Principia mathematica, vol. 1 (of 3)

It follows from the above that a function in which appears as argument requires that "" should not stand for any function which is capable of a given argument, but must be restricted in such a way that none of the functions which are possible values of "" should involve any reference to the totality of such functions. Let us take as an illustration the definition of identity. We might attempt to define " is identical with " as meaning "whatever is true of is true of ," i.e. " always implies ." But here, since we are concerned to assert all values of " implies " regarded as a function of , we shall be compelled to impose upon some limitation which will prevent us from including among values of values in which "all possible values of " are referred to. Thus for example " is identical with " is a function of ; hence, if it is a legitimate value of in " always implies " we shall be able to infer, by means of the above definition, that if is identical with , and is identical with , then is identical with . Although the conclusion is sound, the reasoning embodies a vicious-circle fallacy, since we have taken " implies " as a possible value of , which it cannot be. If, however, we impose any limitation upon , it may happen, so far as appears at present, that with other values of we might have true and false, so that our proposed definition of identity would plainly be wrong. This difficulty is avoided by the "axiom of reducibility," to be explained later.

Let us give the name "-functions" to functions that are significant for a given argument . Then suppose we take any selection of -functions, and consider the proposition " satisfies all the functions belonging to the selection in question." If we here replace by a variable, we obtain an -function; but by the vicious-circle principle this -function cannot be a member of our selection, since it refers to the whole of the selection. Let the selection consist of all those functions which satisfy . Then our new function is where is the argument. It thus appears that, whatever selection of -functions we may make, there will be other -functions that lie outside our selection. Such -functions, as the above instance illustrates, will always arise through taking a function of two arguments, and , and asserting all or some of the values resulting from varying . What is necessary, therefore, in order to avoid vicious-circle fallacies, is to divide our -functions into "types," each of which contains no functions which refer to the whole of that type.

When something is asserted or denied about all possible values or about some (undetermined) possible values of a variable, that variable is called apparent, after Peano. The presence of the words all or some in a proposition indicates the presence of an apparent variable; but often an apparent variable is really present where language does not at once indicate its presence. Thus for example " is mortal" means "there is a time at which will die." Thus a variable time occurs as apparent variable.

The clearest instances of propositions not containing apparent variables are such as express immediate judgments of perception, such as "this is red" or "this is painful," where "this" is something immediately given. In other judgments, even where at first sight no variable appears to be present, it often happens that there really is one. Take (say) "Socrates is human." To Socrates himself, the word "Socrates" no doubt stood for an object of which he was immediately aware, and the judgment "Socrates is human" contained no apparent variable. But to us, who only know Socrates by description, the word "Socrates" cannot mean what it meant to him; it means rather "the person having such-and-such properties," (say) "the Athenian philosopher who drank the hemlock." Now in all propositions about "the so-and-so" there is an apparent variable, as will be shown in Chapter III. Thus in what we have in mind when we say "Socrates is human" there is an apparent variable, though there was no apparent variable in the corresponding judgment as made by Socrates, provided we assume that there is such a thing as immediate awareness of oneself.

Whatever may be the instances of propositions not containing apparent variables, it is obvious that propositional functions whose values do not contain apparent variables are the source of propositions containing apparent variables, in the sense in which the function is the source of the proposition . For the values for do not contain the apparent variable , which appears in ; if they contain an apparent variable , this can be similarly eliminated, and so on. This process must come to an end, since no proposition which we can apprehend can contain more than a finite number of apparent variables, on the ground that whatever we can apprehend must be of finite complexity. Thus we must arrive at last at a function of as many variables as there have been stages in reaching it from our original proposition, and this function will be such that its values contain no apparent variables. We may call this function the matrix of our original proposition and of any other propositions and functions to be obtained by turning some of the arguments to the function into apparent variables. Thus for example, if we have a matrix-function whose values are , we shall derive from it

It is thus plain that all possible propositions and functions are obtainable from matrices by the process of turning the arguments to the matrices into apparent variables. In order to divide our propositions and functions into types, we shall, therefore, start from matrices, and consider how they are to be divided with a view to the avoidance of vicious-circle fallacies in the definitions of the functions concerned. For this purpose, we will use such letters as , , , , , , , to denote objects which are neither propositions nor functions. Such objects we shall call individuals. Such objects will be constituents of propositions or functions, and will be genuine constituents, in the sense that they do not disappear on analysis, as (for example) classes do, or phrases of the form "the so-and-so."

The first matrices that occur are those whose values are of the forms i.e. where the arguments, however many there may be, are all individuals. The functions , , ..., since (by definition) they contain no apparent variables, and have no arguments except individuals, do not presuppose any totality of functions. From the functions , ... we may proceed to form other functions of , such as (, (, (, (, and so on. All these presuppose no totality except that of individuals. We thus arrive at a certain collection of functions of , characterized by the fact that they involve no variables except individuals. Such functions we will call "first-order functions."

We may now introduce a notation to express "any first-order function." We will denote any first-order function by "" and any value for such a function by "." Thus "" stands for any value for any function which involves no variables except individuals. It will be seen that "" is itself a function of two variables, namely and . Thus involves a variable which is not an individual, namely . Similarly "(" is a function of the variable , and thus involves a variable other than an individual. Again, if is a given individual, is a function of , but it is not a function of the form , because it involves an (apparent) variable which is not an individual. Let us give the name "predicate" to any first-order function (This use of the word "predicate" is only proposed for the purposes of the present discussion.) Then the statement " implies with all possible values of " may be read "all the predicates of are predicates of ." This makes a statement about , but does not attribute to a predicate in the special sense just defined.

Owing to the introduction of the variable first-order function , we now have a new set of matrices. Thus "" is a function which contains no apparent variables, but contains the two real variables and . (It should be observed that when is assigned, we may obtain a function whose values do involve individuals as apparent variables, for example if is (. But so long as is variable, contains no apparent variables.) Again, if is a definite individual, is a function of the one variable . If and are definite individuals, " implies " is a function of the two variables , , and so on. We are thus led to a whole set of new matrices, These matrices contain individuals and first-order functions as arguments, but (like all matrices) they contain no apparent variables. Any such matrix, if it contains more than one variable, gives rise to new functions of one variable by turning all its arguments except one into apparent variables. Thus we obtain the functions

We will give the name of second-order matrices to such matrices as have first-order functions among their arguments, and have no arguments except first-order functions and individuals. (It is not necessary that they should have individuals among their arguments.) We will give the name of second-order functions to such as either are second-order matrices or are derived from such matrices by turning some of the arguments into apparent variables. It will be seen that either an individual or a first-order function may appear as argument to a second-order function. Second-order functions are such as contain variables which are first-order functions, but contain no other variables except (possibly) individuals.

We now have various new classes of functions at our command. In the first place, we have second-order functions which have one argument which is a first-order function. We will denote a variable function of this kind by the notation , and any value of such a function by . Like , is a function of two variables, namely and . Among possible values of will be (where is constant), , , and so on. (These result from assigning a value to , leaving to be assigned.) We will call such functions "predicative functions of first-order functions."

In the second place, we have second-order functions of two arguments, one of which is a first-order function while the other is an individual. Let us denote undetermined values of such functions by the notation As soon as is assigned, we shall have a predicative function of . If our function contains no first-order function as apparent variable, we shall obtain a predicative function of if we assign a value to . Thus, to take the simplest possible case, if is , the assignment of a value to gives us a predicative function of , in virtue of the definition of "." But if contains a first-order function as apparent variable, the assignment of a value to gives us a second-order function of .

We may now proceed in exactly the same way to third-order matrices, which will be functions containing second-order functions as arguments, and containing no apparent variables, and no arguments except individuals and first-order functions and second-order functions. Thence we shall proceed, as before, to third-order functions; and so we can proceed indefinitely. If the highest order of variable occurring in a function, whether as argument or as apparent variable, is a function of the th order, then the function in which it occurs is of the th order. We do not arrive at functions of an infinite order, because the number of arguments and of apparent variables in a function must be finite, and therefore every function must be of a finite order. Since the orders of functions are only defined step by step, there can be no process of "proceeding to the limit," and functions of an infinite order cannot occur.

We will define a function of one variable as predicative when it is of the next order above that of its argument, i.e. of the lowest order compatible with its having that argument. If a function has several arguments, and the highest order of function occurring among the arguments is the th, we call the function predicative if it is of the th order, i.e. again, if it is of the lowest order compatible with its having the arguments it has. A function of several arguments is predicative if there is one of its arguments such that, when the other arguments have values assigned to them, we obtain a predicative function of the one undetermined argument.

It is important to observe that all possible functions in the above hierarchy can be obtained by means of predicative functions and apparent variables. Thus, as we saw, second-order functions of an individual are of the form where is a second-order predicative function. And speaking generally, a non-predicative function of the th order is obtained from a predicative function of the th order by turning all the arguments of the th order into apparent variables. (Other arguments also may be turned into apparent variables.) Thus we need not introduce as variables any functions except predicative functions. Moreover, to obtain any function of one variable , we need not go beyond predicative functions of two variables. For the function (, where is given, is a function of and , and is predicative. Thus it is of the form , and therefore ( is of the form (. Thus speaking generally, by a succession of steps we find that, if is a predicative function of a sufficiently high order, any assigned non-predicative function of will be of one of the two forms where is a predicative function of and .

The nature of the above hierarchy of functions may be restated as follows. A function, as we saw at an earlier stage, presupposes as part of its meaning the totality of its values, or, what comes to the same thing, the totality of its possible arguments. The arguments to a function may be functions or propositions or individuals. (It will be remembered that individuals were defined as whatever is neither a proposition nor a function.) For the present we neglect the case in which the argument to a function is a proposition. Consider a function whose argument is an individual. This function presupposes the totality of individuals; but unless it contains functions as apparent variables, it does not presuppose any totality of functions. If, however, it does contain a function as apparent variable, then it cannot be defined until some totality of functions has been defined. It follows that we must first define the totality of those functions that have individuals as arguments and contain no functions as apparent variables. These are the predicative functions of individuals. Generally, a predicative function of a variable argument is one which involves no totality except that of the possible values of the argument, and those that are presupposed by any one of the possible arguments. Thus a predicative function of a variable argument is any function which can be specified without introducing new kinds of variables not necessarily presupposed by the variable which is the argument.

A closely analogous treatment can be developed for propositions. Propositions which contain no functions and no apparent variables may be called elementary propositions. Propositions which are not elementary, which contain no functions, and no apparent variables except individuals, may be called first-order propositions. (It should be observed that no variables except apparent variables can occur in a proposition, since whatever contains a real variable is a function, not a proposition.) Thus elementary and first-order propositions will be values of first-order functions. (It should be remembered that a function is not a constituent in one of its values: thus for example the function " is human" is not a constituent of the proposition "Socrates is human.") Elementary and first-order propositions presuppose no totality except (at most) the totality of individuals. They are of one or other of the three forms where is a predicative function of an individual. It follows that, if represents a variable elementary proposition or a variable first-order proposition, a function is either or or . Thus a function of an elementary or a first-order proposition may always be reduced to a function of a first-order function. It follows that a proposition involving the totality of first-order propositions may be reduced to one involving the totality of first-order functions; and this obviously applies equally to higher orders.

The propositional hierarchy can, therefore, be derived from the functional hierarchy, and we may define a proposition of the nth order as one which involves an apparent variable of the th order in the functional hierarchy. The propositional hierarchy is never required in practice, and is only relevant for the solution of paradoxes; hence it is unnecessary to go into further detail as to the types of propositions.

It remains to consider the "axiom of reducibility." It will be seen that, according to the above hierarchy, no statement can be made significantly about "all -functions," where is some given object. Thus such a notion as "all properties of ," meaning "all functions which are true with the argument ," will be illegitimate. We shall have to distinguish the order of function concerned. We can speak of "all predicative properties of ," "all second-order properties of ," and so on. (If is not an individual, but an object of order , "second-order properties of " will mean "functions of order satisfied by .") But we cannot speak of "all properties of ." In some cases, we can see that some statement will hold of "all th-order properties of ," whatever value may have. In such cases, no practical harm results from regarding the statement as being about "all properties of ," provided we remember that it is really a number of statements, and not a single statement which could be regarded as assigning another property to , over and above all properties. Such cases will always involve some systematic ambiguity, such as that involved in the meaning of the word "truth," as explained above. Owing to this systematic ambiguity, it will be possible, sometimes, to combine into a single verbal statement what are really a number of different statements, corresponding to different orders in the hierarchy.

This is illustrated in the case of the liar, where the statement "all 's statements are false" should be broken up into different statements referring to his statements of various orders, and attributing to each the appropriate kind of falsehood.

The axiom of reducibility is introduced in order to legitimate a great mass of reasoning, in which, prima facie, we are concerned with such notions as "all properties of " or "all -functions," and in which, nevertheless, it seems scarcely possible to suspect any substantial error. In order to state the axiom, we must first define what is meant by "formal equivalence." Two functions , are said to be "formally equivalent" when, with every possible argument , is equivalent to , i.e. and are either both true or both false. Thus two functions are formally equivalent when they are satisfied by the same set of arguments. The axiom of reducibility is the assumption that, given any function , there is a formally equivalent predicative function, i.e. there is a predicative function which is true when is true and false when is false. In symbols, the axiom is: For two variables, we require a similar axiom, namely: Given any function , there is a formally equivalent predicative function, i.e.

If we call a predicate of an object a predicative function which is true of that object, then the predicates of an object are only some among its properties. Take for example such a proposition as "Napoleon had all the qualities that make a great general." We may interpret this as meaning "Napoleon had all the predicates that make a great general." Here there is a predicate which is an apparent variable. If we put "" for " is a predicate required in a great general," our proposition is Since this refers to a totality of predicates, it is not itself a predicate of Napoleon. It by no means follows, however, that there is not some one predicate common and peculiar to great generals. In fact, it is certain that there is such a predicate. For the number of great generals is finite, and each of them certainly possessed some predicate not possessed by any other human being—for example, the exact instant of his birth. The disjunction of such predicates will constitute a predicate common and peculiar to great generals[20]. If we call this predicate , the statement we made about Napoleon was equivalent to . And this equivalence holds equally if we substitute any other individual for Napoleon. Thus we have arrived at a predicate which is always equivalent to the property we ascribed to Napoleon, i.e. it belongs to those objects which have this property, and to no others. The axiom of reducibility states that such a predicate always exists, i.e. that any property of an object belongs to the same collection of objects as those that possess some predicate.

We may next illustrate our principle by its application to identity. In this connection, it has a certain affinity with Leibniz's identity of indiscernibles. It is plain that, if and are identical, and is true, then is true. Here it cannot matter what sort of function may be: the statement must hold for any function. But we cannot say, conversely: "If, with all values of , implies , then and are identical"; because "all values of " is inadmissible. If we wish to speak of "all values of ," we must confine ourselves to functions of one order. We may confine to predicates, or to second-order functions, or to functions of any order we please. But we must necessarily leave out functions of all but one order. Thus we shall obtain, so to speak, a hierarchy of different degrees of identity. We may say "all the predicates of belong to ," "all second-order properties of belong to ," and so on. Each of these statements implies all its predecessors: for example, if all second-order properties of belong to , then all predicates of belong to , for to have all the predicates of is a second-order property, and this property belongs to . But we cannot, without the help of an axiom, argue conversely that if all the predicates of belong to , all the second-order properties of must also belong to . Thus we cannot, without the help of an axiom, be sure that and are identical if they have the same predicates. Leibniz's identity of indiscernibles supplied this axiom.

It should be observed that by "indiscernibles" he cannot have meant two objects which agree as to all their properties, for one of the properties of is to be identical with , and therefore this property would necessarily belong to if and agreed in all their properties. Some limitation of the common properties necessary to make things indiscernible is therefore implied by the necessity of an axiom. For purposes of illustration (not of interpreting Leibniz) we may suppose the common properties required for indiscernibility to be limited to predicates. Then the identity of indiscernibles will state that if and agree as to all their predicates, they are identical. This can be proved if we assume the axiom of reducibility. For, in that case, every property belongs to the same collection of objects as is defined by some predicate. Hence there is some predicate common and peculiar to the objects which are identical with . This predicate belongs to , since is identical with itself; hence it belongs to , since y has all the predicates of ; hence y is identical with . It follows that we may define and as identical when all the predicates of belong to , i.e. when (. We therefore adopt the following definition of identity[21]:

But apart from the axiom of reducibility, or some axiom equivalent in this connection, we should be compelled to regard identity as indefinable, and to admit (what seems impossible) that two objects may agree in all their predicates without being identical.

The axiom of reducibility is even more essential in the theory of classes. It should be observed, in the first place, that if we assume the existence of classes, the axiom of reducibility can be proved. For in that case, given any function of whatever order, there is a class consisting of just those objects which satisfy . Hence "" is equivalent to " belongs to ." But " belongs to " is a statement containing no apparent variable, and is therefore a predicative function of . Hence if we assume the existence of classes, the axiom of reducibility becomes unnecessary. The assumption of the axiom of reducibility is therefore a smaller assumption than the assumption that there are classes. This latter assumption has hitherto been made unhesitatingly. However, both on the ground of the contradictions, which require a more complicated treatment if classes are assumed, and on the ground that it is always well to make the smallest assumption required for proving our theorems, we prefer to assume the axiom of reducibility rather than the existence of classes. But in order to explain the use of the axiom in dealing with classes, it is necessary first to explain the theory of classes, which is a topic belonging to Chapter III. We therefore postpone to that Chapter the explanation of the use of our axiom in dealing with classes.

It is worth while to note that all the purposes served by the axiom of reducibility are equally well served if we assume that there is always a function of the th order (where is fixed) which is formally equivalent to , whatever may be the order of . Here we shall mean by "a function of the th order" a function of the th order relative to the arguments to ; thus if these arguments are absolutely of the th order, we assume the existence of a function formally equivalent to whose absolute order is the th. The axiom of reducibility in the form assumed above takes , but this is not necessary to the use of the axiom. It is also unnecessary that should be the same for different values of ; what is necessary is that should be constant so long as is constant. What is needed is that, where extensional functions of functions are concerned, we should be able to deal with any -function by means of some formally equivalent function of a given type, so as to be able to obtain results which would otherwise require the illegitimate notion of "all -functions"; but it does not matter what the given type is. It does not appear, however, that the axiom of reducibility is rendered appreciably more plausible by being put in the above more general but more complicated form.

The axiom of reducibility is equivalent to the assumption that "any combination or disjunction of predicates[22] is equivalent to a single predicate," i.e. to the assumption that, if we assert that has all the predicates that satisfy a function there is some one predicate which will have whenever our assertion is true, and will not have whenever it is false, and similarly if we assert that has some one of the predicates that satisfy a function ). For by means of this assumption, the order of a non-predicative function can be lowered by one; hence, after some finite number of steps, we shall be able to get from any non-predicative function to a formally equivalent predicative function. It does not seem probable that the above assumption could be substituted for the axiom of reducibility in symbolic deductions, since its use would require the explicit introduction of the further assumption that by a finite number of downward steps we can pass from any function to a predicative function, and this assumption could not well be made without developments that are scarcely possible at an early stage. But on the above grounds it seems plain that in fact, if the above alternative axiom is true, so is the axiom of reducibility. The converse, which completes the proof of equivalence, is of course evident.

That the axiom of reducibility is self-evident is a proposition which can hardly be maintained. But in fact self-evidence is never more than a part of the reason for accepting an axiom, and is never indispensable. The reason for accepting an axiom, as for accepting any other proposition, is always largely inductive, namely that many propositions which are nearly indubitable can be deduced from it, and that no equally plausible way is known by which these propositions could be true if the axiom were false, and nothing which is probably false can be deduced from it. If the axiom is apparently self-evident, that only means, practically, that it is nearly indubitable; for things have been thought to be self-evident and have yet turned out to be false. And if the axiom itself is nearly indubitable, that merely adds to the inductive evidence derived from the fact that its consequences are nearly indubitable: it does not provide new evidence of a radically different kind. Infallibility is never attainable, and therefore some element of doubt should always attach to every axiom and to all its consequences. In formal logic, the element of doubt is less than in most sciences, but it is not absent, as appears from the fact that the paradoxes followed from premisses which were not previously known to require limitations. In the case of the axiom of reducibility, the inductive evidence in its favour is very strong, since the reasonings which it permits and the results to which it leads are all such as appear valid.

But although it seems very improbable that the axiom should turn out to be false, it is by no means improbable that it should be found to be deducible from some other more fundamental and more evident axiom. It is possible that the use of the vicious-circle principle, as embodied in the above hierarchy of types, is more drastic than it need be, and that by a less drastic use the necessity for the axiom might be avoided. Such changes, however, would not render anything false which had been asserted on the basis of the principles explained above: they would merely provide easier proofs of the same theorems. There would seem, therefore, to be but the slenderest ground for fearing that the use of the axiom of reducibility may lead us into error.