Would you mind concisely explaining one example and application of it? I've read (for example) about possibility theory, but I hardly understand what it helps us do that probability theory didn't.
Fuzzy logic reminds me Albert Einstein quote about mathematics. "As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."
So, there is this material that needs to be heated on a curve (time 0 = room temp, 20 minutes = 400C, 120 minutes = room temp - cannot remember after 20 years - it cooled sorta quickly because of fans when heater off). Software package with oven basically turned heaters full blast, the off, then full blast, etc. about half the time material wasn’t cooked properly. I wrote a bunch of rules that computed a % the heater should be at. Worked really well and was an amazingly small program.
Social wise I wrote rules to judge success based on about 20 factors. Never bothered to pursue it beyond my own satisfaction.
Let's say you have are taking a shower with an automated temperature control. The current temperature is set to 28 degrees Celsius, but there is a sudden shift to 24 degrees.
Probability Theory: Based on prior knowledge, I know that a temperature of 24 degrees Celsius is considered "hot" 10% of the time, "warm" 50% of the time, and "cold" 40% of the time. Based on this the current expected value of the probability distribution is still "warm", so I will leave it same until cold > warm.
Fuzzy Theory: Based on my fuzzy set membership, 24 degrees Celsius has 25% membership in the "hot" set, 60% membership in the "warm" set, and 50% membership in the "cold" set. My fuzzy system is setup to maintain a particular relationship between cold and hot (let's say, cold == hot), and therefore increases temperature until balanced.
For example, you can't have a temperature that is 80% hot AND 80% cold. This problem becomes even more apparent as you increase the number of states (lukewarm, warmish, very warm), as each state reduces the probability of another state.
The differences become more profound as you do more complicated set operations, fuzzy relations, fuzzy systems, and such. In fact, there is even fuzzy probability theory, in which you have a probability distribution of different fuzzy sets.
I don't see why this is an issue. You can also define each of the states as its own binary random variable and then the probabilities of two states conditioned on the temperature can add up to more than 1.
I thought your original post was meant to explain a practical application of fuzzy theory and how it differs from probability theory. Perhaps there is another example that better illustrates how fuzzy theory simplifies a problem where using probability theory would be messy / impossible?
Yes, but then you're defining a probability distribution over the space of fuzzy states.
Is one sample of a fuzzy logic based controller of an industrial process. There are many such examples, I have a book here on fuzzy logic that details a whole raft of them.
https://en.wikipedia.org/wiki/Fuzzy_logic#Early_applications