38. The Contradictions (Second Part).



Harold R. (Hal) Foster’s Prince Valiant

© Respective copyright/trademark holders.

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.

Script: Stan Lee  Pencils: & Inks: Steranko  © Respective copyright/trademark holders.

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.

Script and Layouts: Jim Shooter Pencils and Inks: George Papp  Cosmic Boy  Phantom Girl  & LSH^TM ©

Respective copyright/trademark holders.

Harold R. (Hal) Foster’s Prince Valiant

© Respective copyright/trademark holders.

 

37. The Contradictions (first part of two)

<

Harold R. (Hal) Foster’s Prince Valiant

© Respective copyright/trademark holders.

 

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.

Pencils and Inks: George Evans © Respective copyright/trademark holders.

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.

Dan DeCarlo The greatest of all time comic book artist!

Archie Andrews Gang   © Respective copyright/trademark holders.



Harold R. (Hal) Foster’s Prince Valiant

© Respective copyright/trademark holders.

 

36. Reasons for Accepting the Axiom of Reducibility

Harold R. (Hal) Foster’s Prince Valiant=]

© Respective copyright/trademark holders.

VII. Reasons for Accepting the Axiom of Reducibility.

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 premises which were not previously known to require limitations. In the case of the axiom of reducibility, the inductive evidence in its favor is very strong, since the reasoning(s) 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.

Stan Lee (plot); Ernie Hart [H. E. Huntley] (script) Pencils: Jack Kirby Inks: Don Heck

Ant Man and the Wasp  © Respective copyright/trademark holders.




Harold R. (Hal) Foster’s Prince Valiant

© Respective copyright/trademark holders.

35. The Axiom of Reducibility

Harold R. (Hal) Foster’s Prince Valiant

© Respective copyright/trademark holders.

VI. The Axiom of Reducibility.

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 a-functions," where a is some given object. Thus such a notion as "all properties of a," meaning "all functions which are true with the argument a," will be illegitimate. We shall have to distinguish the order of function concerned. We can speak of "all predicative properties of a," "all second-order properties of a," and so on. (If a is not an individual, but an object of order n, "second-order properties of a" will mean "functions of order n + 2 satisfied by a.") But we cannot speak of "all properties of a." In some cases, we can see that some statement will hold of "all nth-order properties of a," whatever value n may have. In such cases, no practical harm results from regarding the statement as being about "all properties of a," 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 a, 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 A'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 a" or "all a-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 φx^, yx^ are said to be "formally equivalent" when, with every possible argument x, φx is equivalent to yx, i.e. φx and yx 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 φx^, there is a formally equivalent predicative function, i.e. there is a predicative function which is true when φx is true and false when φx is false. In symbols, the axiom is:

┠ : (∃y) : φx . ≡_x . y!x.

For two variables, we require a similar axiom, namely: Given any function φ(x^, y^), there is a formally equivalent predicative function, i.e.

┠ : (∃y) : φ(x, y). ≡_x,y . y!x.

In order to explain the purposes of the axiom of reducibility, and the nature of the grounds for supposing it true, we shall first illustrate it by applying it to some particular cases.

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 "ƒ(φ!z^)" for "φ!z^ is a predicate required in a great general," our proposition is

(φ): ƒ(φ!z^) implies φ!(Napoleon).

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*¹. If we call this predicate y!z^, the statement we made about Napoleon was equivalent to y!(Napoleon). 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 x and y are identical, and φx is true, then φy is true. Here it cannot matter what sort of function φx^ may be: the statement must hold for any function. But we cannot say, conversely: "If, with all values of φ, φx implies φy, then x and y are identical"; because "all values of φ" is inadmissible. If we wish to speak of "all values of φ," we must confine ourselves to functions of one order. We may confine φ 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 x belong to y," "all second-order properties of x belong to y," and so on. Each of these statements implies all its predecessors: for example, if all second-order properties of x belong to y, then all predicates of x belong to y, for to have all the predicates of x is a second-order property, and this property belongs to x. But we cannot, without the help of an axiom, argue conversely that if all the predicates of x belong to y, all the second-order properties of x must also belong to y. Thus we cannot, without the help of an axiom, be sure that x and y 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 x is to be identical with x, and therefore this property would necessarily belong to y if x and y 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 x and y 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 x. This predicate belongs to x, since x is identical with itself; hence it belongs to y, since y has all the predicates of x; hence y is identical with x. It follows that we may define x and y as identical when all the predicates of x belong to y, i.e. when (φ): φ!x . ⊃ . φ!y. We therefore adopt the following definition of identity*²:

 x=y . = : (φ) : φ!x . ⊃ . φ!y  Df.

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 φz^ of whatever order, there is a class a consisting of just those objects which satisfy φz. Hence "φz" is equivalent to "x belongs to a." But "x belongs to a" is a statement containing no apparent variable, and is therefore a predicative function of x. 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 nth order (where n is fixed) which is formally equivalent to φx^, whatever may be the order of φx^. Here we shall mean by "a function of the nth order" a function of the nth order relative to the arguments to φx^; thus if these arguments are absolutely of the mth order, we assume the existence of a function formally equivalent to φx^ whose absolute order is the m + nth. The axiom of reducibility in the form assumed above takes n = 1, but this is not necessary to the use of the axiom. It is also unnecessary that n should be the same for different values of m; what is necessary is that n should be constant so long as m is constant. What is needed is that, where extensional functions of functions are concerned, we should be able to deal with any a-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 a-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*³ is equivalent to a single predicate," i.e. to the assumption that, if we assert that x has all the predicates that satisfy a function ƒ(φ!z^), there is some one predicate which x will have whenever our assertion is true, and will not have whenever it is false, and similarly if we assert that x has some one of the predicates that satisfy a function ƒ(φ!z^). 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.

*¹ When a (finite) set of predicates is given by actual enumeration, their disjunction is a predicate, because no predicate occurs as apparent variable in the disjunction.

*² Note that in this definition the second sign of equality is to be regarded as combining with "Df" to form one symbol; what is defined is the sign of equality not followed by the letters "Df."

*³ Here the combination or disjunction is supposed to be given intensionally. If given extensionally (i.e. by enumeration), no assumption is required; but in this case the number of predicates concerned must be finite.



Pencils: Ramona Fradon Inks: Bob Smith  Super Friends^TM © Respective copyright/trademark holders.

The authors’ approach to the Axiom of Reducibility is curious to my thinking. This axiom is subject to the greatest caution in presentation, and alternative proofs are suggested; and in posts ahead we will find further explanation and justification. I believe that Mr. Russell was uneasy about this axiom on two grounds. On the essential aspects of the system, the metaphysics in thinking and understanding it, this axiom is superfluous, because it is no help in these aspects at all. The value of the axiom is in justifying the symbology by which, in use, one will communicate with notation to a remote reader. Secondly, the axiom of reducibility serves to reduce both labor and length of symbolic notation, and Russell understood the clear validity of this, but at the same time, as a classically trained academic he was very uneasy about the final ambiguity inherent in this aspect.

 I believe that this axiom is entirely palliative for Bertrand Russell’s sense of vulnerability with the underlying ambiguity that permeates the system, and he well understood that he could not be disputed in the metaphysics, because if one gets it they get it; but if they don’t get it, still they have no concrete basis to dispute it. But in the notational aspect it is fairly easy to call him out on the small contradictions that are addressed by the axiom of reducibility. It was not enough that the axiom solved by intension a number of problems inherent in the system, it required justification beyond utility, by extension.

 In terms of hanbloglaka, or more broadly North American Indian thought and perception, this axiom is total diversion, but I am not persuaded that it is without value that I haven’t yet perceived. The essence of shaman knowledge is to see with multiple perspectives without preconceived assumptions, and regarding actions and propositions through to their logical conclusion. Still there is always accepted ambiguity, and in the end being wrong is not worse that being right, the only disgrace is in false pride, or claiming knowledge one does not possess.

Quick-Draw  McGraw^TM © Respective copyright/trademark holders.

Harold R. (Hal) Foster’s Prince Valiant

© Respective copyright/trademark holders.