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.
55. In a three-valued logic system as described in Exercise 53, how many rows are needed for a truth table
with n statement letters?
56. In 2003, then U.S. Secretary of Defense Donald Rumsfeld won Britain’s Plain English Campaign 2003
Golden Bull Award for this statement: “Reports that say that something hasn’t happened are always interesting to me, because as we know, there are known knowns, there are things we know we know. We also
know there are known unknowns; that is to say we know there are some things we do not know. But there
are also unknown unknowns—the ones we don’t know we don’t know.”
What possibility did Secretary Rumsfeld omit?
57. Four machines, A, B, C, and D, are connected on a computer network. It is feared that a computer virus
may have infected the network. Your security team makes the following statements:
a. If D is infected, then so is C.
b. If C is infected, then so is A.
c. If D is clean, then B is clean but C is infected.
d. If A is infected, then either B is infected or C is clean.
Assuming that these statements are all true, what can you conclude? Explain your reasoning.
58. The Dillies have five teenaged children, two boys named Ollie and Rollie, and three girls named Mellie,
Nellie, and Pollie. Each is a different number of years old, from 13 to 17. There are three bedrooms for the
children in the Dillie house, so two share the yellow room, two share the white room, and one alone has
the smaller green room. Can you match each one’s name and age, and tell who sleeps where?
a. No one shares a room with a sibling of the opposite sex.
b. Pollie is exactly one year older than Mellie.
c. The two teenagers who share the yellow room are two years apart in age.
d. The two who share the white room are three years apart in age.
e. Rollie is somewhat older than Ollie but somewhat younger than the sibling who has the green room.
2
Determine who sleeps in each room and what their ages are. Explain your reasoning.
59. An advertisement for a restaurant at an exclusive club in Honolulu says, “Members and nonmembers
only.” Give two possible interpretations of this statement.
60. The following newspaper headline was printed during a murder trial:
“I am a liar” says murder defendant!
Can the jury reach any conclusion from this statement?
2 Scott Marley, Dell Logic Puzzles, April, 1998
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.
55. In a three-valued logic system as described in Exercise 53, how many rows are needed for a truth table
with n statement letters?
56. In 2003, then U.S. Secretary of Defense Donald Rumsfeld won Britain’s Plain English Campaign 2003
Golden Bull Award for this statement: “Reports that say that something hasn’t happened are always interesting to me, because as we know, there are known knowns, there are things we know we know. We also
know there are known unknowns; that is to say we know there are some things we do not know. But there
are also unknown unknowns—the ones we don’t know we don’t know.”
What possibility did Secretary Rumsfeld omit?
57. Four machines, A, B, C, and D, are connected on a computer network. It is feared that a computer virus
may have infected the network. Your security team makes the following statements:
a. If D is infected, then so is C.
b. If C is infected, then so is A.
c. If D is clean, then B is clean but C is infected.
d. If A is infected, then either B is infected or C is clean.
Assuming that these statements are all true, what can you conclude? Explain your reasoning.
58. The Dillies have five teenaged children, two boys named Ollie and Rollie, and three girls named Mellie,
Nellie, and Pollie. Each is a different number of years old, from 13 to 17. There are three bedrooms for the
children in the Dillie house, so two share the yellow room, two share the white room, and one alone has
the smaller green room. Can you match each one’s name and age, and tell who sleeps where?
a. No one shares a room with a sibling of the opposite sex.
b. Pollie is exactly one year older than Mellie.
c. The two teenagers who share the yellow room are two years apart in age.
d. The two who share the white room are three years apart in age.
e. Rollie is somewhat older than Ollie but somewhat younger than the sibling who has the green room.
2
Determine who sleeps in each room and what their ages are. Explain your reasoning.
59. An advertisement for a restaurant at an exclusive club in Honolulu says, “Members and nonmembers
only.” Give two possible interpretations of this statement.
60. The following newspaper headline was printed during a murder trial:
“I am a liar” says murder defendant!
Can the jury reach any conclusion from this statement?
2 Scott Marley, Dell Logic Puzzles, April, 1998