In a logic with (finite) partially ordered quantification this is not in general the case.
Branching quantification first appeared in a 1959 conference paper of Leon Henkin.
is It (in fact every formula with a Henkin prefix, not just the simplest one) is equivalent to its second-order Skolemization, i.e.
(i.e. "there are infinitely many") defined as Several things follow from this, including the nonaxiomatizability of first-order logic with
[2] Hintikka in a 1973 paper[6] advanced the hypothesis that some sentences in natural languages are best understood in terms of branching quantifiers, for example: "some relative of each villager and some relative of each townsman hate each other" is supposed to be interpreted, according to Hintikka, as:[7][8] which is known to have no first-order logic equivalent.
[7] The idea of branching is not necessarily restricted to using the classical quantifiers as leaves.
is not closed under negation, Barwise also proposed a practical test to determine whether natural language sentences really involve branching quantifiers, namely to test whether their natural-language negation involves universal quantification over a set variable (a
For instance students were shown undirected bipartite graphs—with squares and circles as vertices—and asked to say whether sentences like "more than 3 circles and more than 3 squares are connected by lines" were correctly describing the diagrams.