Exercises for Chapter 6, part 1

1. Use the FORALLe and FORALLi 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), FORALLx(F(x) -> G(x)), FORALLy H(y) |- G(charlie) ^ H(charlie)

(b) FORALLx F(x), FORALLy(F(y) -> G(y)) |- FORALLz(G(z) ^ F(z))

(c) (FORALLx H(x)) v (FORALLx G(x)) |- FORALLx(G(x) v 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) FORALLx ~F(x) |- ~(FORALLx 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.)