for the norm algebra, three for its interaction with the bridge.

IIIThe laws, checked

The same harness style as the invariant and heuristic pages — each law verified against random inputs, in your browser, on the engine that ships in box-and-box.

IVType sketch

// the status lattice and a norm
type Status = "optional" | "obligatory" | "forbidden" | "conflict";

interface Norm {
  modality:  "obligatory" | "forbidden" | "permitted";
  condition: (ctx: Ctx) => boolean;   // factual detachment
  priority:  number;                  // resolves conflicts
  ctd?:      Norm;                     // contrary-to-duty repair
}

function join(a: Status, b: Status): Status;        // accrue — D1..D5
function resolve(v: Verdict): Resolved;               // priority — D6
function detach(n: Norm, ctx: Ctx,
       opts?: { violated: boolean }): InForce;          // partial — D7,D8

// the three-modality decision: alethic ▸ deontic ▸ axiological
function govern(
  options: { id; value: Value; utility: number; ctx: Ctx }[],
  cfg: { req: Requirements; norms: Norm[]; semiring: Semiring }
): {
  decision: string | null;
  forcedByObligation: boolean;
  escalation: { required: true; repair: string; reason: string } | null;
  deonticallyVetoed: { id; by: string[]; overridable: true }[];
  alethicallyVetoed: { id; failures }[];
};

VWorked example

A regulated workflow: an agent must send a report whose data contains PII. Three norms are in force — forbidden to transmit PII without consent (priority 10); obligatory to obtain consent first, with a contrary-to-duty repair escalate-to-DPO (priority 8); and an explicit permission for the redacted path. Watch all three deontic behaviours appear as the situation changes. Runs live below.

VIStanding on

None of this is new logic — it is the standard apparatus, made executable and wired to the rest of the stack. The triad and the deontic square are von Wright's; the O(p)→P(p) consistency axiom is Standard Deontic Logic's D; the repair-on-violation mechanism is the contrary-to-duty tradition that grew out of Chisholm's paradox — the observation that "you ought not do X, but if you do, you ought to do Y" cannot be modelled by a flat prohibition. The stack's existing governance fields — authority_path, deny_default, audit — were already proto-deontic; this layer gives them an algebra.