analyze the logical forms of the following statements aalice

analyze the logical forms of the following statements:

a)alice and bob are not both in the room

b)alice and bob are both not in the room

c)either alice or bob is not in the room

d)neither alice nor Bob is in the room

Solution

Let A = ”Alice is in the room.” and B = ”Bob is in the room.”

a. ”Alice and Bob are not both in the room.”

The semantic meaning of the original statement is that they are not in the room together or equivalently ”Not Alice is in the room and Bob is in the room.”

Thus the logical form is ¬(A B).

b. ”Alice and Bob are both not in the room.”

The semantic meaning of this statement is that neither of them is in the room or equivalently ”Alice is not in the room and Bob is not in the room.

Thus the logical form is ¬A ¬B.

c. ”Either Alice or Bob is not in the room.”

The semantic meaning of this statment is that one of them is not in the room or equivalently ”Alice is not in the room or Bob is not in the room.”

Thus the logical form is ¬A ¬B.

d. ”Neither Alice nor Bob is in the room.”

The semanic meaning of this statement is the same as (b) (that neither of them is in the room)

so the logical form is ¬A ¬B.