Evaluation of "Not" and "XOR" in Fuzzy logic rules

Question

Consider we have three fuzzy variables A , B and D ,. Also , consider that the fuzzification process has been applied and A has been evaluated to 0.5 and B has been evaluated to 0.1. Now , we know that if A and b linked by " AND " ( if A And B then D ) we will take the minimum value and in this case D = 0.1 . For "OR" , we take the maximum ( D= 0.5) . But for the following cases what will be the value of D?

Case 1 : D = A XOR B

Case 2 : D = Not B

Answer 1

Your question is fuzzy, there are some issues, but I'll try to answer.

First of all, "if A and B then D" is itself a logical statement $(A \land B) \implies D$. Since you don't state the value of D, we cannot evaluate it. But I'll assume you are just asking about the value of D.

Secondly, there are many semantics to fuzzy connectives, the one you mention (minimum and maximum) is Gödel-Dummett Logic, but you could also have probabilistic interpretation ($A \land B = AB$; $A \lor B = A+B-AB$), and many others. In the most often used version of fuzzy semantics, the conjunction $\land$ is represented by a kind of function called T-norm and from it you can derive its dual function called S-norm or T-conorm, which serves as a semantics for disjunction $\lor$ using classical de'Morgan rules $A \lor B = \lnot (\lnot A \land \lnot B)$ where negation is usually $\lnot A = 1 - A$. To be formally sound, T-norm based fuzzy logics usually have two versions of both conjunction and disjunction (strong and weak, GD often being the weak and probabilistic being the strong). Wikipedia has good pages about t-norm formal fuzzy logics and especially T-norms themselves with a lot of visualisations of the resulting values: https://en.wikipedia.org/wiki/T-norm

Implications can be derived and formalized in many ways, you have to choose your priorities (axioms), but the most used way seems to be the residuum (which is different from the classical $A \implies B = \lnot A \lor B$, but that is also an option).

Finally, the XOR. Well this is even more complicated. You can use two classical definitions of XOR, which are equal in crisp logic, but not necessarily in fuzzy logic:

1. $A \operatorname{xor}_S B = (A \land \lnot B) \lor (\lnot A \land B)$ i.e. "one but not the other" which is usually marked with S subscript, since it has S-norm (or) at the top level of the formula (one and not the other OR the other and not the one)
2. $A \operatorname{xor}_T B = (A \lor B) \land \lnot (A \land B)$ i.e. "one of them but not both" which is marked with T subscript due to T-norm (and) at the top level (one or the other AND not both at the same time)

These two definitions can be based on the same basic logic but they don't necessarily give the same results! Think how they become intuitively different once you have three variables to XOR instead of just two. And, in addition, they are not the only ones, there is a lot of space for meaningful axiomatization, it all depends on properties you need (with fuzzy logic, you likely won't have them all as you have them in classical crisp logic).

HOWEVER, if you assume GD-logic as you suggest, both of them are the same:

1. $A \operatorname{xor}_{S_{GD}} B = (A \land_{GD} \lnot B) \lor_{GD} (\lnot A \land_{GD} B)$ gives https://www.wolframalpha.com/input?i=max%28min%28x%2C1-y%29%2Cmin%281-x%2Cy%29%29+from+0+to+1
2. $A \operatorname{xor}_{T_{GD}} B = (A \lor_{GD} B) \land_{GD} \lnot (A \land_{GD} B)$ gives https://www.wolframalpha.com/input?i=min%28max%28x%2Cy%29%2C1-min%28x%2Cy%29%29+from+0+to+1

which accidentally turns out to be the same thing, probably the thing you asked for.

NOTE: for probabilistic semantics these two major definitions of xor are not the same anymore. Compare:

• S-based https://www.wolframalpha.com/input?i=%28x%281-y%29+%2B+%281-x%29y+-+xy%281-x%29%281-y%29%29+from+0+to+1
• T-based https://www.wolframalpha.com/input?i=%28x%2By-xy%29%281-xy%29+from+0+to+1

They are kind of skewed the opposite way. But my intuitive probabilistic understanding of XOR is yet another one, kind of in between them: $A \operatorname{xor}_{{0.5}_P} B = A + B - 2AB$, which similarly to the GD-XOR, gives truth value 0.5 if any of the inputs is 0.5. See https://www.wolframalpha.com/input?i=x%2By-2xy+from+0+to+1

So it's all about what you need to represent.