Differential Operators and Quantifiers
Consider the commutative operators on binary logic: A*B =
B*A These are determined by three values:
T*T,
T*F (which equals F*T), and F*F.
Each of these can take two values; therefore there are
2*2*2 = 8 possible such operators:
T*T=T T*F=T
F*T=T F*F=T X*Y =
T
T*T=T T*F=T
F*T=T F*F=F X*Y = X or
Y
T*T=T T*F=F
F*T=F F*F=T X*Y = X
iff Y
T*T=T T*F=F
F*T=F F*F=F X*Y = X
and Y
T*T=F T*F=T
F*T=T F*F=T X*Y =
Not(X and Y)
T*T=F T*F=T
F*T=T F*F=F X*Y = X
xor Y
T*T=F T*F=F
F*T=F F*F=T X*Y =
Not(X or Y)
T*T=F T*F=F
F*T=F F*F=F X*Y =
F
Two of these are constant: T and F. Two are positive
functions: And and Or. They have these rules:
x and y = y and x ; x
or y =
y or x
x and x = x ; x or x = x
x and T = x ; x or F = x
x and F = F ; x
or T =
T
x and (y and z)
= (x and y) and z
x or (y or z) = (x or y) or z
x or (y and z)
= (x or y) and (x or z)
x and (y or z)
= (x and y) or (x or z)
There are two negative functions: Nand and Nor.
x nand y = not (x and y) =
not(x) or not(y)
x nor y = not (x or y)
= not(x) and not(y)
There are two “differential” functions: Xor and Iff.
x iff y = y iff x ; x
xor y = y xor x
x iff x = T ; x xor x
= F
x iff T = x ; x xor F
= x
x iff F = not(x) ; x xor T
= not(x)
x iff (y iff z)
= (x iff y) iff z = x
xor (y xor z) = (x xor y) xor z
x iff y = not(x xor y) = x xor y xor T
x xor y = not(x iff y)
= x iff y iff F
Iff , Xor, And and Or can be mapped to arithmetic, modulo
2. Mapping F to 0 and T to 1 maps Xor to + and And to *; mapping F to 1 and T
to 0 maps Iff to + and Or to *. Therefore these distribution laws:
x or (y iff z)
= (x or y) iff (x or z)
x and (y xor z)
= (x and y) xor (x or z)
And does not double-distribute over Iff, nor Or over Xor,
but they do triple-distribute:
x or (y xor z xor w)
= (x or y) xor (x or z) xor (x
or w)
x and (y iff z)
= (x and y) iff (x and z) iff (x
and w)
Now consider quantifiers. These are extended versions of
the above operators, operating on a predicate’s values in a universe of
discourse. For instance, given a universe of discourse = {a,b}, and a predicate
P(x) on {a,b}, then define:
A(x)(P(x)) = all x
are P =
P(a) and P(b)
S(x)(P(x)) = some x
is P =
P(a) or P(b)
No(x)(P(x)) = not
one x is P = not (P(a) or P(b))
Na(x)(P(x)) = not
all x are P = not(P(a) and P(b))
Nra(x)(P(x)) = none
or all x are P = P(a) iff P(b)
Sbna(x)(P(x)) = some
but not all x are P = P(a) xor P(b)
There are also two constant quantifiers T and F. Over a
universe of discourse with two elements, the quantifiers are Or, And, Nor, Nand, Iff and Xor on P(a) and
P(b).
Over
a larger universe of discourse {a1, a2, a3…}, define:
A(x)(P(x)) =
P(a1) and P(a2) and P(a3) and …
S(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 … )
Na(x)(P(x)) = not
( P(a1) and P(a2) and P(a3) and … )
Nra(x)(P(x)) = No(x)(P(x))
or A(x)(P(x))
Sbna(x)(P(x)) = S(x)(P(x))
and Na(x)(P(x))
These
equations apply:
A(P) = not
S(not P) ; S(P)
= not A(not P)
No(P) = A(not P) ; Na(P)
= S(not P)
Nra(P) = Nra(not P) ; Sbna(P) =
Sbna(not P)
Nra(P)
= A(x,y)( P(x) iff P(y) )
Sbna(P)
= S(x,y)( P(x) xor P(y))
Nra (“nunnerol”) is the constancy operator; true if P(x) is
equally true for any two x; and Sbna (“sumbunol”) is the variability operator;
true if unequally true for some two x. So an alternate notation for these
“differential” quantifiers is
Sbna = V, for
Variability; “P is variable”.
Nra = C, for
Constancy; “P is constant”.
I also call these quantifiers “Wilsonian”, after the
speculative-fiction writer Robert Anton Wilson, who coined “sumbunol” as a philosophical
corrective to ideological overgeneralization.
Various deduction rules apply to the quantifiers. For any a
and b:
A(P) implies P(a)
P(a) implies S(P)
No(P) implies not P(a)
not P(a) implies Na(P)
P(a) and
Nrl(P) implies P(b)
P(a) and not P(b) implies Sbna(P)
For any Boolean function F(p,q):
Nrl(P) and Nrl(Q) implies Nrl(F(P,Q))
Sbnl(F(P,Q)) implies Sbnl(P)
or Sbnl(Q)
If
the universe of discourse has only one element {a}, then:
A(P) = S(P)
= P(a)
Na(P) =
No(P) =
not P(a)
Sbna(P) = F
Nra(P) = T
And if the universe of discourse is empty, then:
A(P) = T
S(P) = F
Na(P) = F
No(P) = T
Sbna(P) = F
Nra(P) = T
Assuming that the universe of discourse is not empty (i.e.
S(x)(x=x); i.e. ‘something is itself’) then these equations follow:
Sbnl(P) and A(P)
= A(P) and No(P) = No(P)
and Sbnl(P) = F
Nrl(P) or Na(P)
= Na(P) or S(P) = S(P)
or Nrl(P) = T
Sbnl, A and No are pairwise disjoint; Nrl, Na and S are
pairwise exhaustive. They join in pairs:
Sbnl(P) = not Nrl(P)
= S(P) and Na(P)
A(P) = not Na(P)
= Nrl(P) and S(P)
N(P) = not S(P)
= Na(P) and Nrl(P)
Nrl(P) = No(P) or A(P)
Na(P) = Sbna(P) or No(P)
S(P) = A(P) or Sbna(P)
Therefore:
Variability = Existence and Exceptions
Universality = Constancy and Existence
Nonexistence = Exceptions and Constancy
Constancy = Nonexistence or Universality
Exceptions = Variability or Nonexistence
Existence = Universality or Variability
Here are Wilsonian versions of mathematical induction:
A(x)(P(x) iff P(x+1)) = Nrl(x)(P(x))
Sbnl(x)(P(x)) = S(x)(P(x) xor P(x+1))
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