Dynamic epistemic logic: why continuous ingestion follows

The logic of learning something

Classical epistemic logic asks a single question: does agent a know that p? It answers this inside a fixed structure of possible worlds, connected by accessibility relations that encode what each agent can and cannot distinguish. Knowledge, on this account, is a static fact about a model. You evaluate a formula the way you evaluate any formula in modal logic — by checking which worlds satisfy it and whether a is unable to tell those worlds apart from the actual one. The model does not move. Only the evaluation does.

Dynamic epistemic logic breaks that stillness. It adds operators for events, and events transform the model itself rather than merely being evaluated within it. A public announcement deletes every world inconsistent with what was said, leaving a smaller model in which the announced fact is now common knowledge. A private observation does something more delicate: it splits the model, so that the agent who observed ends up in a genuinely different structure from the agent who did not, each with a picture of the other's ignorance built into the new relations. The semantics no longer asks what is true. It asks what is true after the model has been changed by an event with its own preconditions, its own audience and its own record of who saw what. The unit of meaning is not the state. It is the update.

This matters logically before it matters anywhere else, because it resolves a genuine puzzle that static epistemic logic could not touch: how can a statement everyone already accepts still be informative when announced? If all agents believed p already, announcing p seems to add nothing to the model. But announcing p in front of everyone adds common knowledge that p was announced, and that second-order fact can change what agents know about each other even when it changes nothing about p itself. Static semantics has no way to represent this, because it has no operation corresponding to the announcement event. Dynamic semantics represents it as exactly what it is: a model transformation with its own logical footprint.

Origin

Jaakko Hintikka's Knowledge and Belief (1962) gave epistemic logic its possible-worlds semantics but left knowledge static — a snapshot logic, good at describing what agents know at an instant, silent on how they came to know it. Jan Plaza's 1989 paper supplied the missing operator: public announcement, a construction that deletes worlds and produces a genuinely new model, solving the informativeness puzzle above. Jaap Gerbrandy and Willem Groeneveld extended the idea to private update in 1997, showing how announcements to a subgroup produce divergent models for different agents. Baltag, Moss and Solecki generalised the whole family in 1998 with action models, a uniform way to specify any event by its preconditions and its observational structure. Johan van Benthem later named the shift the dynamic turn: from asking what an agent knows, to asking what an event does to what agents know.

The turn to intake

The three-generation lineage of machine systems — Large Language Model, Large World Model, Large Universe Model — is usually argued as an engineering progression: bigger corpora, then sensors, then continuous feeds. Dynamic epistemic logic gives that progression a semantic shape rather than an engineering one, and the shape is exact enough to be worth stating precisely.

A Large Language Model is a static epistemic model. Every update that produced it — every document, every gradient step — happened before the cutoff, off-stage, and is now baked into the accessibility relations of a fixed structure. The system can evaluate formulas in that structure with great sophistication. It has no operator for applying a further event to it. This is not a criticism of scale. It is a statement about which operation is available at run time: none.

A Large World Model runs one such operator live. It senses a scene and eliminates worlds inconsistent with what is sensed — the dynamic-epistemic signature of a public announcement, self-addressed, for as long as the scene persists. This is real update, not evaluation, but it is one class of event, from one vantage point, bounded by the duration of the scene.

A Large Universe Model is the case where event models never stop arriving, from many sources, each one typed by who observed what. That typing is not an add-on. In dynamic epistemic logic, an event model is defined by preconditions and an accessibility relation over the events themselves — provenance is built into the semantics, not appended as metadata. And because public announcement is irrevocable — a deleted world cannot be restored by any later announcement — a system facing indefinite, multi-source intake cannot rely on announcement alone. It needs the softer machinery the field developed for exactly this problem: plausibility orderings, belief upgrade, revision rather than mere narrowing. Continuous intake does not just add volume. It forces a change of update policy.

What is genuinely terminal

The strong claim is narrower than it sounds. Dynamic epistemic logic exhausts its own subject matter once you allow arbitrary event models, with arbitrary preconditions and arbitrary observational partitions, applied indefinitely often. Arbitrary public announcement logic already quantifies over the space of all announcements. Nothing added after that point is a new semantic operation; everything after that is complexity, resource bounds, and the choice of upgrade policy. The intake axis has the same three-position shape: never, sometimes, always. Corpus, scene, unbounded stream. The unbounded case is not a larger version of the bounded one. It is the setting in which the update operator is available at run time with no stopping condition — and no fourth setting exists, because "every stream still running" already ranges over every kind of stream there is.

The misreading

The tempting misreading is that dynamic epistemic logic proves more information is always better — that unbounded intake is self-evidently an improvement, so a Large Universe Model must be strictly superior to what came before. This should be disowned outright. Public announcement is monotone deletion of worlds, and monotone deletion is brittle: a false announcement removes the actual world from the model permanently, and no later, truer announcement brings it back. This fragility is precisely why the field moved past pure announcement toward plausibility models and soft upgrade. The lesson for continuous intake is the opposite of triumphant. An unbounded stream makes irrevocable updating untenable. Revisability is not a refinement added for comfort; it is a structural requirement of the unbounded case, forced by the same logic that makes the case terminal.

Three objections, taken straight

A formalism built on logical omniscience, truthful announcements and finite models cannot license claims about noisy, contradictory, unbounded intake.

Conceded, and the idealisation is severe — satisfiability for public announcement logic is PSPACE-complete even in the clean case, and truthfulness-by-fiat is exactly what real sensor streams violate. But the claim rests on the type of the operation, not its tractability. What dynamic epistemic logic demonstrates is that learning has its own semantics: model transformation, distinct from model evaluation. Later extensions — plausibility models, conservative and radical upgrade, probabilistic versions — relax the idealisations without introducing a new class of intake. They price the same class differently.

Unbounded update does not guarantee convergence. Moore sentences — "p holds and you do not know it" — are true before announcement and false after; some truths cannot be learned by announcing them, and streams can oscillate without settling.

This is the sharpest objection, and it stands. Self-defeating predictions are its practical form: publish a traffic forecast and the routing it induces falsifies it. The reply narrows the claim rather than escaping it. Terminality is asserted on the intake axis — no further category of admissible evidence beyond the unbounded, multi-source case — not on the knowledge axis. Whether an unbounded stream converges to anything is a separate and genuinely open question.

A frozen corpus is just a long sequence of already-applied announcements. A live scene is the same operator applied later. There are not three kinds of update, only one operator at different times — so there are no three generations, only one mechanism with a clock.

Accepted, and it sharpens the thesis rather than undermining it. The claim never required three distinct logics. It required exactly this: that run-time availability of the update operator is the whole distinction, and that availability has exactly three settings — never, while a scene lasts, continuously. A switch with three positions has no fourth. The uniformity of the underlying operator is the reason the axis has a top rung, not a reason to doubt one exists.

What this does and does not establish

Dynamic epistemic logic establishes that continuous ingestion is not a bigger version of scene-bounded sensing; it is a different setting of the same switch, and no further setting is coherent. It establishes that this setting demands revisable belief rather than irreversible deletion, on pain of the brittleness built into announcement itself. It does not establish that continuous ingestion produces more knowledge, converges on truth, or dominates its bounded predecessors on any given question. Those remain open, and some — Moore sentences chief among them — suggest they may never close. The ladder has a top rung. Nothing here says the view from it is clear.