On Wilsonian Quantifiers

          On Wilsonian Quantifiers

          The science-fiction writer Robert Anton Wilson invented a word, sumbunol, meaning “some but not all”. He did so to combat the temptation to over-generalize. Sumbunol’s formal definitions are:

         

                             Sumbunol(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”. Sumbunol = varying; so another symbol or sumbunol is “Var”: Var(x)(P(x))  =  P varies.

          The negation of sumbunol is ollerno, meaning “all or no”, with these formal definitions:

         

                             Ollerno(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; “ollerno things are P” is equivalent to “P is the same for any two things”.  Ollerno = constant; so another symbol for ollerno is “Con”: Con(x)(P(x)) = P is constant.

Given a universe of discourse with two elements {a,b}, and a predicate P(x) on {a,b}, then:

                   All(x)(P(x))  =  P(a) and P(b)

                   Some(x)(P(x))  =  P(a) or P(b)

                   No(x)(P(x))  =  not (P(a) or P(b))

                   NotAll(x)(P(x))  =  not(P(a) and P(b))

                   Con(x)(P(x))  =  P(a) iff P(b)

                   Var(x)(P(x))  =  P(a) xor P(b)

          So these six quantifiers correspond to the six non-constant commutative boolean functions on two inputs.

If the universe of discourse has only one element {a}, then:

                   All(P)  =  Some(P) = P(a)

                   NotAll(P) = No(P)  =  not P(a)

                   Var(P)  =  F

                   Con(P)  =  T

          And if the universe of discourse is empty, then:

                   All(P)  =  T

Some(P) =   F

NotAll(P) = F

No(P)  =  T

                   Var(P)  =  F

                   Con(P)  =  T

Over a larger universe of discourse {a1, a2, a3…}, define:

All(x)(P(x))  =  P(a1) and P(a2) and P(a3) and …

Some(x)(P(x))  =  P(a1) or P(a2) or P(a3) or …

No(x)(P(x))  =  not ( P(a1) or P(a2) or P(a3) or … )

                   NotAll(x)(P(x))  =  not ( P(a1) and P(a2) and P(a3) and … )

                   Con(x)(P(x))  = All(x)(P(x)) or No(x)(P(x))

                   Var(x)(P(x))  = Some(x)(P(x)) and NotAll(x)(P(x))

          In general these equations hold:

          Negation:

          Var(x)(P(x))  =  Var(x)(not P(x))  =  not Con(x)(P(x))

          Con(x)(P(x))  =  Con(x)(not P(x))  =  not Var(x)(P(x))

          Partial Distribution:  

A and Var(x)(P(x))   =  Var(x)( A and P(x) )

          A or Con(x)(P(x))   =  Con(x)( A or P(x) )

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

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

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

          Sumbunol and ollerno have these Equivalence Rules:

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

Implies       Con(x)(P(x) iff R(x))  

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

Con(x)( P(x) iff Q(x) )    and     Con(x)(Q(x))

Implies       Con(x)(P(x))      

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

                   Var(x)(P(x))  and  Con(x)(Q(x))   

implies   Var(x)( P(x) iff Q(x) )

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

Con(x)( P(x) iff Q(x) )

                   Implies       Con(x)(P(x))   iff    Con(x)(Q(x)) 

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

                   Var(x)(P(x))   xor   Var(x)(Q(x))  

implies       Var(x)( P(x) xor Q(x) )

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

          Sumbunol and ollerno also have two Functionality Rules:

         

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

                   Con(x)(P(x))   and   Con(x)(Q(x))

                   Implies       Con(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

                   Var(x) ( F(P(x),Q(x)) )

                   Implies       Var(x)(P(x))   or   Var(x)(Q(x))

          This is “Variability”: varying output implies a varying input.

          Here is “Proof By Constancy Plus Example”:

                   For all a and b,

                   Con(x)(P(x))    and    P(a)              

Implies       P(b)

          Here are “Variation By Opposing Examples”:

                   For all a and b,

                   P(a)    and    not P(b)             

implies       Var(x)(P(x))

          Here is “Constancy and Existence implies Universality”:

                   Con(x)(P(x))   and   Exists(x)(P(x))        

implies                 All(x)(P(x))

          Here is “Existence implies Variation or Universality”:

                   Exists(x)(P(x))   

implies                 Var(x)(P(x))   or   All(x)(P(x))

         

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

          If  Exist(x)(x=x),  then

All(x)(P(x))         iff      Con(x)(P(x))   and   Exists(x)(P(x))        

                   Exists(x)(P(x))    iff      Var(x)(P(x))   or   All(x)(P(x))

No(x)(P(x))         iff      NotAll(x)(P(x))   and   Con(x)(P(x))      

NotAll(x)(P(x))  iff      Var(x)(P(x))   or  No(x)(P(x))       

Var(x)(P(x))        iff      Some(x)(P(x))   and   NotAll(x)(P(x))    

                   Con(x)(P(x))       iff      All(x)(P(x))   or   No(x)(P(x))

          If anything exists, then  

universality = constancy and existence

                    existence = variation or universality

                   nonexistence = exceptions and constancy

                   exceptions = variability or nonexistence

                   variability = existence and exceptions

                   constancy  =  universality or nonexistence

          Here is a Wilsonian Quantifier Troika:

Moe: No frogs are princes.

Larry: Some but not all frogs are princes.

Curly: All frogs are princes.

Moe, Larry and Curly all agree that frogs exist.

By 2/3 majorities each, we get this Wilsonian Trilemma:

LK: Some frogs are princes.

ML: Some frogs are not princes.

KM: All or no frogs are princes.

The last can be read, “All frogs are equally princes.”

          Any two of a trilemma imply the negation of the third. Therefore:

If some frogs are princes, and some frogs are not princes, then sumbunal frogs are princes.

If some frogs are not princes, and ollerno frogs are princes, then no frogs are princes.

If ollerno frogs are princes, and some frogs are princes, then all frogs are princes.

In general a Wilsonian trilemma has the form:

Some A have property P;

Some A do not have property P;

All As have property P equally.

For instance:

Some men are good;

Some men are not good;

All men are equally good.

         

The trilemma implies these three deduction rules:

If some men are good, and some men are not good,

then not all men are equally good.

If some men are not good, and all men are equally good,

          then no men are good.

If all men are equally good, and some men are good,

          then all men are good.

          Here are Wilsonian versions of mathematical induction:

                    All(n)( P(n) iff P(n+1) )       =      Con(n)( P(n) )

                   Var(n)( P(n) )                        =      Some(n)( P(n) xor P(n+1) )

          On the integers, the iffs and xors of ollerno and sumbunol need only be between elements separated by adding one. The integers are deductively linked by succession.

          In nonstandard analysis, where there are infinitesimal quantities, you can express the intermediate value theorem in Wilsonian terms:

          If f(x) is continuous on [a,b], and i is any infinitesimal, then

          Con(x)( f(x)>0 )           =       All(x) ( f(x)>0   iff  f(x+i)>0 )

          f’s sign is constant if it is constant under any infinitesimal change.

          Var(x)( f(x)>0 )           =       Some(x) ( f(x)>0   xor  f(x+i)>0 )

          f’s sign varies if it varies under some infinitesimal change.

Wilsonian quantifiers even apply to hashtag politics. Consider the hashtag #BLM = “Black Lives Matter”. This hashtag denotes an aspiration, not a description. Its implicit protest message is that as things are, black lives do not matter.

          A Wilsonian hashtag would be: #SBNALM = “Some but not all lives matter”. That is a cynical description of political reality. Its aspirational opposite: #ALMOND = “All lives matter or none do”.

          Actually, I think that, in the long run, #SBNALM is an overclass aspirational delusion, and #ALMOND is the gritty reality.