A is necessary for B iff (“if and only if”) B can’t occur without A. Whenever you have B, you have A. Anything B is A.

A is sufficient for B iff A guarantees B. Whenever you have A, you have B. Anything A is B.

Examples:

To build a “formal” or “essential” definition of X (the kind Socrates and Plato are looking for), you need to specify what properties all and only X’s have.  Then you’ll know the Form of X, or X-ness itself. To do this, you simply need each property to be necessary, and the properties together to be sufficient.  For example, here is how we get a formal definition of “square”:

All squares are equilateral, all squares are rectangular, and only squares are equilateral and rectangular.

Plato’s basic idea is that everything in the world fits into some class, and the classes can be precisely defined. In other words, Plato presupposes that all concepts are what Wittgenstein calls closed: i.e., Plato believes it is possible to specify the necessary and sufficient conditions for membership in the class of things denoted by any concept. This basic idea of Plato’s had a profound effect on Western thought; it led people to think we could, just by thinking, say exactly what goodness or beauty or philosophy or persons are. Until quite recently, the idea defined philosophy’s task: figure out those conditions. Ludwig Wittgenstein revolutionized Western philosophy by arguing that this underlying idea is wrong. He points out that most concepts are open: i.e., that we define concepts in terms of paradigms, and argue that a particular thing either is or is not a member of the class in question on the basis of resemblance or lack of resemblance of the thing in question to the paradigm.