Terminology 2: The notion of a class is undefined. Arbitrary classes may be represented by symbols called variables. Given variables x and y , (x \in y) will be interpreted as a membership statement, asserting x is an element (or member) of y . We may also say x is in y , or y contains x , or similar. The parenthesis may be omitted should no confusion arise. Below, we recursively specify which assertions about classes are called statements.
(a) Base cases: Every membership statement is a statement.
(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 natural language assertion with the same meaning as one obtained using the above will also be accepted as expressing a statement.