PREDICATE CALCULUS EXERCISES

10/29/2011

State the rules of inference and replacement used in the following proofs

(1) Example

1. ₳x(Fx → Gx)

2. Ǝx(Fx & Hx) / ˫ Ǝx (Hx & Gx)

3. Fa & Ha [2 ƎE]

4. Fa → Ga [1 ₳E]

5. Fa [3 &E SIMP]

6. Ga [4, 5 →E MP]

7. Ha & Fa [3 COM]

8. Ha [7 &E SIMP]

9. Ha & Ga [8 &I CONJ]

10. ˫ Ǝx (Hx & Gx) [9 ƎI]

(2)

1. ₳x(Fx →  Gx) / ˫ Ǝx[~Fx v (Gx v Hx)]

2. Fa →  Ga

3. ~Fa v Ga

4. (~Fa v Ga) v Ha

5.  ~Fa v (Ga v Ha)

6. ˫ Ǝx[~Fx v (Gx v Hx)]

(3)

1. Ǝx(~Fx)

2. Ǝx (~Gx) / ˫ Ǝx(Fx ↔ Gx)

3. ~Fa

4. ~Ga

5. ~Fa & ~Ga

6. (Fa & Ga) v (~Fa & ~Ga)

7. Fa ↔ Ga

8. ˫ Ǝx(Fx ↔ Gx)

Adding three statements to the premises will produce a formal proof of validity. Supply these statements and indicate the rules of inference and replacement used.

(1) Example

1. ₳x(Fx → Gx)

2. Fa / ˫ Ga

3. Fa → Ga [1 ₳E]

4. ˫ Ga [3, 2 →E MP]

(2)

1. ₳x(~Fx v ~Gx) / ˫ ~(Fa & Ga)

(3)

1. ₳x(Fx → Gx)

2. Fa / ˫ Ǝx(Gx)

(4)

1. ₳x(Fx → Gx)

2. ₳x(Gx → Hx) / ˫ ₳x(Fx → Hx)

3. Fx → Gx [1 ₳E]