The next section of logic is the algebraic stage. Now, the difference between arithmetic and algebra is, that in arithmetic definite numbers are considered, and in algebra symbols--namely, letters--are introduced which stand for any numbers. The idea of a number is also enlarged. These letters, standing for any numbers, are called sometimes variables and sometimes parameters. Their essential characteristic is that they are undetermined, unless, indeed, the algebraic conditions which they satisfy implicitly determine them. Then they are sometimes called unknowns. An algebraic formula with letters is a blank form. It becomes a determinate arithmetic statement when definite numbers are substituted for the letters. The importance of algebra is a tribute to the study of form. Consider now the following proposition-- The specific heat of mercury is 0·033. This is a definite proposition which, with certain limitations, is true. But the truth-value of the proposition does not immediately concern us. Instead of mercury put a mere letter which is the name of some undetermined thing: we get-- The specific heat of _x_ is 0·033. This is not a proposition; it has been called by Russell a propositional function. It is the logical analogy of an algebraic expression. Let us write ƒ(_x_) for any propositional function. We could also generalise still further, and say, The specific heat of _x_ is _y_. We thus get another propositional function, F(_x_, _y_), of two arguments _x_ and _y_, and so on for any number of arguments. Now, consider ƒ(_x_). There is the range of values of _x_, for which ƒ(_x_) is a proposition, true or false. For values of _x_ outside this range, ƒ(_x_) is not a proposition at all, and is neither true nor false. It may have vague suggestions for us, but it has no unit meaning of definite assertion. For example, The specific heat of water is 0·033 is a proposition which is false; and-- The specific heat of virtue is 0·033 is, I should imagine, not a proposition at all; so that it is neither true nor false, though its component parts raise various associations in our minds. This range of values, for which ƒ(_x_) has sense, is called the "type" of the argument _x_. But there is also a range of values of _x_ for which ƒ(_x_) is a true proposition. This is the class of those values of the argument which _satisfy_ ƒ(_x_). This class may have no members, or, in the other extreme, the class may be the whole type of the arguments. We thus conceive two general propositions respecting the indefinite number of propositions which share in the same logical form, that is, which are values of the same propositional function. One of these propositions is, ƒ(_x_) yields a true proposition for each value of _x_ of the proper type; the other proposition is, There is a value of _x_ for which ƒ(_x_) is true.