First Order Logic Questions for UI and EG

First Order Logic Practice Exercises:
Universal Instantiation and Existential Generalization

For each problem below, provide a derivation of the conclusion from the premises using the rules of natural deduction (&E, &I, vE, vI, MP, MT, DN, <–>I, <–>E, –>I, and RAA) and the two new rules in first order logic (UI, EG).

1. ∀x(Fxb –> Gxb), ~Gab ∴ ∃x~Fax

2. Fe v ∀xGx, ∀x~Fx & (Gb –> He) ∴ ∃xHx

3. Gac, ∀x∀y(Hxy <–> Gxy) ∴ Hac v Kde

4. ∀x(Fx –> ~Gx) ∴ ∀xGx –> ∃x~Fx

5. ∃x∃yHxy –> ∀x(Gx & Lx), Gc –> ∀xKx ∴ Hac –> ∃x(Kx & Lc)

6. ∀x∀y(Fxy –> Gxy), Fac, ∃x∃yGxy –> Ha ∴ ∃xHx

7. Fcd <–> Ga, ∀x∀yFxy & He ∴ ∃x∃y(Gx & Hy)

8. ∀x(Fx v Gx), ∃xGx –> Ka, ∀x~Kx & ∀xLx ∴ Fb