As the title suggests, What is the difference between randomness and fuzziness?
My take: They are two-sides of the same coin - they are two different ways of expressing uncertainty. Consider a universal set $\Omega$. Suppose we want to construct a subset $A \subset \Omega$, then
- Randomness: let $E_x = \{x\in A, \text{for some} x \in \Omega\}$ be a singleton event, defined for each $x \in \Omega$. Then randomness can be thought of as the uncertainty in the occurrence of the event $E_x$ for each $x \in X$. However, in the end the outcome is still binary, i.e. for any $x \in X$ we either have $x \in A$ (the event $E_x$ occurs) or $x \notin A$ (the event $E_x$ doesn't occur).
- Fuzziness: here instead of $A$ being a classical-set (like the one defined above), we define a membership-function $\mu_A:\Omega\rightarrow [0,1]$, such that $\mu_A(x)$ gives the degree to which $x$ must belong to $A$.
However, if we define $\mu_A(x)$ to satisfy the Kolmogorov's Probability Axioms, then the fuzzy member-ship function can be thought of as encoding the probability of the event $E_x$, i.e. $\mu_A(x) = P(E_x)$.
Randomness
is a concept of probability theory. It is not a concept of logic. Randomness means that there is a process that selects an element from a set. Each element of this set has an individual probability of being selected. (The sum of all probabilities of all elements is $1$.) This intrinsic probability of the elements is a function of the elements in the set.
Before an element is selected, you have no information about its probability of being selected other than its individual probability. After an element is selected, however, you know with certainty that the selected element was selected, while all other elements were not.
When we speak of randomness, we are not speaking of circumstantial evidence, and we are not speaking of statements that might be true or false. We are talking about a process of selection.
Fuzziness
is a concept of logic. It is not a concept from probability theory. Fuzziness means having circumstantial evidence that suggests that a statement is true and some other circumstantial evidence that suggests that it is false. An omniscient oracle knows exactly whether the statement is true or false. So a fuzziness of 30% does not mean that a statement is 30% true. The statement is in fact either 100% true or 100% false. What is meant is the following: If one considers all available indications, and if one calculates all combinations of all possible still unknown indications, then it will turn out in 30% of these combinations that the statement is true. However, it is still unclear which of these possible combinations correspond to reality.
When you talk about fuzziness, you are not dealing with a process that selects something from a set. You do not have elements that have probabilities. Instead you are dealing with circumstantial evidence for or against a statement being true.