Dynamic Semantics

Origin. Irene Heim's File Change Semantics (dissertation, UMass 1982; "On the projection problem for presuppositions", 1983) and Hans Kamp's DRT independently made meaning a context-change operation. Groenendijk and Stokhof, "Dynamic Predicate Logic" (Linguistics and Philosophy 14, 1991), gave the compositional, representation-free version in which formulas denote relations between assignments. Veltman, "Defaults in update semantics" (Journal of Philosophical Logic 25, 1996), extended the idea to epistemic modality with states as sets of possibilities and might as a test. Stalnaker's (1978) common ground is the pragmatic ancestor.

Models. A meaning is not a truth condition but a function from input contexts to output contexts — a context change potential. In DPL a context is a variable assignment and a formula denotes a binary relation on assignments; in update semantics a context is an information state, a set of world–assignment pairs, and a formula is a partial function shrinking it. Truth is recovered as a derived notion: φ is true at g iff some output is reachable from g. Two kinds of sentence emerge from the same definition — those that genuinely change the state, and tests, which either return the input unchanged or return the absurd state.

Formalism.

Context change potential: [[φ]]: context → context. Meaning is update function. Not static denotation. Dynamic interpretation.

Information states: σ: set of world-assignment pairs. ⟨w, g⟩ ∈ σ. g: variable assignment. Tracks discourse referents.

Update semantics: σ[[∃x.φ]] = ⋃_d σ[x↦d][[φ]]. Existential introduces referent. Available for later anaphora. Extends assignment.

Anaphora: "A man walked in. He sat down." ∃x.man(x) ∧ walked_in(x); sat_down(x). x bound across sentences. Dynamic binding.

DPL (Dynamic Predicate Logic): Groenendijk & Stokhof. [[φ]] ⊆ G × G, a relation between assignments. Dynamic conjunction: φ; ψ = relational composition. [[∃x]] = {⟨g,h⟩ : h differs from g at most at x} — random reset. Existential scope extends rightward without a syntactic scope marker. Truth: φ true at g iff ∃h ⟨g,h⟩ ∈ [[φ]].

Negation is a test: [[¬φ]] = {⟨g,g⟩ : no h with ⟨g,h⟩ ∈ [[φ]]}. Output = input, so nothing binds out of a negation. Consequence: ¬¬φ ≠ φ as a CCP, though the two have the same truth conditions. Dynamic conjunction is associative but not commutative.

Implication and universals are tests too: [[φ → ψ]] = {⟨g,g⟩ : ∀h (⟨g,h⟩ ∈ [[φ]] → ∃k ⟨h,k⟩ ∈ [[ψ]])}. Antecedent referents reach the consequent, nothing reaches past the whole. Donkey anaphora falls out; so does the accessibility pattern of DRT.

Update with might (Veltman): σ[[might φ]] = σ if σ[[φ]] ≠ ∅, else ∅. Test, not update. Epistemic modality. Information-sensitive.

Presupposition: Presupposition = requirement on input context. Must be satisfied before update. Accommodation if not. Context manipulation.

Update consequence: φ₁, …, φₙ ⊨ ψ iff σ[[φ₁]]…[[φₙ]][[ψ]] = σ[[φ₁]]…[[φₙ]] for every σ. Consequence is order-sensitive and non-monotonic once tests are present. "It might be raining; it isn't raining" is consistent. "It isn't raining; it might be raining" is not.

Symbols.

Metatheory. DPL is truth-conditionally equivalent to first-order logic — every formula's truth conditions are expressible statically — so the whole gain is in the CCPs, not in what can be said. What changes is the algebra: dynamic conjunction is relational composition, hence associative but not commutative, and negation is a test, so double negation elimination fails at the level of CCPs even though ¬¬φ and φ agree on truth conditions. Groenendijk and Stokhof (1991) showed this reproduces DRT's accessibility pattern compositionally, without a representational level, and Muskens (1996) obtained the same result inside classical type logic. Veltman's (1996) update consequence is order-sensitive and non-monotonic once tests are in the language, which is a genuine departure from Tarskian consequence rather than a reformulation of it. Rothschild and Yalcin (2016) proved the sharp converse: an update system is equivalent to a static, intersective one exactly when it is idempotent and commutative, so dynamicity is doing work only where those two properties fail.

Applies to. Formal semantics. Discourse analysis. Anaphora. Presupposition.

Limitations. The framework adds no expressive power over first-order logic and its satisfiability problem is undecidable for the same reason, so its only claim is on binding and update, not on content. DPL's random reset is destructive: reusing a variable erases the referent stored there, an artefact of assignment-based states that Vermeulen's referent systems and Dekker's Predicate Logic with Anaphora were built to remove. Because context change potentials are stipulated connective by connective, a static meaning does not determine a unique dynamic one — many CCPs share a truth condition, so the dynamic entries are underdetermined by the data they explain, and Schlenker's local-context theory derives presupposition projection from static meanings without them. The proviso problem shows the accommodation rule is not fully specified either. And the departures from classical logic are real costs: conjunction is not commutative, double negation elimination fails, and update consequence is non-monotonic.

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