Introduction to Mathematical Philosophy

ambiguous descriptions, but rather more complicated. We come now to the main subject of the present chapter, namely, the definition of the word the (in the singular). One very important point about the definition of "a so-and-so" applies equally to "the so-and-so"; the definition to be sought is a definition of propositions in which this phrase occurs, not a definition of the phrase itself in isolation. In the case of "a so-and-so," this is fairly obvious: no one could suppose that "a man" was a definite object, which could be defined by itself. [Pg 172] Socrates is a man, Plato is a man, Aristotle is a man, but we cannot infer that "a man" means the same as "Socrates" means and also the same as "Plato" means and also the same as "Aristotle" means, since these three names have different meanings. Nevertheless, when we have enumerated all the men in the world, there is nothing left of which we can say, "This is a man, and not only so, but it is the 'a man,' the quintessential entity that is just an indefinite man without being anybody in particular." It is of course quite clear that whatever there is in the world is definite: if it is a man it is one definite man and not any other. Thus there cannot be such an entity as "a man" to be found in the world, as opposed to specific man. And accordingly it is natural that we do not define "a man" itself, but only the propositions in which it occurs. In the case of "the so-and-so" this is equally true, though at first sight less obvious. We may demonstrate that this must be the case, by a consideration of the difference between a name and a definite description. Take the proposition, "Scott is the author of Waverley." We have here a name, "Scott," and a description, "the author of Waverley," which are asserted to apply to the same person. The distinction between a name and all other symbols may be explained as follows:— A name is a simple symbol whose meaning is something that can only occur as subject, i.e. something of the kind that, in Chapter XIII., we defined as an "individual" or a "particular." And a "simple" symbol is one which has no parts that are symbols. Thus "Scott" is a simple symbol, because, though it has parts (namely, separate letters), these parts are not symbols. On the other hand, "the author of Waverley" is not a simple symbol, because the separate words that compose the phrase are parts which are symbols. If, as may be the case, whatever seems to be an "individual" is really capable of further analysis, we shall have to content ourselves with what may be called "relative individuals," which will be terms that, throughout the context in question, are never analysed and never occur [Pg 173] otherwise than as subjects. And in that case we shall have correspondingly to content ourselves with "relative names." From the standpoint of our present problem, namely, the definition of descriptions, this problem, whether these are absolute names or only relative names, may be ignored, since it concerns different stages in the hierarchy of "types," whereas we have to compare such couples as "Scott" and "the author of Waverley," which both apply to the same object, and do not raise the problem of types. We may, therefore, for the moment, treat names as capable of being absolute; nothing that we shall have to say will depend upon this assumption, but the wording may be a little shortened by it. We have, then, two things to compare: (1) a name, which is a simple symbol, directly designating an individual which is its meaning, and having this meaning in its own right, independently of the meanings of all other words; (2) a description, which consists of several words, whose meanings are already fixed, and from which results whatever is to be taken as the "meaning" of the description. A proposition containing a description is not identical with what that proposition becomes when a name is substituted, even if the name names the same object as the description describes. "Scott is the author of Waverley" is obviously a different proposition from "Scott is Scott": the first is a fact in literary history, the second a trivial truism. And if we put anyone other than Scott in place of "the author of Waverley," our proposition would become false, and would therefore certainly no longer be the same proposition. But, it may be said, our proposition is essentially of the same form as (say) "Scott is Sir Walter," in which two names are said to apply to the same person. The reply is that, if "Scott is Sir Walter" really means "the person named 'Scott' is the person named 'Sir Walter,'" then the names are being used as descriptions: i.e. the individual, instead of being named, is being described as the person having that name. This is a way in which names are frequently used [Pg 174] in practice, and there will, as a rule, be nothing in the phraseology to show whether they are being used in this way or as names. When a name is used directly, merely to indicate what we are speaking about, it is no part of the fact asserted, or of the falsehood if our assertion happens to be false: it is merely part of the symbolism by which we express our thought. What we want to express is something which might (for example) be translated into a foreign language; it is something for which the actual words are a vehicle, but of which they are no part. On the other hand, when we make a proposition about "the person called 'Scott,'" the actual name "Scott" enters into what we are asserting, and not merely into the language used in making the assertion. Our proposition will now be a different one if we substitute "the person called 'Sir Walter.'" But so long as we are using names as names, whether we say "Scott" or whether we say "Sir Walter" is as irrelevant to what we are asserting as whether we speak English or French. Thus so long as names are used as names, "Scott is Sir Walter" is the same trivial proposition as "Scott is Scott." This completes the proof that "Scott is the author of Waverley" is not the same proposition as results from substituting a name for "the author of Waverley," no matter what name may be substituted. When we use a variable, and speak of a propositional function, say, the process of applying general statements about to particular cases will consist in substituting a name for the letter "," assuming that is a function which has individuals for its arguments. Suppose, for example, that is "always true"; let it be, say, the "law of identity," . Then we may substitute for "" any name we choose, and we shall obtain a true proposition. Assuming for the moment that "Socrates," "Plato," and "Aristotle" are names (a very rash assumption), we can infer from the law of identity that Socrates is Socrates, Plato is Plato, and Aristotle is Aristotle. But we shall commit a fallacy if we attempt to infer, without further premisses, that the author of Waverley is the author of Waverley. This results [Pg 175] from what we have just proved, that, if we substitute a name for "the author of Waverley" in a proposition, the proposition we obtain is a different one. That is to say, applying the result to our present case: If "" is a name, "" is not the same proposition as "the author of Waverley is the author of Waverley," no matter what name "" may be. Thus from the fact that all propositions of the form "" are true we cannot infer, without more ado, that the author of Waverley is the author of Waverley. In fact, propositions of the form "the so-and-so is the so-and-so" are not always true: it is necessary that the so-and-so should exist (a term which will be explained shortly). It is false that the present King of France is the present King of France, or that the round square is the round square. When we substitute a description for a name, propositional functions which are "always true" may become false, if the description describes nothing. There is no mystery in this as soon as we realise (what was proved in the preceding paragraph) that when we substitute a description the result is not a value of the propositional function in question. We are now in a position to define propositions in which a definite description occurs. The only thing that distinguishes "the so-and-so" from "a so-and-so" is the implication of uniqueness. We cannot speak of "the inhabitant of London," because inhabiting London is an attribute which is not unique. We cannot speak about "the present King of France," because there is none; but we can speak about "the present King of England." Thus propositions about "the so-and-so" always imply the corresponding propositions about "a so-and-so," with the addendum that there is not more than one so-and-so. Such a proposition as "Scott is the author of Waverly" could not be true if Waverly had never been written, or if several people had written it; and no more could any other proposition resulting from a propositional function by the substitution of "the author of Waverly" for "." We may say that "the author of Waverly" means "the value of for which ' wrote [Pg 176] Waverly' is true." Thus the proposition "the author of Waverly was Scotch," for example, involves: (1) " wrote Waverly" is not always false; (2) "if and wrote Waverly, and are identical" is always true; (3) "if wrote Waverly, was Scotch" is always true. These three propositions, translated into ordinary language, state: (1) at least one person wrote Waverly; (2) at most one person wrote Waverly; (3) whoever wrote Waverly was Scotch. All these three are implied by "the author of Waverly was Scotch." Conversely, the three together (but no two of them) imply that the author of Waverly was Scotch. Hence the three together may be taken as defining what is meant by the proposition "the author of Waverly was Scotch." We may somewhat simplify these three propositions. The first and second together are equivalent to: "There is a term such that ' wrote Waverly' is true when is and is false when is not ." In other words, "There is a term such that ' wrote Waverly' is always equivalent to ' is .'" (Two propositions are "equivalent" when both are true or both are false.) We have here, to begin with, two functions of , " wrote Waverly" and " is ," and we form a function of by considering the equivalence of these two functions of for all values of ; we then proceed to assert that the resulting function of is "sometimes true," i.e. that it is true for at least one value of . (It obviously cannot be true for more than one value of .) These two conditions together are defined as giving the meaning of "the author of Waverly exists." We may now define "the term satisfying the function exists." This is the general form of which the above is a particular case. "The author of Waverly" is "the term satisfying the function ' wrote Waverly.'" And "the so-and-so" will [Pg 177] always involve reference to some propositional function, namely, that which defines the property that makes a thing a so-and-so. Our definition is as follows:— "The term satisfying the function exists" means: "There is a term such that is always equivalent to ' is .'" In order to define "the author of Waverly was Scotch," we have still to take account of the third of our three propositions, namely, "Whoever wrote Waverly was Scotch." This will be satisfied by merely adding that the in question is to be Scotch. Thus "the author of Waverly was Scotch" is: "There is a term such that (1) ' wrote Waverly' is always equivalent to ' is ,' (2) is Scotch." And generally: "the term satisfying