Intersective update

An update with ${\displaystyle \varphi }$ is called intersective if it amounts to taking the intersection of the input context with the proposition denoted by ${\displaystyle \varphi }$. Crucially, this definition assumes that there is a single fixed proposition that ${\displaystyle \varphi }$ always denotes, regardless of the context.^[8]

• Intersective update: Let ${\displaystyle [\![\varphi ]\!]}$ be the proposition denoted by ${\displaystyle \varphi }$. Then ${\displaystyle \varphi }$ is intersective if and only if for any ${\displaystyle C}$, we have that ${\displaystyle C[\varphi ]=C\cap [\![\varphi ]\!]}$

Intersective update was proposed by Robert Stalnaker in 1978 as a way of formalizing the speech act of assertion.^[9][8] In Stalnaker's original system, a context (or context set) is defined as a set of possible worlds representing the information in the common ground of a conversation. For instance, if ${\displaystyle C=\{w,v,u\}}$ this represents a scenario where the information agreed upon by all participants in the conversation indicates that the actual world must be either ${\displaystyle w}$, ${\displaystyle v}$, or ${\displaystyle u}$. If ${\displaystyle [\![\varphi ]\!]=\{w,v\}}$, then updating ${\displaystyle C}$ with ${\displaystyle \varphi }$ would return a new context ${\displaystyle C[\varphi ]=\{w,v\}}$. Thus, an assertion of ${\displaystyle \varphi }$ would be understood as an attempt to rule out the possibility that the actual world is ${\displaystyle u}$.

From a formal perspective, intersective update can be taken as a recipe for lifting one's preferred static semantics to dynamic semantics. For instance, if we take classical propositional semantics as our starting point, this recipe delivers the following intersective update semantics.^[8]

• Intersective update semantics based on classical propositional logic:

1. ${\displaystyle C[P]=\{w\in C\mid w(P)=1\}}$
2. ${\displaystyle C[\neg \varphi ]=C-C[\varphi ]}$
3. ${\displaystyle C[\varphi \land \psi ]=C[\varphi ]\cap C[\psi ]}$
4. ${\displaystyle C[\varphi \lor \psi ]=C[\varphi ]\cup C[\psi ]}$

The notion of intersectivity can be decomposed into the two properties known as eliminativity and distributivity. Eliminativity says that an update can only ever remove worlds from the context—it can't add them. Distributivity says that updating ${\displaystyle C}$ with ${\displaystyle \varphi }$ is equivalent to updating each singleton subset of ${\displaystyle C}$ with ${\displaystyle \varphi }$ and then pooling the results.^[8]

• Eliminativity: ${\displaystyle \varphi }$ is eliminative iff ${\displaystyle C[\varphi ]\subseteq C}$ for all contexts ${\displaystyle C}$
• Distributivity: ${\displaystyle \varphi }$ is distributive iff ${\displaystyle C[\varphi ]=\bigcup \{\{w\}[\varphi ]\mid w\in C\}}$

Intersectivity amounts to the conjunction of these two properties, as proven by Johan van Benthem.^[8][10]

The test semantics for modals

The framework of update semantics is more general than static semantics because it is not limited to intersective meanings. Nonintersective meanings are theoretically useful because they contribute different information depending on what information is already present in the context. For instance, if ${\displaystyle \varphi }$ is intersective, then it will update any input context with the exact same information, namely the information encoded by the proposition ${\displaystyle [\![\varphi ]\!]}$. On the other hand, if ${\displaystyle \varphi }$ is nonintersective, it could contribute ${\displaystyle [\![\varphi ]\!]}$ when it updates some contexts, but some completely different information when it updates other contexts.^[8]

Many natural language expressions have been argued to have nonintersective meanings. The nonintersectivity of epistemic modals can be seen in the infelicity of epistemic contradictions.^[11][8]

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

These sentences have been argued to be bona fide logical contradictions, unlike superficially similar examples such as Moore sentences, which can be given a pragmatic explanation.^[12][8]

Epistemic contradiction principle: ${\displaystyle \varphi \land \Diamond \neg \varphi \models \bot }$

These sentences cannot be analysed as logical contradictions within purely intersective frameworks such as the relational semantics for modal logic. The Epistemic Contradiction Principle only holds on the class of relational frames such that ${\displaystyle Rwv\Rightarrow (w=v)}$. However, such frames also validate an entailment from ${\displaystyle \Diamond \varphi }$ to ${\displaystyle \varphi }$. Thus, accounting for the infelicity of epistemic contradictions within a classical semantics for modals would bring along the unwelcome prediction that "It might be raining" entails "It is raining".^[12][8] Update Semantics skirts this problem by providing a nonintersective denotation for modals. When given such a denotation, the formula ${\displaystyle \Diamond \neg \varphi }$ can update input contexts differently depending on whether they already contain the information that ${\displaystyle \varphi }$ provides. The most widely adopted semantic entry for modals in update semantics is the test semantics proposed by Frank Veltman.^[1]

• The test semantics for modals: ${\displaystyle C[\Diamond \varphi ]={\begin{cases}C&{\text{if }}C[\varphi ]\neq \varnothing \\\varnothing &{\text{otherwise}}\end{cases}}}$

On this semantics, ${\displaystyle \Diamond \varphi }$ tests whether the input context could be updated with ${\displaystyle \varphi }$ without getting trivialized, i.e. without returning the empty set. If the input context passes the test, it remains unchanged. If it fails the test, the update trivializes the context by returning the empty set. This semantics can handle epistemic contradictions because no matter the input context, updating with ${\displaystyle \varphi }$ will always output a context that fails the test imposed by ${\displaystyle \Diamond \neg \varphi }$.^[8][13]