Nitpicking, but is this any sort of conventional syntax for defining a predicate? I've never seen it before, and I don't find it particularly intuitive...
That's clearly the intended meaning, but what it really says is that there exists a set whose elements are of the form C(x) where "x is a context" and C is left undefined. This, of course, is meaningless. Understanding is left as an exercise for the reader?
The notation (C(x): x is a context) actually means "C(x) is shorthand for the following sentence: 'x is a context'". C is not undefined, it is the template "___ is a context".
I guess the way the brackets are used may not be standard and could be confusing.
I know what the existential operator is, and C is a predicate, not a verb. But I've always seen predicates defined using lambda calculus (or just English)...this notation makes no sense to me.