Philosophical proof

Symbolic Logic Final Makeup

Provide trees to determine whether the following arguments are valid (10 points each).

1. {(∀x)[Gx ↔ Fx], (∃x)[ Fx ∨ Gx ]} ⊢ (∃x)[ Gx ]

2. {(∀x)[Gx ↔ Fx], (∃x)[ ¬Fx ∨ Gx ]} ⊢ (∃x)[ Gx ∨ Fx ]

3. {(∃x)[ (¬Fx ∧ Hx) → Gx ], (∀x)[ Hx → Fx ] } ⊢ (∃x)[ Gx ]

4. {(∀x)[Fx → (Hx ∨ Gx)], (∃x)[ ¬Fx ∧ Gx ]} ⊢ (∃x)[ Hx ]

5. {(∃x)[ Fx ∨ Gx ], (∀x)[ ¬Hx → ¬Gx], (∃x)[ Hx ↔ Fx ] } ⊢ (∃x)[ Fx ]

Provide trees to determine whether the following formulae are tautologies: (10 points each).

6. (∀x)[ (Fx ↔ ¬x=x) → ¬Fx ]

7. (∀x)[ (Fx ∨ ¬(∃y)[y=x ∧ ¬Fy]) → Fx]

8. (∀x)[ Fx ↔ ( (∃y)[y=x ∧ ¬Fy] ∨ Fx) ]

Provide trees to determine whether the following formulae are contradictions (10 points each).

9. (∀x)[ (Fx ∧ ¬x=x) ↔ Fx ]

10. (∀x)[ (∃y)[y=x] ∧ Fx ] ↔ ¬(∀x)[ Fx ∨ Gx ]

Provide derivations to show that the following arguments are valid (10 points each).

11. { } ⊢ (∀x)(∃y)[ y=x ∧ ¬(Fy ↔ ¬Fx) ]

12. { } ⊢ (∀x)(∀y)[ x=y → (Fx → ¬(∃z)[ (¬Fz ∧ Fy) ∧ z=y ]) ]

13. { (∀x)(∀y)[ G(x,y) ↔ ¬G(y,x) ], (∃x)[G(x,b)] }

⊢ (∀x)[x=b → (∃y)[¬G(y,x)]]

14. { (∃x)[ Fx ∨ Gx ], (∀x)[ ¬Hx → ¬Gx], (∀x)[ Hx ↔ Fx ], }

⊢ (∃x)[ Fx ]

15. { (∃x)(∀y)[y=x → ¬(G(y,x) ↔ G(x,y))], (∃x)[G(x,b) ∧ G(b,x)] }

⊢ (∃x)(∃y)[¬x=y]

16. {(∃x)(∀y)[y=x → ¬G(y,x)], (∀x)[(∃y)[¬G(x,y)] → (Hx ∨ G(x,x))], (∀x)(∀y)[¬Hx ↔ ¬[Fy ↔ y=x]]}

⊢ (∃x)[Hx ∧ Fx]