55. The Logical Product Of Two Propositions

Harold R. (Hal) Foster’s Prince Valiant

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Story and Art: John Stanley and Irving Tripp  Little Lulu 

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Illustration: Meyers - 1912 Outdoor Life & Indian Stories

Harold R. (Hal) Foster’s Prince Valiant

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Big Chief Western  Feb 1941

I am a bit slow sometimes. After reading this material over and over I have finally realized what has been bothering me. I understand the material easily on a visceral level, but the descriptions, and most especially the proofs built in by the authors are a great labor for me to digest, and I have been feeling frustrated by this. Now after so many re-readings I managed to penetrate the systems order of proofs; though I am sure it is clear enough to someone with a more mathematical bend of mind.

Still even as I am back to enjoying this material again, I do have to take a bit of time to digest it before making each future post, and it seems for now about every two weeks will be the best I can manage.

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Depictions of North American Indians In Popular Culture

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Northwest Coast Salish Boy circa 1900. © Respective copyright holders.

Artist Unknown (possibly Assiniboin) Man on Horseback Shooting A Woman
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Circa 1885 © Respective copyright holders.

Stephan Mopope (Kiowa) Dancer With Bustle 1929
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Spicy Western Stories Mar 1937.  In the depths of the great depression our Indian brothers joined the ranks of gangsters, spies, Oriental arch-fiends and any other menace to womanhood in Spicy pulp magazines. © Respective copyright holders.

Dagwood Comics 35 Oct 1953 - In these days of television's infancy, all kids played cowboys and Indians from time to time; some were left with stereo-typical perceptions while others more sympathetic sought out more truthful ideas. © Respective copyright holders.

Old Nick Of The Swamp 40 Beadle's Frontier Series 1908 Pulp Magazine. Popular depictions of Indians such as this from this time period are often particularly vicious. Here the white cowboy lassoes the fleeing 'Buck' with a noose, causing him to drop his rifle and scalps. Odds are  the writer and artist never saw the West or an Indian in their lives. © Respective copyright holders.

Red Wolf 1 May 1971 (Gil Kane/John Severin cover art) - At this time the comics publishers were trying out superheroes of all ethnic flavors.  © Respective copyright holders.



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42. The Distinction Between Intensional and Extensional Functions

Harold R. (Hal) Foster’s Prince Valiant

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A function of a function is called intensional when it is not extensional.

The nature and importance of the distinction between intensional and extensional functions will be made clearer by some illustrations. The proposition "'x is a man' always implies 'x is a mortal'" is an extensional function of the function "x^ is a man," because we may substitute, for "x is a man, "'x is a featherless biped," or any other statement which applies to the same objects to which "x is a man "applies, and to no others. But the proposition "A believes that 'x is a man' always implies 'x is a mortal"' is an intensional function of "x^ is a man," because A may never have considered the question whether featherless bipeds are mortal, or may believe wrongly that there are featherless bipeds which are not mortal. Thus even if "x is a featherless biped" is formally equivalent to "x is a man," it by no means follows that a person who believes that all men are mortal must believe that all featherless bipeds are mortal, since he may have never thought about featherless bipeds, or have supposed that featherless bipeds were not always men. Again, the proposition "the number of arguments that satisfy the function φ!z^ is n" is an extensional function of φ!^, because its truth or falsehood is unchanged if we substitute for φ!z^ any other function which is true whenever φ!z^ is true, and false whenever φ!z^ is false. But the proposition "A asserts that the number of arguments satisfying φ!z^ is n" is an intensional function of φ!z^, since, if A asserts this concerning φ!z, he certainly cannot assert it concerning all predicative functions that are equivalent to φ!z^, because life is too short. Again, consider the proposition "two white men claim to have reached the North Pole." This proposition states "two arguments satisfy the function “x^ is a white man who claims to have reached the North Pole." The truth or falsehood of this proposition is unaffected if we substitute for "x^ is a white man who claims to have reached the North Pole" any other statement which holds of the same arguments, and of no others. Hence it is an extensional function. But the proposition "it is a strange coincidence that two white men should claim to have reached the North Pole," which states "it is a strange coincidence that two arguments should satisfy the function 'x^ is a white man who claims to have reached the North Pole,’" is not equivalent to "it is a strange coincidence that two arguments should satisfy the function 'x^ is Dr. Cook or Commander Peary."' Thus "it is a strange coincidence that φ!z^ should be satisfied by two arguments" is an intensional function of φ!z^.

The above instances illustrate  the fact that the functions of functions with which mathematics is especially concerned are extensional, and that intensional functions of functions only occur where non-mathematical ideas are introduced, such as what somebody believes or affirms, or the emotions aroused by some fact. Hence it is natural, in a mathematical logic, to lay special stress on extensional functions of functions.

When two functions are formally equivalent, we may say that they have the same extension. In this definition, we are in close agreement with usage. We do not assume that there is such a thing as an extension: we merely define the whole phrase "having the same extension." We may now say that an extensional function of a function is one whose truth or falsehood depends only upon the extension of its argument. In such a case, it is convenient to regard the statement concerned as being about the extension. Since extensional functions are many and important, it is natural to regard the extension as an object, called a class, which is supposed to be the subject of all the equivalent statements about various formally equivalent functions. Thus e.g. if we say "there were twelve Apostles," it is natural to regard this statement as attributing the property of being twelve to a certain collection of men, namely those who were Apostles, rather than as attributing the property of being satisfied by twelve arguments to the function "x^ was an Apostle." This view is encouraged by the feeling that there is something which is identical in the case of two functions which "have the same extension." And if we take such simple problems as "how many combinations can be made of n things?" it seems at first sight necessary that each "combination" should be a single object which can be counted as one. This, however, is certainly not necessary technically, and we see no reason to suppose that it is true philosophically. The technical procedure by which the apparent difficulty is overcome is as follows.

We have seen that an extensional function of a function may be regarded as a function of the class determined by the argument-function, but that an intensional function cannot be so regarded. In order to obviate the necessity of giving different treatment to intensional and extensional functions of functions, we construct an extensional function derived from any function of a predicative function y!z^, and having the property of being equivalent to the function from which it is derived, provided this function is extensional, as well as the property of being significant (by the help of the systematic ambiguity of equivalence) with any argument φ^ whose arguments are of the same type as those of y!z^. The derived function, written "ƒ{z^(φz)},"is defined as follows: Given a function ƒ(y!z^), our derived function is to be "there is a predicative function which is formally equivalent to φz^ and satisfies ƒ." If φz^ is a predicative function, our derived function will be true whenever ƒ(φz^) is true. If ƒ(φz^)  is an extensional function, and φz^ is a predicative function, our derived function will not be true unless ƒ(φz^) is true; thus in this case, our derived function is equivalent to ƒ(φz^). If ƒ(φz^) is not an extensional function, and if φz^ is a predicative function, our derived function may sometimes be true when the original function is false. But in any case the derived function is always extensional.

In order that the derived function should be significant for any function φz^, of whatever order, provided it takes arguments of the right type, it is necessary and sufficient that ƒ(y!z^) should be significant, where y!z^ is any predicative function. The reason of this is that we only require, concerning an argument φz^, the hypothesis that it is formally equivalent to some predicative function y!z^, and formal equivalence has the same kind of systematic ambiguity as to type that belongs to truth and falsehood, and can therefore hold between functions of any two different orders, provided the functions take arguments of the same type. Thus by means of our derived function we have not merely provided extensional functions everywhere in place of intensional functions, but we have practically removed the necessity for considering differences of type among functions whose arguments are of the same type. This effects the same kind of simplification in our hierarchy as would result from never considering any but predicative functions.

If ƒ(y!z^) can be built up by means of the primitive ideas of disjunction, negation, (x) . φx, and (∃x) . φx, as is the case with all the functions of functions that explicitly occur in the present work, it will be found that, in virtue of the systematic ambiguity of the above primitive ideas, any function φz^  whose arguments are of the same type as those of y!z^ can significantly be substituted for y!z^  in ƒ without any other symbolic change. Thus in such a case what is symbolically, though not really, the same function ƒ can receive as arguments functions of various different types. If, with a given argument φz, the function ƒ(φ!z^), so interpreted, is equivalent to ƒ(y!z^) whenever y!z^ is formally equivalent to φz^, then ƒ{z^(φz)} is equivalent to ƒ(φz^) provided there is any predicative function formally equivalent to φz^. At this point, we make use of the axiom of reducibility, according to which there always is a predicative function formally equivalent to φz^. 

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 We tend to believe mathematical problems have reliable answers that are provable and constant, but that more human questions involving beliefs, emotions, speculations, etc., are a completely different matter. In fact, although the propositional equations we are learning to construct may seem a bit more complex than typical math, they are actually similar; and plainly behave in the same ways, and reveal any errors clearly.

 

Once again these aspects are only in small part about understanding the system, or thinking like an Indian does. These ending posts more directly go towards communicating complex thought clearly to others, and a systematic means of notation. As I have said above I feel that by the time most persons reach an understanding of the system we find that we no longer require the purpose we set out to satisfy. This is because if one understands the system well at every step, she changes in her perception of dominant culture, common use of language, the process of metaphysics, and all of life as it goes on around us. If one is very strongly attached to a stake in dominant culture, earning money as a means of living life, such as responsibility for supporting family or debt; I am not certain that she can learn this system at all, or would even consider investing the time it requires.


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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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