# Homework

PHIL008, Erich Reck UCR, Spring 2020

Homework Set 3

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

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(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:

P ® ( Q ® R) Q ® ( P ® R)