[page 181]
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 α is said to be a “unit” class if the propositional function “‘x is an α’ is always equivalent to ‘x is c’” (regarded as a function of c) is not always false, i.e., in more ordinary language, if there is a term c such that x will be a member of α when x is c 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 [page 182] 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 “a” is a name for the same particular, a proposition in which “a” occurs is not (as we saw in the preceding chapter) identical with what this proposition becomes when for “a” we substitute “the last thing Cæsar saw before he died.” If our language does not contain the name “a,” or some other name for the same particular, we shall have no means of expressing the proposition which we expressed by means of “a” as opposed to the one that [page 183] 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 [page 184] “equivalent” when both are true or both false; two propositional functions φx, ψx are “formally equivalent” when φx is always equivalent to ψx. 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 [page 185] 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 φx, we say that a is a “member” of the class if φa 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 n things m at a time” represents a class of classes, namely, the class of all classes of m terms that can be selected out of a given class of n 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 [page 186] 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 Phœnix 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 “x is human” and “x 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 “x is human” or for “x is mortal” any formally equivalent function. But, as we have just seen, the statement “I believe that all men are mortal” cannot be regarded as being about the class determined by either function, because its truth-value may be changed by the substitution of a formally equivalent function (which leaves the class unchanged). We will call a statement involving a function φx an “extensional” function of the function φx, if it is like “all men are mortal,” i.e. if its truth-value is unchanged by the substitution of any formally equivalent function; and when a function of a function is not extensional, we will call it “intensional,” so that “I believe that all men are mortal” is an intensional function of “x is human” or “x is mortal.” Thus extensional functions of a function φx may, for practical [page 187] purposes, be regarded as functions of the class determined by φx, while intensional functions cannot be so regarded.
It is to be observed that all the specific functions of functions that we have occasion to introduce in mathematical logic are extensional. Thus, for example, the two fundamental functions of functions are: “φx is always true” and “φx is sometimes true.” Each of these has its truth-value unchanged if any formally equivalent function is substituted for φx. In the language of classes, if α is the class determined by φx, “φx is always true” is equivalent to “everything is a member of α,” and “φx is sometimes true” is equivalent to “α has members” or (better) “α has at least one member.” Take, again, the condition, dealt with in the preceding chapter, for the existence of “the term satisfying φx.” The condition is that there is a term c such that φx is always equivalent to “x is c.” This is obviously extensional. It is equivalent to the assertion that the class defined by the function φx is a unit class, i.e. a class having one member; in other words, a class which is a member of 1.
Given a function of a function which may or may not be extensional, we can always derive from it a connected and certainly extensional function of the same function, by the following plan: Let our original function of a function be one which attributes to φx the property f; then consider the assertion “there is a function having the property f and formally equivalent to φx.” This is an extensional function of φx; it is true when our original statement is true, and it is formally equivalent to the original function of φx if this original function is extensional; but when the original function is intensional, the new one is more often true than the old one. For example, consider again “I believe that all men are mortal,” regarded as a function of “x is human.” The derived extensional function is: “There is a function formally equivalent to ‘x is human’ and such that I believe that whatever satisfies it is mortal.” This remains true when we substitute “x is a rational animal” [page 188] for “x is human,” even if I believe falsely that the Phœnix is rational and immortal.
We give the name of “derived extensional function” to the function constructed as above, namely, to the function: “There is a function having the property f and formally equivalent to φx,” where the original function was “the function φx has the property f.”
We may regard the derived extensional function as having for its argument the class determined by the function φx, and as asserting f of this class. This may be taken as the definition of a proposition about a class. I.e. we may define:
To assert that “the class determined by the function φx has the property f” is to assert that φx satisfies the extensional function derived from f.
This gives a meaning to any statement about a class which can be made significantly about a function; and it will be found that technically it yields the results which are required in order to make a theory symbolically satisfactory.1
What we have said just now as regards the definition of classes is sufficient to satisfy our first four conditions. The way in which it secures the third and fourth, namely, the possibility of classes of classes, and the impossibility of a class being or not being a member of itself, is somewhat technical; it is explained in Principia Mathematica, but may be taken for granted here. It results that, but for our fifth condition, we might regard our task as completed. But this condition—at once the most important and the most difficult—is not fulfilled in virtue of anything we have said as yet. The difficulty is connected with the theory of types, and must be briefly discussed.2
We saw in Chapter XIII. that there is a hierarchy of logical types, and that it is a fallacy to allow an object belonging to one of these to be substituted for an object belonging to another. [page 189] Now it is not difficult to show that the various functions which can take a given object a as argument are not all of one type. Let us call them all a-functions. We may take first those among them which do not involve reference to any collection of functions; these we will call “predicative a-functions.” If we now proceed to functions involving reference to the totality of predicative a-functions, we shall incur a fallacy if we regard these as of the same type as the predicative a-functions. Take such an every-day statement as “a is a typical Frenchman.” How shall we define a “typical Frenchman”? We may define him as one “possessing all qualities that are possessed by most Frenchmen.” But unless we confine “all qualities” to such as do not involve a reference to any totality of qualities, we shall have to observe that most Frenchmen are not typical in the above sense, and therefore the definition shows that to be not typical is essential to a typical Frenchman. This is not a logical contradiction, since there is no reason why there should be any typical Frenchmen; but it illustrates the need for separating off qualities that involve reference to a totality of qualities from those that do not.
Whenever, by statements about “all” or “some” of the values that a variable can significantly take, we generate a new object, this new object must not be among the values which our previous variable could take, since, if it were, the totality of values over which the variable could range would only be definable in terms of itself, and we should be involved in a vicious circle. For example, if I say “Napoleon had all the qualities that make a great general,” I must define “qualities” in such a way that it will not include what I am now saying, i.e. “having all the qualities that make a great general” must not be itself a quality in the sense supposed. This is fairly obvious, and is the principle which leads to the theory of types by which vicious-circle paradoxes are avoided. As applied to a-functions, we may suppose that “qualities” is to mean “predicative functions.” Then when I say “Napoleon had all the qualities, etc.,” I mean [page 190] “Napoleon satisfied all the predicative functions, etc.” This statement attributes a property to Napoleon, but not a predicative property; thus we escape the vicious circle. But wherever “all functions which” occurs, the functions in question must be limited to one type if a vicious circle is to be avoided; and, as Napoleon and the typical Frenchman have shown, the type is not rendered determinate by that of the argument. It would require a much fuller discussion to set forth this point fully, but what has been said may suffice to make it clear that the functions which can take a given argument are of an infinite series of types. We could, by various technical devices, construct a variable which would run through the first n of these types, where n is finite, but we cannot construct a variable which will run through them all, and, if we could, that mere fact would at once generate a new type of function with the same arguments, and would set the whole process going again.
We call predicative a-functions the first type of a-functions; a-functions involving reference to the totality of the first type we call the second type; and so on. No variable a-function can run through all these different types: it must stop short at some definite one.
These considerations are relevant to our definition of the derived extensional function. We there spoke of “a function formally equivalent to φx.” It is necessary to decide upon the type of our function. Any decision will do, but some decision is unavoidable. Let us call the supposed formally equivalent function ψ. Then ψ appears as a variable, and must be of some determinate type. All that we know necessarily about the type of φ is that it takes arguments of a given type—that it is (say) an a-function. But this, as we have just seen, does not determine its type. If we are to be able (as our fifth requisite demands) to deal with all classes whose members are of the same type as a, we must be able to define all such classes by means of functions of some one type; that is to say, there must be some type of a-function, say the nth, such that any a-function is formally [page 191] equivalent to some a-function of the nth type. If this is the case, then any extensional function which holds of all a-functions of the nth type will hold of any a-function whatever. It is chiefly as a technical means of embodying an assumption leading to this result that classes are useful. The assumption is called the “axiom of reducibility,” and may be stated as follows:—
“There is a type (τ say) of a-functions such that, given any a-function, it is formally equivalent to some function of the type in question.”
If this axiom is assumed, we use functions of this type in defining our associated extensional function. Statements about all a-classes (i.e. all classes defined by a-functions) can be reduced to statements about all a-functions of the type τ. So long as only extensional functions of functions are involved, this gives us in practice results which would otherwise have required the impossible notion of “all a-functions.” One particular region where this is vital is mathematical induction.
The axiom of reducibility involves all that is really essential in the theory of classes. It is therefore worth while to ask whether there is any reason to suppose it true.
This axiom, like the multiplicative axiom and the axiom of infinity, is necessary for certain results, but not for the bare existence of deductive reasoning. The theory of deduction, as explained in Chapter XIV., and the laws for propositions involving “all” and “some,” are of the very texture of mathematical reasoning: without them, or something like them, we should not merely not obtain the same results, but we should not obtain any results at all. We cannot use them as hypotheses, and deduce hypothetical consequences, for they are rules of deduction as well as premisses. They must be absolutely true, or else what we deduce according to them does not even follow from the premisses. On the other hand, the axiom of reducibility, like our two previous mathematical axioms, could perfectly well be stated as an hypothesis whenever it is used, instead of being assumed to be actually true. We can deduce [page 192] its consequences hypothetically; we can also deduce the consequences of supposing it false. It is therefore only convenient, not necessary. And in view of the complication of the theory of types, and of the uncertainty of all except its most general principles, it is impossible as yet to say whether there may not be some way of dispensing with the axiom of reducibility altogether. However, assuming the correctness of the theory outlined above, what can we say as to the truth or falsehood of the axiom?
The axiom, we may observe, is a generalised form of Leibniz’s identity of indiscernibles. Leibniz assumed, as a logical principle, that two different subjects must differ as to predicates. Now predicates are only some among what we called “predicative functions,” which will include also relations to given terms, and various properties not to be reckoned as predicates. Thus Leibniz’s assumption is a much stricter and narrower one than ours. (Not, of course, according to his logic, which regarded all propositions as reducible to the subject-predicate form.) But there is no good reason for believing his form, so far as I can see. There might quite well, as a matter of abstract logical possibility, be two things which had exactly the same predicates, in the narrow sense in which we have been using the word “predicate.” How does our axiom look when we pass beyond predicates in this narrow sense? In the actual world there seems no way of doubting its empirical truth as regards particulars, owing to spatio-temporal differentiation: no two particulars have exactly the same spatial and temporal relations to all other particulars. But this is, as it were, an accident, a fact about the world in which we happen to find ourselves. Pure logic, and pure mathematics (which is the same thing), aims at being true, in Leibnizian phraseology, in all possible worlds, not only in this higgledy-piggledy job-lot of a world in which chance has imprisoned us. There is a certain lordliness which the logician should preserve: he must not condescend to derive arguments from the things he sees about him.
[page 193]
Viewed from this strictly logical point of view, I do not see any reason to believe that the axiom of reducibility is logically necessary, which is what would be meant by saying that it is true in all possible worlds. The admission of this axiom into a system of logic is therefore a defect, even if the axiom is empirically true. It is for this reason that the theory of classes cannot be regarded as being as complete as the theory of descriptions. There is need of further work on the theory of types, in the hope of arriving at a doctrine of classes which does not require such a dubious assumption. But it is reasonable to regard the theory outlined in the present chapter as right in its main lines, i.e. in its reduction of propositions nominally about classes to propositions about their defining functions. The avoidance of classes as entities by this method must, it would seem, be sound in principle, however the detail may still require adjustment. It is because this seems indubitable that we have included the theory of classes, in spite of our desire to exclude, as far as possible, whatever seemed open to serious doubt.
The theory of classes, as above outlined, reduces itself to one axiom and one definition. For the sake of definiteness, we will here repeat them. The axiom is:
There is a type τ such that if φ is a function which can take a given object a as argument, then there is a function ψ of the type τ which is formally equivalent to φ.
The definition is:
If φ is a function which can take a given object a as argument, and τ the type mentioned in the above axiom, then to say that the class determined by φ has the property f is to say that there is a function of type τ, formally equivalent to φ, and having the property f.
[Chapter XVII notes]
1. See Principia Mathematica, vol. i. pp. 75–84 and *20.
2. The reader who desires a fuller discussion should consult Principia Mathematica, Introduction, chap. ii.; also *12.