  Generalised quantifiers, which include Henkin's branching quantifiers, have
been introduced by Mostowski and Lindstr\"om and developed as a substantial
topic application of logic, especially model theory, to linguistics with work
by Barwise, Cooper, Keenan.
  In this paper, we mainly study the proof theory of some non-standard
quantifiers as second order formulae . Our first example is the usual pair of
first order quantifiers (for all / there exists) when individuals are viewed as
individual concepts handled by second order deductive rules. Our second example
is the study of a second order translation of the simplest branching
quantifier: ``A member of each team and a member of each board of directors
know each other", for which we propose a second order treatment.
