Question

Provide natural deductions to demonstrate the following. (The only new rules needed for these are 3-Introduction and V-Elimination. You may also use the Derived Rules from TFL [useful for #2!).) [5pts each] 1. Vx(FxGx) + 3x (GxA3yFy) 2. Vx(FxVGx), Vy-Gy+ 3z(FzA-Gz) 3. VxTyRxy + 3y=xRxy 4. Vx(Fx→ VyHy) = x(VyFyHx)

Answer 1

|vx(FX GX) VE 1 AE 2 NE 2 2 Fa A Ga 3 Ga 4 Fa ByFy 6 Ga A ayFy 7 = x(Gx 1 3yFy) 314 AI 3,5 31 6

1vx(Fx V Gx) 2 Vy-Gy Fa v Ga 4 -Ga 5 Fa 6 Fa 1 -Ga 7 z(Fz A-G) VE 1 VE 2 DS 3,4 AI 4,5 31 6

1 xWyRxy 2 VyRay 3 Rab 3xRxb 5 yaxRxy VE 1 VE 2 313 31 4

vx(Fx → VyHy) WyFy VE2 VE 1 Fa - WyHy VyHy HD 7 | vyғy — нь 8 Ex(VyFy — Hx) -E 3,4 VE 5 -I 2-6 31 7