80-211                                                                                                             Spring 2003

 

 

Assignment #2

Due on Friday, January 31st.

 

 

1.  Prove the following sequents using only the 10 basic inference rules (see page 39-40)

 

            (a)        P ├ Q → (P &Q)

(b)        (P→Q) & (P→R) ├ P→ (Q &R)

(c)        P & Q ├ P v Q

(d)        P→Q, R→S ├ (P v R) → (Q v S)

(e)        ~P→P ├ P

 

2. Prove the following sequents using only the 10 basic inference rules and the definition of the biconditional.

 

(a)        P↔Q ├ ~P ↔ ~Q

(b)        (P ↔ ~Q), (Q ↔ ~R) ├ P↔R

 

3.   (a)  Prove the following sequent:

 

P, ~P   Q

 

      (b) What does this sequent suggest about what follows from a contradiction?

 

 

4. Define @ by the following, A@B = ~(A & B).  Using df-@ in a way parallel Df. ↔, find a proof for the following sequent.

 

            (a)  P @ P ├ ~P