Wednesday, June 4, 2014

More Trilemmas

Jury Trilemma:

1. No punishing juries for a decision not based on the law and evidence.
2. No double jeopardy.
3. No jury nullifiication.

A legal system can have any two of these but not all three.

***

Spinning numbers:

Voter A:  u = -1, v = 0,  w = 1
Voter B:  u = 0,  v = 1,  w = -1
Voter C:  u = 1,  v = -1, w = 0

Therefore this trilemma:
u < v ;
v < w ;
w < u
though all agree that the order is linear, within {-1,0,1}.

***

1/3 of two things:

Let x equal sqrt(u) when this is defined on the reals;
then x is 0 or 1 or nil (meaning not existing).
Similarly let y = sqrt(v)  ; and z = sqrt(w) :

Voter A:  x = nil, y = 0,   z = 1
Voter B:  x = 0,   y = 1,   z = nil
Voter C:  x = 1,   y = nil, z = 0

Unanimous:
{x,y,z} has two elements; 0, 1 and nil.
the three are distinct because one doesn't exist

Majorities:

x exists
y exists
z exists
x and y do not co-exist
y and z do not co-exist
z and x do not co-exist
x is not 0
x is not 1
x is 0 or 1
y is not 0
y is not 1
y is 0 or 1
z is not 0
z is not 1
z is 0 or 1

***

2/3 of one thing:

Let x = sqrt(u(1-u)), y = sqrt(v(1-v)), z = sqrt(w(1-w)), when these are defined in the reals.

Voter A:  x = nil, y = 0,   z = 0
Voter B:  x = 0,   y = 0,   z = nil
Voter C:  x = 0,   y = nil, z = 0

Unanimous:
{x,y,z} has one element; 0 twice, and nil.

Majorities:

x exists
y exists
z exists
x and y do not co-exist
y and z do not co-exist
z and x do not co-exist
x = 0
y = 0
z = 0
x not = y
y not = z
z not = x

***

1/3 of one thing:

Let x = sqrt(u-1), y = sqrt(v-1), z = sqrt(w-1), when these are defined in the reals.

Voter A:  x = nil, y = nil, z = 0
Voter B:  x = nil, y = 0,   z = nil
Voter C:  x = 0,   y = nil, z = nil

Unanimous:
{x,y,z} has one element; 0, and nil twice.

Majorities:

x does not exist and x is not 0
y does not exist and y is not 0
z does not exist and z is not 0
x or y exists and equals 0
y or z exists and equals 0
z or x exists and equals 0
x and y do not equally exist
y and z do not equally exist
z and x do not equally exist