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. ˫