Well-formed Formula (wff)

Not all strings can represent propositions of predicate logic. Those that produce a proposition when their symbols are interpreted are called well-formed formulas of the first order predicate logic. A predicate name followed by a list of variables such as P(x, y), where P is a predicate name, and x and y are variables, is called an atomic formula. Wffs are constructed using the following rules:

1. True and False are wffs.
2. Each propositional constant (i.e. specific proposition).
3. Each atomic formula (i.e. a specific predicate with variables) is a wff.
4. If A and B are wffs, then so are A, (A Ù B), (AÚ B), (A ® B), and (A « B).
5. If x is a variable (representing objects of the universe of discourse), and A is a wff, then so are "x A and $x A.

Examples: (all of these are wffs but only the last three examples are sentences)

Between(b, x, y) - block b is between x and y

ØLeftOf(x, c) - it is not the case that x is to the left of c

"x Small(x) - everyone is small.

Tet(b) - b is a tetrahedron.

$x $y(Cube(x) Ù Large(y)) - There exists a cube and something large.