Resources: Chapter 8 of HLW

A.

For each of the following sequents, provide a counterexample to show that it is invalid.

1. ∀xFx → ∀xGx ⊢ ∀x(Fx→Gx)

2. ∀x(Fx→Gx) ⊢ ∃x(Fx∧Gx)

3. ⊢ ∀xFx ∨ ∀x¬Fx

4. ∃x(Fx→P) ⊢ ∃xFx → P

B.

For each of the following sentences, provide one interpretation in which it is true and one interpretation in which it is false. An interpretation may be presented by giving a set M and a subset R^M of M × M, or it may be presented as an arrow diagram.

1. ∀x∀y∃z(Rxz∧Ryz)

2. ∀x(∃yRyx→∀zRzx)