The sentence ``x + 2 = 5'' cannot be assigned a truth value and so it is technically not a statement. However, if we knew the value of x we could determine its truth value. We say that it is a statement involving a variable, namely x. If we allow x to have value 3 then we get a true statement whereas, if we let x have value 1 we get a false statement. Symbolically, we might represent the above statement as p(x). We see that p(3) is true while p(1) is false. Similarly, the statement xy = 4 could be represented symbolically as q(x,y) since it is a statement involving two variables. Again, to assign a truth value to q(x,y) we would have to know the values of both x and y. If x had value 2 and y had value 5 then q(2,5) would have truth value F. Note that q(2,2) has truth value T.
Another way to be able to assign truth values to statements involving variables is through the use of quantifiers. The universal quantifier, , in English means ``For all possible values of the variable''. The existential quantifier, , in English means ``There is a value of the variable''. Thus, with the examples above, the following are statements: ; . The first states that for all possible values of the variable x that x + 2 = 5. The second states that there is a value of x for which x + 2 = 5. Clearly, we have made each of these a statement (and we might believe the first to be false and the second to be true). The problem we are left with is what do we mean by the possible values of the variable? We assume that the variable in our statement may be replaced by a quantity from a set of known objects. This set of objects is called the universal set of discourse, or more briefly, the universal set. Usually, this universal set will be understood from the context. However, if necessary, we shall define the universal set for each variable in the statement and write something like . Here, the notation translates into English as ``x an element of the set U''. Note that for quantified statements involving more than one variable, the universal set for each variable may be different.
Alternatively, we may think about a statement involving variables as a statement about the universal set U. The truth set of the statement consists of those elements of U for which the statement is true. For example, if the universal set is the set of integers then the truth set of the statement `` '' is . A statement about U is a tautology if and only if its truth set is U and a contradiction if its truth set is the empty set. Thus a statement such as is true if its truth set is U. If its truth set is a proper subset of U then it is a false statement. A statement of the form is true if its truth set is non empty and false if its truth set is the empty set.