Terminology 2: The notion of a class is taken as undefined. Symbols may be declared to be variables, thought of as representing arbitrary classes. Below, we recursively specify what we mean by a statement, each one interpreted as expressing a particular assertion about classes:
(a) Base cases: Given variables x and y , the following is a statement: (x \in y). (b) Recursion: If \phi and \psi are statements, so is a claim of both of these, i.e. \textup{"}\phi \textup{ and } \psi \textup{"}. (c) Recursion: If \phi is a statement, then so the denial thereof, i.e. \textup{"not } \phi \textup{"}. (d) Recursion: For a variable x , if \phi is a statement then so is the claim \textup{"there exists } x \textup{ such that } \phi \textup{"}.
Any assertion using synonymous phrasing or simplified notation, which has the same intuitive meaning as one obtained using the specification above, is also to be accepted as expressing a statement. In particular, we typically drop outermost parentheses, such as in (x \in y) , should no confusion arise.