39. Incomplete Symbols

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 (1) Descriptions. By an "incomplete " symbol we mean a symbol which is not supposed to have any meaning in isolation, but is only defined in certain contexts. In ordinary mathematics, for example,

d_   and ∫^b

dx              ^a

are incomplete symbols: something has to be supplied before we have anything significant. Such symbols have what may be called a "definition in use." Thus if we put

    ∇² = ¶² + ¶² + ¶²     

           ¶x²    ¶y²  ¶z²        Df,

we define the use of ∇², but ∇² by itself remains without meaning. This distinguishes such symbols from what (in a generalized sense) we may call proper names: "Socrates," for example, stands for a certain man, and therefore has a meaning by itself, without the need of any context. If we supply a context, as in "Socrates is mortal," these words express a fact of which Socrates himself is a constituent: there is a certain object, namely Socrates, which does have the property of mortality, and this object is a constituent of the complex fact which we assert when we say "Socrates is mortal." But in other cases, this simple analysis fails us. Suppose we say: "The round square does not exist." It seems plain that this is a true proposition, yet we cannot regard it as denying the existence of a certain object called "the round square." For if there were such an object, it would exist: we cannot first assume that there is a certain object, and then proceed to deny that there is such an object. Whenever the grammatical subject of a proposition can be supposed not to exist without rendering the proposition meaningless, it is plain that the grammatical subject is not a proper name, i.e. not a name directly representing some object. Thus in all such cases, the proposition must be capable of being so analyzed that what was the grammatical subject shall have disappeared. Thus when we say "the round square 'does not exist," we may, as a first attempt at such analysis, substitute " it is false that there is an object x which is both round and square." Generally, when "the so-and-so" is said not to exist, we have a proposition of the form*¹

   "~E!(⍳x)(φx),"

i.e.     ~{(∃^C): φx . ≡_x . x = c},

or some equivalent. Here the apparent grammatical subject (⍳x)(φx) has completely disappeared; thus in

   "~E!(⍳x)(φx),” (⍳x)(φx) is an incomplete symbol.

By an extension of the above argument, it can easily be shown that (⍳x)(φx) is always an incomplete symbol. Take, for example, the following proposition: "Scott is the author of Waverley." [Here "the author of Waverley" is (⍳x)(x wrote Waverley).] This proposition expresses an identity; thus if "the author of Waverley" could be taken as a proper name, and supposed to stand for some object c, the proposition would be "Scott is c." But if c is anyone except Scott, this proposition is false; while if c is Scott, the proposition is "Scott is Scott," which is trivial, and plainly different from "Scott is the author of Waverley." Generalizing, we see that the proposition

     a = (⍳x)(φx)

is one which may be true or may be false, but is never merely trivial, like a = a; whereas, if (⍳x)(φx) were a proper name, a = (⍳x)(φx) would necessarily be either false or the same  as the trivial proposition a = a. We may express this by saying that a = (⍳x)(φx) is not a value of the propositional function a = y,  from which it follows that (⍳x)(φx) is not a value of y. But since y may be anything, it follows that (⍳x)(φx) is nothing. Hence, since in use it has meaning, it must be an incomplete symbol.

It might be suggested that "Scott is the author of Waverley" asserts that "Scott" and "the author of Waverley" are two names for the same object. But a little reflection will show that this would be a mistake. For if that were the meaning of "Scott is the author of Waverley,"what would be required for its truth would be that Scott should have been called the author of Waverley: if he had been so called, the proposition would be true, even if someone else had written Waverley; while if no one called him so, the proposition would be false, even if he had written Waverley. But in fact he was the author of Waverley at a time when no one called him so, and he would not have been the author if everyone had called him so but someone else had written Waverley. Thus the proposition "Scott is the author of Waverley "is not a proposition about names, like "Napoleon is Bonaparte"; and this illustrates the sense in which "the author of Waverley "differs from a true proper name.

Thus all phrases (other than propositions) containing the word the (in the singular) are incomplete symbols: they have a meaning in use, but not in isolation. For "the author of Waverley" cannot mean the same as "Scott," or "Scott is the author of Waverley" would mean the same as "Scott is Scott," which it plainly does not; nor can "the author of Waverley" mean anything other than "Scott," or "Scott is the author of Waverley" would be false. Hence "the author of Waverley" means nothing.

It follows from the above that we must not attempt to define "(⍳x)(φx)," but must define the uses of this symbol, i.e. the propositions in whose symbolic expression it occurs. Now in seeking to define the uses of this symbol, it is important to observe the import of propositions in which it occurs. Take as an illustration: "The author of Waverley was a poet." This implies (1) that Waverley was written, (2) that it was written by one man, and not in collaboration, (3) that the one man who wrote it was a poet. If any one of these fails, the proposition is false. Thus "the author of 'Slawkenburgius on Noses' was a poet" is false, because no such book was ever written; "the author of 'The Maid's Tragedy' was a poet" is false, because this play was written by Beaumont and Fletcher jointly. These two possibilities of falsehood do not arise if we say "Scott was a poet." Thus our interpretation of the uses of (⍳x)(φx) must be such as to allow for them. Now taking φx to replace "x wrote Waverley," it is plain that any statement apparently about (⍳x)(φx)  requires (1) (∃x).(φx) and (2) φx . φ .⊃_x,y . x = y; here (1) states that at least one object satisfies φx, while (2) states that at most one object satisfies φx. The two together are equivalent to

     (∃c) : φx . ≡_x . x = c,

which we defined as     E! (⍳x)(φx).

Thus "E! (⍳x)(φx)" must be part of what is affirmed by any proposition about (⍳x)(φx). If our proposition is ƒ{(⍳x)(φx)}, what is further affirmed is ƒc, if φx . ≡_x . x = c. Thus we have

     ƒ{(⍳x)(φx) . = : (∃^C) : φx . ≡_x . x = c : ƒc  Df,

i.e. "the x satisfying φx satisfies ƒx" is to mean: "There is an object c such that φx is true when, and only when, x is c, and ƒc is true, "or, more exactly: " There is a c such that 'φx' is always equivalent to 'x is c,' and ƒc." In this, " (⍳x)(φx)" has completely disappeared; thus " (⍳x)(φx) " is merely symbolic, and does not directly represent an object, as single small Latin letters are assumed to do*².

*¹  see post 22.

*²  We shall generally write "ƒ(⍳x)(φx)" rather than "ƒ{(⍳x)(φx)}" in future.

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  This discussion of incomplete symbols, particularly those used in description, in this post and in those immediately following, shed light on some simple aspects that commonly adversely affect communication with language. Our premise stated previously in post 30, is that complexity begins with just two significant parts of speech. Our experience in life and conversation shows us that misunderstandings are very common, unfortunately almost the norm. Here we learn to spot the aspects of language that lead to misunderstanding, and by attempting to avoid making these ourselves, and carefully practicing to see them in others language it is possible to clarify problem usage kindly. This way more precise understanding may be possible.

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38. The Contradictions (Second Part).



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The Contradictions (Part two). 🔑

(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*¹.

<<(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 φ!x^ is a first-order function, "the term which satisfies φ!x^" 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 n" occurs is necessarily of a higher order than the nth. Thus the paradox disappears.

(6) Among trans-finite ordinals some can be defined, while others cannot; for the total number of possible definitions is À₀*², while the number of trans-finite ordinals exceeds À₀. 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*³.

(7) Richard's paradox*⁴ 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 E be the class of such decimals. Then E has À₀ terms; hence its members can be ordered as the 1st, 2nd, 3rd, .... Let N be a number defined as follows: If the nth figure in the nth decimal is p, let the nth figure in N be p + 1 (or 0, if p = 9). Then N is different from all the members of E, since, whatever finite value n may have, the nth figure in N is different from the nth figure in the nth of the decimals composing E, and therefore N is different from the nth decimal. Nevertheless we have defined N in a finite number of words, and therefore N ought to be a member of E. Thus N both is and is not a member of E.

<<(6,7) The solutions of the paradox about the least indefinable ordinal and of Richard's paradox are closely analogous to the above. The notion of "definable," which occurs in both, is nearly the same as "nameable," which occurs in our fifth paradox: "definable" is what "nameable" becomes when elementary names are excluded, i.e. "definable" means "nameable by a name which is not elementary." But here there is the same ambiguity as to type as there was before, and the same need for the addition of words which specify the type to which the definition is to belong. And however the type may be specified, " the least ordinal not definable by definitions of this type" is a definition of a higher type; and in Richard's paradox, when we confine ourselves, as we must, to decimals that have a definition of a given type, the number N, which causes the paradox, is found to have a definition which belongs to a higher type, and thus not to come within the scope of our previous definitions.

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 w, this must also apply to w; and similarly for the analogous relational contradiction. In the cases of names and definitions, the paradoxes result from considering non-name-ability 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.

An indefinite number of other contradictions, of similar nature to the above seven, can easily be manufactured. In all of them, the solution is of the same kind. In all of them, the appearance of contradiction is produced by the presence of some word which has systematic ambiguity of type, such as truth, falsehood, function, property, class, relation, cardinal, ordinal, name, definition. Any such word, if it’s typical ambiguity is overlooked, will apparently generate a totality containing members defined in terms of itself, and will thus give rise to vicious-circle fallacies. In most cases, the conclusions of arguments which involve vicious-circle fallacies will not be self-contradictory, but wherever we have an illegitimate totality, a little ingenuity will enable us to construct a vicious-circle fallacy leading to a contradiction, which disappears as soon as the typically ambiguous words are rendered typically definite, i.e. are determined as belonging to this or that type.

Thus the appearance of contradiction is always due to the presence of words embodying a concealed typical ambiguity, and the solution of the apparent contradiction lies in bringing the concealed ambiguity to light.

In spite of the contradictions which result from unnoticed typical ambiguity, it is not desirable to avoid words and symbols which have typical ambiguity. Such words and symbols embrace practically all the ideas with which mathematics and mathematical logic are concerned: the systematic ambiguity is the result of a systematic analogy. That is to say, in almost all the reasoning which constitutes mathematics and mathematical logic, we are using ideas which may receive any one of an infinite number of different typical determinations, any one of which leaves the reasoning valid. Thus by employing typically ambiguous words and symbols, we are able to make one chain of reasoning applicable to any one of an infinite number of different cases, which would not be possible if we were to forego the use of typically ambiguous words and symbols.

Among propositions wholly expressed in terms of typically ambiguous notions practically the only ones which may differ, in respect of truth or falsehood, according to the typical determination which they receive, are existence-theorems. If we assume that the total number of individuals is n, then the total number of classes of individuals is 2ⁿ", the total number of classes of classes of individuals is 2²", and so on. Here n may be either finite or infinite, and in either case 2ⁿ >n. Thus cardinals greater than n but not greater than 2ⁿ exist as applied to classes, but not as applied to classes of individuals, so that whatever may be supposed to be the number of individuals, there will be existence-theorems which hold for higher types but not for lower types. Even here, however, so long as the number of individuals is not asserted, but is merely assumed hypothetically, we may replace the type of individuals by any other type, provided we make a corresponding change in all the other types occurring in the same context. That is, we may give the name "relative individuals" to the members of an arbitrarily chosen type τ, and the name "relative classes of individuals" to classes of "relative individuals," and so on. Thus so long as only hypotheticals are concerned, in which existence-theorems for one type are shown to be implied by existence-theorems for another, only relative types are relevant even in existence-theorems. This applies also to cases where the hypothesis (and therefore the conclusion) is asserted, provided the assertion holds for any type, however chosen. For example, any type has at least one member; hence any type which consists of classes, of whatever order, has at least two members. But the further pursuit of these topics must be left to the body of the work.

*¹ This contradiction was suggested to us by Mr G. G. Berry of the Bodleian Library.

*² À₀ is the number of finite integers. _*123.

*³ Cf. Konig, "Ueber die Grundlagen der Mengenlehre und das Kontinuumproblem," Math. Annalen, Vol. LXI. (1905); A. C. Dixon, "On 'well-ordered' aggregates," Proc. London Math. Soc. Series 2, Vol. iv. Part i. (1906); and E. W. Hobson, "On the Arithmetic Continuum," ibid. The solution offered in the last of these papers depends upon the variation of the " apparatus of definition," and is thus in outline in agreement with the solution adopted here. But it does not invalidate the statement in the text, if " definition " is given a constant meaning.

*⁴ Cf. Poincare, "Les mathematiques et la logique," Revue de Metaphysique et de Morale, Mai 1906, especially sections vii. and ix.; also Peano, Revista de Mathematica, Vol. viii. No. 5 (1906), p. 149 ff.

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This part, addressing contradictions, is key to translating the use and real world value of the system into practice. This is also the clearest explanation thus far of the basic understanding of shaman sight; or seeing through externalities that distract most human beings from using their ability to recognize complex and difficult problems, let alone thinking them through and separating their causes and solutions.

When we want to discern reality in contradictory intentions no matter how large or small, sorting it out is a matter of looking at all sides of the issue without prejudice; propositions placed in evidence that include arguments that are self-referential are the ones that unravel as vicious-circle Fallacies. Don’t presume these gremlins: truth, falsehood, function, name, reality, relation, etc. must be intentionally removed, though perhaps they may be, these typical ambiguities may foul the water but conversely they allow us to open the tap of creativity that makes this system and all mathematics happen in the first place.

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37. The Contradictions (first part of two)

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VIII. The Contradictions (for example).

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.

<<(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 p, and p 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 p is a proposition of the nth order, a proposition in which p occurs as an apparent variable is not of the nth order, but of a higher order. Hence the kind of truth or falsehood which can belong to the statement "there is a proposition p which I am affirming and which has falsehood of the nth order" is truth or falsehood of a higher order than the nth. 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 2n + 1" is false, while the statement "I am making a false statement of order 2n" is true. But in this state of things there is no contradiction.

(2) Let w be the class of all those classes which are not members of themselves. Then, whatever class x may be, "x is a w" is equivalent to "x is not an x." Hence, giving to x the value w, "w is a w" is equivalent to "w is not a w."

<<(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, (x). φx presupposes the function φx^. 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 “z^(φz)" the class defined by φz^, the symbol "φ{z^(φz)}" 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 a is a class, the statement "a is not a member of a" 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) Let T be the relation that subsists between two relations R and S whenever R does not have the relation R to S. Then, whatever relations R and S may be, "R has the relation T to S" is equivalent to "R does not have the relation R to S." Hence, giving the value T to both R and S, "T has the relation T to T" is equivalent to "T does not have the relation T to T."

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

(4) Burali-Forti's contradiction 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, Ω say. But in that case the series of all ordinals including Ω has the ordinal number Ω + 1, which must be greater than Ω. Hence Ω is not the ordinal number of all ordinals.

<<(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.

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It is clear that our system prevents contradictions by understanding how they may arise and steering clear. As we have seen above in the authors’ treatment of four classic mathematical contradictions, their resolution presented is not the result of any external changes in the problems themselves; but solely in our perception of the problems in terms of our system. Nearly all insolvable problems can find a resolution with a new approach, a fresh look or a change in attitude.

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