WEEK 10 :: Homework

Construct countermodels demonstrating that the following arguments are invalid. (Work out the answers on paper first, and then input into ∃LOGIC to check.)


C11. Fa ∴ Gb

C12. ∴ ∀x(Fx→∀xFx)

C13. (∀xFx → P) ∴ ∀x(Fx → P)

C14. ∃x(Fx → P) ∴ (∃xFx → P)

C15. ∴ ¬(∃x¬Fx ∧ (Fa ∧ Fb))

C16. ∀x∃y(Fx ↔ Gy) ∴ ∃y∀x(Fx ↔ Gy)

C17. ∃xGx. ∀x(Gx → Hx) ∴ ∀xHx

C18. Fa. Fb. ∃xFx ∴ ∀xFx

C19. ∀y(Fy → ∀xGx). ∃xFx ∴ ∀xFx

C20 ∴ ∀y∃x(Fy ∧ Gx) → ∃x(Gx ∧ ¬Fx)

C21. ∃xGx → ∀x(Gx → Hx). ∀x(Hx ∨ (Jx → Fx)) ∴ Ga → ∃xFx


For each of the following arguments either construct a derivation of the conclusion from the premises or show that it is invalid by constructing a relevant countermodel (Work out the answers on paper first, and then input into ∃LOGIC to check.)


A1. ∃y∀x(Fx ↔ Fy). ∃xFx ∴ ∀xFx

A2. ∃y(Gy → Fy). ∃xFx ∴ (∀xFx ∨ ∃xGx)

A3. (∃x(Fx ∧ ¬Gx) → ∀x(Fx → Hx)). ∃x(Fx ∧ Jx) ∴ (∀x(Fx ∧ ¬Hx) → ∃x(Jx ∧ Gx))

A4. ∴ (∀y∃x(Fy ∧ Gx) ∨ ∃x(Gx ∧ Fx))

   

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