Read https://plato.stanford.edu/entries/logic-higher-order/ https://plato.stanford.edu/entries/logic-higher-order/ if you wish. I view the "for all properties"

Read https://plato.stanford.edu/entries/logic-higher-order/ https://plato.stanford.edu/entries/logic-higher-order/ if you wish. I view the "for all properties" bit as handwaving. See particularly section 6 of that article, which at one point gives: The semantics of second-order logic depending on these hard questions of set theory puts second-order logic into the same “basket” with set theory. Criticism of set theory becomes criticism of second-order logic and vice versa. My Constructivist tendencies are due to my discomfort with accepting the validity of set-theoretic constructions with decisions based on unverifiable truths. Accepting second order logic requires accepting quantifying over similar collections of unverifiable claims. I'm going to be no more sympathetic.