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.
