In this theory Sets are nothing but object extensions of some predicate. This theory propose that for every first order predicate there is an object extending it defined after some extensional relation.
This goes in the following manner:
Define: E is extensional iff for all x,y: (for all z. z E x iff z E y) -> x=y
where E is a primitive binary relation symbol.
Now sets are defined as
x is a set iff Exist E,P: E is extensional & for all y. y E x <-> P(y)
 If E, D are primitive binary relation symbols then:
E,D are extensional -> (For all x,y: (for all z. z E x iff z D y) -> x=y).
is an axiom.
 If E, D are primitive binary relation symbols; P,Q are first order language formulas in which x do not occur free, then
E,D are extensional -> for all x ((for all y. y E x iff Q) & (for all y. y D x iff P)) -> (for all y. P<->Q)
is an axiom
 If P is first order predicate, then
Exist E,x: E is extensional & for all y. y E x iff P(y)
is an axiom.
where E range over primitive binary relations only.
It is possible that this might interpret PA?
The whole motivation beyond this theory is to extend any first order predicate by objects. It is a purely logical motivation. If this does interpret PA and no inconsistency is shown with it, then PA can in a sense be seen as a kind of a logical theory. IF we extend  to allow infinitely long formulas, then possibly second order arithmetic would be provable? if so then it would be a kind of a logical theory also.