Exercises for Chapter 6, part 1

1. Use the ∀e and ∀i deduction rules to prove these sequents. Use the Natural-deduction-proof-checker tool, and submit your answers within a zipped folder, like you did for the previous exercise sets.

(a) F(charlie), ∀x(F(x) —> G(x)), ∀y H(y) |− G(charlie) ∧ H(charlie)

(b) ∀x F(x), ∀y(F(y) —> G(y)) |− ∀z(G(z) ∧ F(z))

(c) (∀x H(x)) ∨ (∀x G(x)) |− ∀x(G(x) ∨ H(x))
(Note that the converse cannot be proved. For example, let H be ``is left-handed'' and G be ``is right-handed'' and say that x ranges over the domain of humans.)

(d) ∀x ¬F(x) |− ¬(∀x F(x))
(Note that the converse cannot be proved. For example, let F be ``is left-handed'' and say that x ranges over the domain of humans.)