Fuzzy Logic is used in artificial intelligence. In fuzzy logic, a proposition has a truth value that is a number between 0 and 1 inclusive. A proposition with a truth value of 0 is false and one with truth value of 1 is true. Truth values that are between 0 and 1 indicate varying degrees of truth. For instance, the truth value 0.8 can be assigned to the statement "Fred is happy.'' since Fred is happy most of the time, and the truth value 0.35 can be assigned to the statement "John is happy.'' since John is happy slightly less than half the time. The truth value of the negation of a proposition in fuzzy logic is 1 minus the truth value of the proposition. The truth value of a conjunction of two propositions in fuzzy logic is the minimum of the truth values of the two propositions. What are the truth value of the statements: (a) ``Fred and John are happy.'' and (b) ``Neither Fred nor John is happy.''

We know that the truth value of a conjunction of two propositions (in fuzzy logic) is the minimum of the truth values of each proposition. In our case, the truth value that corresponds to the statement "Fred is happy" is 0.8 and the truth value that corresponds to "John is happy" is 0.35. The minimum of these values (0.8, 0.35), is 0.35, which corresponds to the truth value of the statement "Fred and John are happy".

For (b) part:

In order to answer this part of the problem, first we need to find the truth values of the negations of the initial propositions ("Fred is not happy" and "John is not happy"), which we do by calculating 1 minus the truth value of the initial proposition. Therefore, the truth value of the negations are:

Truth value of "Fred is not happy" (negation of "Fred is happy") is 1 - 0.8 = 0.2

Truth value of "John is not happy" (negation of "John is happy") is 1 - 0.35 = 0.65

Finally, in order to find the conjunction of these negations, we just find the minimum of these values, which is: 0.2.