Logic for Philosophers
Problem Set 2
Due April 18th
Name: ___________________________________________________
Student ID#_______________________________________________
All of the following sequents are derivable. Provide derivations for them.
1. ∀xFx, ∀x(Fx → Gx) ├ ∀xGx
1
2
1
2
1,2
1,2
1. ∀xFx
2. ∀x(Fx → Gx)
3. Fa
4. (Fa → Ga)
5. Ga
6. ∀xGx
A
A
1 ∀E
2 ∀E
3, 4 →E
5 ∀I
2. Fa, ∀x(Gx → ~Fx) ├ ~Ga
1
2
3
2
2,3
1,2,3
1,2
1. Fa
2. ∀x(Gx → ~Fx)
3. Ga
4. (Ga → ~Fa)
5. ~Fa
6. (Fa & ~Fa)
7. ~Ga
A
A
A
2 ∀E
3, 4 →E
1, 5 &E
3, 6 ~I
3. ∀x(Fx → Gx), ∃xFx ├ ∃xGx
1
2
3
1
1. ∀x(Fx → Gx)
2. ∃xFx
3. Fa
4. (Fa → Ga)
A
A
A
1 ∀E
1,3
1,3
1,2
5. Ga
6. ∃xGx
7. ∃xGx
3, 4 →E
5 ∃I
2, 3, 6 ∃E
4. ∀x(Fx → Gx) ├ (Fa → Ga)
1
1
1. ∀x(Fx → Gx)
1. (Fa → Ga)
A
1 ∀E
5. ∀x(Fx v Gx), ∀x~Fx ├ ∀xGx
1
2
1
2
1,2
1,2
1. ∀x(Fx v Gx)
2. ∀x~Fx
3. (Fa v Ga)
4. ~Fa
5. Ga
6. ∀xGx
A
A
1 ∀E
2 ∀E
3, 4 vE
5 ∀I
6. ∀x(Fx → Gx), ∃x(Fx & Hx) ├ ∃x(Fx & Gx)
1
2
3
3
1
1,3
1,3
1,3
1,2
1. ∀x(Fx → Gx)
2. ∃x(Fx & Hx)
3. (Fa & Ha)
4. Fa
5. (Fa → Ga)
6. Ga
7. (Fa & Ga)
8. ∃x(Fx & Gx)
9. ∃x(Fx & Gx)
A
A
A
3 &E
1 &E
4, 5 →E
4, 6 &I
7 ∃I
2, 3, 8 ∃E
7. ∀x(Fx → Gx) ├ (~∃xGx → ~∃xFx)
1
2
3
4
1. ∀x(Fx → Gx)
2. ~∃xGx
3. ∃xFx
4. Fa
A
A
A
A
1
1,4
1,4
1,2,4
1,2, 3
1,2
1
5. (Fa → Ga)
6. Ga
7. ∃xGx
8. (∃xGx & ~∃xGx)
9. (∃xGx & ~∃xGx)
10. ~∃xFx
11. (~∃xGx → ~∃xFx)
1 ∀E
4, 5 →E
6 ∃I
2, 7 &I
3, 4, 8 ∃E
3, 9 ~I
2, 10 →I
8. (∀xFx & ∀xGx) ├ ∀x(Fx & Gx)
1
1
1
1
1
1
1
1. (∀xFx & ∀xGx)
2. ∀xFx
3. ∀xGx
4. Fa
5. Ga
6. (Fa & Ga)
7. ∀x(Fx & Gx)
A
1 &E
1 &E
2 ∀E
3 ∀E
4, 5 &I
6 ∀I
9. (∃xFx v ∃xGx) ├ ∃x(Fx v Gx)
1
2
3
1,3
5
5
5
1,3
1,2,3
1,2
11
11
11
1,2
1,2
1
1. (∃xFx v ∃xGx)
2. ~∃x(Fx v Gx)
3. ~∃xFx
4. ∃xGx
5. Ga
6. (Fa v Ga)
7. ∃x(Fx v Gx)
8. ∃x(Fx v Gx)
9. (∃x(Fx v Gx) & ~∃x(Fx v Gx))
10. ∃xFx
11. Fa
12. (Fa v Ga)
13. ∃x(Fx v Gx)
14. ∃x(Fx v Gx)
15. (∃x(Fx v Gx) & ~∃x(Fx v Gx))
16. ∃x(Fx v Gx)
A
A
A
1, 3 vE
A
5 vI
6 ∃I
4, 5, 7 ∃E
2, 8 &I
3, 9 ~E
A
11 vI
12 ∃I
10, 11, 13 ∃E
2, 14 &I
2, 15 ~E
10. ∀xFx ├ ~∃x~Fx
1
2
3
1
1,3
3
1,3
1,2
1
1. ∀xFx
2. ∃x~Fx
3. ~Fa
4. Fa
5. (Fa & ~Fa)
6. ~∀xFx
7. (∀xFx & ~∀xFx)
8. (∀xFx & ~∀xFx)
9. ~∃x~Fx
A
A
A
1 ∀E
3, 4 &I
1 ~I
1, 6 &I
2, 3, 7 ∃E
2, 8 ~I