dmargol1

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.

Both are ways of dealing with uncertainty, but probability theory is more about the uncertainty around which state a variable has, while fuzzy theory is the uncertainty around the states themselves.

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.