Introduction to Mathematical Philosophy

IN the present chapter we shall be concerned with the in the plural: the inhabitants of London, the sons of rich men, and so on. In other words, we shall be concerned with classes. We saw in Chapter II. that a cardinal number is to be defined as a class of classes, and in Chapter III. that the number 1 is to be defined as the class of all unit classes, i.e. of all that have just one member, as we should say but for the vicious circle. Of course, when the number 1 is defined as the class of all unit classes, "unit classes" must be defined so as not to assume that we know what is meant by "one"; in fact, they are defined in a way closely analogous to that used for descriptions, namely: A class is said to be a "unit" class if the propositional function "' is an ' is always equivalent to ' is '" (regarded as a function of ) is not always false, i.e., in more ordinary language, if there is a term such that will be a member of when is but not otherwise. This gives us a definition of a unit class if we already know what a class is in general. Hitherto we have, in dealing with arithmetic, treated "class" as a primitive idea. But, for the reasons set forth in Chapter XIII., if for no others, we cannot accept "class" as a primitive idea. We must seek a definition on the same lines as the definition of descriptions, i.e. a definition which will assign a meaning to propositions in whose verbal or symbolic expression words or symbols apparently representing classes occur, but which will assign a meaning that altogether eliminates all mention of classes from a right analysis [Pg 181] of such propositions. We shall then be able to say that the symbols for classes are mere conveniences, not representing objects called "classes," and that classes are in fact, like descriptions, logical fictions, or (as we say) "incomplete symbols." The theory of classes is less complete than the theory of descriptions, and there are reasons (which we shall give in outline) for regarding the definition of classes that will be suggested as not finally satisfactory. Some further subtlety appears to be required; but the reasons for regarding the definition which will be offered as being approximately correct and on the right lines are overwhelming. The first thing is to realise why classes cannot be regarded as part of the ultimate furniture of the world. It is difficult to explain precisely what one means by this statement, but one consequence which it implies may be used to elucidate its meaning. If we had a complete symbolic language, with a definition for everything definable, and an undefined symbol for everything indefinable, the undefined symbols in this language would represent symbolically what I mean by "the ultimate furniture of the world." I am maintaining that no symbols either for "class" in general or for particular classes would be included in this apparatus of undefined symbols. On the other hand, all the particular things there are in the world would have to have names which would be included among undefined symbols. We might try to avoid this conclusion by the use of descriptions. Take (say) "the last thing Cæsar saw before he died." This is a description of some particular; we might use it as (in one perfectly legitimate sense) a definition of that particular. But if "" is a name for the same particular, a proposition in which "" occurs is not (as we saw in the preceding chapter) identical with what this proposition becomes when for "" we substitute "the last thing Cæsar saw before he died." If our language does not contain the name "" or some other name for the same particular, we shall have no means of expressing the proposition which we expressed by means of "" as opposed to the one that [Pg 182] we expressed by means of the description. Thus descriptions would not enable a perfect language to dispense with names for all particulars. In this respect, we are maintaining, classes differ from particulars, and need not be represented by undefined symbols. Our first business is to give the reasons for this opinion. We have already seen that classes cannot be regarded as a species of individuals, on account of the contradiction about classes which are not members of themselves (explained in Chapter XIII.), and because we can prove that the number of classes is greater than the number of individuals. We cannot take classes in the pure extensional way as simply heaps or conglomerations. If we were to attempt to do that, we should find it impossible to understand how there can be such a class as the null-class, which has no members at all and cannot be regarded as a "heap"; we should also find it very hard to understand how it comes about that a class which has only one member is not identical with that one member. I do not mean to assert, or to deny, that there are such entities as "heaps." As a mathematical logician, I am not called upon to have an opinion on this point. All that I am maintaining is that, if there are such things as heaps, we cannot identify them with the classes composed of their constituents. We shall come much nearer to a satisfactory theory if we try to identify classes with propositional functions. Every class, as we explained in Chapter II., is defined by some propositional function which is true of the members of the class and false of other things. But if a class can be defined by one propositional function, it can equally well be defined by any other which is true whenever the first is true and false whenever the first is false. For this reason the class cannot be identified with any one such propositional function rather than with any other—and given a propositional function, there are always many others which are true when it is true and false when it is false. We say that two propositional functions are "formally equivalent" when this happens. Two propositions are "equivalent" [Pg 183] when both are true or both false; two propositional functions , are "formally equivalent" when is always equivalent to . It is the fact that there are other functions formally equivalent to a given function that makes it impossible to identify a class with a function; for we wish classes to be such that no two distinct classes have exactly the same members, and therefore two formally equivalent functions will have to determine the same class. When we have decided that classes cannot be things of the same sort as their members, that they cannot be just heaps or aggregates, and also that they cannot be identified with propositional functions, it becomes very difficult to see what they can be, if they are to be more than symbolic fictions. And if we can find any way of dealing with them as symbolic fictions, we increase the logical security of our position, since we avoid the need of assuming that there are classes without being compelled to make the opposite assumption that there are no classes. We merely abstain from both assumptions. This is an example of Occam's razor, namely, "entities are not to be multiplied without necessity." But when we refuse to assert that there are classes, we must not be supposed to be asserting dogmatically that there are none. We are merely agnostic as regards them: like Laplace, we can say, "je n'ai pas besoin de cette hypothèse." Let us set forth the conditions that a symbol must fulfil if it is to serve as a class. I think the following conditions will be found necessary and sufficient:— (1) Every propositional function must determine a class, consisting of those arguments for which the function is true. Given any proposition (true or false), say about Socrates, we can imagine Socrates replaced by Plato or Aristotle or a gorilla or the man in the moon or any other individual in the world. In general, some of these substitutions will give a true proposition and some a false one. The class determined will consist of all those substitutions that give a true one. Of course, we have still to decide what we mean by "all those which, etc." All that [Pg 184] we are observing at present is that a class is rendered determinate by a propositional function, and that every propositional function determines an appropriate class. (2) Two formally equivalent propositional functions must determine the same class, and two which are not formally equivalent must determine different classes. That is, a class is determined by its membership, and no two different classes can have the same membership. (If a class is determined by a function , we say that is a "member" of the class if is true.) (3) We must find some way of defining not only classes, but classes of classes. We saw in Chapter II. that cardinal numbers are to be defined as classes of classes. The ordinary phrase of elementary mathematics, "The combinations of things at a time" represents a class of classes, namely, the class of all classes of terms that can be selected out of a given class of terms. Without some symbolic method of dealing with classes of classes, mathematical logic would break down. (4) It must under all circumstances be meaningless (not false) to suppose a class a member of itself or not a member of itself. This results from the contradiction which we discussed in Chapter XIII. (5) Lastly—and this is the condition which is most difficult of fulfilment,—it must be possible to make propositions about all the classes that are composed of individuals, or about all the classes that are composed of objects of any one logical "type." If this were not the case, many uses of classes would go astray—for example, mathematical induction. In defining the posterity of a given term, we need to be able to say that a member of the posterity belongs to all hereditary classes to which the given term belongs, and this requires the sort of totality that is in question. The reason there is a difficulty about this condition is that it can be proved to be impossible to speak of all the propositional functions that can have arguments of a given type. We will, to begin with, ignore this last condition and the problems which it raises. The first two conditions may be [Pg 185] taken together. They state that there is to be one class, no more and no less, for each group of formally equivalent propositional functions; e.g. the class of men is to be the same as that of featherless bipeds or rational animals or Yahoos or whatever other characteristic may be preferred for defining a human being. Now, when we say that two formally equivalent propositional functions may be not identical, although they define the same class, we may prove the truth of the assertion by pointing out that a statement may be true of the one function and false of the other; e.g. "I believe that all men are mortal" may be true, while "I believe that all rational animals are mortal" may be false, since I may believe falsely that the Phoenix is an immortal rational animal. Thus we are led to consider statements about functions, or (more correctly) functions of functions. Some of the things that may be said about a function may be regarded as said about the class defined by the function, whereas others cannot. The statement "all men are mortal" involves the functions " is human" and " is mortal"; or, if we choose, we can say that it involves the classes men and mortals. We can interpret the statement in either way, because its truth-value is unchanged if we substitute for " is human" or for " is mortal" any formally equivalent function. But, as we have just