PREDICATE CALCULUS IN-CLASS EXERCISES

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

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]

(1)

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)]

(2)

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 one or two statements to the premises will produce a formal proof of validity. Supply the statement and indicate the rule:

Example

1. Fy / ˫ x(Fx)

2. ˫ x(Fx) [1 I (UG)]

(3)

1. x(Fx) / ˫ Fy

2. ˫

(4)

1. Ǝx(Fx) / ˫ Fa

2. ˫

(5)

1. x(Fx) / ˫ Ǝx(Fx)

2.

3. ˫

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

Example

1. x(Fx → Gx)

2. Fa / ˫ Ga

3. Fa → Ga [1 E]

4. ˫ Ga [3, 2 MP (→E)]

(6)

1. x (Fx → Gx) / ˫  ~Ga → ~Fa

2.

3. ˫

(7)

1. x (Fx → Gx) / ˫ x(~Gx → ~Fx)

2.

3.

4. ˫

(8)

1. x(Fx → Gx)

2. Fy / ˫ x(Gx)

3.

4.

5. ˫

(9)

1. x(Fx → Gx) / ˫ x(~Fx v Gx)

2.

3.

4. ˫

(10)

1. x(Fx → Gx)

2. x(Gx → Hx) / ˫ x(Fx → Hx)

3. Fy → Gy [1 E]

4.

5.

6. ˫

(11)

1. x (~Fx → ~Gx) / ˫ Ǝx (~Gx v Fx)

2.

3.

4.

5. ˫ Ǝx (~Gx v Fx) [3 ƎI (EG)]

(12)

1. Ǝx(~Fx v ~Gx) / ˫ Ǝx~(Fx & Gx)

2.

3.

4. ˫

(13)

1. Ǝx(~Fx) / ˫ Ǝx~(Fx & Gx)

2.

3.

4.

5. ˫ Ǝx~(Fx & Gx) [4 ƎI (EG)]

(14)

1. Ǝx (Fx & Gx)

2. Fa → (Ga → Ha) / ˫ Ǝx (Hx)

3.

4.

5.

6. ˫ Ǝx(Hx) [5 ƎI (EG)]

(15)

1. x(~Gx → ~Fx)

2. Ǝx(Fx) / ˫ Ǝx(Gx)

3. ~Ga → ~Fa [1 E]

4.

5.

6.

7. ˫ Ǝx(Gx) [6 ƎI (EG)]

The following examples involve relational predicates. 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.

Example

1. Ǝx Ǝy(Fxy & Fyx) / ˫ Fba

2. Ǝy(Fay & Fya) [1 ƎE (EI)]

3. Fab & Fba [2 ƎE (EI)]

4. Fba [3 SIMP (&E)]

(16)

1. x y(Fxy → Fyx)

2. Fab / ˫ Fba

3.

4.

5. ˫                  ]

(18)

1. x(Fx → Gx)

2. Fa

3. a=b / ˫ Gb

4.

5.

6. ˫