Propositional wffs and truth tables belong to a system of

Chapter 1, Problem 54

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Propositional wffs and truth tables belong to a system of two-valued logic because everything has one of two values, F or T, which we can think of as 0 or 1. In fuzzy logic, or many-valued logic, statement letters are assigned values in a range between 0 and 1 to reflect some probability to which they are false or true. A statement letter with a truth value of 0.9 is mostly true or has a high probability of being true while a statement letter with a truth value of 0.05 has a very high probability of being false. Fuzzy logic Show that every compound wff is equivalent to a wff using only the connective 0 . (Hint: Use Exercise 47 and find equivalent statements for A ` B and A in terms of 0 .) Show that every compound statement is equivalent to a statement using only the connective T. (Hint: See Exercise 51.) 24 Formal Logic is used to manage decisions in many imprecise situations such as robotics, manufacturing, or instrument control. Truth values for compound statements are determined as follows. A has the truth value 1 A. A ` B has the truth value that is the minimum of the values of A and of B. A ~ B has the truth value that is the maximum of the values of A and B. a. Explain why these are reasonable assignments for the truth values of A, A ` B, and A ~ B. Suppose the statement, Flight 237 is on time, is estimated to have a truth value of 0.84 and the statement, Runway conditions are icy, is estimated to have a truth value of 0.12. Find the truth values of the following statements. b. Runway conditions are not icy. c. Runway conditions are icy and flight 237 is on time. d. Runway conditions are icy or flight 237 is not on time.