Derivation Exercises Unit 3 Part 2 Selected Answers

Derivation Exercises Unit 3 Part 2 Selected Answers - Unit...

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1 Answers to selected derivations for Unit 3 part 2 Unit 3 Part 2: Derivations Exercises Derivations with AND, OR and BICONDITIONAL . SECTION 1: use only MP, MT, R, DN, ADD, S, ADJ, MTP, BC and CB 1. (P Q) (R S). (P S) Q 1 Show (P S) Q show conc 2 P Q pr1 S or SL - must use S on the main ‘ 3 R S pr1 S or SR 4 P 2 S 5 Q 2 S 6 S 3 S 7 P S 4 6 ADJ must adjoin P S first 8 (P S) Q 5 7 ADJ 9 8 dd 2. P Q. ~Q. P R 1 Show P R show conc 2 P Q pr1 3 ~Q pr2 4 P 2 3 mtp 5 P R 4 add 6 5 dd 3. P Q. Q R. R P 1 Show R P show conc 2 Show R P show one direction of the 3 R ass cd show line 2 is Ass ant., show cons. 4 R Q pr2 BC 5 Q 3 4 mp 6 Q P pr1 BC 7 P 5 6 mp, cd now 2 is shown and available to use 8 Show P R show the other direction of the 9 P ass cd show line 8 is . Ass ant, show cons. 10 P Q pr1 BC 11 Q 9 10 mp 12 Q R pr2 BC 13 R 11 12 mp, cd now 8 is shown and available to use 14 R P 2 8 CB dd the derivation is direct (but used two conditional subderivations).
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2 Answers to selected derivations for Unit 3 part 2 4. W P. P S Q. ~W Q 1 Show ~W Q show conc 2 ~W ass cd goal: Q 3 P 2 pr1 mtp 4 P S 3 add Disjoin ‘P’ to ‘S’ so it matches ant. of pr2. 5 Q 4 pr2 mp 6 5 cd 5. P Q (~P Q S) 1 Show P Q (~P Q S) show conc 2 P Q ass cd goal: ~P Q S 3 show ~P Q S) show consequence of 1 4 ~P ass cd goal: Q S 5 Q 2 4 mtp 6 Q S 5 add 7 6 cd 8 3 cd 6. P (Q R). R S. ~S P. W Q R. T 1 Show T show conc 2 ~T ass id premises must be inconsistent, since no T in them. 3 ~S pr3 s 4 ~R pr2 3 mt 5 ~(W Q) pr4 4 mt 6 P pr3 s 7 P (Q R) pr1 bc 8 Q R 6 7 mp 9 Q 4 8 mtp 10 W Q 9 add 11 5 10 id 7. R ~S. S ~T. W T. R ~W 1 Show R ~W show conc: goal R, ~W 2 S pr2 s 3 ~~S 2 dn 4 R 3 pr1 mtp 5 ~T pr2 s 6 W T pr3 bc 7 ~W 5 6 mt 8 R ~W 4 7 adj, dd
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3 Answers to selected derivations for Unit 3 part 2 8. P (Q R). Q S. ~P (~R S) 1 Show ~P (~R S) show conc 2 ~P ass cd assume ant. of 1, show cons. 3 show ~R S 4 ~R ass cd assume ant of 3, new goal: cons. S 5 Q R 2 pr1 mtp line 2 is available to use with pr1 6 Q 4 5 mtp 7 Q S pr2 BC choose direction of to match line 6 8 S 6 7 mp, cd cd complete, box back to 4 & cancel 3 9 3 cd 3 is cons. of 1, cd done, box & cancel. 9. ( ~S T) (W S). (~T ~X) (X W) 1 Show (~T ~X) (X W) show conc 2 ~T ~X ass cd assume ant, show cons. goals: X, W 3 ~S T pr1 sl 4 W S pr1 sr 5 ~S 3 sl 6 T 3 sr 7 W 4 5 mtp one half of the goal!
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Derivation Exercises Unit 3 Part 2 Selected Answers - Unit...

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