Unit 3 Part 2 2011 Answers

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

This preview shows pages 1–3. Sign up to view the full content.

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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 19

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online