Unit 3 Part 2 2011 Answers

Unit 3 Part 2 2011 Answers - DERIVATIONS NATURAL DEDUCTION...

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Unit 3 Part 2: Derivations with Conjunction, Disjunction and Biconditional Answers Niko Scharer 1 DERIVATIONS NATURAL DEDUCTION Part 2 Solutions 3.11 E1 Which inference rule if any justifies the following arguments? (S, ADJ, ADD, MTP, BC, CB or none) a) R (P ~S) (P ~S) R ____________ (P ~S) R CD b) (P Q) (S T) ______________ S T sl c) (P (S ~Q)) ________________ ~P (P (S ~Q)) add d) P R S ___________ P R S none e) Q ~(S P) (S P) _____________ Q none (2 steps: pr2 dn, pr1 mtp) f) (V Z) (~W Y) ~(~W Y) _______________ V Z mtp g) S ~R S ~R __________ S ~R none is not a symbol of SL h) P P R ____________ P R P adj 3.11 EG1 Let’s try out the new rules P Q. (R P) ~S . S T. (V W) Q. W V. T (V W) 1 Show T (V W) First line is always a show line. Show conclusion! The conclusion is a conjunction. Thus, we probably want to derive each conjunct separately (T and V W) and then use ADJ to join them and achieve our goal. 2 P pr1 S Premise 1 is a conjunction. We can use S to free up the conjuncts – in this instance, to infer P. 3 Q pr1 S We can use S again to free up the other conjunct – this time to infer Q. 4 R P 2 ADD The antecedent of premise 2 is (R P) – we can easily derive that from line 2 using ADD. I make sure I’m putting the two disjuncts in the order that they appear in premise 2. 5 ~S 4 pr2 MP Now I can use MP with lines 4 and the second premise to get ~S. 6 T 5 pr3 MTP Premise 3 is a disjunction and line 5 is the negation of one of the disjuncts. Thus, I can use MTP to derive the other disjunct, T. That will give me one half of my goal –the first conjunct.
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Unit 3 Part 2: Derivations with Conjunction, Disjunction and Biconditional Answers Niko Scharer 2 Now I just need to derive the other half of my goal – (V W). P Q. (R P) ~S . S T. (V W) Q. W V. T (V W) 1 Show T (V W) My goal is to get both conjuncts: T and V W. Then I’ll use ADJ. 2 P pr1 SL Here I am using the alternate justification for S. 3 Q pr1 SR Here I am using the alternate justification for S. 4 R P 2 ADD 5 ~S 4 pr2 MP 6 T 5 pr3 MTP Now I have half my goal. Now I need the other half: V W If I had both V W and W V then I could achieve my goal by using CB. I already have W V – that’s the last premise. So all I really want is V W. 7 Q (V W) pr4 BC Premise 4 is a biconditional. One side is the same as line 3. The other side is the sentences that I want: V W. By using BC on premise 4, I can turn it into a usable conditional. I could derive either Q (V W) OR (V W) Q with BC. I make sure that I derive the former so that Q is the antecedent. 8
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Unit 3 Part 2 2011 Answers - DERIVATIONS NATURAL DEDUCTION...

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