On Diagonal Quantifiers

          On Diagonal Quantifiers

          By Nathaniel Hellerstein

 

1.    Variation and Constancy

 

          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 things have property P

                   =       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 is the variation quantifier; so the symbol for sumbunol is “Var”: Var(x)(P(x))  =  P varies.

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

                             Ollerno things have property P

                    =       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 is the constancy quantifier; so the symbol for ollerno is “Con”: Con(x)(P(x)) = P is constant.

          I call variation and constancy diagonal quantifiers. I also call them Wilsonian quantifiers, in honor of Robert Anton Wilson.

 

 

 

2.    Diagonal and Other Quantifiers

 

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

         

If the universe of discourse is empty, then:

                   All(P)  =  No(P)  = Con(P)  =  T

Some(P) =  NotAll(P) =  Var(P)  = F

 

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))

 

 

 3.    Constancy and Equality Compared

 

Compare and contrast:

 

(x = y)       =       For any property P,   P(x) iff P(y).

Con(P)       =       For any objects x and y,  P(x) iff P(y).

 

Equality and constancy are complementary. Both start from objects having a property equally; equality generalizes the property, constancy generalizes the objects. Equality defines entities; constancy defines laws.

          Their negations are also similar:

 

(x  y)       =       For some property P,   P(x) xor P(y).

Var(P)        =       For some objects x and y,  P(x) xor P(y).

        

 

4.    Diagonal Quantifier Laws

 

          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(p₁, p₂,… p_n) is any function on Boolean logic, then

                   Con(x)(P₁(x))  and Con(x)(P₂(x)) and …  Con(x)(P_n(x))

                    Implies       Con(x) ( F(P₁(x), P₂(x), … P_n(x))

          This is “Constancy”: constant inputs imply a constant output.

 

If F(p₁, p₂,… p_n) is any function on Boolean logic, then

Var (x) ( F(P₁(x), P₂(x), … P_n(x))

                   Implies    Var(x)(P₁(x))  or Var(x)(P₂(x)) or …  Var(x)(P_n(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 is “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

 

  

5.    Limit Diagonal Quantifiers

 

          On the infinite ordered set {1,2,3…}, define the cofinity and infinity quantifiers thus:

 

          Cof(n)(P(n))       =       P eventually stays true

                                      =       P is true for all but finitely many n

                                      =       Exists(N)All(n) ( n>N  implies P(n) )

=        ( P(1) and P(2) and P(3) and P(4) and ... )

                    or  (P(2) and P(3) and P(4) and ...)

                          or  (P(3) and P(4) and ...)

                                                                                         or  (P(4) and ... )

                                                         or  ....

          Inf(n)(P(n))        =       P is persistently true

                                      =       P is true for infinitely many n

                                      =       All(N)Exists(n) ( n>N  and P(n) )

=        ( P(1) or P(2) or P(3) or P(4) or ... )

                and  (P(2) or P(3) or P(4) or ...)

                   and  (P(3) or P(4) or ...  )

                                                                               and  (P(4) or ... )

                                           and  ....

 

          Cof and Inf have these identities:

          Not Cof(x)(P(x))   =    Inf(x)(not P(x))

          Not Inf(x)(P(x))   =    Cof(x)(not P(x))

          A and Cof(x)(P(x))  =   Cof(x)(A and P(x))

          A or Cof(x)(P(x))    =   Cof(x)(A or P(x))

          A and Inf(x)(P(x))  =   Inf(x)(A and P(x))

          A or Inf(x)(P(x))     =   Inf(x)(A or P(x))

 

          Here are definitions of the convergence and divergence quantifiers:

 

          Conv(n)(P(n))    =       P is convergent

=       Cof(n)(P(n))   or   not Inf(n)(P(n))

          =       Exists(N)All(m,n) ((m>N and n>N) implies (P(m) iff P(n) )

          =       P is eventually constant

 

          Div(n)(P(n))       =       P is divergent

=       Inf(n)(P(n))   and  not Cof(n)(P(n))

          =       All(N)Exists(m,n) (m>N and n>N and (P(m) xor P(n))

          =       P is persistently variable

 

          These equations apply:

          Cof(n)(P(n))       =       Conv(n)(P(n)) and Inf(n)(P(n))

                   Cofinite equals convergent and persistently true.

          Inf(n)(P(n))         =       Div(n)(P(n)) or Cof(n)(P(n))

                   Persistently true equals divergent or cofinite.

         

          Not Conv(P)  =  Div(P)

          Conv(Not P)   =  Conv(P)

Not Div(P)  = Conv(P)

          Div(Not P)   =  Div(P)

         

          Partial Distribution:

A and Div(x)(P(x))   =  Div(x)( A and P(x) )

          A or Conv(x)(P(x))   =  Conv(x)( A or P(x) )

          “And” distributes over divergence, and “or” distributes over convergence; but “and” does not distribute over convergence; nor does “or” distribute over divergence:

T or Div(P(x)) = T;   but Div(T or P(x)) = F.

F and Conv(P(x)) = F;  but  Conv(F and P(x)) = T.

 

          Divergence and convergence have these Equivalence Rules:

 

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

Implies       Conv(x)(P(x) iff R(x))         

If “P iff Q” and “Q iff R” converge, then so does “P iff R”.

 

Conv(x)( P(x) iff Q(x) )    and     Conv(x)(Q(x))

Implies       Conv(x)(P(x))   

If “P iff Q” and “Q” converge, then so does “P”.

 

Div(x)(P(x))  and     Conv(x)(Q(x))

Implies  Div(x)(P(x) iff Q(x))      

          If “P” diverges and “Q” converges, then “P iff Q” diverges.

 

Conv(x)( P(x) iff Q(x) )

                   Implies       Conv(x)(P(x))   iff    Conv(x)(Q(x))     

          If “P iff Q” converges, then “P” and “Q” are equally convergent.

 

                   Div(x)(P(x))   xor   Div(x)(Q(x)) 

implies      Div(x)( P(x) xor Q(x) )

 

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

 

          Convergence and divergence have two Continuity Rules:

         

If F(p₁, p₂,… p_n) is any function on Boolean logic, then

                   Conv(x)(P₁(x))  and Conv(x)(P₂(x)) and …  Conv(x)(P_n(x))

                   Implies       Conv(x) ( F(P₁(x), P₂(x), … P_n(x))

          Convergent inputs imply a convergent output.

 

If F(p₁, p₂,… p_n) is any function on Boolean logic, then

Div(x) ( F(P₁(x), P₂(x), … P_n(x) )

                   Implies    Div(x)(P₁(x))  or Div(x)(P₂(x)) or …  Div(x)(P_n(x))

          Divergent output implies a divergent input.

 

        6.    Diagonal Quantifiers in Higher Math

 

Here are diagonal-quantifier 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.

 

          Conditional equations are variable, singular equations are constant:

          Var(x)(x=1)

          Con(x)(x=x+1)

          Con(x)(x=x)

 

          If a real number R equals 0. R₁ R₂ R₃ R₄ … in base 2, then

          R is dyadic                  =       Conv(n)(R_n=0)

 

 

7.    Diagonal Quantifier Troikas and Trilemmas

 

          Here is a Diagonal 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.

When the Stooges vote, each of the following propositions passes by 2/3 majorities each:

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.”

          Call this a Diagonal Quantifier Trilemma.

 

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

If some frogs are princes, and some frogs are not princes, then some but not all frogs are princes.

If some frogs are not princes, and all or no frogs are princes, then no frogs are princes.

If all or no frogs are princes, and some frogs are princes, then all frogs are princes.

 

In general a diagonal-quantifier trilemma has the form:

Some A have property P;

Some A do not have property P;

All As have property P equally;

-         choose two!

For instance:

Some men are good;

Some men are not good;

All men are equally good;

-         choose two!

         

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.

 

 

8.    Diagonal Quantifiers in Hashtag Politics

 

Diagonal quantifiers 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. Eventually, the value of human life isn’t variable, it’s constant.

          The value of human life converges.

 

          9.    Diagonal Quantifiers in Theology

 

Consider the concept of a Holy Land. For this phrase to bear non-zero information, then there must be some land that is holy, and some land that is not holy. Sumbunol!         

If you don’t care to argue about which lands are holy, and which aren’t, then you should affirm the opposite of “sumbunall lands are holy”; namely “ollerno lands are holy”.

“All or no lands are holy”: equivalently, “All lands are equally holy”. This is equivalent to “God is present everywhere or nowhere”, i.e. “God is equally present everywhere”. Call the property of being present everywhere or nowhere “equipresence”. The equipresence of God implies that any land is just as holy as any other.

Divine equipresence appeals to both theists and atheists, for opposite reasons. Taken as an ambiguous balance, it appeals to agnostics. It does not appeal to theocrats, who have an institutional need for some lands to be holier than others. Equipresence is anti-theocratic.

          The opposite of equipresence is varipresence; the property of being present some places but not all. I am varipresent; so are you; and according to theocrats, so is God.

          What other equiproperties might one attribute to God? How about power, knowledge and compassion?

          An equipotent God has all power, or no power.

          An equiscient God knows everything or nothing.

          An equicompassionate God loves everyone or no-one.

          Theist, atheist, and agnostic can agree that God is equipotent, equiscient, and equicompassionate; though for different reasons.

          Their opposites are varipotence, variscience, and varicompassion. I have all these attributes, as do you. Theocrats tend to assign at least one of these attributes to God, usually varicompassion. So the equi-properties are anti-theocratic.

I’ve discovered that there’s equiscience in my work-life. At the start of every semester, I tell the students in my classes that I will run review sessions for each chapter of the book; and during those sessions they may ask any questions about that chapter; and that I welcome all questions; and that the only thing that I do not want to hear during question-time is silence. “Because if you have no questions, then you know either everything or nothing, and in neither case can I teach you anything!” Equiscience vs. teachability; education requires variscience.

Any proposed theory of physics is equipresent and equipotent. It applies everywhere or nowhere; it’s equidescriptive.

         Here is an Equipresence Troika:

Moe: “God is present nowhere.”

Larry: “God is present everywhere.”

Curly: “God is present somewhere but not everywhere.”

 

When the Stooges vote, they pass by 2/3 majorities the Equipresence Trilemma:

God is present somewhere;

God is absent somewhere;

God is present everwhere or nowhere.