Resources: Chapter 8 of HLW
For each of the following sequents, provide a counterexample to show that it is invalid.
∀xFx → ∀xGx ⊢ ∀x(Fx→Gx)
∀x(Fx→Gx) ⊢ ∃x(Fx∧Gx)
⊢ ∀xFx ∨ ∀x¬Fx
∃x(Fx→P) ⊢ ∃xFx → P
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 RM of M × M, or it may be presented as an arrow diagram.
∀x∀y∃z(Rxz∧Ryz)
∀x(∃yRyx→∀zRzx)