open formula also called open sentence, a sentence with a free occurrence of a variable. A closed sentence, sometimes called a statement, has no free occurrences of variables. In a language whose only variable-binding operators are quantifiers, an occurrence of a variable in a formula is bound provided that occurrence either is within the scope of a quantifier employing that variable or is the occurrence in that quantifier. An occurrence of a variable in a formula is free provided it is not bound. The formula ‘xy ( O’ is open because both ‘x’ and ‘y’ occur as free variables. In ‘For some real number y, xy ( O’, no occurrence of ‘y’ is free; but the occurrence of ‘x’ is free, so the formula is open. The sentence ‘For every real number x, for some real number y, xy ( O’ is closed, since none of the variables occur free. Semantically, an open formula such as ‘xy ( 0’ is neither true nor false but rather true of or false of each assignment of values to its free-occurring variables. For example, ‘xy ( 0’ is true of each assignment of two positive or two negative real numbers to ‘x’ and to ‘y’ and it is false of each assignment of 0 to either and false at each assignment of a positive real to one of the variables and a negative to the other. See also QUANTIFICATION, SCOPE. C.S.