Squiggle operator

In formal semantics, the squiggle operator $\sim$ is an operator which constrains the occurrence of focus. On its most common definition, the squiggle operator takes a syntactic argument $\alpha$ and a discourse salient argument $C_{i}$ and introduces a presupposition that the ordinary semantic value of $C_{i}$ is either a subset or an element of the focus semantic value of $\alpha$ . The squiggle was first introduced by Mats Rooth in 1992 as part of his treatment of focus within the framework of Alternative semantics. It has become one of the standard tools in formal work on focus, playing a key role in accounts of contrastive focus, ellipsis, deaccenting, and question-answer congruence.

Empirical motivation

The empirical motivation for the squiggle operator comes from cases where focus marking requires a salient antecedent in discourse which stands in some particular relation with the focused constituent. One notable instance of this phenomenon is question-answer congruence. Informally, a focused constituent in an answer must represent the part of the utterance which provides the answer to the question. For instance, the following pair of dialogues show that in response to a question of who likes stroopwafel, focus must be placed on the name of the person who likes stroopwafel rather than on the word "stroopwafel" itself.^[1]

Q: Who likes stroopwafel?
A: HELEN likes stroopwafel.
Q: Who likes stroopwafel?
A: #Helen likes STROOPWAFEL.

The judgments reverse when the question is what Helen likes.

Q: What does Helen like?
A: #HELEN likes stroopwafel.
Q: What does Helen like?
A: Helen likes STROOPWAFEL.

Similar phenomena arise with other instances of focus. For instance, contrastive focus requires each focused constituent to contrast with another discourse salient constituent of the same semantic type.^[2]

An AMERICAN farmer was talking to a CANADIAN farmer.
?? An AMERICAN farmer was talking to a Canadian FARMER.

Formal details

The squiggle operator is what allows the focus denotation and the ordinary denotation to interact. In the Alternative Semantics approach to focus, each constituent $\alpha$ has both an ordinary denotation $[\![\alpha ]\!]_{o}$ and a focus denotation $[\![\alpha ]\!]_{f}$ which are composed by parallel computations. The ordinary denotation of $\alpha$ is simply whatever denotation it would have in a non-alternative-based system. The focus denotation of a constituent is typically the set of all ordinary denotations one could get by substituting a focused constituent for another expression of the same type.^[3]

Sentence: HELEN likes stroopwafel.
Ordinary denotation: $[\![{\text{HELEN likes stroopwafel}}]\!]_{o}=\lambda w\,.{\text{ Helen likes stroopwafel in }}w$
Focus denotation: $[\![{\text{HELEN likes stroopwafel}}]\!]_{f}=\{\lambda w\,.x{\text{ likes stroopwafel in }}w\,|\,x\in {\mathcal {D}}_{e}\}$

Sentence: Helen likes STROOPWAFEL.
Ordinary denotation: $[\![{\text{Helen likes STROOPWAFEL}}]\!]_{o}=\lambda w\,.{\text{ Helen likes stroopwafel in }}w$
Focus denotation: $[\![{\text{Helen likes STROOPWAFEL}}]\!]_{f}=\{\lambda w\,.{\text{ Helen likes }}x{\text{ in }}w\,|\,x\in {\mathcal {D}}_{e}\}$

The squiggle operator takes two arguments, a contextually provided antecedent $C_{i}$ and an overt argument $\alpha$ . It introduces a presupposition that $C_{i}$ 's ordinary denotation is a subset of $\alpha$ 's focus denotation, or in other words that $[\![C_{i}]\!]_{o}\subseteq [\![\alpha ]\!]_{f}$ . If the presupposition is satisfied, it passes along its overt argument's ordinary denotation while "resetting" its focus denotation. In other words, when the presupposition is satisfied, $[\![\alpha \sim C_{i}]\!]_{o}=[\![\alpha ]\!]_{o}$ and $[\![\alpha \sim C_{i}]\!]_{f}=\{[\![\alpha ]\!]_{o}\}$