Now, that you have seen the Common Math applied to Boolean States.

Let's look at a Quasi Boolean State, where the boolean state is not just an absolute 0 or 1, but a value greater than 0 and less than 1; some Probable state between 0 and 1.

Let's say we can only be 50% certain of A and B are either 0 or 1, then A = 0.5 and B = 0.5.

Then (A And B) = A × B = 0.5 × 0.5 = 0.25

From the truth table in the absolute states we have

(A And B) = A × B

A	B	A And B
0	0	0
0	1	0
1	0	0
1	1	1

If we average the total outcomes it becomes, (0 + 0 + 0 + 1) / 4 = 1 / 4 = 0.25; just what we got when we took a 50/50 guess on their states.

Let's try the Or condition: (A Or B) = A - (A × B) + B = 0.5 - (0.5 × 0.5) + 0.5 = 0.5 - 0.25 + 0.5 = 0.75

Looking at the truth table for Or,

(A Or B) = A - (A × B) + B

A	B	A Or B
0	0	0
0	1	1
1	0	1
1	1	1

Now, average the outcomes, (0 + 1 + 1 + 1) / 4 = 3 / 4 = 0.75

And one more for Xor.

(A Xor B) = A - (A × B) - (A × B) + B = 0.5 - (0.5 × 0.5) - (0.5 × 0.5) + 0.5 = 0.5 - 0.25 - 0.25 + 0.5 = 0.5

Truth table for Xor

(A Xor B) = A - (A × B) - (A × B) + B

A	B	A Xor B
0	0	0
0	1	1
1	0	1
1	1	0

Average the outcomes, (0 + 1 + 1 + 0) / 4 = 2 / 4 = 1 / 2 = 0.5