In The Oxford Companion to Philosophy, the entry on existence includes this paragraph.
It is often held that ‘exist’ is not a firstlevel predicate. What this means is that ‘exist’ does not express a property of objects, as verbs like ‘shine’ and ‘fall’ do. According to Frege and Russell, ‘exist’ is a second-level predicate, expressing a property of properties. Thus ‘God exists’ does not have the same logical form as ‘Sirius shines’, predicating a property of a particular object. Rather, it is equivalent to ‘Godhood is instantiated’, asserting that the property of being divine has at least one instance, or that there is at least one thing possessing that property.
We may notate the proposition 'Sirius shines' thus: $ \exists s(Ss) $
However, I don't know how to notate the proposition 'the property of Godhood is instantiated'. I presume it involves second order logic.
How do logicians notate a proposition that posits the instantiation of a property?