ANSWERS FOR EXERCISES IN VOLUME I, CHAPTER 5 5-1. a) 1 |-P>-D P 2 |-D>-F P 3 |-P P +--------------------- 4 |-D 1,3,>E 5 |-F 2,4,>E 5-1. b) 1 |-C>-D P 2 |-C P +--------------------- 3 |-D 1,2,>E 4 |-DvE 3,vI 5-1. c) 1 |Fv-G P 2 |-F P 3 |GvK P +--------------------- 4 |-G 1,2,vE 5 |K 3,4,vE 5-1. d) 1 |A>B P 2 |A P 3 |B>-C P 4 |CvD P +--------------------- 5 |B 1,2,>E 6 |-C 3,5,>E 7 |D 4,6,vE 8 |DvE 7,vI 5-1. e) 1 |Lv-M P 2 |-L P 3 |MvD P 4 |D>H P +--------------------- 5 |-M 1,2,vE 6 |D 3,5,vE 7 |H 4,6,>E 5-1. f) 1 |C P 2 |C>(HvA) P 3 |-H P +--------------------- 4 |HvA 1,2,>E 5 |A 3,4,vE 6 |Av-K 5,vI 5-1. g) 1 |(Kv-D)>F P 2 |K P +--------------------- 3 |Kv-D 2,vI 4 |F 1,3,>E 5 |FvD 4,vI 5-1. h) 1 |D P 2 |(DvB)>-G P 3 |(-Gv-H)>(GvQ) P +--------------------- 4 |DvB 1,vI 5 |-G 2,4,>E 6 |-Gv-H 5,vI 7 |GvQ 3,6,>E 8 |Q 5,7,vE 9 |Qv-A 8,vI 5-1. i) 1 |(Mv-T)>(AvJ) P 2 |-A P 3 |BvM P 4 |-A>-B P +--------------------- 5 |-B 2,4,>E 6 |M 3,5,vE 7 |Mv-T 6,vI 8 |AvJ 1,7,>E 9 |J 2,8,vE 10 |JvD 9,vI 5-2. a) 1 |A>B P 2 |B>C P 3 |C>D P +--------------------- 4 | |A A | +--------------------- 5 | |A>B 1,R 6 | |B 4,5,>E 7 | |B>C 2,R 8 | |C 6,7,>E 9 | |C>D 3,R 10 | |D 8,9,>E 11 |A>D 4-10,>I 5-2. b) 1 |NvP P +--------------------- 2 | |-N A | +--------------------- 3 | |NvP 1,R 4 | |P 2,3,vE 5 |-N>P 2-4,>I 5-2. c) 1 |B P +--------------------- 2 | |A A | +--------------------- 3 | |B 1,R 4 |A>B 2-3,>I 5-2. d) 1 |-B P +--------------------- 2 | |BvC A | +--------------------- 3 | |-B 1,R 4 | |C 2,3,vE 5 |(BvC)>C 2-4,>I 5-2. e) 1 |K>-D P 2 |DvH P +--------------------- 3 | |K A | +--------------------- 4 | |K>-D 1,R 5 | |-D 3,4,>E 6 | |DvH 2,R 7 | |H 5,6,vE 8 |K>H 3-7,>I 5-2. f) 1 |A>B P +--------------------- 2 | |A A | +--------------------- 3 | |A>B 1,R 4 | |B 2,3,>E 5 | |BvC 4,vI 6 |A>(BvC) 2-5,>I 5-2. g) 1 |F>(CvM) P 2 |-C P +--------------------- 3 | |F A | +--------------------- 4 | |F>(CvM) 1,R 5 | |CvM 3,4,>E 6 | |-C 2,R 7 | |M 5,6,vE 8 |F>M 3-7,>I 5-2. h) 1 |(DvB)>J P +--------------------- 2 | |D A | +--------------------- 3 | |DvB 2,vI 4 | |(DvB)>J 1,R 5 | |J 3,4,>E 6 |D>J 2-5,>I 5-2. i) 1 |A>K P 2 |(KvP)>L P +--------------------- 3 | |A A | +--------------------- 4 | |A>K 1,R 5 | |K 3,4,>E 6 | |KvP 5,vI 7 | |(KvP)>L 2,R 8 | |L 6,7,>E 9 |A>L 3-8,>I 5-2. j) 1 |Q>-S P 2 |Q>(SvF) P +--------------------- 3 | |Q A | +--------------------- 4 | |Q>-S 1,R 5 | |-S 3,4,>E 6 | |Q>(SvF) 2,R 7 | |SvF 3,6,>E 8 | |F 5,7,vE 9 |Q>F 3-8,>I 5-2. k) 1 |P P 2 |(-DvK)>B P 3 |(Fv-D)>-K P 4 |P>(Kv-F) P +--------------------- 5 | |Fv-D A | +--------------------- 6 | |(Fv-D)>-K 3,R 7 | |-K 5,6,>E 8 | |P 1,R 9 | |P>(Kv-F) 4,R 10 | |Kv-F 8,9,>E 11 | |-F 7,10,vE 12 | |-D 5,11,vE 13 | |-DvK 12,vI 14 | |(-DvK)>B 2,R 15 | |B 13,14,>E 16 | |Bv-P 15,vI 17 |(Fv-D)>(Bv-P) 5-16,>I 5-3. a) 1 |B&(B>-A) P +--------------------- 2 |B 1,&E 3 |B>-A 1,&E 4 |-A 2,3,>E 5-3. b) 1 |-C<->(AvB) P 2 |A P +--------------------- 3 |(AvB)>-C 1,<>E 4 |AvB 2,vI 5 |-C 3,4,>E 5-3. c) 1 |A>-B P 2 |BvC P +--------------------- 3 | |A A | +--------------------- 4 | |A>-B 1,R 5 | |-B 3,4,>E 6 | |BvC 2,R 7 | |C 5,6,vE 8 |A>C 3-7,>I 5-3. d) 1 |D P 2 |(D&A)>C P +--------------------- 3 | |A A | +--------------------- 4 | |D 1,R 5 | |D&A 3,4,&I 6 | |(D&A)>C 2,R 7 | |C 5,6,>E 8 |A>C 3-7,>I 5-3. e) 1 |AvB P +--------------------- 2 | |-A&-B A | +--------------------- 3 | |-A 2,&E 4 | |-B 2,&E 5 | |AvB 1,R 6 | |B 3,5,vE 7 |-(-A&-B) 2-6,-I 5-3. f) 1 |A&B P +--------------------- 2 | |A A | +--------------------- 3 | |A&B 1,R 4 | |B 3,&E 5 |A>B 2-4,>I 6 | |B A | +--------------------- 7 | |A&B 1,R 8 | |A 7,&E 9 |B>A 6-8,>I 10 |A<->B 5,9,<>I 5-3. g) 1 |-A>B P 2 |-A>-B P +--------------------- 3 | |-A A | +--------------------- 4 | |-A>B 1,R 5 | |B 3,4,>E 6 | |-A>-B 2,R 7 | |-B 3,6,>E 8 |--A 3-7,-I 9 |A 8,-E 5-4. a) 1 |C&-H P +--------------------- 2 |-H 1,&E 5-4. b) 1 |JvD P 2 |---D P +--------------------- 3 |-D 2,-E 4 |J 1,3,vE 5-4. c) 1 |A&B P +--------------------- 2 |A 1,&E 3 |B 1,&E 4 |B&A 2,3,&I 5-4. d) 1 |A>-D P 2 |--A P +--------------------- 3 |A 2,-E 4 |-D 1,3,>E 5-4. e) 1 |G>D P 2 |G>-D P +--------------------- 3 | |G A | +--------------------- 4 | |G>D 1,R 5 | |D 3,4,>E 6 | |G>-D 2,R 7 | |-D 3,6,>E 8 |-G 3-7,-I 5-4. f) 1 |A<->-B P +--------------------- 2 |-B>A 1,<>E 5-4. g) 1 |M P 2 |Rv-H P +--------------------- 3 |M&(Rv-H) 1,2,&I 5-4. h) 1 |A&(B&C) P +--------------------- 2 |A 1,&E 3 |B&C 1,&E 4 |B 3,&E 5 |C 3,&E 6 |A&B 2,4,&I 7 |(A&B)&C 5,6,&I 5-4. i) 1 |-C>D P 2 |D>-C P +--------------------- 3 |D<->-C 1,2,<>I 5-4. j) 1 |A<->-B P 2 |-B P +--------------------- 3 |-B>A 1,<>E 4 |A 2,3,>E 5-4. k) 1 |-C>---A P 2 |-C P +--------------------- 3 |---A 1,2,>E 4 |-A 3,-E 5-4. l) 1 |K>-B P 2 |B&F P +--------------------- 3 | |K A | +--------------------- 4 | |K>-B 1,R 5 | |-B 3,4,>E 6 | |B&F 2,R 7 | |B 6,&E 8 |-K 3-7,-I 5-4. m) 1 |-P P 2 |-Q P +--------------------- 3 | |PvQ A | +--------------------- 4 | |-P 1,R 5 | |Q 3,4,vE 6 | |-Q 2,R 7 |-(PvQ) 3-6,-I 5-4. n) 1 |(N>K)&(N>L) P +--------------------- 2 | |N A | +--------------------- 3 | |(N>K)&(N>L) 1,R 4 | |N>K 3,&E 5 | |N>L 3,&E 6 | |K 2,4,>E 7 | |L 2,5,>E 8 | |K&L 6,7,&I 9 |N>(K&L) 2-8,>I 5-4. o) 1 |D>(AvF) P 2 |D>-F P 3 |-A P +--------------------- 4 | |D A | +--------------------- 5 | |D>(AvF) 1,R 6 | |AvF 4,5,>E 7 | |-A 3,R 8 | |F 6,7,vE 9 | |D>-F 2,R 10 | |-F 4,9,>E 11 |-D 4-10,-I 12 |-D&-A 3,11,&I 5-4. p) 1 |H<->J P 2 |H<->K P +--------------------- 3 |H>J 1,<>E 4 |J>H 1,<>E 5 |H>K 2,<>E 6 |K>H 2,<>E 7 | |J A | +--------------------- 8 | |J>H 4,R 9 | |H 7,8,>E 10 | |H>K 5,R 11 | |K 9,10,>E 12 |J>K 7-11,>I 13 | |K A | +--------------------- 14 | |K>H 6,R 15 | |H 13,14,>E 16 | |H>J 3,R 17 | |J 15,16,>E 18 |K>J 13-17,>I 19 |J<->K 12,18,<>I 5-5. X Y X&Y&Z X&Y&Z X&Y&Z Z X &E Y &E Z &E X&Y&Z &I X Y Z XvYvZ -X XvYvZ vI XvYxZ vI XvYxZ vI YvZ vE XvYvZ XvYvZ -Y -Z XvZ vE XvY vE 5-6. For most valid arguments, some truth assignments to the sentence letters which make one or more premise false will make the conclusion true and some such assignments will make the conclusion false. For example, the argument, "A, A->B. Therefore B." is valid. Making A true and B false makes one premise, `A->B',and the conclusions, `B' false. Making `A' false and `B' true makes one premise, `A' false and the conclusion `B' true. Some valid arguments do not have so much freedom. "B. Therefore B." is valid. Obviously in this example the premise and conclusion always have the same truth value. "B. Therefore Av-A." is also valid. In this case the conclusion is a logical truth and so always true. But these are special cases. In general, anything can happen, depending on the details of the case.