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# fill in the missing rules

PHIL008, Erich Reck UCR, Spring 2020

Homework Set 3

Solutions for the following problems are due on Friday, May 22, 6pm (usual procedure).

(1) For each of the following two (incorrect) formal proofs, explain what is defective, i.e., which inference rule has been misapplied, in which line, and what exactly is wrong: a) 1 P1 b) 1 Q2 Ù Q4

2 P3 2 Q1 « Q2 3 (P1 Ù P2) ® (P4 Ù P5) 3 Q3 ® Q1

4 P1 Ù P2 ÙI 1 4 Q2 ÙE 1 5 (P4 Ù P5) ®E 3, 4 5 Q1 «E 2, 4

6 P5 ÙE 5 6 Q3 ®E 3, 5 7 Q2 Ù Q3 ÙI 4, 6

(2) For each of the following two (essentially correct but incomplete) formal proofs, fill in the missing rules and line numbers on the right side:

a) 1 P1 b) 1 Q1 Ù Q2 2 P2 Ù P3 2 Q1 ® Q3 3 (P1 Ù P3 ) ® P4 3 Q3 « Q4

4 P3 4 Q1 5 P1 Ù P3 5 Q3 6 P4 6 Q4 7 Q4 Ú Q5

(3) Provide a formal proof (with full notation) for each of the following arguments:

a) P1 Ù P5 , P2 Ù (P3 Ù P4) P1 Ù P3

b) (Q1 Ù Q2) Ù Q3 , Q2 ® Q4 , (Q3 Ù Q4) « Q5 Q1 Ù (Q3 Ù Q5)

c) P1 , P2 ® P3 , (P1 Ù P3) « ¬P4 , ¬P4 ® P5 P2 ® P5

d) Q1 « Q3 , Q2 « ¬Q4 , ¬Q4 ® (Q5 Ù Q6) (Q1 Ù Q2) ® (Q3 Ù Q5 )

e) (P1 Ú P2) ® P3 , (P2 Ú P3) ® P1 P1 « P3

f) Q1 Ú Q2 , Q1 ® (Q5 Ù Q6) , Q2 ® (Q3 Ù Q4), Q4 « Q6 Q6

(*) EXTRA CREDIT PROBLEM (OPTIONAL)

Provide a formal proof (with full notation) for the following argument:

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