Introduction to Mathematical Philosophy

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 an "extensional" function of the function , 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 " is human" or " is mortal." Thus extensional functions of a function may, for practical [Pg 186] purposes, be regarded as functions of the class determined by , 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: " is always true" and " is sometimes true." Each of these has its truth-value unchanged if any formally equivalent function is substituted for . In the language of classes, if is the class determined by , " is always true" is equivalent to "everything is a member of ," and " is sometimes true" is equivalent to " has members" or (better) " has at least one member." Take, again, the condition, dealt with in the preceding chapter, for the existence of "the term satisfying ." The condition is that there is a term such that is always equivalent to " is ." This is obviously extensional. It is equivalent to the assertion that the class defined by the function 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 the property ; then consider the assertion "there is a function having the property and formally equivalent to ." This is an extensional function of ; it is true when our original statement is true, and it is formally equivalent to the original function of 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 " is human." The derived extensional function is: "There is a function formally equivalent to ' is human' and such that I believe that whatever satisfies it is mortal." This remains true when we substitute " is a rational animal" [Pg 187] for " is human," even if I believe falsely that the Phoenix 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 and formally equivalent to ," where the original function was "the function has the property ." We may regard the derived extensional function as having for its argument the class determined by the function , and as asserting 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 has the property " is to assert that satisfies the extensional function derived from . 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.[41] [41]See Principia Mathematica, vol. I. pp. 75-84 and * 20. 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.[42] [42]The reader who desires a fuller discussion should consult Principia Mathematica, Introduction, chap. II.; also * 12. 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. [Pg 188] Now it is not difficult to show that the various functions which can take a given object as argument are not all of one type. Let us call them all -functions. We may take first those among them which do not involve reference to any collection of functions; these we will call "predicative -functions." If we now proceed to functions involving reference to the totality of predicative -functions, we shall incur a fallacy if we regard these as of the same type as the predicative -functions. Take such an everyday statement as " 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 French men." 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 -functions, we may suppose that "qualities" is to mean "predicative functions." Then when I say "Napoleon had all the qualities, etc.," I mean [Pg 189] "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 of these types, where 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 -functions the first type of -functions; -functions involving reference to the totality of the first type we call the second type; and so on. No variable -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 ." 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 . Then appears as a variable, and must be of some determinate type. All that we know necessarily about the type of is that it takes arguments of a given type—that it is (say) an -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 , 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 -function, say the , such that any -function is formally [Pg 190] equivalent to some -function of the type. If this is the case, then any extensional function which holds of all -functions of the type will hold of any -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 ( say) of -functions such that, given any -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 -classes (i.e. all classes defined by -functions) can be reduced to statements about all -functions of the type . 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 -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 [Pg 191] 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