The sixth rung · what the agent knows

Epistemic Arithmetic

Every rung so far governs what an agent should do. None of them say what it knows — yet the whole premise of this stack is that intelligence is accumulated knowledge. This rung is the missing modality: knowledge and graded belief over possible worlds. Knowing is truth across every world you still consider possible; learning is crossing worlds off; and the difference between not knowing and knowing that you don't know is exactly the κ signal that tells the system when to stop and deliberate.

IThe algebra

A model is a set of possible worlds — each a way things might be — and, for each agent, an accessibility relation giving the worlds it can't yet rule out. Knowledge is the universal: K φ holds iff φ is true in every accessible world. Belief is the same shape over a weaker relation, and possibility is the dual (¬K¬φ).

Knowledge vs belief — the relation makes the difference

Both operators read "true in all accessible worlds." What separates them is the shape of the relation, and that single structural fact decides whether the modality is factive.

The other two operations are dynamics and pooling. Learning is a truthful public announcement: announce(ψ) deletes every world where ψ is false, shrinking the possibilities — so knowledge only grows. Across agents, everyone-knows is per-agent knowledge, common knowledge is its fixpoint (everyone knows that everyone knows…, computed as reachability over the union), and distributed knowledge pools information by intersecting what each agent can rule out — the group knows more than any member.

IIThe laws

Eight for the modal core, three for the bridge to the rest of the ladder. The highlighted cross-check is the defining split — factivity holds for knowledge and fails for belief.

IIIThe laws, checked

Each law against random models in your browser — knowledge laws over random S5 partitions, belief laws over KD45 doxastic sets — on the engine that ships in box-and-box.

IVType sketch

// a model: possible worlds + per-agent accessibility
interface Model { worlds: World[]; actual: World; access: Record<Agent,(w:World)=>World[]>; }

function knows(m: Model, a: Agent, φ: Formula): boolean;      // true in all accessible worlds
function believes(m: Model, a: Agent, φ: Formula, θ?: number): boolean; // graded — the β link
function knowsItDoesntKnow(m: Model, a: Agent, φ: Formula): boolean; // K¬Kφ — the κ signal
function route(m: Model, a: Agent, φ: Formula): "act"|"deliberate"|"uncertain";

// learning = truthful public announcement (delete the ruled-out worlds)
function announce(m: Model, ψ: Formula): Model;

// multi-agent
function everyone(m: Model, ags: Agent[], φ: Formula): boolean;
function common(m: Model, ags: Agent[], φ: Formula): boolean;       // fixpoint of everyone-knows
function distributed(m: Model, ags: Agent[], φ: Formula): boolean;   // pooled

VWorked example

An agent considers two worlds possible — {p,q} and {¬p,q}. It can't act on p, but it can detect that it can't, and route to deliberation. Then it learns p and the gap closes. Finally, two agents who each know half pool into distributed knowledge of the whole. Runs live.

VIStanding on

Modelling knowledge as truth across accessible possible worlds is Hintikka's (1962); the multi-agent machinery — everyone-knows, common knowledge, distributed knowledge — is Fagin, Halpern, Moses & Vardi, Reasoning About Knowledge (1995); common knowledge as the coordination prerequisite is Aumann (1976) and Halpern & Moses's coordinated-attack result; and learning as world-eliminating announcement is Plaza's public announcement logic (1989). The connection to confidence (β as graded belief) and to κ (a known-unknown as the deliberation trigger) is what wires this rung to the rest of the stack. None of the logic is new; the contribution is the executable, property-tested synthesis.