Exercises for Chapter 6, part 2

Use the ∃i and ∃e rules to prove these sequents. Use the Natural-deduction-proof-checker tool, and submit your answers within a zipped folder.

1. F(charlie), ∀x(F(x) —> G(x)) |− ∃x( F(x) ∧ G(x))
(some intuition: F == healthy; G == happy)

2. ∀x (F(x) —> (∃y H(y,x))), ∃x F(x) |− ∃x∃y H(y,x)
(some intution: F == hasAjob; H == isBossOf)

3. ∃x(F(x) ∧ G(x)) |− (∃y F(y)) ∧ (∃z G(z))
(The converse cannot be proved: Let F == healthy; G == happy. Perhaps there are two people, one healthy and one happy, but no one is both healthy and happy at the same time.)

4. ∃x F(x) |− ¬(∀x ¬F(x))
(some intuition: F == isLeftHanded)