Principia mathematica, vol. 1 (of 3)

We are now in a position to show how the theory of types affects the solution of the contradictions which have beset mathematical logic. For this purpose, we shall begin by an enumeration of some of the more important and illustrative of these contradictions, and shall then show how they all embody vicious-circle fallacies, and are therefore all avoided by the theory of types. It will be noticed that these paradoxes do not relate exclusively to the ideas of number and quantity. Accordingly no solution can be adequate which seeks to explain them merely as the result of some illegitimate use of these ideas. The solution must be sought in some such scrutiny of fundamental logical ideas as has been attempted in the foregoing pages.

(1) The oldest contradiction of the kind in question is the Epimenides. Epimenides the Cretan said that all Cretans were liars, and all other statements made by Cretans were certainly lies. Was this a lie? The simplest form of this contradiction is afforded by the man who says "I am lying"; if he is lying, he is speaking the truth, and vice versa.

(3) Let be the relation which subsists between two relations and whenever does not have the relation to . Then, whatever relations and may be, " has the relation to " is equivalent to " does not have the relation to ." Hence, giving the value to both and , " has the relation to " is equivalent to " does not have the relation to ."

(4) Burali-Forti's contradiction[23] may be stated as follows: It can be shown that every well-ordered series has an ordinal number, that the series of ordinals up to and including any given ordinal exceeds the given ordinal by one, and (on certain very natural assumptions) that the series of all ordinals (in order of magnitude) is well-ordered. It follows that the series of all ordinals has an ordinal number, say. But in that case the series of all ordinals including has the ordinal number , which must be greater than . Hence is not the ordinal number of all ordinals.

(5) The number of syllables in the English names of finite integers tends to increase as the integers grow larger, and must gradually increase indefinitely, since only a finite number of names can be made with a given finite number of syllables. Hence the names of some integers must consist of at least nineteen syllables, and among these there must be a least. Hence "the least integer not nameable in fewer than nineteen syllables" must denote a definite integer; in fact, it denotes 111,777. But "the least integer not nameable in fewer than nineteen syllables" is itself a name consisting of eighteen syllables; hence the least integer not nameable in fewer than nineteen syllables can be named in eighteen syllables, which is a contradiction[24].

(6) Among transfinite ordinals some can be defined, while others can not; for the total number of possible definitions is [25], while the number of transfinite ordinals exceeds . Hence there must be indefinable ordinals, and among these there must be a least. But this is defined as "the least indefinable ordinal," which is a contradiction[26].

(7) Richard's paradox[27] is akin to that of the least indefinable ordinal. It is as follows: Consider all decimals that can be defined by means of a finite number of words; let be the class of such decimals. Then has terms; hence its members can be ordered as the 1st, 2nd, 3rd,.... Let be a number defined as follows. If the th figure in the th decimal is , let the th figure in be (or 0, if ). Then is different from all the members of , since, whatever finite value may have, the th figure in is different from the th figure in the nth of the decimals composing , and therefore is different from the th decimal. Nevertheless we have defined in a finite number of words, and therefore ought to be a member of . Thus both is and is not a member of .

In all the above contradictions (which are merely selections from an indefinite number) there is a common characteristic, which we may describe as self-reference or reflexiveness. The remark of Epimenides must include itself in its own scope. If all classes, provided they are not members of themselves, are members of , this must also apply to ; and similarly for the analogous relational contradiction. In the cases of names and definitions, the paradoxes result from considering non-nameability and indefinability as elements in names and definitions. In the case of Burali-Forti's paradox, the series whose ordinal number causes the difficulty is the series of all ordinal numbers. In each contradiction something is said about all cases of some kind, and from what is said a new case seems to be generated, which both is and is not of the same kind as the cases of which all were concerned in what was said. But this is the characteristic of illegitimate totalities, as we defined them in stating the vicious-circle principle. Hence all our contradictions are illustrations of vicious-circle fallacies. It only remains to show, therefore, that the illegitimate totalities involved are excluded by the hierarchy of types which we have constructed.

(1) When a man says "I am lying," we may interpret his statement as: "There is a proposition which I am affirming and which is false." That is to say, he is asserting the truth of some value of the function "I assert , and is false." But we saw that the word "false" is ambiguous, and that, in order to make it unambiguous, we must specify the order of falsehood, or, what comes to the same thing, the order of the proposition to which falsehood is ascribed. We saw also that, if is a proposition of the th order, a proposition in which occurs as an apparent variable is not of the th order, but of a higher order. Hence the kind of truth or falsehood which can belong to the statement "there is a proposition which I am affirming and which has falsehood of the th order" is truth or falsehood of a higher order than the th. Hence the statement of Epimenides does not fall within its own scope, and therefore no contradiction emerges.

If we regard the statement "I am lying" as a compact way of simultaneously making all the following statements: "I am asserting a false proposition of the first order," "I am asserting a false proposition of the second order," and so on, we find the following curious state of things: As no proposition of the first order is being asserted, the statement "I am asserting a false proposition of the first order" is false. This statement is of the second order, hence the statement "I am making a false statement of the second order" is true. This is a statement of the third order, and is the only statement of the third order which is being made. Hence the statement "I am making a false statement of the third order" is false. Thus we see that the statement "I am making a false statement of order " is false, while the statement "I am making a false statement of order " is true. But in this state of things there is no contradiction.

(2) In order to solve the contradiction about the class of classes which are not members of themselves, we shall assume, what will be explained in the next Chapter, that a proposition about a class is always to be reduced to a statement about a function which defines the class, i.e. about a function which is satisfied by the members of the class and by no other arguments. Thus a class is an object derived from a function and presupposing the function, just as, for example, presupposes the function . Hence a class cannot, by the vicious-circle principle, significantly be the argument to its defining function, that is to say, if we denote by "" the class defined by , the symbol "" must be meaningless. Hence a class neither satisfies nor does not satisfy its defining function, and therefore (as will appear more fully in Chapter III) is neither a member of itself nor not a member of itself. This is an immediate consequence of the limitation to the possible arguments to a function which was explained at the beginning of the present Chapter. Thus if is a class, the statement " is not a member of " is always meaningless, and there is therefore no sense in the phrase "the class of those classes which are not members of themselves." Hence the contradiction which results from supposing that there is such a class disappears.

(3) Exactly similar remarks apply to "the relation which holds between and whenever does not have the relation to ." Suppose the relation is defined by a function , i.e. holds between and whenever is true, but not otherwise. Then in order to interpret " has the relation to " we shall have to suppose that and can significantly be the arguments to . But (assuming, as will appear in Chapter III, that presupposes its defining function) this would require that should be able to take as argument an object which is defined in terms of , and this no function can do, as we saw at the beginning of this Chapter. Hence " has the relation to " is meaningless, and the contradiction ceases.

(4) The solution of Burali-Forti's contradiction requires some further developments for its solution. At this stage, it must suffice to observe that a series is a relation, and an ordinal number is a class of series. (These statements are justified in the body of the work.) Hence a series of ordinal numbers is a relation between classes of relations, and is of higher type than any of the series which are members of the ordinal numbers in question. Burali-Forti's "ordinal number of all ordinals" must be the ordinal number of all ordinals of a given type, and must therefore be of higher type than any of these ordinals. Hence it is not one of these ordinals, and there is no contradiction in its being greater than any of them[28].

(5) The paradox about "the least integer not nameable in fewer than nineteen syllables" embodies, as is at once obvious, a vicious-circle fallacy. For the word "nameable" refers to the totality of names, and yet is allowed to occur in what professes to be one among names. Hence there can be no such thing as a totality of names, in the sense in which the paradox speaks of "names." It is easy to see that, in virtue of the hierarchy of functions, the theory of types renders a totality of "names" impossible. We may, in fact, distinguish names of different orders as follows: (a) Elementary names will be such as are true "proper names," i.e. conventional appellations not involving any description. (b) First-order names will be such as involve a description by means of a first-order function; that is to say, if is a first-order function, "the term which satisfies " will be a first-order name, though there will not always be an object named by this name. (c) Second-order names will be such as involve a description by means of a second-order function; among such names will be those involving a reference to the totality of first-order names. And so we can proceed through a whole hierarchy. But at no stage can we give a meaning to the word "nameable" unless we specify the order of names to be employed; and any name in which the phrase "nameable by names of order " occurs is necessarily of a higher order than the th. Thus the paradox disappears.