If you proved that '∈' were logical, then you could do away with second-order logic. It seems that identity comes out to be a logical symbol and elementhood doesn't seem to be a logical symbol. How does P relate to x in the statement Px? They are not related via 'the elementhood relation'; rather, elementhood defines P.

(x = y) <-> equiv(x, y) // a binary relation S = #{a b} <-> ∀x(Sx → (x=a ∨ x=b)) a ∈ S <-> Sa

Intensions are functions that assign sets to predicates at each possible world (or related set-theoretic devices that encode the same information). On such accounts, for example, the semantic value of ‘red’ is the function that maps each possible world to the set of things in that world that are red. Finely individuated properties are more useful in semantics than intensions as used by Montague because intensions are still too coarse-grained to explain many semantic phenomena involving intensional idioms. For example, semantic accounts that employ intensions would most naturally treat ‘circle’ and ‘locus of points equidistant from a point’ as having the same meaning (since they have the same intension). The "paradox of analysis ... asks us to explain how it could be true, e.g., that ‘Tom believes something is a circle, but does not believe that it is a locus of points equidistant from a point’. If ‘circle’ and ‘locus of points equidistant from a point’ have the same meaning an explanation is hard to find."

The two phrases, “morning star” and “evening star” may designate the same object, but they do not have the same meaning. In classical first-order logic intension plays no role.

connotation|sense|intension|meaning and denotation|reference|extension|designata