Principia mathematica, vol. 1 (of 3)

The primitive idea "" has been explained to mean " is always true," i.e. "all values of are true." But whatever function may be, there will be arguments with which is meaningless, i.e. with which as arguments does not have any value. The arguments with which has values form what we will call the "range of significance" of . A "type" is defined as the range of significance of some function. In virtue of *9·14, if , , and are significant, i.e. either true or false, so is . From this it follows that two types which have a common member coincide, and that two different types are mutually exclusive. Any proposition of the form , i.e. any proposition containing an apparent variable, determines some type as the range of the apparent variable, the type being fixed by the function .

The division of objects into types is necessitated by the vicious-circle fallacies which otherwise arise[52]. These fallacies show that there must be no totalities which, if legitimate, would contain members defined in terms of themselves. Hence any expression containing an apparent variable must not be in the range of that variable, i.e. must belong to a different type. Thus the apparent variables contained or presupposed in an expression are what determines its type. This is the guiding principle in what follows.

As explained in *9, propositions containing variables are generated from propositional functions which do not contain these apparent variables, by the process of asserting all or some values of such functions. Suppose is a proposition containing ; we will give the name of generalization to the process which turns into or , and we will give the name of generalized propositions to all such as contain apparent variables. It is plain that propositions containing apparent variables presuppose others not containing apparent variables, from which they can be derived by generalization. Propositions which contain no apparent variables we call elementary propositions[53], and the terms of such propositions, other than functions, we call individuals. Then individuals form the first type.

It is unnecessary, in practice, to know what objects belong to the lowest type, or even whether the lowest type of variable occurring in a given context is that of individuals or some other. For in practice only the relative types of variables are relevant; thus the lowest type occurring in a given context may be called that of individuals, so far as that context is concerned. It follows that the above account of individuals is not essential to the truth of what follows; all that is essential is the way in which other types are generated from individuals, however the type of individuals may be constituted.

By applying the process of generalization to individuals occurring in elementary propositions, we obtain new propositions. The legitimacy of this process requires only that no individuals should be propositions. That this is so, is to be secured by the meaning we give to the word individual. We may explain an individual as something which exists on its own account; it is then obviously not a proposition, since propositions, as explained in Chapter II of the Introduction (p. 46), are incomplete symbols, having no meaning except in use. Hence in applying the process of generalization to individuals we run no risk of incurring reflexive fallacies. We will give the name of first-order propositions to such as contain one or more apparent variables whose possible values are individuals, but contain no other apparent variables. First-order propositions are not all of the same type, since, as was explained in *9, two propositions which do not contain the same number of apparent variables cannot be of the same type. But owing to the systematic ambiguity of negation and disjunction, their differences of type may usually be ignored in practice. No reflexive fallacies will result, since no first-order proposition involves any totality except that of individuals.

Let us denote by "" or "" or etc. an elementary function whose argument or arguments are individual. We will call such a function a predicative function of an individual. Such functions, together with those derived from them by generalization, will be called first-order functions. In practice we may without risk of reflexive fallacies treat first-order functions as a type, since the only totality they involve is that of individuals, and, by means of the systematic ambiguity of negation and disjunction, any function of a first-order function which will concern us will be significant whatever first-order function is taken as argument, provided the right meanings are given to the negations and disjunctions involved.

For the sake of clearness, we will repeat in somewhat different terms our account of what is meant by a first-order function. Let us give the name of matrix to any function, of however many variables, which does not involve any apparent variables. Then any possible function other than a matrix is derived from a matrix by means of generalization, i.e. by considering the proposition which asserts that the function in question is true with all possible values or with some value of one of the arguments, the other argument or arguments remaining undetermined. Thus e.g. from the function we shall be able to derive the four functions of which the two first are functions of , while the two last are functions of . (All propositions, with the exception of such as are values of matrices, are also derived from matrices by the above process of generalization. In order to obtain a proposition from a matrix containing variables, without assigning values to any of the variables, it is necessary to turn all the variables into apparent variables. Thus if is a matrix, ( is a proposition.) We will give the name first-order matrices to such as have only individuals for their arguments, and we will give the name of first-order functions (of any number of variables) to such as either are first-order matrices or are derived from first-order matrices by generalization applied to some (not all) of the arguments to such matrices. First-order propositions will be such as result from applying generalization to all the arguments to a first-order matrix.

As we have already stated, the notation "" is used for any elementary function of one variable. Thus "" represents any value of any elementary function of one variable. It will be seen that "" is a function of two variables, namely and . Since it contains no apparent variable, it is a matrix, but since it contains a variable (namely which is not an individual, it is not a first-order matrix. The same applies to , where a is some definite constant. We can build up a number of new matrices, such as All these are matrices which involve first-order functions among their arguments. Such matrices we will call second-order matrices. From these matrices, by applying generalization to their arguments, whether to such as are functions or to such (if any) as are individuals, we obtain new functions and propositions. Such functions (together with second-order matrices) will be called second-order functions, and such propositions will be called second-order propositions. Thus we are led to the following definitions:

A second-order matrix of one variable will be called a predicative second-order function of one variable or a predicative function of a first-order matrix. Thus , and are predicative functions of . Similarly a function of several variables of which at least one is a first-order matrix, while the rest are either individuals or first-order matrices, will be called predicative if it is a matrix.

It will be seen, however, that a second-order function may have only individuals for its arguments; instances were given just now under the heading (3). Such functions we shall not call predicative, since predicative functions of individuals have already been defined as being such as are of the first order. Thus the order of a function is not determined by the order of its argument or arguments; indeed, the function may be of any order superior to the order or orders of its arguments.

A variable matrix whose argument is will be denoted by , and generally, a matrix whose arguments are , , ... , , ... (where there is at least one function among the arguments) will be denoted by Such a matrix is not of the first or second order, since it contains the new variable whose values are second-order matrices. We proceed to construct new matrices as we did with the matrix ; these constitute third-order matrices. These together with the functions derived from them by generalization are called third-order functions, and the propositions derived from third-order matrices by generalization are called third-order propositions.

The variables occurring in the present work, from this point onwards, will all be either individuals or matrices of some order in the above hierarchy. Propositions, which have occurred hitherto as variables, will no longer do so except in a few isolated cases of which no subsequent use is made. In practice, for the reasons explained on p. 169, a function of a matrix may be regarded as capable of any argument which is a function of the same order and takes arguments of the same type.

In practice, we never need to know the absolute types of our variables, but only their relative types. That is to say, if we prove any proposition on the assumption that one of our variables is an individual, and another is a function of order , the proof will still hold if, in place of an individual, we take a function of order , and in place of our function of order we take a function of order , with corresponding changes for any other variables that may be involved. This results from the assumption that our primitive propositions are to apply to variables of any order.

We shall use small Latin letters (other than , , , for variables of the lowest type concerned in any context. For functions, we shall use the letters , , , , , , (except that, at a later stage, will be defined as a constant relation, and will be defined as the order-type of the continuum).

When any predicative function, say , occurs as apparent variable, it would be strictly more correct to indicate the fact by placing "(" before what follows, as thus: "(." But for the sake of brevity we write simply "(" instead of "(." Since what follows the in brackets must always contain with arguments supplied, no confusion can result from this practice.

It should be observed that, in virtue of the manner in which our hierarchy of functions was generated, non-predicative functions always result from such as are predicative by means of generalization. Hence it is unnecessary to introduce a special notation for non-predicative functions of a given order and taking arguments of a given order. For example, second-order functions of an individual are always derived by generalization from a matrix where the functions , , , ... are predicative. It is possible, therefore, without loss of generality, to use no apparent variables except such as are predicative.

We require, however, a means of symbolising a function whose order is not assigned. We shall use "" or "" or etc. to express a function ( or ) whose order, relatively to its argument, is not given. Such a function cannot be made into an apparent variable, unless we suppose its order previously fixed. As the only purpose of the notation is to avoid the necessity of fixing the order, such a function will not be used as an apparent variable; the only functions which will be so used will be predicative functions, because, as we have just seen, this restriction involves no loss of generality.

It is important to observe that, since there are various types of propositions and functions, and since generalization can only be applied within some one type (or, by means of systematic ambiguity, within some well-defined and completed set of types), all phrases referring to "all propositions" or "all functions," or to "some (undetermined) proposition" or "some (undetermined) function," are prima facie meaningless, though in certain cases they are capable of an unobjectionable interpretation. Contradictions arise from the use of such phrases in cases where no innocent meaning can be found.

If mathematics is to be possible, it is absolutely necessary (as explained in the Introduction, Chapter II) that we should have some method of making statements which will usually be equivalent to what we have in mind when we (inaccurately) speak of "all properties of ." (A "property of " may be defined as a propositional function satisfied by .) Hence we must find, if possible, some method of reducing the order of a propositional function without affecting the truth or falsehood of its values. This seems to be what common-sense effects by the admission of classes. Given any propositional function of whatever order, this is assumed to be equivalent, for all values of , to a statement of the form " belongs to the class ." Now assuming that there is such an entity as the class , this statement is of the first order, since it involves no allusion to a variable function. Indeed its only practical advantage over the original statement is that it is of the first order. There is no advantage in assuming that there really are such things as classes, and the contradiction about the classes which are not members of themselves shows that, if there are classes, they must be something radically different from individuals. It would seem that the sole purpose which classes serve, and one main reason which makes them linguistically convenient, is that they provide a method of reducing the order of a propositional function.

We shall, therefore, not assume anything of what may seem to be involved in the common-sense admission of classes, except this, that every propositional function is equivalent, for all its values, to some predicative function of the same argument or arguments.

This assumption with regard to functions is to be made whatever may be the type of their arguments. Let be a function, of any order, of an argument , which may itself be either an individual or a function of any order. If is a matrix, we write the function in the form ; in such a case we call a predicative function. Thus a predicative function of an individual is a first-order function; and for higher types of arguments, predicative functions take the place that first-order functions take in respect of individuals. We assume, then, that every function of one variable is equivalent, for all its values, to some predicative function of the same argument. This assumption seems to be the essence of the usual assumption of classes; at any rate, it retains as much of classes as we have any use for, and little enough to avoid the contradictions which a less grudging admission of classes is apt to entail. We will call this assumption the axiom of classes, or the axiom of reducibility.

We shall assume similarly that every function of two variables is equivalent, for all its values, to a predicative function of those variables, i.e. to a matrix. This assumption is what seems to be meant by saying that any statement about two variables defines a relation between them. We will call this assumption the axiom of relations or (like the previous axiom) the axiom of reducibility.