Wilsonian Quantifiers

          Wilsonian Quantifiers

          The science-fiction writer Robert Anton Wilson invented a word, sumbunol, meaning “some but not all”. He did so as a way to combat the temptation to over-generalize.

Sumbunol’s formal definition is:

          Sbn(x)(P(x))   =    Exists(x)(P(x))  and  Exists(y)(not P(y))

                                 =       Exists(x,y)( P(x) xor P(y) )

          The last equation says; “sumbunol things are P” is equivalent to “P differs on some two things”. So sumbunol = varying.

          The negation of sumbunol is nunnerol, meaning “none or all”, with this formal definition:

          Nrl(x)(P(x))        =       All(x)(P(x))  or  All(y)(not P(y))

                                      =       All(x,y)( P(x) iff P(y) )

          The last equation says; “nunnerol things are P” is equivalent to “P is the same for any two things”.  So nunnerol = constant.

          These equations hold:

          Negation:

          Sbn(x)(P(x))  =  Sbn(x)(not P(x))  =  not Nrl(x)(P(x))

          Nrl(x)(P(x))  =  Nrl(x)(not P(x))  =  not Sbn(x)(P(x))

          Partial Distribution:  

A and Sbn(x)(P(x))   =  Sbn(x)( A and P(x) )

          A or Nrl(x)(P(x))   =  Nrl(x)( A or P(x) )

          “And” distributes over sumbunol, and “or” distributes over nunnerol; but “and” does not distribute over nunnerol; nor does “or” distribute over sumbunol:

True  or  Sbn(P(x)) = True;   but Sbn( True or P(x) ) = False.

False and Nrl(P(x)) = False;  but  Nrl( False and P(x) ) = True.

          Sumbunol and nunnerol have these Equivalence Rules:

Nrl(x)( P(x) iff Q(x) )    and     Nrl(x)(Q(x) iff R(x))

Implies       Nrl(x)(P(x) iff R(x))    
If “P iff Q” and “Q iff R” are constant, then “P iff R” is constant.

Nrl(x)( P(x) iff Q(x) )    and     Nrl(x)(Q(x))

Implies       Nrl(x)(P(x))       

If “P iff Q” is constant, and Q is constant, then P is constant.

                   Sbn(x)(P(x))  and  Nrl(x)(Q(x))     

implies   Sbn(x)( P(x) iff Q(x) )

          If P varies and Q is constant, then “P iff Q” varies.  

Nrl(x)( P(x) iff Q(x) )

                   Implies       Nrl(x)(P(x))   iff    Nrl(x)(Q(x))    

          If “P iff Q” is constant, then P and Q are equally constant.

                   Sbn(x)(P(x))   xor   Sbn(x)(Q(x))  

implies       Sbn(x)( P(x) xor Q(x) )

If P varies or else Q varies, then “P or else Q” varies.

          If F(p,q) is any function on Boolean logic, then

                   Nrl(x)(P(x))   and   Nrl(x)(Q(x))

                   Implies       Nrl(x) ( F(P(x),Q(x)) 
     This is “Constancy”: constant inputs imply a constant output.

If F(p,q) is any function on Boolean logic, then

                   Sbn(x) ( F(P(x),Q(x)) )

                   Implies       Sbn(x)(P(x))   or   Sbn(x)(Q(x))
     This is “Variability”: varying output implies a varying input.

                    For all a and b,

                   Nrl(x)(P(x)) and P(a)            

Implies       P(b)

          This is “Proof By Constancy Plus Example”.

                   For all a and b,

                   P(a)    and    not P(b)             

implies       Sbn(x)(P(x))

          This is “Opposing Examples”.

                   Nrl(x)(P(x))   and   Exists(x)(P(x))

implies                 All(x)(P(x))

          This is “Constancy and Existence implies Universality”.

                   Exists(x)(P(x))   

implies                 Sbn(x)(P(x))   or   All(x)(P(x))

          This is “Existence implies Variation or Universality”.

         

          The reverse implications require that the universe of discourse of the quantifiers be not empty; i.e. that something exists:  Exist(x)(x=x)

          Given this assumption, then the above implications are equations.

          If  Exist(x)(x=x),  then
              All(x)(P(x))        iff     Nrl(x)(P(x))   and   Exists(x)(P(x))

              Exists(x)(P(x))   iff     Sbn(x)(P(x))   or   All(x)(P(x)) 
           If anything exists, then    
                          universality = constancy and existence

                and    existence = variation or universality