Phil 3 proofs test 5

Dom’t think if i see a quantifier i have to eliminate it

Two incomplete proofs, fill in the blanks

1. label premises

2. AS

3. etc.

3 proofs

1. longest is 8 lines

Ax~F(x) |- ~ExF(x)

1. Ax~F(x) Pr

   2. Ex F(x) As

      3. F(a) As

      4. ~F(a) AE, 1

      5. |_ ~E, 3,4

   6. |_ EE, 2, 3-5

7. ~ExF(x) ~I, 2-6

for EE, cite the existential line, as well as the subproof used

ExEy(F(x) ^ G(y)) |- Ex(F(x) ^ ExG(X)

• two Existential, means two subproofs

1. ExEy(F(x) ^ G(y))

   2. Ey ( F (a) & G (y) ) As

      3. F(a) ^ G(b) As (b because new name, EE req)

      4. F(a) ^E3

      5. G(b) ^E3

      6. ExF(x) EI,4

      7. ExG(x) EI,4 ( would have done a Ey if need)

      8. ExF(x) ^ ExG(x) ^I,7,6

   9. Ex F(x) ^ ExG(x) EE. 2, 3-8

10. Ex F(x) ^ ExG(x) EE. 1, 2-9

• Allowed to discharge from 2, 3-8 because of one of y

Ax[EyF(y,x) → AzF(x,z)] |- AyAx ( F(y,x) → F)x,y))

1. Ax[EyF(y,x) → AzF(x,z)] Pr

   2. F(b,a) As

   3. 3

   4. 3

   5. 3

   6. 3

   7. F(a,b)

2. F(b,a) → F(a,b) → I, 2-

6. AxF(b,x) → F(x, b))