34. Simple Matrices Form the Hierarchy of Functions and Propositions

Harold R. (Hal) Foster’s Prince Valiant

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The first matrices that occur are those whose values are of the forms

     φx, y(x, y) c(x, y, z...)

i.e. where the arguments, however many there may be, are all individuals. The functions φ, y, c..., since (by definition) they contain no apparent variables, and have no arguments except individuals, do not presuppose any totality of functions. From the functions y, c... we may proceed to form other functions of x, such as (y) . y (x, y), (∃y) . y(x, y), (y, z) . c(x, y, z), (y) : (∃z) . c(x, y, z), and so on. All these presuppose no totality except that of individuals. We thus arrive at a certain collection of functions of x, characterized by the fact that they involve no variables except individuals. Such functions we will call "first-order functions."

We may now introduce a notation to express "any first-order function." We will denote any first-order function by "φ!x^" and any value for such a function by

"φ!x." Thus "φ!x" stands for any value for any function which involves no variables except individuals. It will be seen that "φ!x" is itself a function of two variables, namely φ!z^ and x. Thus φ!x involves a variable which is not an individual, namely φ!z^. Similarly " (x) . φ!x" is a function of the variable φ!z^, and thus involves a variable other than an individual. Again, if a is a given individual,

     "φ!x implies φ!a with all possible values of φ"

is a function of x, but it is not a function of the form φ!x, because it involves an (apparent) variable φ which is not an individual. Let us give the name "predicate" to any first-order function φ!x^. (This use of the word "predicate" is only proposed for the purposes of the present discussion.) Then the statement  "φ!x implies φ!a with all possible values of φ" may be read "all the predicates of x are predicates of a." This makes a statement about x, but does not attribute to x a predicate in the special sense just defined.

Owing to the introduction of the variable first-order function φ!z^, we now have a new set of matrices. Thus "φ!x" is a function which contains no apparent variables, but contains the two real variables φ!z^ and x. (It should be observed that when φ is assigned, we may obtain a function whose values do involve individuals as apparent variables, for example if φ!x is (y) . y (x, y). But so long as φ is variable, φ!x contains no apparent variables.) Again, if a is a definite individual, φ!a is a function of the one variable φ!z^. If a and b are definite individuals, " φ!a^ implies y!b^" is a function of the two variables φ!z^, y!z^, and so on. We are thus led to a whole set of new matrices,

     ƒ(φ!z^), g(φ!z^, y!z^), F(φ!z^, x), and so on.

These matrices contain individuals and first-order functions as arguments, but (like all matrices) they contain no apparent variables. Any such matrix, if it contains more than one variable, gives rise to new functions of one variable by turning all its arguments except one into apparent variables. Thus we obtain the functions

     (φ) . g (φ!z^, y!z^), which is a function of y!z^.

     (x) . F(φ!z^, x), which is a function of φ!z^.

     (φ). F(φ!z^, x), which is a function of x.

We will give the name of second-order matrices to such matrices as have first-order functions among their arguments, and have no arguments except first-order functions and individuals. (It is not necessary that they should have individuals among their arguments.) We will give the name of second-order functions to such as either are second-order matrices or are derived from such matrices by turning some of the arguments into apparent variables. It will be seen that either an individual or a first-order function may appear as argument to a second-order function. Second-order functions are such as contain variables which are first-order functions, but contain no other variables except (possibly) individuals.

We now have various new classes of functions at our command. In the first place, we have second-order functions which have one argument which is a first-order function. We will denote a variable function of this kind by the notation ƒ!(φ^!z^), and any value of such a function by ƒ!(φ!z^). Like φ!x,    ƒ!(φ!z^) is a function of two variables, namely ƒ!(φ^!z^) and φ!z^. Among possible values of      ƒ!(φ!x^) will be φ!a (where a is constant), (x) . φ!x,    (∃x) . φ!x, and so on. (These result from assigning a value to ƒ, leaving φ to be assigned.) We will call such functions "predicative functions of first-order functions."

In the second place, we have second-order functions of two arguments, one of which is a first-order function while the other is an individual. Let us denote undetermined values of such functions by the notation

     ƒ!(φ!z^, x).

As soon as x is assigned, we shall have a predicative function of φ!z^. If our function contains no first-order function as apparent variable, we shall obtain a predicative function of x if we assign a value to φ!z^. Thus, to take the  simplest possible case, if ƒ!(φ!z^, x) is φ!x, the assignment of a value to φ gives us a predicative function of x, in virtue of the definition of "φ!x." But if   ƒ!(φ!z^, x) contains a first-order function as apparent variable, the assignment of a value to φ!z^ gives us a second-order function of x.

In the third place, we have second-order functions of individuals. These will all be derived from functions of the form ƒ!(φ!z^, x) by turning φ into an apparent variable. We do not, therefore, need a new notation for them.

We have also second-order functions of two first-order functions, or of two such functions and an individual, and so on.

We may now proceed in exactly the same way to third-order matrices, which will be functions containing second-order functions as arguments, and containing no apparent variables, and no arguments except individuals and first-order functions and second-order functions. Thence we shall proceed, as before, to third-order functions; and so we can proceed indefinitely. If the highest order of variable occurring in a function, whether as argument or as apparent variable, is a function of the nth order, then the function in which it occurs is of the n + 1th order. We do not arrive at functions of an infinite order, because the number of arguments and of apparent variables in a function must be finite, and therefore every function must be of a finite order. Since the orders of functions are only defined step by step, there can be no process of "proceeding to the limit," and functions of an infinite order cannot occur.

We will define a function of one variable as predicative when it is of the next order above that of its argument, i.e. of the lowest order compatible with its having that argument. If a function has several arguments, and the highest order of function occurring among the arguments is the nth, we call the function predicative if it is of the n + 1th order, i.e. again, if it is of the lowest order compatible with its having the arguments it has. A function of several arguments is predicative if there is one of its arguments such that, when the other arguments have values assigned to them, we obtain a predicative function of the one undetermined argument.

It is important to observe that all possible functions in the above hierarchy can be obtained by means of predicative functions and apparent variables. Thus, as we saw, second-order functions of an individual x are of the form

     (φ) . ƒ!(φ!z^, x) or (∃φ) . ƒ!(φ!z^, x) or

          (φ, y) . ƒ!(φ!z^, y!z^, x) or etc.,

where ƒ is a second-order predicative function. And speaking generally, a non-predicative function of the nth order is obtained from a predicative function of the nth order by turning all the arguments of the n - 1th order into apparent variables. (Other arguments also may be turned into apparent variables.) Thus we need not introduce as variables any functions except predicative functions. Moreover, to obtain any function of one variable x, we need not go beyond predicative functions of two variables. For the function (y) . ƒ!(φ!z^, φ!z^, x), where ƒ is given, is a function of φ!z^ and x, and is predicative. Thus it is of the form F!(φ!z^, x), and therefore (φ, y) . ƒ!(φ!z^, y!z^, x) is of the form  (φ) . F!(φ!z^, x). Thus speaking generally, by a succession of steps we find that, if φ!u^ is a predicative function of a sufficiently high order, any assigned non-predicative function of x will be of one of the two forms

     (φ) . F(φ!u^, x), (∃φ) . F!(φ!u^, x),

where F is a predicative function of φ!u^ and x.

The nature of the above hierarchy of functions may be restated as follows. A function, as we saw at an earlier stage, presupposes as part of its meaning the totality of its values, or, what comes to the same thing, the totality of its possible arguments. The arguments to a function may be functions or propositions or individuals. (It will be remembered that individuals were defined as whatever is neither a proposition nor a function.) For the present we neglect the case in which the argument to a function is a proposition. Consider a function whose argument is an individual. This function presupposes the totality of individuals; but unless it contains functions as apparent variables, it does not presuppose any totality of functions. If, however, it does contain a function as apparent variable, then it cannot be defined until some totality of functions has been defined. It follows that we must first define the totality of those functions that have individuals as arguments and contain no functions as apparent variables. These are the predicative functions of individuals. Generally, a predicative function of a variable argument is one which involves no totality except that of the possible values of the argument, and those that are presupposed by any one of the possible arguments. Thus a predicative function of a variable argument is any function which can be specified without introducing new kinds of variables not necessarily presupposed by the variable which is the argument.

A closely analogous treatment can be developed for propositions. Propositions which contain no functions and no apparent variables may be called elementary propositions. Propositions which are not elementary, which contain no functions, and no apparent variables except individuals, may be called first-order propositions. (It should be observed that no variables except apparent variables can occur in a proposition, since whatever contains a real variable is a function, not a proposition.) Thus elementary and first-order propositions will be values of first-order functions. (It should be remembered that a function is not a constituent in one of its values: thus for example the function "x^ is human" is not a constituent of the proposition "Socrates is human.") Elementary and first-order propositions presuppose no totality except (at most) the totality of individuals. They are of one or other of the three forms

     φ!x; (x) . φ!x; (∃x) . φ!x,

where φ!x is a predicative function of an individual. It follows that, if p represents a variable elementary proposition or a variable first-order proposition, a function ƒp is either ƒ(φ!x) or ƒ{(x) . φ!x} or {(∃x). φ!x}. Thus a function of an elementary or a first-order proposition may always be reduced to a function of a first-order function. It follows that a proposition involving the totality of first-order propositions may be reduced to one involving the totality of first-order functions; and this obviously applies equally to higher orders. The propositional hierarchy can, therefore, be derived from the functional hierarchy, and we may define a proposition of the nth order as one which involves an apparent variable of the n - 1th order in the functional hierarchy. The propositional hierarchy is never required in practice, and is only relevant for the solution of paradoxes; hence it is unnecessary to go into further detail as to the types of propositions. 





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“Well it ain’t no use to sit and wonder why babe.

If yuh don’t know by now.”

-Bob Dylan

The matrix is the womb, the rock where precious minerals are extracted; and here basis for the construction of the Hierarchy of Functions and Propositions. This very long post text above is deceiving and intimidating, in that what meaning it conveys is dwarfed here by the authors’ explanation of that meaning’s expression and notation. Or how it may be discussed and written. These distinctions are well understood by all shamans and used exclusively among themselves and our spirit helpers. Such use of language for us is merely intensional, and rarely spoken aloud, only in oblique indication for points of clarity. In our system the authors had no thought or intention to address or define native American Indian thought or perception, this is clear by the systems structure, and the use that they chose to demonstrate its practical extensional use; the principle text of the work Principia Mathematica.

The authors' intention was to create a metaphysical end run past language to a notational symbolic system that can convey extreme complexity in a clear, compact manner and without error. Ironically once the system is assimilated clearly in a reader's mind, her perception alters to a state where language regains useful function. Unfortunately it works only one way, just as one understands everything involving language, expression, intension and aspiration in all cultures, she can only respond obliquely and mysteriously; or by means of artistic expression to be understood. This should become clearer in posts to follow.

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Harold R. (Hal) Foster’s Prince Valiant

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33. The Hierarchy of Functions and Propositions.

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Harold R. (Hal) Foster’s Prince Valiant

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V. The Hierarchy of Functions and Propositions.

We are thus led to the conclusion, both from the vicious-circle principle and from direct inspection, that the functions to which a given object a can be an argument are incapable of being arguments to each other, and that they have no term in common with the functions to which they can be arguments. We are thus led to construct a hierarchy. Beginning with a and the other terms which can be arguments to the same functions to which a can be argument*, we come next to functions to which a is a possible argument, and then to functions to which such functions are possible arguments, and so on. But the hierarchy which has to be constructed is not as simple as might at first appear. The functions which can take a as argument form an illegitimate totality, and themselves require division into a hierarchy of functions. This is easily seen as follows. Let ƒ(φz^, x) be a function of the two variables φz^ and x. Then if, keeping x fixed for the moment, we assert this with all possible values of φ, we obtain a proposition:

     (φ) . ƒ(φz, x).

Here, if x is variable, we have a function of x; but as this function involves a totality of values of φz^*², it cannot itself be one of the values included in the totality, by the vicious-circle principle. It follows that the totality of values of φz^ concerned in (φ) . ƒ(φz^, x) is not the totality of all functions in which x can occur as argument, and that there is no such totality as that of all functions in which x can occur as argument.

It follows from the above that a function in which φz^ appears as argument requires that " φz^" should not stand for any function which is capable of a given argument, but must be restricted in such a way that none of the functions which are possible values of "φz^" should involve any reference to the totality of such functions. Let us take as an illustration the definition of identity. We might attempt to define "x is identical with y" as meaning "whatever is true of x is true of y," i.e. "φx always implies φy." But here, since we are concerned to assert all values of "φx implies φy" regarded as a function of φ, we shall be compelled to impose upon φ some limitation which will prevent us from including among values of φ values in which "all possible values of φ" are referred to. Thus for example "x is identical with a" is a function of x; hence, if it is a legitimate value of φ in "φx always implies φy," we shall be able to infer, by means of the above definition, that if x is identical with a, and x is identical with y, then y is identical with a. Although the conclusion is sound, the reasoning embodies a vicious-circle fallacy, since we have taken "(φ) . φx implies φa" as a possible value of φx, which it cannot be. If, however, we impose any limitation upon φ, it may happen, so far as appears at present, that with other values of φ we might have φx true and φy false, so that our proposed definition of identity would plainly be wrong. This difficulty is avoided by the "axiom of reducibility," to be explained later. For the present, it is only mentioned in order to illustrate the necessity and the relevance of the hierarchy of functions of a given argument.

Let us give the name "a-functions" to functions that are significant for a given argument a. Then suppose we take any selection of a-functions, and consider the proposition "a satisfies all the functions belonging to the selection in question." If we here replace a by a variable, we obtain an a-function; but by the vicious-circle principle this a-function cannot be a member of our selection, since it refers to the whole of the selection. Let the selection consist of all those functions which satisfy ƒ(φz^). Then our new function is

      (φ). {ƒ(φz^) implies φx},

where x is the argument. It thus appears that, whatever selection of a-functions we may make, there will be other a-functions that lie outside our selection. Such a-functions, as the above instance illustrates, will always arise through taking a function of two arguments, φz^ and x, and asserting all or some of the values resulting from varying φ. What is necessary, therefore, in order to avoid vicious-circle fallacies, is to divide our a-functions into " types," each of which contains no functions which refer to the whole of that type.

When something is asserted or denied about all possible values or about some (undetermined) possible values of a variable, that variable is called apparent, after Peano. The presence of the words all or some in a proposition indicates the presence of an apparent variable; but often an apparent variable is really present where language does not at once indicate its presence. Thus for example "A is mortal" means "there is a time at which A will die." Thus a variable time occurs as apparent variable.

The clearest instances of propositions not containing apparent variables are such as express immediate judgments of perception, such as "this is red" or " this is painful," where "this" is something immediately given. In other judgments, even where at first sight no variable appears to be present, it often happens that there really is one. Take (say) "Socrates is human." To Socrates himself, the word "Socrates" no doubt stood for an object of which he was immediately aware, and the judgment "Socrates is human" contained no apparent variable. But to us, who only know Socrates by description, the word "Socrates" cannot mean what it meant to him; it means rather "the person having such-and-such properties," (say) "the Athenian philosopher who drank the hemlock." Now in all propositions about "the so-and-so" there is an apparent variable, as will be shown in Chapter III. Thus in what we have in mind when we say "Socrates is human" there is an apparent variable, though there was no apparent variable in the corresponding judgment as made by Socrates, provided we assume that there is such a thing as immediate awareness of oneself.

Whatever may be the instances of propositions not containing apparent variables, it is obvious that propositional functions whose values do not contain apparent variables are the source of propositions containing apparent variables, in the sense in which the function φx^ is the source of the proposition (x) . φx. For the values for φx^ do not contain the apparent variable x, which appears in (x) . φx; if they contain an apparent variable y, this can be similarly eliminated, and so on. This process must come to an end, since no proposition which we can apprehend can contain more than a finite number of apparent variables, on the ground that whatever we can apprehend must be of finite complexity. Thus we must arrive at last at a function of as many variables as there have been stages in reaching it from our original proposition, and this function will be such that its values contain no apparent variables. We may call this function the matrix of our original proposition and of any other propositions and functions to be obtained by turning some of the arguments to the function into apparent variables. Thus for example, if we have a matrix-function whose values are φ(x, y), we shall derive from it

      (y). φ(x, y), which is a function of x,

      (x). φ(x, y), which is a function of y,

      (x, y). φ(x, y), meaning " φ(x, y) is true with all possible values of x and y."

This last is a proposition containing no real variable, i.e. no variable except apparent variables.

It is thus plain that all possible propositions and functions are obtainable from matrices by the process of turning the arguments to the matrices into apparent variables. In order to divide our propositions and functions into types, we shall, therefore, start from matrices, and consider how they are to be divided with a view to the avoidance of vicious-circle fallacies in the definitions of the functions concerned. For this purpose, we will use such letters as a, b, c, x, y, z, w, to denote objects which are neither propositions nor functions. Such objects we shall call individuals. Such objects will be constituents of propositions or functions, and will be genuine constituents, in the sense that they do not disappear on analysis, as (for example) classes do, or phrases of the form "the so-and-so."

* Cf. Chapter III.

*²When we speak of "values of φz^" it is φ, not z, that is to be assigned. This follows from the explanation in the note on post (link)27. When the function itself is the variable, it is possible and simpler to write φ rather than φz^, except in positions where it is necessary to emphasize that an argument must be supplied to secure significance.

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 Just a few days ago I awoke from a dream and marveled at it. In the dream I was talking to some people, and realized that I knew what I was talking about. I thought it was just a joke with the punchline “I woke up and in the dream I knew what I was talking about.” I saw the humor but could not connect the meaning to the truth of the humor until the dream I had today.
I have been puzzling about what to write about today’s post. I knew it is important, but how do I express its importance in words? My dream today gave me clarity on this question, and it lies in the relation of variables and apparent variables. In the statement “The first part of each and every previous post that is quoted from Principia Mathematica is definitively without error and therefore correct.” This is a definite statement that obviously contains no ‘real’ variables. Still, one may infer two apparent variables in the previous sentence, 1) future posts may contain error in the first part of the post; 2) other parts of the previous posts are or may not be without error. Although this example seems obscure, in some form it arises in every use of language beyond simple declarative sentences. The element of time acts to degrade everything and language structure is not exempt from the effect. This points out the meaning of my dream: not that I knew what I was saying, but that others understood my meaning.



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32. Why a Given Function requires Arguments of a Certain Type

Harold R. (Hal) Foster’s Prince Valiant

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Why a Given Function requires Arguments of a Certain Type



The considerations so far adduced in favor of the view that a function cannot significantly have as argument anything defined in terms of the function itself have been more or less indirect. But a direct consideration of the kinds of functions which have functions as arguments and the kinds of functions which have arguments other than functions will show, if we are not mistaken, that not only is it impossible for a function φz^ to have itself or anything derived from it as argument, but that, if yz^ is another function such that there are arguments a with which both "φa" and "ya" are significant, then yz^ and anything derived from it cannot significantly be argument to φz^. This arises from the fact that a function is essentially an ambiguity, and that, if it is to occur in a definite proposition, it must occur in such a way that the ambiguity has disappeared, and a wholly unambiguous statement has resulted. A few illustrations will make this clear. Thus "(x) . φx," which we have already considered, is a function of φx; as soon as φx^ is assigned, we have a definite proposition, wholly free from ambiguity. But it is obvious that we cannot substitute for the function something which is not a function: "(x) . φx" means "φx in all cases," and depends for its significance upon the fact that there are "cases" of φx, i.e. upon the ambiguity which is characteristic of a function. This instance illustrates the fact that, when a function can occur significantly as argument, something which is not a function cannot occur significantly as argument. But conversely, when something which is not a function can occur significantly as argument, a function cannot occur significantly. Take, e.g. "x is a man," and consider "φx^ is a man." Here there is nothing to eliminate the ambiguity which constitutes φx^; there is thus nothing definite which is said to be a man. A function, in fact, is not a definite object, which could be or not be a man; it is a mere ambiguity awaiting determination, and in order that it may occur significantly it must receive the necessary determination, which it obviously does not receive if it is merely substituted for something determinate in a proposition*. This argument does not, however, apply directly as against such a statement as "{(x) . φx} is a man." Common sense would pronounce such a statement to be meaningless, but it cannot be condemned on the ground of ambiguity in its subject. We need here a new objection, namely the following: A proposition is not a single entity, but a relation of several; hence a statement in which a proposition appears as subject will only be significant if it can be reduced to a statement about the terms which appear in the proposition. A proposition, like such phrases as "the so-and-so," where grammatically it appears as subject, must be broken up into its constituents if we are to find the true subject or subjects*². But in such a statement as "p is a man," where p is a proposition, this is not possible. Hence "{(x). φx)} is a man" is meaningless.

* Note that statements concerning the significance of a phrase containing "φx^" concern the symbol "φz^," and therefore do not fall under the rule that the elimination of the functional ambiguity is necessary to significance. Significance is a property of signs.

*² C.f. Chapter III

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This small patch of ‘mental driver’ is deceptively simple, but serves as a giant step towards comprehension of some aspects of the system that we have, until now largely had to accept on face value. Speaking for myself, the rules regarding the use of functions, and the broad restrictions of the vicious-circle principle seemed to be very restrictive and cumbersome obstacles that will arise at every turn. This post goes very far to shrinking that field by providing important specific arguments to guide us in using propositions, and seeing more specifically how functions fit into useful calculations.

Still this post is mainly perceived in the abstract, and our comfort with its ambiguity at this point may be indicative of our confidence in Principia Mathematica. To study this system calls for a leap of faith, for even as the rational perception of value may take hold early, there is so much that must be absorbed, with no clear application beyond ambiguity, in sight. This is a trip into the shaman mind. Very likely anyone reading this work is interested from a philosophical field of study. I suspect few mathematicians in the post war era ever bothered to pursue the intensional aspects of the system, instead seeking some arcane advantage by perusing an obscure text that everyone knew of, but few had actually read.

 

Two world wars changed the field of mathematics radically; from a few Cambridge professors corresponding with their counterparts in Berlin or Rome with papers of pure and universal mathematics. War, industry and big profits changed the Mathematicscape in every way imaginable, so that by 1950, the mind of the mathematician was so far removed from that resembling Bertrand Russell’s struggling to finish developing our system from 1905-1910, as to be seen as completely divergent in both process and intent.

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