Fuzzy Logic Operators: AND, OR and NOT

1Combining vague conditions

Real rules rarely test one thing. A nurse at a health centre in Nyagatare decides how urgently to see a child by combining several observations: the fever is high AND the child is very sleepy; the cough is long-lasting OR the breathing is fast. Each observation is a matter of degree. To build such rules into a machine we need fuzzy versions of AND, OR and NOT that work on degrees between 0 and 1, and that still give the ordinary answers when the degrees are exactly 0 or 1.

2Why min, max and 1 − a?

A chain is only as strong as its weakest link: a statement "A and B" cannot be truer than its less true part, so AND takes the minimum. "A or B" is as true as its truer part, so OR takes the maximum. NOT turns "fully true" into "fully false" and leaves 0.5 where it is.

These operators extend ordinary logic. When the inputs are only 0 or 1 they give exactly the Boolean truth table:

The last row is something Boolean logic cannot express at all.

3The operators as pictures

On a graph, AND is the region under both curves (the lower envelope, shaded), and OR is the region under either curve (the upper envelope, the grey band). The two curves cross at 20 °C, where $\frac{22-x}{5}=\frac{x-18}{5}\backslash frac\{22\; -\; x\}\{5\}\; =\; \backslash frac\{x\; -\; 18\}\{5\}$ gives $x=20x\; =\; 20$ and $\mu =0.4\backslash mu\; =\; 0.4$, so the peak of "cool AND comfortable" is 0.4.

NOT flips each curve upside down about $\mu =0.5\backslash mu\; =\; 0.5$: wherever comfortable is high, NOT comfortable is low, and the two always add to 1.

4Which laws survive?

Most laws of ordinary logic still hold for min, max and $1-a1\; -\; a$: AND and OR are commutative and associative, and De Morgan's laws hold:

$$1-\min (a,b)=\max (1-a,1-b)1-\max (a,b)=\min (1-a,1-b)1\; -\; \backslash min(a,\; b)\; =\; \backslash max(1\; -\; a,\backslash ;\; 1\; -\; b)\; \backslash qquad\; 1\; -\; \backslash max(a,\; b)\; =\; \backslash min(1\; -\; a,\backslash ;\; 1\; -\; b)$$

Two famous laws do not hold. In Boolean logic, "A OR NOT A" is always true and "A AND NOT A" is always false. In fuzzy logic, with $a=0.5a\; =\; 0.5$:

$$\max (0.5,1-0.5)=0.5\min (0.5,1-0.5)=0.5\backslash max(0.5,\backslash ;\; 1\; -\; 0.5)\; =\; 0.5\; \backslash qquad\; \backslash min(0.5,\backslash ;\; 1\; -\; 0.5)\; =\; 0.5$$

A 25 °C day can be hot to degree 0.5 and not hot to degree 0.5 at the same time. This is not a mistake: it is exactly the borderline case that a vague word creates.

5Other choices of operator

Min and max are not the only possibilities. The product $a\times ba\; \backslash times\; b$ is another common AND, and the probabilistic sum $a+b-aba\; +\; b\; -\; ab$ is another OR. For hot 0.7 and humid 0.4 they give 0.28 and 0.82. Product AND lets every input affect the result, while min ignores the larger input completely. Whichever pair you choose, use it consistently throughout one system.

6In code

This program, run with python3, computes the worked example, the alternative operators, and checks De Morgan's law on every pair of values from 0 to 1 in steps of 0.1.

from itertools import product

def AND(a, b): return min(a, b)      # Zadeh intersection
def OR(a, b):  return max(a, b)      # Zadeh union
def NOT(a):    return 1 - a          # complement

hot, humid = 0.7, 0.4
print("hot AND humid     ", AND(hot, humid))
print("hot OR humid      ", OR(hot, humid))
print("NOT humid         ", round(NOT(humid), 2))
print("hot AND NOT humid ", round(AND(hot, NOT(humid)), 2))
print("product AND       ", round(hot * humid, 2))
print("probabilistic OR  ", round(hot + humid - hot * humid, 2))

# De Morgan: NOT(a AND b) == NOT a OR NOT b, for every pair on a 0.1 grid
grid = [i / 10 for i in range(11)]
print("De Morgan holds:", all(abs(NOT(AND(a, b)) - OR(NOT(a), NOT(b))) < 1e-12
                              for a, b in product(grid, grid)))
# Excluded middle: a OR NOT a is not always 1
print("0.5 OR NOT 0.5 =", OR(0.5, NOT(0.5)))

Output:

hot AND humid      0.4
hot OR humid       0.7
NOT humid          0.6
hot AND NOT humid  0.6
product AND        0.28
probabilistic OR   0.82
De Morgan holds: True
0.5 OR NOT 0.5 = 0.5

7Hands-on: test the laws yourself

1. Copy the program and run it.
2. Add a check of the second De Morgan law, $1-\max (a,b)=\min (1-a,1-b)1\; -\; \backslash max(a,\; b)\; =\; \backslash min(1\; -\; a,\; 1\; -\; b)$, over the same grid.
3. Find, by trying values, the value of $aa$ at which $\max (a,1-a)\backslash max(a,\; 1\; -\; a)$ is smallest. What is that smallest value?
4. Write a function AND_prod(a, b) and test whether De Morgan's law holds when product AND is paired with probabilistic-sum OR.

Check: the program prints True for both De Morgan laws with min and max, and you can state for which $aa$ "A OR NOT A" is least true.