CSc-180 (Gordon)
Week 3B notes
Fuzzy Logic Truth Tables
Consider standard Boolean logic truth table for OR:
A
0
0
1
1
B
0
1
0
1
Boolean truth table for NOT:
^
0
1
1
1
A
0
1
¬
1
0
Given that in Fuzzy logic, A^B = max(A,B),
a corresponding partial truth table for Fuzzy OR would be:
A
0
0
0
.5
.5
.5
1
1
1
B
0
.5
1
0
.5
1
0
.5
1
^
0
.5
1
.5
.5
1
1
1
1
Fuzzy logic for NOT(A) is 1-A, ∴ partial fuzzy table for NOT:
A
0
.5
1
note that when membership=0.5, A=¬A
What about implication? In Boolean logic, A
A
0
0
0
.5
.5
.5
1
1
1
B max((1-A),B)
0
1
.5
1
1
1
0
.5
.5
.5
1
1
0
0
.5
.5
1
1
A
B
0
0
0
.5
.5
.5
1
1
1
0
.5
1
0
.5
1
0
.5
1
¬
1
.5
0
In Fuzzy Logic, that would be max((1-A),B)
“Godel” implication (A≤B)νB
1
1
1
0
1
1
0
.5
1
Modus Ponens
≡ max((1-A),B)
1
1
1
.5
.5
1
1
.5
1
B = ¬A ν B
Which definition of implication is “better”?
One test would be to see which obeys Modus Ponens.
Modus Ponens is: ((A
Modus Ponens
≡ (A≤B)νB
1
1
1
1
1
1
1
1
1
B) ^A)
B