One-many relations are involved in all phrases of the form "the so-and-so of such-and-such." "The King of England," [Pg 45] "the wife of Socrates," "the father of John Stuart Mill," and so on, all describe some person by means of a one-many relation to a given term. A person cannot have more than one father, therefore "the father of John Stuart Mill" described some one person, even if we did not know whom. There is much to say on the subject of descriptions, but for the present it is relations that we are concerned with, and descriptions are only relevant as exemplifying the uses of one-many relations. It should be observed that all mathematical functions result from one-many relations: the logarithm of , the cosine of , etc., are, like the father of , terms described by means of a one-many relation (logarithm, cosine, etc.) to a given term (). The notion of function need not be confined to numbers, or to the uses to which mathematicians have accustomed us; it can be extended to all cases of one-many relations, and "the father of " is just as legitimately a function of which is the argument as is "the logarithm of ." Functions in this sense are descriptive functions. As we shall see later, there are functions of a still more general and more fundamental sort, namely, propositional functions; but for the present we shall confine our attention to descriptive functions, i.e. "the term having the relation to ," or, for short, "the of ," where is any one-many relation. It will be observed that if "the of " is to describe a definite term, must be a term to which something has the relation , and there must not be more than one term having the relation to , since "the," correctly used, must imply uniqueness. Thus we may speak of "the father of " if is any human being except Adam and Eve; but we cannot speak of "the father of " if is a table or a chair or anything else that does not have a father. We shall say that the of "exists" when there is just one term, and no more, having the relation to . Thus if is a one-many relation, the of exists whenever belongs to the converse domain of , and not otherwise. Regarding "the of " as a function in the mathematical [Pg 46] sense, we say that is the "argument" of the function, and if is the term which has the relation to , i.e. if is the of , then is the "value" of the function for the argument . If is a one-many relation, the range of possible arguments to the function is the converse domain of , and the range of values is the domain. Thus the range of possible arguments to the function "the father of " is all who have fathers, i.e. the converse domain of the relation father, while the range of possible values for the function is all fathers, i.e. the domain of the relation. Many of the most important notions in the logic of relations are descriptive functions, for example: converse, domain, converse domain, field. Other examples will occur as we proceed.