Dynamic semantics

Dynamic semantics is a framework in logic and natural language semantics which treats the meaning of a sentence as its potential to update a context. In static semantics, knowing the meaning of a sentence amounts to knowing when it is true; in dynamic semantics, knowing the meaning of a sentence means knowing "the change it brings about in the information state of anyone who accepts the news conveyed by it." In dynamic systems, sentences are mapped to functions called context change potentials which take an input context and return an output context. Dynamic semantics was originally developed by Irene Heim and Hans Kamp in 1981 to model anaphora, but has since been applied widely to phenomena including presupposition, plurals, questions, discourse relations, and modality.

Dynamics of Anaphora

See also: Discourse Representation Theory and Donkey anaphora

The first systems of dynamic semantics were the closely related File Change Semantics and Discourse Representation Theory, developed simultaneously and independently by Irene Heim and Hans Kamp. These systems were intended to capture donkey anaphora, which resists an elegant compositional treatment in classic approaches to semantics such as Montague grammar.

Donkey Sentence: Every farmer who owns a donkey beats it.

To capture the empirically observed truth conditions of this sentence in first order logic, one would need to translate the indefinite noun phrase "a donkey" as a universal quantifier scoping over the variable corresponding to the pronoun "it". Rather than allow such black magic, Heim and Kamp proposed that the indefinite introduces a new discourse referent which remains available outside the syntactic scope of the operator that introduced it. As a consequence, these systems validate Egli's Theorem.

Egli's Theorem: $(\exists x\phi )\land \psi \Leftrightarrow \exists x(\phi \land \psi )$

This consequence is additionally useful to capture other cases of variables which are semantically but not syntactically bound.

Intersentential Anaphora: A farmer owns a donkey. He beats it.

Update Semantics

The term Update Semantics refers to a collection of related systems which apply the dynamic framework to worldly content. In the classic Update Semantics, a formula $\phi$ is interpreted as a function $[\phi ]$ which takes and returns a context $C$. In the classic Update Semantics, a context $C$ is defined as a set of possible worlds representing the information in the common ground of a discourse. Then the semantics can be given recursively as follows:

• $C[p]=\{w\in C\,|\,w(p)=1\}$
• $C[\neg \phi ]=C-C[\phi ]$
• $C[\phi \land \psi ]=C[\phi ][\psi ]$
• $C[\phi \lor \psi ]=C[\phi ]\cup C[\psi ]$
• $C[\Diamond \phi ]=\varnothing$ if $C[\phi ]=\varnothing$ and $C$ otherwise

One motivation for this system comes from the infelicity of epistemic contradictions.

Epistemic Contradiction: #It's raining and it might not be raining.

These sentences cannot be treated as logical contradictions within relational semantics for modal logic without bringing along the unwelcome prediction that "It might be raining" entails "It is raining". (To see why, observe that the formula $P\land \Diamond \neg P$ is only a contradiction on the class of relational frames such that $Rwv\Rightarrow w=v$.) However, Update Semantics skirts this problem by providing an information sensitive semantics for modals in which the information provided by first conjunct of an epistemic contradiction determines what possibilities are live when the second conjunct is interpreted.

See also

• Donkey anaphora
• Discourse Representation Theory
• Modal logic