Since they belong to the same probability distribution, the states are not independently defined. A change that increases the probability of hot MUST decrease the probability of cold, since the probability of all states must sum to 100%.
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.
"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"
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?
> 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.
Yes, but then you're defining a probability distribution over the space of fuzzy states.
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.