Squiggle operator

In formal semantics, the squiggle operator formula_1 is an operator which constrains the occurrence of focus. On one common definition, the squiggle operator takes a syntactic argument formula_2 and a discourse salient argument formula_3 and introduces a presupposition that the "ordinary semantic value" of formula_3 is either a subset or an element of the "focus semantic value" of formula_2. 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 expression. For instance, the following pairs shows that contrastive focus is only felicitous when there is a salient "focus antecedent" which contrasts with the focused expression.
Another instance of this phenomenon is "question-answer congruence", also known as "answer focus". Informally, a focused constituent in an answer to a question must represent the part of the utterance which resolves the issue raised by 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. When focus is instead placed on the word "stroopwafel" itself, the answer is infelicitous, as indicated by the # sign.
If instead the question is what Helen likes, the word "stroopwafel" will be the expression which resolves the issue. Thus, focus will belong on "stroopwaffel" instead of "Helen".
Formal details.
In the Roothian Squiggle Theory, formula_1 is what requires a focused expression to have a suitable focus antecedent. In doing so, it also allows the "focus denotation" and the "ordinary denotation" to interact. In the alternative Semantics approach to focus, each constituent formula_2 has both an ordinary denotation formula_8 and a focus denotation formula_9 which are composed by parallel computations. The ordinary denotation of formula_2 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.
The squiggle operator takes two arguments, a contextually provided antecedent formula_3 and an overt argument formula_2. In the above examples, formula_3 is a variable which can be valued as formula_2's focus antecedent, while formula_2 itself could be the constituent [HELEN likes stroopwafel]. 
On one common definition, formula_1 introduces a presupposition that formula_3's ordinary denotation is either a subset or an element of formula_22's focus denotation, or in other words that either formula_23 or formula_24. If this presupposition is satisfied, formula_1 passes along its overt argument's ordinary denotation while "resetting" its focus denotation. In other words, when the presupposition is satisfied, formula_26 and formula_27.