 For Henkin’s second style of language, at least in Hintikka’s notation (see the entry on  independence friendly logic),  the syntax is well-founded, but the displacement of the quantifier scopes means that the usual quantifier clauses in the definition of satisfaction no longer work. To get a compositional and fully abstract semantics, one has to ask not what assignments of variables satisfy a formula, but what sets of assignments satisfy the formula ‘uniformly’, where ‘uniformly’ means ‘independent of assignments to certain variables, as shown by the slashes on quantifiers inside the formula’. (Further details of revisions of Tarski’s truth definition along these lines are in the entry on  dependence logic.)  Henkin’s second example is of more than theoretical interest, because clashes between the semantic and the syntactic scope of quantifiers occur very often in natural languages.