Question 1: Using the predicate symbols shown and appropriate quantifiers, write each English language statement in predicate logic. The domain is the whole world. In the definition of a predicate, the variables are placeholders. They will get instantiated within the scope of the quantifiers.
P(x) is "x is a person."
T(x) is "x is a time."
F(x,y) is "x is fooled at y."
1. You can fool some of the people all of the time.
2. You can fool all of the people some of the time.
3. You can't fool all of the people all of the time.
Question 2: Which of the following is the correct negation for "Nobody is perfect." Check only one
1. Everyone is imperfect.
2. Everyone is perfect.
3. Someone is perfect.
Question 3: Write the negation of each of the following (in English) (hint: do not give trival translations by prefixing the statement with "it is false that ..." or "Not all ..."):
1. Some farmer grows only corn.
2. All farmers grow corn.
3. Corn is grown only by farmers.
Question 4: Write the negation of the following quantified statement. Move the negation symbol as far inside the predicate as possible. Assume that x, y, and z are all in the universe U.
∃x∀y∀z [(F(x, y) ∧ G(x, z)) → H(y, z)]
Question 5: Prove that the expression below is a valid argument using the deduction method (that is using equivalences and rules of inference in a proof sequence)
(∃x)[P(x) → Q(x)] ∧ (∀y)[Q(y) → R(y)] ∧ (∀x)P(x) → (∃x)R(x)