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.
The first one is simply the idea of multiple quantification in connection with polyadic predicates, so that (in modern terminology) we can have propositions with quantifiers occurring within the scope of other quantifiers.
Also as in possible world semantics, a quantifier (in effect) ranging over individuals that occurs in the scope of a quantifier (in effect) ranging over worlds — '∀x' and '∀w', respectively, in (18) — is, for each value w of the bound world variable, restricted to the objects existing in w.
(In such a wff, α is said to be the scope of the quantifier).
Thus, in (∀x)(ϕx ∨ ϕy) the x in ϕx is bound, since it occurs within the scope of a quantifier containing x, but y is free.
The reason for restricting the replacement variable to one not occurring elsewhere in the scope of the quantifier can be seen from an example: If ϕxy is taken as before to mean "x loves y," the wff (∀x)ϕxy expresses the proposition form "Everyone loves y," in which the identity of y is left unspecified, and so does its bound alphabetical variant (∀z)ϕzy.
For the crucial restriction that ¬ may not occur in the scope of a quantifier, see Hintikka (1991: 49; 1996: 148).
The important point to notice is that the anaphoric pronoun/variable in (15a) is not within the syntactic scope of its quantifier antecedent.
Because the only predicate symbol, R, is in the scope of this quantifier, the sentence has no predicate symbols open to interpretation.
An occurrence of a variable x in a formula A is bound if it is within the scope of a quantifier ∀x or ∃x, otherwise free.
