Thursday, December 20, 2012

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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Monday, December 17, 2012

41. The Scope of Propositions and Classes in Notation



Harold R. (Hal) Foster’s Prince Valiant
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In the case when the smallest proposition enclosed in dots or other brackets contains two or more descriptions, we shall assume, in the absence of any indication to the contrary, that one which typographically occurs earlier has a larger scope than one which typographically occurs later. Thus

     (x)(φx) = (x)(yx)

will mean  (c) : φx . ≡x . x = c : [(x)(yx)] . c = (x)(yx),

while    (x)(yx) = (x)(φx)

will mean (d) : yx . ≡x . x = d : [(x)(φx)] . (x)(φx) = d.

These two propositions are easily shown to be equivalent.

 (2) Classes. The symbols for classes, like those for descriptions, are, in our system, incomplete symbols: their uses are defined, but they themselves are not assumed to mean anything at all. That is to say, the uses of such symbols are so defined that, when the definiens is substituted for the definiendum, there no longer remains any symbol which could be supposed to represent a class. Thus classes, so far as we introduce them, are merely symbolic or linguistic conveniences, not genuine objects as their members are if they are individuals.

 It is an old dispute whether formal logic should concern itself mainly with intensions or with extensions. In general, logicians whose training was mainly philosophical have decided for intensions, while those whose training was mainly mathematical have decided for extensions. The facts seem to be that, while mathematical logic requires extensions, philosophical logic refuses to supply anything except intensions. Our theory of classes recognizes and reconciles these two apparently opposite facts, by showing that an extension (which is the same as a class) is an incomplete symbol, whose use always acquires its meaning through a reference to intension.

In the case of descriptions, it was possible to prove that they are incomplete symbols. In the case of classes, we do not know of any equally definite proof, though arguments of more or less cogency can be elicited from the ancient problem of the One and the Many*1. It is not necessary for our purposes, however, to assert dogmatically that there are no such things as classes. It is only necessary for us to show that the incomplete symbols which we introduce as representatives of classes yield all the propositions for the sake of which classes might be thought essential. When this has been shown, the mere principle of economy of primitive ideas leads to the non-introduction of classes except as incomplete symbols.

To explain the theory of classes, it is necessary first to explain the distinction between extensional and intensional functions. This is seen in effect 'by the following definitions:

The truth-value of a proposition is truth if it is true, and falsehood if it is false. (This expression is due to Frege.)

Two propositions are said to be equivalent when they have the same truth-value, i.e. when they are both true or both false.

Two propositional functions are said to be formally equivalent when they are equivalent with every possible argument, i.e. when any argument which satisfies the one satisfies the other, and vice versa. Thus "x^ is a man" is formally equivalent to " x^  is a featherless biped"; "x^ is an even prime" is formally equivalent to "x^ is identical with 2."

A function of a function is called extensional when its truth-value with any argument is the same as with any formally equivalent argument. That is to say, ƒ(φz^) is an extensional function of φz^ if, provided yz^ is formally equivalent to φz^, ƒ(φz^) is equivalent to ƒ(yz^). Here the apparent variables φ and y are necessarily of the type from which arguments can significantly be supplied to ƒ. We find no need to use as apparent variables any functions of non-predicative types; accordingly in the sequel all extensional functions considered are in fact functions of predicative functions*2.

*1 Briefly, these arguments reduce to the following: If there is such an object as a class, it must be in some sense one object. Yet it is only of classes that many can be predicated. Hence, if we admit classes as objects, we must suppose that the same object can be both one and many, which seems impossible.

*2 Cf post 34.


Story and Art: John Stanley and Irving Tripp  Little Lulu © Respective copyright/trademark holders.
Here the authors expand on ways of recognizing and reconciling the seemingly opposite natures of mathematics and philosophy. As with all intractable differences, reconciliation can come without changing a problem, or even by compromise for that matter. The trick to working things out is to change our perception out the issues involved. Looking around for aspects that agree together can be a road to seeing a formal equivalence of value. So many obstacles arise in simple misunderstandings from the imprecise use of language, even after multiple attempts at clarification.

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Harold R. (Hal) Foster’s Prince Valiant
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Saturday, December 8, 2012

40. Defeating Ambiguity by Notation



Harold R. (Hal) Foster’s Prince Valiant

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The proposition "a = (x)(φx)" is easily shown to be equivalent to "φx . ≡x . x = a." For, by the definition, it is
         (c) : φx . ≡x. x = c : a = c,
i.e. "there is a c for which φx . ≡x . x = c, and this c is a," which is equivalent to "φx. ≡x. x = a." Thus "Scott is the author of Waverley" is equivalent to:
 "'x wrote Waverley' is always equivalent to 'x is Scott,’"
i.e. "x wrote Waverley" is true when x is Scott and false when x is not Scott.
Thus although “(x)(φx)" has no meaning by itself, it may be substituted for y in any propositional function fy, and we get a significant proposition, though not a value of fy.
When ƒ{(x)(φx)}, as above defined, forms part of some other proposition, we shall say that (x)(φx) has a secondary occurrence. When (x)(φx) has a  secondary occurrence, a proposition in which it occurs may be true even when (x)(φx) does not exist. This applies, e.g. to the proposition: "There is no such person as the King of France." We may interpret this as
          ~{E!(x)(φx)},
or as      ~{(c) . c = (x)(φx)},
if "φx" stands for "x is King of France." In either case, what is asserted is that a proposition p in which (x)(φx) occurs is false, and this proposition p is thus part of a larger proposition. The same applies to such a proposition as the following: "If France were a monarchy, the King of France would be of the House of Orleans."
It should be observed that such a proposition as
         ~ƒ{(x)(φx)}
is ambiguous; it may deny ƒ{(x)(φx)}, in which case it will be true if  (x)(φx) does not exist, or it may mean
         (c) : φx . ≡x. x = c : ~ƒc,
in which case it can only be true if (x)(φx) exists. In ordinary language, the latter interpretation would usually be adopted. For example, the proposition "the King of France is not bald" would usually be rejected as false, being held to mean "the King of France exists and is not bald," rather than "it is false that the King of France exists and is bald." When (x)(φx) exists, the two interpretations of the ambiguity give equivalent results; but when (x)(φx) does not exist, one interpretation is true and one is false. It is necessary to be able to distinguish these in our notation; and generally, if we have such propositions as,
         y(x)(φx). . p,
         p . . y(x)(φx),
         y (x)(φx) . . c (x)(φx),
and so on, we must be able by our notation to distinguish whether the whole or only part of the proposition concerned is to be treated as the "ƒ(x)(φx)" of our definition. For this purpose, we will put "[(x)(φx)]" followed by dots at the beginning of the part (or whole) which is to be taken as ƒ(x)(φx), the dots  being sufficiently numerous to bracket off the ƒ(x)(φx); i.e. ƒ(x)(φx) is to be everything following the dots until we reach an equal number of dots not signifying a logical product, or a greater number signifying a logical product, or the end of the sentence, or the end of a bracket enclosing "[(x)(φx)]."
Thus
           [(x)(φx)]. y(x)(φx) . . p,
 will mean      (c) : ≡x . x = c : yc: . p,
but       [(x)(φx)] : y(x)(φx)) . . p
will mean      (c): φx . ≡x . x = c : yc . . p.
It is important to distinguish these two, for if (x)(φx) does not exist, the first is true and the second false.  Again
         [(x)(φx) . ~ y(x)(φx)
will mean       (c) : φx . ≡x. x = c : ~ yc,
while           ~{[(x)(φx)] . y(x)(φx)}
will mean       ~{(c) : φx . ≡x . x = c : yc}.
Here again, when (x)(φx)  does not exist, the first is false and the second true.
 In order to avoid this ambiguity in propositions containing (x)(φx), we amend our definition, or rather our notation, putting
[(x)(φx)] . ƒ(x)(φx) . = : (c) : φx . ≡x . x = c : ƒc  Df.
By means of this definition, we avoid any doubt as to the portion of our whole asserted proposition which is to be treated as the "ƒ(x)(φx)" of the definition. This portion will be called the scope of (x)(φx).   Thus in
        [(x)(φx)] . ƒ(x)(φx) . . p
the scope of     (x)(φx) is ƒ(x)(φx);  but in
        [(x)(φx)] : ƒ(x)(φx) . . p
the scope is (x)(φx) . . p;
in      ~{[(x)(φx)] . ƒ(x)(φx)}   
the scope is ƒ(x)(φx) but in
        [(x)(φx)] . ~ ƒ(x)(φx)
the scope is       ~ƒ(x)(φx).
It will be seen that when (x)(φx) has the whole of the proposition concerned for its scope, the proposition concerned cannot be true unless E!(x)(φx); but when (x)(φx) has only part of the proposition concerned for its scope, it may often be true even when (x)(φx) does not exist. It will be seen further that  when E!(x)(φx), we may enlarge or diminish the scope of (x)(φx) as much as we please without altering the truth-value of any proposition in which it occurs.
If a proposition contains two descriptions, say (x)(φx) and (x)(yx), we have to distinguish which of them has the larger scope, i.e. we have to distinguish  
(1) [(x)(φx)] : [(x)(yx)] . ƒ{(x)(φx), (x)(yx)},
(2) [(x)(yx)] :  [(x)(φx)] . ƒ{(x)(φx), (x)(yx)}.  
  The first of these, eliminating (x)(φx), becomes
(3) (c) :.  φx . ≡x . x = c : [(x)(yx)]. ƒ{c, (x)(yx)},
which, eliminating (x)(yx), becomes
(4) (c):. φx . ≡x . x = c :. (d) : yxx . x = c : ƒ(c, d),
and the same proposition results if, in (1), we eliminate first (x)(yx) and then (x)(φx). Similarly (2) becomes, when (x)(φx)  and (x)(yx) are eliminated,
  (5) (d) :. yx. ≡x . x = d :. (c) :. φx . ≡x . x = c : ƒ(c, d).
(4) and (5) are equivalent, so that the truth-value of a proposition containing two descriptions is independent of the question which has the larger scope. It will be found that, in most cases in which descriptions occur, their scope is, in practice, the smallest proposition enclosed in dots or other brackets in which they are contained. Thus for example
[(x)(φx)] . y (x)(φx) . . [(x)(φx)] . c (x)(φx)
will occur much more frequently than
    [(x)(φx)] : y (x)(φx) . . c(x)(φx).
For this reason it is convenient to decide that, when the scope of an occurrence of (x)(φx)  is the smallest proposition, enclosed in dots or other brackets, in   which the occurrence in question is contained, the scope need not be indicated by "[(x)(φx)]." Thus e.g.
             p . . a = (x)(φx)
will mean   p . . (x)(φx)] . a = (x)(φx);
and           p. . (a) . a = (x)(φx)
will mean      p . . (a) [(x)(φx)] . a = (x)(φx);
and            p. . a ≠ (x)(φx)
will mean    p . . [(x)(φx)] . ~ {a = (x)(φx)};
but             p . . ~ {a = (x)(φx)}
will mean    p . ~ {[(x)(φx)] . a = (x)(φx)}.
This convention enables us, in the vast majority of cases that actually occur, to dispense with the explicit indication of the scope of a descriptive symbol; and it will be found that the convention agrees very closely with the tacit conventions of ordinary language on this subject. Thus for example, if "(x)(φx)” is "the so-and-so," "a ≠ (x)(φx)" is to be read "a is not the so-and-so," which would ordinarily be regarded as implying that "the so-and-so" exists; but "~{a = (x)(φx)}" is to be read "it is not true that a is the so-and-so," which would generally be allowed to hold if "the so-and-so" does not exist. Ordinary language is, of course, rather loose and fluctuating in its implications on this matter; but subject to the requirement of definiteness, our convention seems to keep as near to ordinary language as possible.

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This post digs in and offers real clarity to the ambiguity in everyday language. We can begin to see here the structure to conversation and written language’s undoing. These aspects are mostly or completely unknown in Indian languages in North America. It is only in the shaman spirit aspect that metaphysical thought or communication is required; these served the function of healing, sciences, religion, agriculture and even a very useful and developed sort of computer technology very similar to memory based techniques used by the greatest engineers, architects, artists and clerics in the West for centuries.

 
Indian cultures were concerned with life, a life connected to the land and the seasons and human concerns in which physical possessions were minor, trivial matters to most people. Society structure was complex and tribal to be sure; but everyone had their place, and these roles were nearly always voluntary and clearly evident. In these societies languages were rudimentary by Western standards, but they held, and still hold great variety and complexity for native speakers. They are brilliantly functional because they are entirely separate from the metaphysical, grammatical and rhetorical aspects that are the province of Chiefs and shamans. A final important aspect to help understand how this worked is to recall that these cultures were non-material and that money was unknown.


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Harold R. (Hal) Foster’s Prince Valiant

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