23. Comic Book Shaman’s Infinite Intelligence Theorem

Harold R. (Hal) Foster’s Prince Valiant

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If R is any relation, the converse of R is the relation which holds between y and x whenever R holds between x and y. Thus greater is the converse of less, before of after, cause of effect, husband of wife, etc. The converse of R is written*¹ CnvʻR   or  R˘.

The definition is   R˘= x^,y^(yRx) Df,

CnvʻR = R˘          Df.

The second of these is not a formally correct definition, since we ought to define "Cnv" and deduce the meaning of CnvʻR. But it is not worthwhile to adopt this plan in our present introductory account, which aims at simplicity rather than formal correctness.

A relation is called symmetrical if R = R, i.e. if it holds between y and x whenever it holds between x and y (and therefore vice versa). Identity, diversity, agreement or disagreement in any respect, are symmetrical relations. A relation is called asymmetrical when it is incompatible with its converse, i.e. when R ⩀ R˘ = L, or, what is equivalent,

     xRy . É_x,y . ~(yRx).

Before and after, greater and less, ancestor and descendant, are asymmetrical, as are all other relations of the sort that lead to series. But there are many asymmetrical relations which do not lead to series; for instance, that of a wife's brother*². A relation may be neither symmetrical nor asymmetrical; for example, this holds of the relation of inclusion between classes: a ⊂ b, and,  b ⊂ a will both be true if a = b, but otherwise only one of them, at most, will be true. The relation brother is neither symmetrical nor asymmetrical, for if x is the brother of y, y may be either the brother or the sister of x.

In the propositional function xRy, we call x the referent and y the relatum. The class x^(xRy), consisting of all the x's which have the relation R to y, is called the class of referents of y with respect to R; the class  x^(xRy), consisting of all the y's to which x has the relation R, is called the class of relata of x with respect to R. These two classes are denoted respectively by Rʻy⃗ and Rʻx ⃖. Thus

      Rʻy⃗ = x^(xRy)  Df,

     Rʻx⃖ = y^(Rx)    Df.

The arrow runs towards y in the first case, to show that we are concerned with things having the relation R to y; it runs away from x in the second case to show that the relation R goes from x to the members of Rʻx. It runs in fact from a referent and towards a relatum.

The notations Rʻy⃗, Rʻx⃖ are very important, and are used constantly. If R is the relation of parent to child, Rʻy = the parents of y, Rʻx = the children of x. We have

          ┠ : x Î Rʻy. ≡ . xRy  

and    ┠ : y Î Rʻx. ≡ . xRy.

These equivalences are often embodied in common language. For example, we say indiscriminately "x is an inhabitant of London" or "x inhabits London." If we put "R" for "inhabits," "x inhabits London" is "x R London," while "x is an inhabitant of London "is " x Î R' London."

Instead of R⃗ and R⃖ we sometimes use sgʻR, gsʻR, where "sg" stands for "sagitta," and "gs" is " sg" backwards. Thus we put

       sgʻR = R⃗     Df,

       gsʻR = R⃖    Df.

These notations are sometimes more convenient than an arrow when the relation concerned is represented by a combination of letters, instead of a single letter such as R. Thus e.g. we should write sgʻ(R ⍝ S), rather than put an arrow over the whole length of (R ⍝ S).

The class of all terms that have the relation R to something or other is called the domain of R. Thus if R is the relation of parent and child, the domain of R will be the class of parents. We represent the domain of R by "DʻR." Thus we put

DʻR = x^{(∃y) . xRy}   Df.

*¹ the second of these notations is taken from Schroder's Algebra und Logik der Relative. R. & W. 3

*² this relation is not strictly asymmetrical, but is so except when the wife's brother is also the sister's husband. In the Greek Church the relation is strictly asymmetrical.

Script: J Thomas Art: W Mortimer, M Esposito © Respective copyright/trademark holders.

 This post is a further step on the path of fine distinction. Recognizing symmetry and
asymmetry in relations is one of the most complex and difficult aspects of our system examined so far. These may or may not affect a given proposition, or propositional equation, but it is helpful to understand that symmetry and asymmetry exists in these aspects, because failure to do so may occasionally lead us to error.

 

As I began studying Principia Mathematica the following proposition immediately leapt out at me as applicable to all human interactions.

        ┠ . (x) . x = x 

or, I assert  “x is true of all values of x = equals x.” or “everything equals itself.” as a definite proposition. This led me to devise my very first propositional equation as follows

given:

     φ = the class of all persons as individuals

      x^ = the expression denoting each individual’s intelligence

     φx^ = expresses the set (x^₁, x^₂, …x^_n) containing the aggregate representing  each human being’s intelligence

      x^ = . _~∞. = each person’s intelligence is less than infinite since each person’s      intelligence is  _~∞ it follows

        ┠ . (x) . φx^ = . _~∞

or my definite assertion that no person can rightly be seen more intelligent than another person. I call this Comic Book Shaman’s Infinite Intelligence theorem. This is a fine tool to demonstrate the essential function of our system. You may agree or disagree with my theorem’s conclusion, you may have made a thousand observations that seem to prove without question that in your experience its opposite must be true. But this assertion, and all assertions that follow are not determined by truth or falsehood; only that they will never be proven in error.

James Childress’ Conchy ©Respective copyright/trademark holders.

Harold R. (Hal) Foster’s Prince Valiant

© Respective copyright/trademark holders.

22. Specific Aspects of Constructing and Using Symbols

Harold R. (Hal) Foster’s Prince Valiant

© Respective copyright/trademark holders

Every use of "(ɿx)(φx),” where it apparently occurs as a constituent of a proposition in the place of an object, is defined in terms of the primitive ideas already on hand. An example of this definition in use is given by the proposition "E ! (ɿx)(φx)" which is considered immediately. The whole subject is treated more fully ahead in Chapter III.

The symbol should be compared and contrasted with "x^(φx)" which in use can always be read as "the x's which satisfy φx." Both symbols are incomplete symbols defined only in use, and as such are discussed in Chapter III. The symbol " x^(φx)" always has an application, namely to the class determined by φx; but "(ɿx)(φx)" only has an application when φx^ is only satisfied by one value of x, neither more nor less. It should also be observed that the meaning given to the symbol by the definition, given immediately below, of E ! (ɿx)(φx). does not presuppose that we know the meaning of "one." This is also characteristic of the definition of any other use of (ɿx)(φx).

  

We now proceed to define "E! (ɿx)(φx)” so that it can be read "the x satisfying φx exists." (It will be observed that this is a different meaning of existence from that which we express by “∃.") Its definition is

    E ! ((ɿx) (φx) . = :  (∃c) : φx . º_x . x =c  Df,

i.e. "the x satisfying φx^ exists" is to mean "there is an object c such that φx is true when x is c but not otherwise."

The following are equivalent forms:

    ┠ :. E! (ɿx) (φx) . º : (∃c) : φc : φx. É . x = c,

    ┠ :. E! (ɿx) (φx) . º : (∃c) . φc : φx φy É_xy .    x = y,

    ┠ :. E! (ɿx) (φx) . º : (∃c) : φc : x ≠ c . É_x .     ~ φx.

The last of these states that "the x satisfying φ^ exists" is equivalent to "there is an object c satisfying φ^, and every object other than c does not satisfy φ^."

The kind of existence just defined covers a great many cases. Thus for example "the most perfect Being exists" will mean:

    (∃c) : x is most perfect. . º_x . x = c,

which, taking the last of the above equivalences, is equivalent to

 (∃c): c is most perfect : x ≠ c . É_x . x is not most perfect.

A proposition such as "Apollo exists" is really of the same logical form, although it does not explicitly contain the word the. For "Apollo" means really "the object having such-and-such properties," say "the object having the properties enumerated in the Classical Dictionary*. "If these properties make up the propositional function φx, then "Apollo" means "(ɿx)(φx)," and "Apollo exists" means "E! "(ɿx)(φx)." To take another illustration, "the author of Waverley" means "the man who (or rather, the object which) wrote Waverley." Thus "Scott is the author of Waverley" is

    Scott = (ɿx)(x wrote Waverley).   

Here (as we observed before) the importance of identity in connection with descriptions plainly appears.

The notation “(ɿx) (φx),” which is long and inconvenient, is seldom used, being chiefly required to lead up to another notation, namely "Rʻy," meaning "the object having the relation R to y." That is, we put

    Rʻy = (ɿx) (xRy)  Df.

The inverted comma may be read "of." Thus "Rʻy" is read "the R of y." Thus if R is the relation of father to son, " Rʻy" means "the father of y"; if R is the relation of son to father, "Rʻy" means "the son of y," which will only "exist" if y has one son and no more. R'y is a function of y, but not a propositional function; we shall call it a descriptive function. All the ordinary functions of mathematics are of this kind, as will appear more fully in the sequel. Thus in our notation, " sin y" would be written " sin ʻy," and "sin" would stand for the relation which sin ʻy has to y. Instead of a variable descriptive function ƒy, we put Rʻy, where the variable relation R takes the place of the variable function ƒ. A descriptive function will in general exist while y belongs to a certain domain, but not outside that domain; thus if we are dealing with positive rationals, √y will be significant if y is a perfect square, but not otherwise; if we are dealing with real numbers, and agree that "√y" is to mean the positive square root (or, is to mean the negative square root), √y will be significant provided y is positive, but not otherwise; and so on.

Thus every descriptive function has what we may call a "domain of definition" or a "domain of existence," which may be thus defined: If the function in question is Rʻy, its domain of definition or of existence will be the class of those arguments y for which we have E! Rʻy, i.e. for which E!(ɿx)(xRy), i.e. for which there is one x, and no more, having the relation R to y.

If R is any relation, we will speak of Rʻy as the "associated descriptive function." A great many of the constant relations which we shall have occasion to introduce are only or chiefly important on account of their associated descriptive functions. In such cases, it is easier (though less correct) to begin by assigning the meaning of the descriptive function, and to deduce the meaning of the relation from that of the descriptive function. This will be explored in the explanations of notation in the next post.

* The same principle applies to many uses of the proper names of existent objects, e.g. to all uses of proper names for objects known to the speaker only by report, and not by personal acquaintance (i.e. the Holy Grail, the North Pole, the elephants graveyard).

Script: Richard Hughes  Pencils and inks: Chic Stone

© Respective copyright/trademark holders.

The above part begins to address the process of constructing symbols that can serve to represent thoughts, expressions, objects, relations, etc. with which we may construct our propositional equations. Aspects of this system appear akin to binary language in nature. In fact we will, in future posts examine the construction of similar even more compact equations by means of ‘the stroke,’ that are again a step towards even more simplification. Clearly this system may be translated into binary code and function on a computer in some fashion; but it would have no useful function. The purpose of this system is to bring order to thought, improve the processing of stored data of the mind and integrate it with new perceptions, in real time. This is accomplished by understanding the nature of this data and systems of combining it in increasingly more efficient ways, with avoiding error as the main priority.

With regard to my thoughts in the last previous post here, on Evident Reality as a logical tool to add equational ‘proof’ to Einstein’s Theories of Special and General Relativity, applying physical and cultural data derived from our shared and collected human experience; by means of Whitehead and Russell’s system of logical philosophy. Below is an initial framing of the equation I laid out this afternoon over three cups of coffee, using mainly elements we have already encountered in previous posts. As we see below, with our system, great complexity begins with a relatively simple set of divisions in which great complexity may be divined.

I summed my scribbling up for my waiter with the quote at the top of the sheet paraphrasing the authors: ‘analysis applied to process equals division. ’

An example of this blogger’s regrettable handwriting.

Scripted (perhaps by): Al Capp  Pencils and Inks: Frank Frazetta  © Respective copyright/trademark holders.

 

 

Harold R. (Hal) Foster’s Prince Valiant

© Respective copyright/trademark holders