28. Possible Arguments for Functions

Harold R. (Hal) Foster’s Prince Valiant

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Possible arguments for functions   

When it is said that e.g. "φ(φz^)" is meaningless, and therefore neither true nor false, it is necessary to avoid a misunderstanding. If “φ(φz^)" were interpreted as meaning "the value for φz^ with the argument φz^ is true," that would be not meaningless, but false. It is false for the same reason for which "the King of France is bald" is false, namely because there is no such thing as "the value for φz^ with the argument φz^." But when, with some argument a, we assert φa we are not meaning to assert "the value for φx^ with the argument a is true"; we are meaning to assert the actual proposition which is the value for φx^ with the argument a. Thus for example if φx^ is "x^ is a man," φ(Socrates) will be "Socrates is a man,” not "the value for the function ‘x^ is a man,' with the argument Socrates, is true." Thus in accordance with our principle that "φ(φz^)" is meaningless, we cannot legitimately deny "the function 'x is a man' is a man," because this is nonsense, but we can  legitimately deny " the value for the function 'x^ is a man' with the argument 'x^ is a man' is true," not on the ground that the value in question is false, but on the ground that there is no such value for the function.

We will denote by the symbol "(x). φx" the proposition " φx always*," i.e. the proposition which asserts all the values for φx^. This proposition involves the function φx^, not merely an ambiguous value of the function. The assertion of φx, where x is unspecified, is a different assertion from the one which asserts all values for φx^ for the former is an ambiguous assertion, whereas the latter is in no sense ambiguous. It will be observed that "(x) . φx" does not assert     "φx with all values of x," because, as we have seen, there must be values of x with which "φx " is meaningless. What is asserted by "(x) . φx" is all propositions which are values for φx^; hence it is only with such values of x as make "φx" significant, i.e. with all possible arguments, that φx is asserted when we assert "(x) . φx." Thus a convenient way to read "(x) . φx" is "φx is  true with all possible values of x." This is, however, a less accurate reading than " φx always," because the notion of truth is not part of the content of what is judged. When we judge "all men are mortal," we judge truly, but the notion of truth is not necessarily in our minds, any more than it need be when we judge "Socrates is mortal."

* We use "always" as meaning "in all cases," not "at all times." Similarly "sometimes" will mean "in some cases."

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Studying Principia Mathematica is as a path to ambiguity. The system that we may learn is individual, and I infer it to be of various use to each who attempts to learn it. It is rater unique because the authors Alfred North Whitehead and Bertrand Russell saw that the system was a vehicle rather than a destination, and that some readers would be defeated, some readers would be right with them, and some readers would continue on with modifications they hadn’t anticipated. Each of the posts up to here, and also those to come, are layers of fine distinction in both thought (perception) and expression, by whatever form it takes. There are aspects, such as the vicious-circle principle, that are challenging to conscious perception. I have experienced great feelings of joy and amusement in reading and re-reading various passages of this text, delighted at the amazing crazy fine distinctions beyond anything I expected.

The nature of taking in our system cannot be subject to conscious recitation, instead a better gauge is to turn back and read the posts again, and when you find that you can breeze thru them easily and understand them as you do a comic strip that is the key to this bus. There are two aspects at play in our text, ‘the system of symbols and their form in use’ and ‘the fine distinctions that are the reasons for the necessity of the system.’ The forms allow the construction of any sort of propositions, functions, objects, classes and more in a form that immediately (visually) reveals any errors within the structure. This is only possible by the systematic application of this metaphysical system.

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27. The Nature of Propositional Functions and Indian's Perception

Harold R. (Hal) Foster’s Prince Valiant

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II.  The Nature of Propositional Functions.

By a "propositional function" we mean something which contains a variable x, and expresses a proposition as soon as a value is assigned to x. That is to say, it differs from a proposition solely by the fact that it is ambiguous: it contains a variable of which the value is unassigned. It agrees with the ordinary functions of mathematics in the fact of containing an unassigned variable: where it differs is in the fact that the values of the function are propositions. Thus e.g. "x is a man" or "sin x = 1 " is a propositional function. We shall find that it is possible to incur a vicious-circle fallacy at the very outset, by admitting as possible arguments to a propositional function terms which presuppose the function. This form of the fallacy is very instructive, and its avoidance leads, as we shall see, to the hierarchy of types. The question as to the nature of a function^†1 is by no means an easy one. It would seem, however, that the essential characteristic of a function is ambiguity. Take, for example, the law of identity in the form "A is A," which is the form in which it is usually enunciated. It is plain that, regarded psychologically, we have here a single judgment. But what are we to say of the object of the judgment? We are not judging that Socrates is Socrates, nor that Plato is Plato, nor any other of the definite judgments that are instances of the law of identity. Yet each of these judgments is, in a sense, within the scope of our judgment. We are in fact judging an ambiguous instance of the propositional function " A is A." We appear to have a single thought which does not have a definite object, but has as its object an undetermined one of the values of the function "A is A." It is this kind of ambiguity that constitutes the essence of a function. When we speak of "φx," where x is not specified, we mean one value of the function, but not a definite one. We may express this by saying that "φx" ambiguously denotes φa, φb, φc, etc., where φa, φb, φc, etc., are the various values of "φx."

When we say that "φx" ambiguously denotes φa, φb, φc, etc., we mean that  “φx" means one of the objects φa, φb, φc, etc., though not a definite one, but an undetermined one. It follows that "φx" only has a well-defined meaning (well-defined, that is to say, except in so far as it is of its essence to be ambiguous) if the objects φa, φb, φc, etc., are well-defined. That is to say, a function is not a well-defined function unless all its values are already well defined. It follows from this that no function can have among its values anything which presupposes the function, for if it had, we could not regard the objects ambiguously denoted by the function as definite until the function was definite, while conversely, as we have just seen, the function cannot be definite until its values are definite. This is a particular case, but perhaps the most fundamental case, of the vicious-circle principle. A function is what ambiguously denotes some one of a certain totality, namely the values of the function; hence this totality cannot contain any members which involve the function, since, if it did, it would contain members involving the totality, which, by the vicious-circle principle, no totality can do.

It will be seen that, according to the above account, the values of a function are presupposed by the function, not vice versa. It is sufficiently obvious, in any particular case, that a value of a function does not presuppose the function. Thus for example the proposition "Socrates is human" can be perfectly apprehended without regarding it as a value of the function "x is human." It is true that, conversely, a function can be apprehended without its being necessary to apprehend its values severally and individually. If this were not the case, no function could be apprehended at all, since the number of values (true and false) of a function is necessarily infinite and there are necessarily possible arguments with which we are unacquainted. What is necessary is not that the values should be given individually and extensionally, but that the totality of the values should be given intentionally, so that, concerning any assigned object, it is at least theoretically determinate whether or not the said object is a value of the function.

It is necessary practically to distinguish the function itself from an undetermined value of the function. We may regard the function itself as that which ambiguously denotes, while an undetermined value of the function is that which is ambiguously denoted. If the undetermined value is written "φx," we will write the function itself "φx^." (Any other letter may be used in place of x.) Thus we should say " φx is a proposition," but "φx^ is a propositional function." When we say "φx is a proposition," we mean to state something which is true for every possible value of x, though we do not decide what value x is to have. We are making an ambiguous statement about any value of the function. But when we say "φx^ is a function," we are not making an ambiguous statement. It would be more correct to say that we are making a statement about an ambiguity, taking the view that a function is an ambiguity. The function itself, φx^ is the single thing which ambiguously denotes its many values; while φx, where x is not specified, is one of the denoted objects, with the ambiguity belonging to the manner of denoting.

We have seen that, in accordance with the vicious-circle principle, the values of a function cannot contain terms only definable in terms of the function. Now given a function φx^, the values for the function^†2 are all pro positions of the form φx. It follows that there must be no propositions, of the form φx, in which x has a value which involves φx^. (If this were the case, the values of the function would not all be determinate until the function was determinate, whereas we found that the function is not determinate unless its values are previously determinate.) Hence there must be no such thing as the value for φx^ with the argument φx^, or with any argument which involves φx. That is to say, the symbol "φ(φx^)" must not express a proposition, as " φa" does if φa is a value for φx^. In fact "φ(φx^)" must be a symbol which does not express anything: we may therefore say that it is not significant. Thus given any function φx^, there are arguments with which the function has no value, as well as arguments with which it has a value. We will call the arguments with which φx^ has a value "possible values of x." We will say that φx^ is "significant with the argument x" when φx^ has a value with the argument x.

^†1 When the word "function" is used in the sequel, "propositional function" is always meant. Other functions will not be in question in the present Chapter.

^†2 We shall speak in this Chapter of "values for φx^" and of "values of φx," meaning in each case the same thing, namely φa, φb, φc, etc. The distinction of phraseology serves to avoid ambiguity where several variables are concerned, especially when one of them is a function.

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This examination of propositional functions is so important in understanding our system, but also the intent of the system. In our culture we seek to define all of our surroundings; and then live within these understandings, using them as a shield against the unknown. It is the way of Indian peoples to exist comfortably in ambiguity; where a decision is made ad hoc, only as it is called for, without pre-conceived supposition about every person or situation encountered.

Like all metaphysical systems, the details are of little use as memorized and repeated on a conscious level. Their value is as absorbed into our being, and our understanding intentional within the ambiguity that is all of life. Although all of dominant culture seeks rules and laws, principles and moral judgments of what is right and wrong in rigid structures that apply to all persons in all circumstances. The problem with this is that these rigid structures are always only approximate at best. Formulas of agreed upon equations that are compromises, because ‘truths’ are mostly impossible to agree upon or to pin down. Inevitably the intended results are subverted to the forms invented to define them. It is always the case that a definition is not the object defined, and vice versa.

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26. The Vicious-Circle Principle & The theory of logical types



Harold R. (Hal) Foster’s Prince Valiant

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CHAPTER II.

THE THEORY OF LOGICAL TYPES

The theory of logical types, to be explained in the present Chapter, recommended itself to us in the first instance by its ability to solve certain contradictions, of which the one best known to mathematicians is Burali-Forti's concerning the greatest ordinal. But the theory in question is not wholly dependent upon this indirect recommendation: it has also a certain consonance with common sense which makes it inherently credible. In what follows, we shall therefore first set forth the theory on its own account, and then apply it to the solution of the contradictions.

I.  The Vicious-Circle Principle.

An analysis of the paradoxes to be avoided shows that they all result from a certain kind of vicious circle. The vicious circles in question arise from supposing that a collection of objects may contain members which can only be defined by means of the collection as a whole. Thus, for example, the collection of propositions will be supposed to contain a proposition stating that "all propositions are either true or false." It would seem, however, that such a statement could not be legitimate unless "all propositions" referred to some already definite collection, which it cannot do if new propositions are created by statements about "all propositions." We shall, therefore, have to say that statements about "all propositions' are meaningless. More generally, given any set of objects such that, if we suppose the set to have a total, it will contain members which presuppose this total, then such a set cannot have a total. By saying that a set has "no total," we mean, primarily, that no significant statement can be made about "all its members." Propositions, as the above illustration shows, must be a set having no total. The same is true, as we shall shortly see, of propositional functions, even when these are restricted to such as can significantly have as argument a given object a. In such cases, it is necessary to break up our set into smaller sets, each of which is capable of a total. This is what the theory of types aims at effecting.

The principle which enables us to avoid illegitimate totalities may be stated as follows: "Whatever involves all of a collection must not be one of the collection"; or, conversely: "If, provided a certain collection had a total, it would have members only definable in terms of that total, then the said collection has no total." We shall call this the "vicious-circle principle," because it enables us to avoid the vicious circles involved in the assumption of illegitimate totalities. Arguments which are condemned by the vicious-circle principle will be called "vicious-circle fallacies." Such arguments, in certain circumstances, may lead to contradictions, but it often happens that the conclusions to which they lead are in fact true, though the arguments are fallacious. Take, for example, the law of excluded middle, in the form "all propositions are true or false." If from this law we argue that, because the law of excluded middle is a proposition, therefore the law of excluded middle is true or false, we incur a vicious-circle fallacy. "All propositions" must be in some way limited before it becomes a legitimate totality, and any limitation which makes it legitimate must make any statement about the totality fall outside the totality.

Similarly, the imaginary skeptic, who asserts that he knows nothing, and is refuted by being asked if he knows that he knows nothing, has asserted nonsense, and has been fallaciously refuted by an argument which involves a vicious-circle fallacy. In order that the skeptic's assertion may become significant, it is necessary to place some limitation upon the things of which he is asserting his ignorance, because the things of which it is possible to be ignorant form an illegitimate totality. But as soon as a suitable limitation has been placed by him upon the collection of propositions of which he is asserting his ignorance, the proposition that he is ignorant of every member of this collection must not itself be one of the collection. Hence any significant skepticism is not open to the above form of refutation.

The paradoxes of symbolic logic concern various sorts of objects: propositions, classes, cardinal and ordinal numbers, etc. All these sorts of objects, as we shall show, represent illegitimate totalities, and are therefore capable of giving rise to vicious-circle fallacies. But by means of the theory (to be explained in Chapter III) which reduces statements that are verbally concerned with classes and relations to statements that are concerned with propositional functions, the paradoxes are reduced to such as are concerned with propositions and propositional functions. The paradoxes that concern propositions are only indirectly relevant to mathematics, while those that more nearly concern the mathematician are all concerned with propositional functions. We shall therefore proceed in the next post to the consideration of propositional functions.

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The vicious-circle principle goes very far to explain so many of society’s greatest problems; and especially those that seek to place people into categories convenient to political, economic, religious or criminal purposes. Reading this post through a few times is useful to begin to see how the principle applies in the world you live in. It helps to see ahead, to contradictions and to error in lines of thought; to help decide which are worth pursuing, and which should be abandoned.

Right and wrong, true and false are subjective parameters in framing any complex proposition. It can easily be seen that with these forming the basis of any proposition, the measure of such will alter in small or great degree in each individual’s estimation. Therefore right and wrong, true and false or as are normally held to be ‘moral’ considerations are dangerous to take into account in propositional equations, because they will inevitably lead to error or ‘nonsense’ that is, propositions that cannot be proven true or false, but which inevitably lead to error.

I would not care to end this post with an impression that faith, religion, belief in God or any devotion is wrong and a certain source of error. This is not the case, and indeed without faith no one would endeavor to learn and practice a system such as Whitehead and Russell set forth in Principia Mathematica. Only by clarity do we see that thoughts contain errors that we cannot trace, and most often they arise from vicious-circle fallacies.  

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25. Plural descriptive functions * end of chapter I

 

Harold R. (Hal) Foster’s Prince Valiant

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Plural descriptive functions. The class of terms x which have the relation R to some member of a class a is denoted by Rʻʻa or R_єʻa. The definition is

Rʻʻa = x {(∃y). y є a . xRy} Df.

Thus for example let R be the relation of inhabiting, and a the class of towns; then Rʻʻa = inhabitants of towns. Let R be the relation "less than" among rationals, and a the class of those rationals which are of the form 1 - 2^-n, for integral values of n; then Rʻʻa will be all rationals less than some member of a, i.e. all rationals less than 1. If P is the generating relation of a series, and a is any class of members of the series, Pʻʻa will be predecessors of a's, i.e. the segment defined by a. If P is a relation such that Pʻy always exists when y є a, P"a will be the class of all terms of the form Pʻy for values of y which are members of a; i.e.

Pʻʻa = x^{(∃y) . y є a . x = Pʻy}.

Thus a member of the class "fathers of great men" will be the father of y, where y is some great man. In other cases, this will not hold; for instance, let P be the relation of a number to any number of which it is a factor; then Pʻʻ (even numbers) = factors of even numbers, but this class is not composed of terms of the form "the factor of x," where x is an even number, because numbers do not have only one factor apiece.

Unit classes. The class whose only member is x might be thought to be identical with x, but Peano and Frege have shown that this is not the case. (The reasons why this is not the case will be explained in a preliminary way in Chapter II of the Introduction.) We denote by " ɿʻx "the class whose only member is x: thus

ɿʻx = y^(y=x)  Df,

i.e. " ɿʻx " means "the class of objects which are identical with x."

The class consisting of x and y will be ɿʻx ◡ ɿʻx; the class got by adding x to a class a will be a ◡ ɿʻx; the class got by taking away x from a class a will be    a- ɿʻx. (We write a - b as an abbreviation for a ◠ -b.)

It will be observed that unit classes have been defined without reference to the number 1; in fact, we use unit classes to define the number 1. This number is defined as the class of unit classes,

1 = a^ {(∃x). a = ɿʻx }  Df.

This leads to

┠ :. a є 1 . ≡ : (∃x): y є a . ≡_y . y = x.

From this it appears further that

┠ : a є 1 . ≡ . E! (ɿʻx) (x є a),

whence ┠ : z^(φz) є 1 . ≡. E! (ɿʻx) (φx),

i.e. "z^(φz) is a unit class" is equivalent to "the x satisfying φx^ exists."

If a є 1, ɿ^⏑ʻa is the only member of a, for the only member of a is the only term to which a has the relation ɿ. Thus "ɿʻa" takes the place of “(ɿʻx) (φx)," if a stands for z^(φz). In practice, "ɿʻa" is a more convenient notation than “(ɿʻx) (φx)," and is generally used instead of “(ɿʻx) (φx).”

The above account has explained most of the logical notation employed in the present work. In the applications to various parts of mathematics, other definitions are introduced; but the objects defined by these later definitions belong, for the most part, rather to mathematics than to logic. The reader who has mastered the symbols explained above will find that any later formulae can be deciphered by the help of comparatively few additional definitions.

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Throughout the previous 24 posts we have seen so many symbols, equations and definitions, but in all there was not a single number used in any notation until finally the number 1 is elaborately defined here at the end of chapter one of the introduction of Principia Mathematica. I can’t help reading a bit of tongue-in-cheek in the author’s putting this definition of 1 at the end of this chapter.

With the tools we have seen so far we have the bare bones of the process, and may be able to begin thinking of and ordering our outlines and propositional equations that are immediately more useful and quantifiable than before. In the posts ahead we shall see many of layers of fine distinction added to our present study.

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