How to solve this fuzzy logic inference problem?

Question

I'm having trouble solving an exercise involving fuzzy logic. It basically says something like that (pardon me if I made some mistake, I'm basing only on my poor memory)

John goes to park if it is a beautiful day and it is not too hot, or if it isn't raining. Assuming that:

1. It is a beautiful day with 0.6 degree

2. It is hot with 0.4 degree

3. It is raining with 0.8 degree

With which degree John will go to park?

Although I've solved a lot of exercises including fuzzy number or fuzzy set, I basically have no idea on how to solve this involving fuzzy logic.

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EDIT

It seems that the solution to the problem is something like the minimum values of the 3 degree; the negative one should be complemented to one. So, $$min(0.6, 0.4, 0.2) = 0.2.$$

but I'm not too sure.

Answer 1

Not an expert by any means, but the way I understand it is this (someone please correct me if I'm wrong):

In fuzzy logic, instead of saying something is either entirely true or entirely false, we say that it is entirely false, mostly false, equally true and false, mostly true, or entirely true. To do so, we represent these "degrees of truth" as numbers between $0$ and $1$. So, for example, if I were to say that my soup was cold with a degree of $0.7$, I'd be saying that it's fairly cold (certainly more cold than it is hot), but that it could have been colder. It's mostly cold.

Now, it is a beautiful day with a degree of $0.6$ (mostly beautiful, but we've seen better days), hot with a degree of $0.4$ (more cool than it is hot), and raining with a degree of $0.8$ (raining fairly hard, but we're not underwater yet). John will go to the park if it is beautiful, not hot, and not raining. Taking the complements of the negative conditions (hot, raining), we have the following degrees of truth: $0.6$, $0.6$, $0.2$.

So what is our degree of truth that John is going to the park? $0.2$ (John is going to the park, but probably not for very long). This is because the "not raining" condition is the condition that most limits John's trip to the park, as it is the least true. So the degree of truth that John is going to the park is $\min(0.6, 0.6, 0.2)$. Thinking about it more intuitively, if it were incredibly beautiful out and not at all hot, John would be no more eager to go to the park; it's still raining heavily, and that's really all that is necessary to deter John.

Answer 2

Ok, not sure by any means, but I think this is the right solution. We have to consider the two logic conjunctives $OR$ and $AND$. The isomorphism with the theory of sets specifies that these two operations are mapped respectively by $\cup$ and $\cap$. In fuzzy logic, we have these three primitives:

• $\neg a = 1 - a $
• $a \land b = min(a, b)$
• $a \lor b = max (a, b)$

Basing on these assumptions, the problem can be specified as the sequent $$P = (B \land H) \lor \neg R$$ $$ = max(min(B,H), \neg R)$$ $$ = max(min(0.6, 0.4), 0.2)$$ $$ = 0.4$$

Haven't found official answers yet btw.