Logistic
Zero is a problem for reduction, just as
infinity is a problem for addition. By the Rabbit Rule;
1/0 + 1/0 = (0*1+1*0)/(0*0) = 0/0,
the indefinite ratio; and by the Stool
Rule;
0 <+>
0 =
0/1 <+> 0/1
= (0*0)/(0*1+1*0) = 0/0.
This makes sense if infinity and zero are
unsigned, for then
1/0 + 1/0 = 1/0 – 1/0 = 0/0 , by common denominators;
0/1 <+> 0/1 = 0/1 <-> 0/1 = 0/0
, by common numerators.
If we give signs to infinity and zero, then
we can reduce some of the indefiniteness:
+∞
+ +∞ = +∞
-∞
+ -∞ = -∞
+∞
+ -∞ = 0/0
+0 <+> +0
= +0
-0 <+> -0
= -0
+0 <+> -0
= 0/0
That implies these tables:
+
| +0 +∞ <+>|
+0 +∞
x | 1/x
-------------- --------------- ----------
+0 |
0 ∞ +0
| 0
0 +0 | ∞
+∞ |
∞ ∞ +∞
| 0
∞ +∞ | 0
This is isomorphic to Boolean logic, under
this matching:
0
--------------- true
∞ --------------- false
+ --------------- and
<+> --------------- or
1/x --------------- not
Let F be the generic positive finite
quotient. (So F = n/m, with n and m both positive integers.) Then we get these
tables:
+ | 0
F ∞ <+>| 0
F ∞ x | 1/x
------------------ ------------- ---------
0 | 0
F ∞ 0 | 0
0 0 0 | ∞
F
| F F ∞ F
| 0 F F F | F
∞
| ∞ ∞ ∞ ∞
| 0 F ∞ ∞ | 0
This duplicates 3-valued Kleenean logic,
under this matching:
0 ------------ true
F ------------ intermediate
∞ ------------ false
+ ------------ and
<+> ------------
or
1/x ------------
not
Addition, reduction and reciprocal are a
fragment of arithmetic; when applied to {0, finite, ∞}, they are isomorphic to
Kleenean logic; when applied to the extended positives, they resemble a fuzzy
logic.
Under this matching, the DeMorgan laws:
not(x and y) =
(not x) or (not y)
not(x or y) =
(not x) and (not y)
have these arithmetical counterparts:
1/(x + y) = (1/x) <+> (1/y)
1/(x <+>
y) = (1/x)
+ (1/y)
Therefore the name, "De Morgan
distribution".
Define the “Mediant” operator (x@y@z) this
way:
x @ y @ z = (x*y*z)/((x+y+z)(x<+>y<+>z))
This implies:
x*y*z = (x+y+z)(x@y@z)(x<+>y<+>z)
x@y@z = (xy+yz+zx)/(x+y+z)
x@y@z = (xy<+>yz<+>zx)/(x<+>y<+>z)
w*(x@y@z) = (w*x)@(w*y)@(w*z)
1/(x@y@z) = (1/x)@(1/y)@(1/z)
x@0@y = x <+> y
x@∞@y = x + y
The last three laws show that the mediant
corresponds to the majority operator in logic, M(x,y,z); since
Not(M(x,y,z)) = M(not x, not y, not z)
M(x,T,y) = x or y
M(x,F,y) = x and y
Now consider the “equivalence” operator:
x iff y = (x
or not y) and (y or not x)
=
(x and y) or (not x and not y)
It has this arithmetical counterpart:
(x <+> 1/y) + (y <+>
1/x)
=
(x + y) <+> (1/x + 1/y)
= (1<+>xy)/(x<+>y)
=
(x+y)/(1+xy)
- otherwise known as relativistic velocity
addition! Denote it by “x~y”,
and call it “equivalence”, or “Einstein addition”. Its laws are:
x
~ 0 = x
x
~
-x = 0
x~y = y~x
(x~y)~z = x~(y~z)
= x~y~z
x~y~z = (x+y+z+xyz)/(xy+yz+zx+1)
x~y~z = (x<+>y<+>z<+>xyz)/(xy<+>yz<+>zx<+>1)
x ~ ∞ = 1/x
x
~
-1/x = ∞
∞ ~ ∞ = 0
1/(x~y) = (1/x)~y = x~(1/y) = x~y~∞
1/(x~y) = (x<+>y)+(1/x<+>1/y) =
(x+1/y)<+>(y+1/x)
1/(x~y) = (x<+>y)/(1<+>xy) = (1+xy)/(x+y)
(1/x)~(1/y) = x~y
-(x~y) = (-x)~(-y)
-(x~y~z) = (-x)~(-y)~(-z)
1/(x~y~z) = (1/x)~(1/y)~(1/z)
x
~ 1 = 1
x
~
-1 = -1
1
~
-1 = 0/0
(A~x) = B if
and only if x = (-A~B)
tanh(x) ~ tanh(y) = tanh
(x + y)
x ~ y = tanh (tanh-1x + tanh-1y)
So Einstein addition is addition conjugated
by hyperbolic tangent:
(x+y)/(1+xy) =
tanh( arctanh(x) + arctanh(y) )
The left-side
definition works even for x and y not in the domain of arctanh; so equiv is
more than just an addition. Einstein addition is more like a multiplication
than an addition, for both have order-two elements:
x~∞~∞
= x
; x*-1*-1 = x
Now consider these
‘diagonal turn’ functions:
b(x) = tan(arctan(x) - 45o)
= (x -
1) / (x + 1)
q(x) = tan(arctan(x) + 45o)
= (x + 1) / (-x + 1)
If x is the slope of a line, then b(x) is
the slope of that line turned clockwise 45 degrees; and q(x) is the slope of
that line turned counterclockwise 45 degrees. b sends 0 to -1 to ∞
to 1 to 0; and q sends 0 to 1 to ∞ to -1 to 0; a period
4 cycle.
The diagonal turn
operators b and q are also Einstein additions conjugated by complex
multiplication:
b(x) = i * ( (-1/i) ~ (x/i) )
q(x) = i * ( (1/i)
~ (x/i) )
In general;
b(q(x)) = q(b(x)) = x
b(b(x)) = q(q(x)) =
-1/x
b(b(b(x))) = q(x) ;
q(q(q(x))) = b(x)
b(b(b(b(x)))) =
q(q(q(q(x)))) = x
b and q are inverses,
both with period 4. They are also negative reciprocals: q(x) * b(x) = -1
We can derive these
identities and conjugations:
b(x*y) = b(x)~b(y)
b(q(x)*q(y)) = x~y
b(x~y) = -b(x)*b(y)
b(q(x)~q(y)) = -x*y
q(-x*y) = q(x)~q(y)
q(-b(x)*b(y)) = x~y
q(x~y) = q(x)*q(y)
q(b(x)~b(y)) = x*y
b(-x) = -
q(x) = 1/b(x)
b(1/x) = 1/q(x) =
- b(x)
b(-q(x))
= 1/x
b(1/q(x)) =
- x
q(-x) = -
b(x) = 1/q(x)
q(1/x) = 1/b(x) =
- q(x)
q(-b(x)) = 1/x
q(1/b(x)) =
- x
Like exponentials, b
and q turn minus into reciprocal; like logarithms, they turn reciprocal into
minus. These 45-degree turns also turn multiplication into Einstein addition,
and vice versa. So reciprocal is diagonal minus, and vice versa; and Einstein
addition is diagonal multiplication, and vice versa.
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