Sunday, September 30, 2012

19. Logical Product and Logical Sum of Classes

Harold R. (Hal) Foster’s Prince Valiant
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Both classes and relations have properties analogous to most of those of propositions that result from negation and the logical sum. The logical product of two classes a and b is their common part, i.e. the class of terms which are members of both. This is represented by a Ç b. Thus we put
 
 a Ç b = x^(x є a . x є b)  Df.


This gives us      
        
: x є a Ç b . º . x є a . x є b,

 i.e. "x is a member of the logical product of a and b" is equivalent to the logical product of "x is a member of a" and "x is a member of b."

 

Similarly the logical sum of two classes a and b is the class of terms which are members of either; we denote it by a È b. The definition is
 
a È b = x^(x є a . v . x є b)  Df,


and the connection with the logical sum of propositions is given by
:. x є a È b . º : x є a . v . x є b.


The negation of a class a consists of those terms x for which " x є a " can be significantly and truly denied. We shall find that there are terms of other types for which "x є a" is neither true nor false, but nonsense. These terms are not members of the negation of a.

Thus the negation of a class a is the class of terms of suitable type which are not members of it, i.e. the class x^(x ~ є a). We call this class "—a" (read "not-a"); thus the definition is
 
--a  = x^(x ~є a)  Df,

and the connection with the negation of propositions is given by
 
: x є --a . º . x ~є a.

 In place of implication we have the relation of inclusion. A class a is said to be included or contained in a class b if all members of a are members of b, i.e. if   x є a Éx . x є b. We write a Ì b" for "a is contained in b." Thus we put
 
a Ì b . =:x є a . É. x є b  Df.


Most of the formulae concerning ‘p . q,’ ‘p v q,’ ‘~p,’ ‘p É q’ remain true if we substitute ‘a Ç b,’ ‘a È b,’ ‘--a,’ ‘a Ì b.’ In place of equivalence, we substitute identity; for ‘p º q’ was defined as ‘p É q . q É p,’  but ‘a Ì b . b Ì a’  
gives  
    
 ‘x є a . º . x є b,’  whence  ‘a = b.’

Script: Stan Lee  Pencils Pencils: Jack Kirby Inks: Vince Colletta  The Mighty ThorÔ  ©  
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In considering two strings of ponies ‘p and q,’  p’ having two Pintos and six palominos; and ‘q’ having four Pintos, two Palominos and three Appaloosas. Additionally the p string includes two stallions and six mares while the q string is made of three stallions and six mares.

The class of ponies considered at hand is named:  a;  ø represents the category Pintos;  y represents  the category Palaminos;  χ represents the category Appaloosas; each pony is an individual, having specific variable characteristics (x1 (stallion) , x2  (mare) , x3, …). The above class may be expressed
a Î p { φ(x1, x2) . ψ ( x1 . x2 )} . V .  q { φ(x1, x2) . ψ(x1 . x2) .χ(x1 . x2)}

p ∩ q = x^(φx1 .  φx2  . ψx2 -3)

 or 3 Pintos and 2 Palominos expresses the logical sum, or the ponies in common between the two strings of ponies.

p ∪ q = x^(φx1 .  φx2  . ψ . ψx2  . χ1 . χ2)

  or 5 Pintos, 9 Palominos and 3 Appaloosas expresses the logical product, or total number of both of the two groups.

This example is really only illustrative of the system, and seems sort of shooting a mosquito with an elephant gun, but I intend that it show the utility of thinking in terms of propositional equations. One could continue listing any number of additional x variables describing the individual animals for any purpose desired, and we can see that this allows comparisons and perhaps exclusions much more accurately and concisely than is possible using language alone.

 John Wayne Adventure Comics Pencils: Frank Frazetta Inks: Al Williamson © Respective copyright/trademark holders.

Harold R. (Hal) Foster’s Prince Valiant
© Respective copyright/trademark holders.

Wednesday, September 26, 2012

18. Classes in Symbols

Harold R. (Hal) Foster’s Prince Valiant
© Respective copyright/trademark holders.

In an exactly analogous manner we introduce dual or dyadic relations, i.e. relations between two terms. Such relations will be called simply "relations"; relations between more than two terms will be distinguished as multiple relations, or (when the number of their terms is specified) as triple, quadruple,... relations, or as triadic, tetrad-al,... relations. Such relations will not concern us until we come to Geometry. For the present, the only relations we are concerned with are dual relations.

Relations, like classes, are to be taken in extension, i.e. if R and S are relations which hold between the same pairs of terms, R and S are to be identical. We may regard a relation, in the sense in which it is required for our purposes, as a class of couples; i.e. the couple (x, y) is to be one of the class of couples constituting the relation R if x has the relation R to y*. This view of relations as classes of couples will not, however, be introduced into our symbolic treatment, and is only mentioned in order to show that it is possible, so to understand the meaning of the word relation; that a relation shall be determined by its extension.

Any function φ(x, y) determines a relation R between x and y. If we regard a relation as a class of couples, the relation determined by φ(x, y) is the class of couples (x, y) for which φ(x, y) is true. The relation determined by the function φ(x, y) will be denoted by

x^y^φ (x, y).

We shall use a capital letter for a relation when it is not necessary to specify the determining function. Thus whenever a capital letter occurs, it is to be understood that it stands for a relation. The propositional function "x has the relation R to y" will be expressed by the notation

xRy.

This notation is designed to keep as near as possible to common language, which, when it has to express a relation, generally mentions it between its terms, as in "x loves y," "x equals y," "x is greater than y," and so on. For "relation" we shall write "Rel"; thus "R є Rel" means "R is a relation.

"Owing to our taking relations in extension, we shall have

:. x^y^φ (x, y) = x^y^y (x, y). φ(x,y) . ºx,y y(x,y),

i.e. two functions of two variables determine the same relation when, and only when, the two functions are formally equivalent.

We have . z {x^y^φ (x, y)} w . º . φ (z, w),

i.e. "z has to w the relation determined by the function φ (x, y)" is equivalent to φ (z,w);

:. R = x^y^φ (x, y) º : xRy . ºx,y . φ(x,y),

:. R = S. =: xRy. ºx,y . xSy,

. x^y^(xRy) = R,

. {x^y^φ (x, y)} є Rel.

These propositions are analogous to those previously given for classes. It results from them that any function of two variables is formally equivalent to some function of the form xRy; hence, in extensional functions of two variables, variation of relations can replace variation of functions of two variables.

 
 
* Such a couple has a sense, i.e. the couple (x, y) is different from the couple (y, x), unless x=y. We shall call it a "couple with sense," to distinguish it from the class consisting of x and y. It may also be called an ordered couple.

Pencils and Inks: Jay Scott Pike © Respective copyright/trademark holders.

With relations between classes we reach a new complexity that is so often the cause of plans running of the rails. Whether it is of the wind, the sky, the buffalo herd and an enemy’s intentions, a calculus of variables in the human mind may discern the playing out of events with uncanny accuracy born of an easy understanding and appreciation of a life’s totality in which all critical factors are known.
In today’s dominant culture this all-encompassing perception is certainly elusive, but by means of this logical metaphysical system we are exploring, one may cover all the bases in constructing propositional equations in which we may test our hypotheses in such a manner as to clearly see if they are in error, or indeed nonsense, and perceive the difference.

Milton Caniff’s Terry and the Pirates © Respective copyright/trademark holders.



Harold R. (Hal) Foster’s Prince Valiant
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Friday, September 21, 2012

17. Definitions Which Occur in theTheory of Classes


Harold R. (Hal) Foster’s Prince Valiant
© Respective copyright/trademark holders.

The definitions which occur in the theory of classes, by which the idea of a class (at least in use) is based on the other ideas assumed as primitive, cannot be understood without a fuller discussion than can be given now (cf. Chapter II of this Introduction and also *20). Accordingly, in this preliminary survey, we proceed to state the more important simple propositions which result from those definitions, leaving the reader to employ in his mind the ordinary unanalyzed idea of a class of things. Our symbols in their usage conform to the ordinary usage of this idea in language. It is to be noticed that in the systematic exposition our treatment of classes and relations requires no new primitive ideas and only two new primitive propositions, namely the two forms of the "Axiom of Reducibility" (cf. next Chapter) for one and two variables respectively. The propositional function "x is a member of the class a " will be expressed, following Peano, by the notation

x є a.

Here є is chosen as the initial of the word є’τσί (so). "x є a " may be read " x is an a." Thus "x є man" will mean "x is a man," and so on. For typographical convenience we shall put

x ~ є a . = . ~ (x є a)   Df,

x,y є a . =. x є a . y є a  Df.

For "class" we shall write "Cls"; thus "a є Cls" means" a is a class."

We have-:

: x є z^ (φz). º . φx,

 i.e. "'x is a member of the class determined by φz^' is equivalent to 'x satisfies φz^,' or to 'φx is true.’"

A class is wholly determinate when its membership is known, that is, there cannot be two different classes having the same membership. Thus if φx, yx are formally equivalent functions, they determine the same class; for in that case, if x is a member of the class determined by φx^, and therefore satisfies φx, it also satisfies yx, and is therefore a member of the class determined by yx^. Thus we have

:. z^ (φz) = z^(yz) . º : φx . ºx . yx.

The following propositions are obvious and important:

:. a = z^(φz) . º : x є a . ºx . φx,

i.e. a is identical with the class determined by φz^ when, and only when, "x is an a" is formally equivalent to φx;

:. a = b . º : x є a . ºx . x є b,

i.e. two classes a and b are identical when, and only when, they have the same membership;

. x^(x є a) = a,

i.e. the class whose determining function is "x is an a" is a, in other words, a is the class of objects which are members of a;

. z^(φz) є Cls,

i.e. the class determined by the function φz^ is a class.

It will be seen that, according to the above, any function of one variable can be replaced by an equivalent function of the form "x є a." Hence any extensional function of functions which holds when its argument is a function of the form "z^ є a," whatever possible value a may have, will hold also when its argument is any function φz^. Thus variation of classes can replace variation of functions of one variable in all the propositions of the sort with which we are concerned.


Dan DeCarlo The greatest of all time comic book artist!Archie Andrews Gang   
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Class is kind of a dirty word today in North America, with its connotation of snooty judgments of persons based on seemingly random criteria. In our system here class is one of the keys to building propositions in equations that can be handled and crunched like numbers.    




Script: Gaylord Du Bois  Pencils and Inks: Russ Manning © Respective copyright/trademark holders.



Harold R. (Hal) Foster’s Prince Valiant
© Respective copyright/trademark holders.