This Boolean algebra, known as the *Lindenbaum algebra*, is a countable atomless Boolean algebra — it is atomless because we can always take the conjunction of any formula with a new atom or its negation — and all such Boolean algebras are isomorphic by a standard back-and-forth argument. So your particular presentation of the algebra is inessential, since there are many presentations of this algebra. 

The back-and-forth argument also shows that the structure is homogeneous — any finite partial isomorphism will extend to an automorphism of the whole algebra. Any finitely many parameters are contained in a finite Boolean subalgebra, and so any two elements that stand in the same relation to the atoms of that subalebra will be automorphic images. This severely limits the definable sets and will support a classification. 

In particular, the set of propositional atoms will not be definable with parameters, since any two atoms no appearing the expressions of the parameters will be automorphic over those parameters.