Update Semantics

Origin. Frank Veltman, "Defaults in update semantics" (Journal of Philosophical Logic 25, 1996), which fixes the system and its slogan; the update conception itself is Stalnaker's (1978) and Heim's (1982), and Veltman's earlier work on data semantics (1985) is the direct ancestor. Groenendijk, Stokhof, and Veltman (1996) combined the epistemic and anaphoric dynamics. Yalcin (2007) and Rothschild and Yalcin (2016, 2017) supply the modern reassessment.

Models. "You know the meaning of a sentence if you know the change it brings about in the information state of anyone who accepts the news conveyed by it." An information state is a set of worlds and a meaning is a function from states to states, so content is what the sentence does rather than what it rules out. Most sentences eliminate worlds, but the epistemic modals do not: might φ returns the state unchanged if φ is compatible with it and the absurd state otherwise, so it is a test on the state rather than an operation within it. This is what makes "It might be raining and it isn't raining" defective while its converse order is fine, and it is why the resulting consequence relation is order-sensitive and non-monotonic.

Formalism.

Information state: σ ⊆ W (set of worlds). Represents what's known/assumed. Eliminative updates. Common ground.

Update function: σ[φ]: result of updating σ with φ. Meaning = update function. Dynamic.

Atomic update: σ[p] = {w ∈ σ : w ⊨ p}. Eliminate p-false worlds. Intersection.

Conjunction: σ[φ ∧ ψ] = σ[φ][ψ]. Sequential update. Order matters sometimes.

Negation: σ[¬φ] = σ - σ[φ]. Worlds not eliminated by φ. Complement in σ.

Might (test): σ[◇φ] = σ if σ[φ] ≠ ∅, else ∅. Test: passes or crashes. Non-eliminative. Epistemic possibility.

Must (test): σ[□φ] = σ if σ[φ] = σ, else ∅. φ holds throughout σ. Epistemic necessity.

Support: σ supports φ (σ ⊨ φ) iff σ[φ] = σ. φ adds no information to σ. Already known.

Absurdity: ∅ = absurd state. σ[⊥] = ∅. Crash/failure.

Consequence: φ₁, …, φₙ ⊨ ψ iff σ[φ₁]…[φₙ][ψ] = σ[φ₁]…[φₙ] for every σ. Update with the premises, then test the conclusion. Order-sensitive, non-monotonic, and not reflexive in general.

Epistemic contradictions: "It's raining and it might not be raining" — defective. "It might not be raining and it's raining" — fine. The asymmetry follows from sequential update, not from a pragmatic rule.

Eliminativity and distributivity: Eliminative: σ[φ] ⊆ σ. Distributive: σ[φ] = ⋃{{w}[φ] : w ∈ σ}. Might is eliminative but not distributive: it looks at the whole state at once. This is exactly what a worldwise truth definition cannot do.

When dynamics is idle: Rothschild and Yalcin (2016): an update system is equivalent to a static intersective one precisely when its updates are idempotent and commute. Veltman's system fails both, so its dynamics is not eliminable.

Symbols.

Metatheory. Every update in Veltman's system is eliminative — none adds worlds — but the modal tests are not distributive: σ[◇φ] is not the union of the results of updating each singleton, because might inspects the whole state at once, which is exactly what a worldwise truth definition cannot do. Rothschild and Yalcin (2016, 2017) proved the general form of this: an update system is equivalent to a static intersective semantics precisely when its updates are idempotent and commutative, and Veltman's are neither, so the dynamics is not eliminable. Consequence is update-then-test, hence order-sensitive and non-monotonic: "It isn't raining" followed by "It might be raining" crashes while the reverse order does not, and adding a premise can destroy an inference. Yalcin (2007) showed that the epistemic contradictions can also be derived statically by relativizing truth to an information parameter, so that phenomenon alone does not establish dynamism.

Applies to. Epistemic modals. Discourse. Conditionals. Common ground.

Limitations. The system is propositional; the first-order and quantified extensions are not part of it, and combining the epistemic tests with anaphoric dynamics requires a further construction. The must-test makes must φ and plain φ equivalent in acceptance — a state supports one iff it supports the other — which contradicts Karttunen's observation that "It must be raining" is evidentially weaker than "It's raining" — a wrong prediction the semantics is built to make. States are single-agent: there is one body of information, so disagreement, presupposed common ground, and the difference between a speaker's information and a group's cannot be represented. Failure of reflexivity and monotonicity means the consequence relation has no ordinary structural proof theory. And the meanings are functions on states rather than compositionally assembled objects, so embedding under attitude verbs and quantifiers is not straightforward.

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