23. Comic Book Shaman’s Infinite Intelligence Theorem

Harold R. (Hal) Foster’s Prince Valiant

© Respective copyright/trademark holders.

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.