19. Logical Product and Logical Sum of Classes

Harold R. (Hal) Foster’s Prince Valiant

© Respective copyright/trademark holders.

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 É_x . 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  ©  

Respective copyright/trademark holders.

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^(φx_{1 .}  φx_{2  .} 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^(φx_{1 .}  φx_{2  .} 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.