Question

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.''

Answer 1

The truth value for the statement 'Fred and John are happy' is 0.35 in fuzzy logic, representing the minimum of their individual happiness levels. The truth value for the statement 'Neither Fred nor John is happy' is 0.2, obtained by negating each statement and selecting the minimum value.

Step-by-step explanation:

In fuzzy logic, a mathematical model of logic in which a proposition can have a value anywhere between completely true (1) and completely false (0), we assess truth values for compound statements based on specific operations. Given the truth values for the propositions "Fred is happy" (0.8) and "John is happy" (0.35), we can evaluate the combined statements.

(a) Fred and John are happy

The truth value of the conjunction ("and") of two propositions in fuzzy logic is the minimum of the individual truth values. Thus, the truth value for "Fred and John are happy" is the minimum of 0.8 and 0.35, which is 0.35.

(b) Neither Fred nor John is happy

The negation of a proposition has a truth value of 1 minus the original truth value. To find the truth value for "Neither Fred nor John is happy" we need to take the negation of each individual truth value and then apply the conjunction rule. The negation of Fred's happiness is 1 - 0.8 = 0.2, and the negation of John's happiness is 1 - 0.35 = 0.65. The truth value of the combined statement is the minimum of these two negations, which is 0.2.

Answer 2

Truth value for "Fred and John are happy" : 0.35

Truth value for "Neither Fred nor John is happy" : 0.2

Step-by-step explanation:

For (a) part:

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.