Representing Logic

Interpretation Methods

The amount of possible binary interpretation methods is related to the amount of input data.

m = 2 ^ ( 2 ^ b )

Where m is the number of interpretation methods and b is the number of input bits.

One Input Bit

Interpretation Methods: 4

A       10
----------
0       00  always false
1       01  not A
2       10  A
3       11  always true

Example

A light bulb hangs on a wire in the middle of an enclosed room with four subjects in the room. Each subject represents a unique interpretation method. The subjects are numbered 0 to 3 and will react by being either happy or sad depending on the state of the light bulb.

  • Subject 0 will be unhappy no matter what. It dislikes both the light and the dark.
  • Subject 1 will only be happy when the light is off. It enjoys the dark and dislikes the light.
  • Subject 2 will only be happy when the light is on. It enjoys the light and dislikes the dark.
  • Subject 3 will be happy no matter what the lightbulb’s state is. It enjoys both the light and the dark.

Even if the room were filled with 10 animals, each must adopt one of the four interpretation methods. There are no other options in a one-bit macrocosm.

Two Input Bits

Interpretation Methods: 16

A       1100
B       1010
------------
0       0000  always false
1       0001  not A or B
2       0010  not A if B
3       0011  not A
4       0100  not B if A
5       0101  not B
6       0110  not A iff B
7       0111  not A and B
8       1000  A and B
9       1001  A iff B
10      1010  B
11      1011  B if A
12      1100  A
13      1101  A if B
14      1110  A or B
15      1111  always true

Example

Two light bulbs hang in an enclosed room (a green light and a red light) with sixteen subjects in the room. Each subject represents a unique interpretation method. The subjects are numbered 0 to 16 and will react by being either happy or sad depending on the state of the light bulbs.

  • Subject 14 is happy as long as any one of the two lights are on.
  • Subject 6 is happy if any one light is on, and is unhappy when no lights are on or if both lights are on.
  • Subject 11 is happy unless the green light is on when the red light is off.

Three Input Bits

Interpretation Methods: 256

A       11110000
B       11001100
C       10101010
----------------
0       00000000  always false
1       00000001  not A or B or C
...
15      00001111  not A
...
51      00110011  not B
...
85      01010101  not C
...
126     01111110  not A iif B iif C
127     01111111  not A and B and C
128     10000000  A and B and C
129     10000001  A iif B iif C
...
170     10101010  C
...
204     11001100  B
...
224     11100000  A and (B or C)
...
234     11101010  (A and B) or C
...
240     11110000  A
...
254     11111110  A or B or C
255     11111111  always true

Four Input Bits

Interpretation Methods: 65536

A       1111111100000000
B       1111000011110000
C       1100110011001100
D       1010101010101010
------------------------
0       0000000000000000  always false
1       0000000000000001  not A or B or C or D
2       0000000000000010
...
65535

Five Input Bits

Interpretation Methods: 4294967296

A       11111111111111110000000000000000
B       11111111000000001111111100000000
C       11110000111100001111000011110000
D       11001100110011001100110011001100
E       10101010101010101010101010101010
----------------------------------------
0       00000000000000000000000000000000  always false
1       00000000000000000000000000000001  not A or B or C or D or E
2       00000000000000000000000000000010
...
4294967295

Six Input Bits

Interpretation Methods: 18446744073709551616

A       1111111111111111111111111111111100000000000000000000000000000000
B       1111111111111111000000000000000011111111111111110000000000000000
C       1111111100000000111111110000000011111111000000001111111100000000
D       1111000011110000111100001111000011110000111100001111000011110000
E       1100110011001100110011001100110011001100110011001100110011001100
F       1010101010101010101010101010101010101010101010101010101010101010
------------------------------------------------------------------------
0       0000000000000000000000000000000000000000000000000000000000000000 always false
1       0000000000000000000000000000000000000000000000000000000000000001 not A or B or C or D or E or F
...
18446744073709551615