Examples Sheet: Proof Theory 2 (1) Complete the proof of the following sequent by adding in the missing propositions. P R , Q ~ R├ ~(P & Q) 1 (1) Ass 2 (2) Ass 3 (3) P & Q Ass 3 (4) P &E 3 1,3 (5) E 1,4 3 (6) Q &E 3 2,3 (7) E 2,6 1,2,3 (8) ~E 5,7 1,2 (9) ~I 3,8 (2) Complete the proof of the following sequent by writing in the rules that should appear on the right (the relevant line numbers on which the rules were applied have been given). P S, Q S ├ (P v Q) S 1 (1) P S 2 (2) Q S 3 (3) (P v Q) 4 (4) P 1,4 (5) S 1,4 6 (6) Q 2,6 (7) S 2,6 1,2,3 (8) S 3,4,5,6,7 1,2 (9) (P v Q) S 3,8 (3) Complete the proof of the following sequent by writing in the numbers that should appear on the left. ├ (PQ) (~Q ~P) (1) (PQ) Ass (2) ~Q Ass (3) P Ass (4) Q E 1,3 (5) ^ ~E 2,4 (6) ~P ~I 3,5 (7) (~Q ~P) I 2,6 (8) (PQ) (~Q ~P) I 1,7 (4) Complete the proof of the following sequent by adding in any missing elements. P (Q v R) , ~Q, ~R ├ ~P 1 (1) P (Q v R) 2 (2) Ass 3 (3) Ass 4 (4) P Ass 1,4 (5) E 1,4 6 (6) Q Ass 2,6 (7) ^ ~E 8 (8) Ass 3,8 (9) 3,8 (10) ^ v E 5,6,7,8,9 1,2,3 (11) ~P ~I 4,10 (5) Construct proofs for the following sequents. i) P Q , ~Q, ├ ~P ii) (P v Q) R├ (P R) & (Q R) iii) P v (Q & R) ├ (P v Q) & (P v R) iv) P S , Q ~ S├ ~(P & Q)
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