FOM: Branching quantifiers

[JoeShipman@aol.com]

M. Mostowski writes:

<<You take the following composition rules
 1. for each variable x, the following are prefixes: Ex, Ax;
 2. if Q and Q' are prefixes then the following are prefixes: (QQ') - horizontal composition, (Q/Q') - vertical composition, and Q^d - the dualization of Q.

 The dualization means that Q^d is equivalent to ~Q~, which trivially is first-order definable, but not in a context of vertical compositions.
 Without dualization you will get the logic with Henkin quantifiers, but with dualization you will obtain just the logic equivalent to the second order logic.>>

Can someone please give an ordinary-language semantic interpretation of these quantifiers?  I don't understand  what they're supposed to mean.

-- Joe Shipman