1. (February 29) Translate the following phrases into formal logic:

1.1. A truly pious person loves everyone who hates him.
In translating this phrase, use the following predicates:

• T(x) meaning x is a truly pious person
• L(x, y): x loved y
• H(x, y): x hates y

1.2. You can cheat all people some time, but you cannot cheat them all the time.
Use the predicate C(x, y, t) meaning that x can cheat y at time t.

1.3. Dolphins east fish all the time, eat each other sometimes.
Use the following predicates:

• D(x): x is a dolphin.
• F(x): x is a fish.
• E(x, y, t): x east y at time t.

2. (March 2) Use natural deduction to derive the formula a ∨ b → c from the formulas a → c and b → c.

3. (March 2) Use natural deduction to derive, from the formula ∀x (A(x) → B(x)), the formula ∀x (¬B(x) → ¬A(x)).