ANSWERS TO EXERCISES IN VOLUME I, CHAPTER 7 These solutions come from Teller's answer book. They differ slightly from the conventions for Weston's computer program. These differences include the following: In the computer program, the ST rule generates a line number, but not in these answers. In these answers the biconditional introduction rule is called <>I. 7-1. a) 1 |AvB P +--------------------- 2 | |A A | +--------------------- 3 | |BvA 2,vI | 4 | |B A | +--------------------- 5 | |BvA 4,vI 6 |BvA 1,2-3,4-5,AC 7-1. b) 1 |Av(BvC) P +--------------------- 2 | |A A | +--------------------- 3 | |AvB 2,vI 4 | |(AvB)vC 3,vI | 5 | |BvC A | +--------------------- 6 | | |B A | | +--------------------- 7 | | |AvB 6,vI 8 | | |(AvB)vC 7,vI | | 9 | | |C A | | +--------------------- 10 | | |(AvB)vC 9,vI 11 | |(AvB)vC 5,6-8,9-10,AC 12 |(AvB)vC 1,2-4,5-11,AC 7-1. c) 1 |(AvB)&(B>C) P +--------------------- 2 |AvB 1,&E 3 |B>C 1,&E 4 | |A A | +--------------------- 5 | |AvC 4,vI | 6 | |B A | +--------------------- 7 | |B>C 3,R 8 | |C 6,7,>E 9 | |AvC 8,vI 10 |AvC 2,4-5,6-9,AC 7-1. d) 1 |(A&B)v(A&C) P +--------------------- 2 | |A&B A | +--------------------- 3 | |A 2,&E 4 | |B 2,&E 5 | |BvC 4,vI 6 | |A&(BvC) 3,5,&I | 7 | |A&C A | +--------------------- 8 | |A 7,&E 9 | |C 7,&E 10 | |BvC 9,vI 11 | |A&(BvC) 8,10,&I 12 |A&(BvC) 1,2-6,7-11,AC 7-1. e) 1 |A&(BvC) P +--------------------- 2 |A 1,&E 3 |BvC 1,&E 4 | |B A | +--------------------- 5 | |A 2,R 6 | |A&B 4,5,&I 7 | |(A&B)v(A&C) 6,vI | 8 | |C A | +--------------------- 9 | |A 2,R 10 | |A&C 8,9,&I 11 | |(A&B)v(A&C) 10,vI 12 |(A&B)v(A&C) 3,4-7,8-11,AC 7-1. f) 1 |Av(B&C) P +--------------------- 2 | |A A | +--------------------- 3 | |AvB 2,vI 4 | |AvC 2,vI 5 | |(AvB)&(AvC) 3,4,&I | 6 | |B&C A | +--------------------- 7 | |B 6,&E 8 | |C 6,&E 9 | |AvB 7,vI 10 | |AvC 8,vI 11 | |(AvB)&(AvC) 9,10,&I 12 |(AvB)&(AvC) 1,2-5,6-11,AC 7-1. g) 1 |(AvB)&(AvC) P +--------------------- 2 |AvB 1,&E 3 |AvC 1,&E 4 | |A A | +--------------------- 5 | |Av(B&C) 4,vI | 6 | |B A | +--------------------- 7 | |AvC 3,R 8 | | |A A | | +--------------------- 9 | | |Av(B&C) 8,vI | | 10 | | |C A | | +--------------------- 11 | | |B 6,R 12 | | |B&C 10,11,&I 13 | | |Av(B&C) 12,vI 14 | |Av(B&C) 7,8-9,10-13,AC 15 |Av(B&C) 2,4-5,6-14,AC 7-1. h) 1 |KvL P 2 |K<->L P +--------------------- 3 | |K A | +--------------------- 4 | |K<->L 2,R 5 | |K>L 4,<>E 6 | |L 3,5,>E 7 | |K&L 3,6,&I | 8 | |L A | +--------------------- 9 | |K<->L 2,R 10 | |L>K 9,<>E 11 | |K 8,10,>E 12 | |K&L 8,11,&I | 13 |K&L 1,3-7,8-12,AC 7-1. i) 1 |(D>G)v(D>I) P +--------------------- 2 | |D>G A | +--------------------- 3 | | |D A | | +--------------------- 4 | | |D>G 2,R 5 | | |G 3,4,>E 6 | | |GvI 5,vI 7 | |D>(GvI) 3-6,>I | 8 | |D>I A | +--------------------- 9 | | |D A | | +--------------------- 10 | | |D>I 8,R 11 | | |I 9,10,>E 12 | | |GvI 11,vI 13 | |D>(GvI) 9-12,>I 14 |D>(GvI) 1,2-7,8-13,AC 7-1. j) 1 |-CvK P 2 |A>D P +--------------------- 3 | |-C A | +--------------------- 4 | | |AvC A | | +--------------------- 5 | | |-C 3,R 6 | | |A 4,5,vE 7 | | |A>D 2,R 8 | | |D 6,7,>E 9 | | |KvD 8,vI 10 | |(AvC)>(KvD) 4-9,>I | 11 | |K A | +--------------------- 12 | | |AvC A | | +--------------------- 13 | | |K 11,R 14 | | |KvD 13,vI 15 | |(AvC)>(KvD) 12-14,>I 16 |(AvC)>(KvD) 1,3-10,11-15,AC 7-1. k) 1 |-HvM P 2 |-M>-C P +--------------------- 3 | |HvC A | +--------------------- 4 | | |H A | | +--------------------- 5 | | |-HvM 1,R 6 | | | |-H A | | | +--------------------- 7 | | | |H 4,R 8 | | |--H 6-7,-I 9 | | |M 5,8,vE | | 10 | | |C A | | +--------------------- 11 | | | |-M A | | | +--------------------- 12 | | | |-M>-C 2,R 13 | | | |-C 11,12,>E 14 | | | |C 10,R 15 | | |--M 11-14,-I 16 | | |M 15,-E 17 | |M 3,4-9,10-16,AC 18 |(HvC)>M 3-17,>I 7-1. l) 1 |(S&J)v(-S&-J) P +--------------------- 2 | |S&J A | +--------------------- 3 | | |S A | | +--------------------- 4 | | |S&J 2,R 5 | | |J 4,&E 6 | |S>J 3-5,>I 7 | | |J A | | +--------------------- 8 | | |S&J 2,R 9 | | |S 8,&E 10 | |J>S 7-9,>I 11 | |S<->J 6,10,<>I | 12 | |-S&-J A | +--------------------- 13 | | |S A | | +--------------------- 14 | | | |-J A | | | +--------------------- 15 | | | |-S&-J 12,R 16 | | | |-S 15,&E 17 | | | |S 13,R 18 | | |--J 14-17,-I 19 | | |J 18,-E 20 | |S>J 13-19,>I 21 | | |J A | | +--------------------- 22 | | | |-S A | | | +--------------------- 23 | | | |-S&-J 12,R 24 | | | |-J 23,&E 25 | | | |J 21,R 26 | | |--S 22-25,-I 27 | | |S 26,-E 28 | |J>S 21-27,>I 29 | |S<->J 20,28,<>I 30 |S<->J 1,2-11,12-29,AC 7-1. m) 1 |K>(FvC) P 2 |J>(CvD) P 3 |-C P +--------------------- 4 | |-(FvD) A | +--------------------- 5 | | |KvJ A | | +--------------------- 6 | | | |K A | | | +--------------------- 7 | | | |K>(FvC) 1,R 8 | | | |FvC 6,7,>E 9 | | | |-C 3,R 10 | | | |F 8,9,vE 11 | | | |FvD 10,vI | | | 12 | | | |J A | | | +--------------------- 13 | | | |J>(CvD) 2,R 14 | | | |CvD 12,13,>E 15 | | | |-C 3,R 16 | | | |D 14,15,vE 17 | | | |FvD 16,vI 18 | | |FvD 5,6-11,12-17,AC 19 | | |-(FvD) 4,R 20 | |-(KvJ) 5-19,-I 21 |-(FvD)>-(KvJ) 4-20,>I 7-2. We have to show that, making free use of all rules except vE, if we in addition assume vE, then we can prove AC, and if we assume AC we can prove vE. The first half of this was done in the text. To show the second half, suppose we have a derivation on which `XvY' and `-X' already appear. We need to derive Y, using AC and any other rules except vE. 1 |XvY P 2 |-X P +--------------------- 3 | |X A | +--------------------- 4 | | |-Y A | | +--------------------- 5 | | |X 3,R 6 | | |-X 2,R 7 | |--Y 4-6,-I 8 | |Y 7,-E | 9 | |Y A | +--------------------- 10 | |Y 9,R 11 |Y 1,3-8,9-10,AC 7-3. a) <->I | 1 | |X A -- | +--------------------- | 2 | |Y | | |Input for derived rule 3 | |Y A | | +--------------------- | 4 | |X -- | 5 |X>Y 1,2,>I 6 |Y>X 3,4,>I 7 |X<->Y 5,6,<>I 7-3. b) <->E | -- 1 |X<->Y |Input for derived rule 2 |X -- | 3 |X>Y 1,<>E 4 |Y 2,3,>E 7-3. c) <->E | -- 1 |X<->Y |Input for derived rule 2 |Y -- | 3 |Y>X 1,<>E 4 |X 2,3,>E 7-3. d) vE | -- 1 |-XvY |Input for derived rule 2 |X -- | 3 | |-Y A | +--------------------- 4 | |-XvY 1,R 5 | |-X 3,4,vE 6 | |X 2,R 7 |--Y 3-6,-I 8 |Y 7,-E 7-3. e) vE | -- 1 |Xv-Y |Input for derived rule 2 |Y -- | 3 | |-X A | +--------------------- 4 | |Xv-Y 1,R 5 | |-Y 3,4,vE 6 | |Y 2,R 7 |--X 3-6,-I 8 |X 7,-E 7-3. f) DC | -- 1 |X>Y |Input for derived rule 2 |-Y -- | 3 | |X A | +--------------------- 4 | |X>Y 1,R 5 | |Y 3,4,>E 6 | |-Y 2,R 7 |-X 3-6,-I 7-3. g) DC | -- 1 |X>-Y |Input for derived rule 2 |Y -- | 3 | |X A | +--------------------- 4 | |X>-Y 1,R 5 | |-Y 3,4,>E 6 | |Y 2,R 7 |-X 3-6,-I 7-3. h) RD | -- 1 | |-X A | | +--------------------- |Input for derived rule 2 | |Y | 3 | |-Y -- | 4 |--X 1-3,-I 5 |X 4,-E 7-3. i) DM | 1 |-(XvY) Input for derived | rule 2 | |X A | +--------------------- 3 | |XvY 2,vI 4 | |-(XvY) 1,R 5 |-X 2-4,-I 6 | |Y A | +--------------------- 7 | |XvY 6,vI 8 | |-(XvY) 1,R 9 |-Y 6-8,-I 10 |-X&-Y 5,9,&I 7-3. j) DM | 1 |-X&-Y Input for | derived rule 2 | |XvY A | +--------------------- 3 | |-X&-Y 1,R 4 | |-X 3,&E 5 | |Y 2,4,vE 6 | |-Y 3,&E 7 |-(XvY) 2-6,-I 7-3. k) DM | 1 |-(X&Y) Input for | derived rule 2 | |-(-Xv-Y) A | +--------------------- 3 | |--X&--Y 2,DM 4 | |--X 3,&E 5 | |--Y 3,&E 6 | |X 4,-E 7 | |Y 5,-E 8 | |X&Y 6,7,&I 9 | |-(X&Y) 1,R 10 |--(-Xv-Y) 2-9,-I 11 |-Xv-Y 10,-E 7-3. l) DM | 1 |-Xv-Y Input for | derived rule 2 | |X&Y A | +--------------------- 3 | |X 2,&E 4 | |Y 2,&E 5 | |-Xv-Y 1,R 6 | |-Y 3,5,vE 7 |-(X&Y) 2-6,-I 7-3. m) CP | 1 |X>Y Input for | derived rule 2 | |-Y A | +--------------------- 3 | | |X A | | +--------------------- 4 | | |X>Y 1,R 5 | | |Y 3,4,>E 6 | | |-Y 2,R 7 | |-X 3-6,-I 8 |-Y>-X 2-7,>I 7-3. n) CP | 1 |-X>Y Input for | derived rule 2 | |-Y A | +--------------------- 3 | | |-X A | | +--------------------- 4 | | |-X>Y 1,R 5 | | |Y 3,4,>E 6 | | |-Y 2,R 7 | |--X 3-6,-I 8 | |X 7,-E 9 |-Y>X 2-8,>I 7-3. o) CP | 1 |X>-Y Input for | derived rule 2 | |Y A | +--------------------- 3 | | |X A | | +--------------------- 4 | | |X>-Y 1,R 5 | | |-Y 3,4,>E 6 | | |Y 2,R 7 | |-X 3-6,-I 8 |Y>-X 2-7,>I 7-3. p) C | 1 |X>Y Input for | derived rule 2 | |-(-XvY) A | +--------------------- 3 | |--X&-Y 2,DM 4 | |--X 3,&E 5 | |-Y 3,&E 6 | |X 4,-E 7 | |X>Y 1,R 8 | |Y 6,7,>E 9 |--(-XvY) 2-8,-I 10 |-XvY 9,-E 7-3. q) C | 1 |-XvY Input for | derived rule 2 | |X A | +--------------------- 3 | |-XvY 1,R 4 | | |-X A | | +--------------------- 5 | | |X 2,R 6 | |--X 4-5,-I 7 | |Y 3,6,vE 8 |X>Y 2-7,>I 7-3. r) C | 1 |-(X>Y) Input for | derived rule 2 | |-X A | +--------------------- 3 | |-Y>-X 2,W 4 | |X>Y 3,CP 5 | |-(X>Y) 1,R 6 |X 2-5,RD 7 | |Y A | +--------------------- 8 | |X>Y 7,W 9 | |-(X>Y) 1,R 10 |-Y 7-9,-I 11 |X&-Y 6,10,&I 7-3. s) C | 1 |X&-Y Input for | derived rule 2 | |X>Y A | +--------------------- 3 | |X&-Y 1,R 4 | |X 3,&E 5 | |Y 2,4,>E 6 | |-Y 3,&E 7 |-(X>Y) 2-6,-I 7-4. a) 1 |M&(-BvC) P +--------------------- 2 |-BvC 1,&E 3 |B>C 2,C 7-4. b) 1 |M>(D>P) P 2 |M>D P +--------------------- 3 | |M A | +--------------------- 4 | |M>D 2,R 5 | |D 3,4,>E 6 | |M>(D>P) 1,R 7 | |D>P 3,6,>E 8 | |P 5,7,>E 9 |M>P 3-8,>I 7-4. c) 1 |(K>S)>(S>H) P 2 |S P +--------------------- 3 |K>S 2,W 4 |S>H 1,3,>E 5 |H 2,4,>E 6 |K>H 5,W 7-4. d) 1 |F>O P 2 |L>J P +--------------------- 3 | |FvL A | +--------------------- 4 | | |F A | | +--------------------- 5 | | |F>O 1,R 6 | | |O 4,5,>E 7 | | |OvJ 6,vI | | 8 | | |L A | | +--------------------- 9 | | |L>J 2,R 10 | | |J 8,9,>E 11 | | |OvJ 10,vI 12 | |OvJ 3,4-7,8-11,AC 13 |(FvL)>(OvJ) 3-14,>I 7-4. e) 1 |-((F&H)v-F) P +--------------------- 2 |-(F&H)&--F 1,DM 3 |-(F&H) 2,&E 4 |--F 2,&E 5 |-Fv-H 3,DM 6 |-H 4,5,vE 7-4. f) 1 |B>(HvR) P +--------------------- 2 | |B A | +--------------------- 3 | |B>(HvR) 1,R 4 | |HvR 2,3,>E 5 | | |-H A | | +--------------------- 6 | | |HvR 4,R 7 | | |R 5,6,vE 8 | |-H>R 5-7,>I 9 |B>(-H>R) 2-8,>I 7-4. g) 1 |-(-M&D) P 2 |F>-M P 3 |Fv-D P +--------------------- 4 |--Mv-D 1,DM 5 | |--M A | +--------------------- 6 | |F>-M 2,R 7 | |-F 5,6,DC 8 | |Fv-D 3,R 9 | |-D 7,8,vE | 10 | |-D A | +--------------------- 11 | |-D 10,R 12 |-D 4,5-9,10-11,AC 7-4. h) 1 |(A>B)&(D>-B) P 2 |(CvD)&(C>-B) P +--------------------- 3 | |A A | +--------------------- 4 | |(A>B)&(D>-B) 1,R 5 | |A>B 4,&E 6 | |B 3,5,>E 7 | |D>-B 4,&E 8 | |-D 6,7,DC 9 | |(CvD)&(C>-B) 2,R 10 | |CvD 9,&E 11 | |C 8,10,vE 12 | |C>-B 9,&E 13 | |-B 11,12,>E 14 |-A 3-13,-I 7-4. i) 1 |P>(DvM) P +--------------------- 2 |-Pv(DvM) 1,C 3 | |-P A | +--------------------- 4 | |-PvD 3,vI 5 | |P>D 4,C 6 | |(P>D)v(P>M) 5,vI | 7 | |DvM A | +--------------------- 8 | | |D A | | +--------------------- 9 | | |P>D 8,W 10 | | |(P>D)v(P>M) 9,vI | | 11 | | |M A | | +--------------------- 12 | | |P>M 11,W 13 | | |(P>D)v(P>M) 12,vI 14 | |(P>D)v(P>M) 7,8-10,11-13,AC 15 |(P>D)v(P>M) 2,3-6,7-14,AC 7-4. j) 1 |(G&-M)>(-M&K) P 2 |K>-G P +--------------------- 3 | |G A | +--------------------- 4 | | |-M A | | +--------------------- 5 | | |G 3,R 6 | | |G&-M 4,5,&I 7 | | |(G&-M)>(-M&K) 1,R 8 | | |-M&K 6,7,>E 9 | | |K 8,&E 10 | | |K>-G 2,R 11 | | |-G 9,10,>E 12 | |M 4-11,RD 13 |G>M 3-12,>I 7-4. k) 1 |AvB P 2 |-B<->(CvD) P 3 |(D&E)v(D&(F>G)) P +--------------------- 4 | |-A A | +--------------------- 5 | |AvB 1,R 6 | |B 4,5,vE 7 | |-B<->(CvD) 2,R 8 | |(CvD)>-B 7,<>E 9 | |-(CvD) 6,8,DC 10 | |-C&-D 9,DM 11 | |-D 10,&E 12 | |(D&E)v(D&(F>G)) 3,R 13 | | |D&E A | | +--------------------- 14 | | |D 13,&E | | 15 | | |D&(F>G) A | | +--------------------- 16 | | |D 15,&E 17 | |D 12,13-14,15-16,AC 18 |A 4-17,RD 7-4. l) 1 |S<->J P +--------------------- 2 |S>J 1,<>E 3 |J>S 1,<>E 4 |-SvJ 2,C 5 |-JvS 3,C 6 | |-S A | +--------------------- 7 | |-JvS 5,R 8 | |-J 6,7,vE 9 | |-S&-J 6,8,&I 10 | |(S&J)v(-S&-J) 9,vI | 11 | |J A | +--------------------- 12 | |-JvS 5,R 13 | |S 11,12,vE 14 | |S&J 11,13,&I 15 | |(S&J)v(-S&-J) 14,vI 16 |(S&J)v(-S&-J) 4,6-10,11-15,AC 7-4. m) 1 |-C>(Fv-(DvN)) P 2 |-N>D P +--------------------- 3 | |-F A | +--------------------- 4 | | |-C A | | +--------------------- 5 | | |-C>(Fv-(DvN)) 1,R 6 | | |Fv-(DvN) 4,5,>E 7 | | |-F 3,R 8 | | |-(DvN) 6,7,vE 9 | | |-D&-N 8,DM 10 | | |-D 9,&E 11 | | |-N 9,&E 12 | | |-N>D 2,R 13 | | |D 11,12,>E 14 | |C 4-13,RD 15 |-F>C 3-14,>I 7-4. n) 1 |(GvA)>(H>B) P 2 |(H>(H&B))>K P +--------------------- 3 | |G A | +--------------------- 4 | |GvA 3,vI 5 | |(GvA)>(H>B) 1,R 6 | |H>B 4,5,>E 7 | | |H A | | +--------------------- 8 | | |H>B 6,R 9 | | |B 7,8,>E 10 | | |H&B 7,9,&I 11 | |H>(H&B) 7-10,>I 12 | |(H>(H&B))>K 2,R 13 | |K 11,12,>E 14 |G>K 3-13,>I 7-4. o) 1 |F>(KvB) P 2 |(-FvG)&(-Gv-K) P +--------------------- 3 |-FvG 2,&E 4 |-Gv-K 2,&E 5 | |F A | +--------------------- 6 | |F>(KvB) 1,R 7 | |KvB 5,6,>E 8 | |-FvG 3,R 9 | |G 5,8,vE 10 | |-Gv-K 4,R 11 | |-K 9,10,vE 12 | |B 7,11,vE 13 |F>B 5-12,>I 7-4. p) 1 |Dv(M>J) P 2 |(M>(M&J))>(PvK) P 3 |(P>D)&(K>F) P +--------------------- 4 |P>D 3,&E 5 |K>F 3,&E 6 | |D A | +--------------------- 7 | |DvF 6,vI | 8 | |M>J A | +--------------------- 9 | | |M A | | +--------------------- 10 | | |M>J 8,R 11 | | |J 9,10,>E 12 | | |M&J 9,11,&I 13 | |M>(M&J) 9-12,>I 14 | |(M>(M&J))>(PvK) 2,R 15 | |PvK 13,14,>E 16 | | |P A | | +--------------------- 17 | | |P>D 4,R 18 | | |D 16,17,>E 19 | | |DvF 18,vI | | 20 | | |K A | | +--------------------- 21 | | |K>F 5,R 22 | | |F 20,21,>E 23 | | |DvF 22,vI 24 | |DvF 15,16-19,20-23,AC 25 |DvF 1,6-7,8-24,AC 7-4. q) 1 |Q<->-(A&F) P 2 |-(MvA)>-H P 3 |-(Q&A)vF P +--------------------- 4 | |Q A | +--------------------- 5 | | |H A | | +--------------------- 6 | | |-(MvA)>-H 2,R 7 | | |--(MvA) 5,6,DC 8 | | |MvA 7,-E 9 | | |Q 4,R 10 | | |Q<->-(A&F) 1,R 11 | | |-(A&F) 9,10,<>E 12 | | |-Av-F 11,DM 13 | | | |-A A | | | +--------------------- 14 | | | |MvA 8,R 15 | | | |M 13,14,vE | | | 16 | | | |-F A | | | +--------------------- 17 | | | |-(Q&A)vF 3,R 18 | | | |-(Q&A) 16,17,vE 19 | | | |-Qv-A 18,DM 20 | | | |Q 9,R 21 | | | |-A 19,20,vE 22 | | | |MvA 8,R 23 | | | |M 21,22,vE 24 | | |M 12,AC 25 | |H>M 5-24,>I 26 |Q>(H>M) 4-25,>I 7-4. r) 1 |(I&-T)>P P 2 |-A>-T P 3 |-TvC P 4 |C>D P +--------------------- 5 | |-P A | +--------------------- 6 | | |I A | | +--------------------- 7 | | |-P 5,R 8 | | |(I&-T)>P 1,R 9 | | |-(I&-T) 7,8,DC 10 | | |-Iv--T 9,DM 11 | | |--T 6,10,vE 12 | | |-A>-T 2,R 13 | | |--A 11,12,DC 14 | | |A 13,-E 15 | | |-TvC 3,R 16 | | |C 11,15,vE 17 | | |C>D 4,R 18 | | |D 16,17,>E 19 | | |D&A 14,18,&I 20 | |I>(D&A) 6-19,>I 21 |-P>(I>(D&A)) 5-20,>I 7-4. s) 1 |B>(NvM) P 2 |N>(C&K) P 3 |C>(K>P) P 4 |-(P&B) P +--------------------- 5 | |B A | +--------------------- 6 | |B>(NvM) 1,R 7 | |NvM 5,6,>E 8 | | |-M A | | +--------------------- 9 | | |NvM 7,R 10 | | |N 8,9,vE 11 | | |N>(C&K) 2,R 12 | | |C&K 10,11,>E 13 | | |C 12,&E 14 | | |K 12,&E 15 | | |C>(K>P) 3,R 16 | | |K>P 13,15,>E 17 | | |P 14,16,>E 18 | | |B 5,R 19 | | |P&B 17,18,&I 20 | | |-(P&B) 4,R 21 | |M 8-20,RD 22 |B>M 5-21,>I 7-5. Suppose we have a derivation with X as its only premise and Y and -Y as conclusions. Relabel X as an assumption, and make the whole derivation the sub-derivation of a premiseless outer-derivation. The sub-derivation licenses drawing -X as final conclusion of its outer-derivation by applying RD. Thus any instance of the new test for contradiction can be converted to an instance of the old test. 7-6. a) 1 | |AvB A | +--------------------- 2 | | |-B A | | +--------------------- 3 | | |AvB 1,R 4 | | |A 2,3,vE 5 | |-B>A 2-4,>I 6 |(AvB)>(-B>A) 1-5,>I 7-6. b) 1 | |-(Mv-(M&N)) A | +--------------------- 2 | |-M&--(M&N) 1,DM 3 | |-M 2,&E 4 | |--(M&N) 2,&E 5 | |M&N 4,-E 6 | |M 5,&E 7 |Mv-(M&N) 1-6,RD 7-6. c) 1 | |H>(O>N) A | +--------------------- 2 | | |H&O A | | +--------------------- 3 | | |H 2,&E 4 | | |O 2,&E 5 | | |H>(O>N) 1,R 6 | | |O>N 3,5,>E 7 | | |N 4,6,>E 8 | |(H&O)>N 2-7,>I 9 |(H>(O>N))>((H&O)>N) 1-8,>I 7-6. d) 1 | |D>B A | +--------------------- 2 | | |D>T A | | +--------------------- 3 | | | |D A | | | +--------------------- 4 | | | |D>T 2,R 5 | | | |T 3,4,>E 6 | | | |D>B 1,R 7 | | | |B 3,6,>E 8 | | | |B&T 5,7,&I 9 | | |D>(B&T) 3-8,>I 10| |(D>T)>(D>(B&T)) 2-9,>I 11|(D>B)>((D>T)>(D>(B&T))) 1-10,>I 7-6. e) 1 | |K>F A | +--------------------- 2 | | |K&P A | | +--------------------- 3 | | |K 2,&E 4 | | |K>F 1,R 5 | | |F 3,4,>E 6 | |(K&P)>F 2-5,>I 7 | |-F>-(K&P) 6,CP 8 |(K>F)>(-F>-(K&P)) 1-7,>I 7-6. f) 1 | |(FvG)>(P&Q) A | +--------------------- 2 | | |F A | | +--------------------- 3 | | |FvG 2,vI 4 | | |(FvG)>(P&Q) 1,R 5 | | |P&Q 3,4,>E 6 | | |Q 5,&E 7 | |F>Q 2-6,>I 8 | |-Q>-F 7,CP 9 |((FvG)>(P&Q))>(-Q>-F) 1-8,>I 7-6. g) 1 | |L>(M>N) A | +--------------------- 2 | | |L>M A | | +--------------------- 3 | | | |L A | | | +--------------------- 4 | | | |L>M 2,R 5 | | | |M 3,4,>E 6 | | | |L>(M>N) 1,R 7 | | | |M>N 3,6,>E 8 | | | |N 5,7,>E 9 | | |L>N 3-8,>I 10| |(L>M)>(L>N) 2-9,>I 11|(L>(M>N))>((L>M)>(L>N)) 1-10,>I 7-6. h) 1 | |(SvT)>F A | +--------------------- 2 | | |(FvG)>H A | | +--------------------- 3 | | | |S A | | | +--------------------- 4 | | | |SvT 3,vI 5 | | | |(SvT)>F 1,R 6 | | | |F 4,5,>E 7 | | | |FvG 6,vI 8 | | | |(FvG)>H 2,R 9 | | | |H 7,8,>E 10| | |S>H 3-9,>I 11| |((FvG)>H)>(S>H) 2-10,>I 12|((SvT)>F)>(((FvG)>H)>(S>H)) 1-11,>I 7-6. i) 1 | |-((I&-J)v((J&K)v-(K&I))) A | +--------------------- 2 | |-(I&-J)&-((J&K)v-(K&I)) 1,DM 3 | |-(I&-J) 2,&E 4 | |-((J&K)v-(K&I)) 2,&E 5 | |-(J&K)&--(K&I) 4,DM 6 | |--(K&I) 5,&E 7 | |K&I 6,-E 8 | |K 7,&E 9 | |I 7,&E 10| |-Iv--J 3,DM 11| |--J 9,10,vE 12| |-(J&K) 5,&E 13| |-Jv-K 12,DM 14| |-K 11,13,vE 15|(I&-J)v((J&K)v-(K&I)) 1-15,RD 7-6. j) 1 | |-(((C&(AvD))v-(C&F))v-(A&-G)) A | +--------------------- 2 | |-((C&(AvD))v-(C&F))&--(A&-G) | | 1,DM 3 | |-((C&(AvD))v-(C&F)) 2,&E 4 | |--(A&-G) 2,&E 5 | |A&-G 4,-E 6 | |A 5,&E 7 | |-(C&(AvD))&--(C&F) 3,DM 8 | |-(C&(AvD)) 7,&E 9 | |--(C&F) 7,&E 10| |C&F 9,-E 11| |C 10,&E 12| |-Cv-(AvD) 8,DM 13| |-(AvD) 11,12,vE 14| |AvD 6,vI 15|((C&(AvD))v-(C&F))v-(A&-G) 1-15,RD 7-7. a) 1 |A&-A P +--------------------- 2 |A 1,&E 3 |-A 1,&E 7-7. b) 1 |(Hv-B)&((-B>H)&-H) P +--------------------- 2 |Hv-B 1,&E 3 |(-B>H)&-H 1,&E 4 |-B>H 3,&E 5 |-H 3,&E 6 |B 4,5,DC 7 |H 2,6,vE 7-7. c) 1 |((H&F)>C)&-(H>(F>C)) P +--------------------- 2 |(H&F)>C 1,&E 3 |-(H>(F>C)) 1,&E 4 |H&-(F>C) 3,C 5 |H 4,&E 6 |-(F>C) 4,&E 7 |F&-C 6,C 8 |F 7,&E 9 |-C 7,&E 10 |H&F 5,8,&I 11 |C 2,10,>E 7-7. d) 1 |(-(GvQ)&(K>G))&-(Pv-K) P +--------------------- 2 |-(GvQ)&(K>G) 1,&E 3 |-(Pv-K) 1,&E 4 |-(GvQ) 2,&E 5 |K>G 2,&E 6 |-G&-Q 4,DM 7 |-G 6,&E 8 |-K 5,7,DC 9 |-P&--K 3,DM 10 |--K 9,&E 7-7. e) 1 |(K>(D>P))&((-KvD)&-(K>P)) P +--------------------- 2 |K>(D>P) 1,&E 3 |(-KvD)&-(K>P) 1,&E 4 |-KvD 3,&E 5 |-(K>P) 3,&E 6 |K&-P 5,C 7 |K 6,&E 8 |-P 6,&E 9 |D>P 2,7,>E 10 |D 4,7,vE 11 |P 9,10,>E 7-7. f) 1 |-(-(Nv-R)>(N<->-R)) P +--------------------- 2 |-(Nv-R)&-(N<->-R) 1,C 3 |-(Nv-R) 2,&E 4 |-(N<->-R) 2,&E 5 |-N&--R 3,DM 6 |-N 5,&E 7 |--R 5,&E 8 |R 7,-E 9 | |-N A | +--------------------- 10 | |R 8,R 11 |-N>R 9-10,>I 12 |-R>N 11,CP 13 | |R A | +--------------------- 14 | |-N 6,R 15 |R>-N 13-14,>I 16 |N>-R 15,CP 17 |N<->-R 12,16,<>I 7-7. g) 1 |(FvG)<->(-F&-G) P +--------------------- 2 | |-F&-G A | +--------------------- 3 | |(FvG)<->(-F&-G) 1,R 4 | |FvG 2,3,<>E 5 | |-(FvG) 2,DM 6 |-(-F&-G) 2-5,-I 7 | |FvG A | +--------------------- 8 | |(FvG)<->(-F&-G) 1,R 9 | |-F&-G 7,8,<>E 10 | |-(FvG) 9,DM 11 |-(FvG) 7-10,-I 12 |-F&-G 11,DM 7-7. h) 1 |(-(FvG)v(P&Q))&-(-Q>-F) P +--------------------- 2 |-(FvG)v(P&Q) 1,&E 3 |-(-Q>-F) 1,&E 4 |-Q&--F 3,C 5 |-Q 4,&E 6 |--F 4,&E 7 |F 6,-E 8 |FvG 7,vI 9 |P&Q 2,8,vE 10 |Q 9,&E 7-7. i) 1 |(A>D)&(((A&-B)v(A&-C))&((B&-D)v(B&C))) P +--------------------- 2 |A>D 1,&E 3 |((A&-B)v(A&-C))&((B&-D)v(B&C)) 1,&E 4 |(A&-B)v(A&-C) 3,&E 5 |(B&-D)v(B&C) 3,&E 6 |A&(-Bv-C) 4,DS 7 |B&(-DvC) 5,DS 8 |A 6,&E 9 |-Bv-C 6,&E 10 |B 7,&E 11 |-DvC 7,&E 12 |-C 9,10,vE 13 |-D 11,12,vE 14 |D 2,8,>E I've cheated in this problem-using the derived rule for the distributive law (DS), which I did not introduce.You have really done the work of proving the distributive law as a derived rule in problem 7-1d. 7-7. j) 1 |(A<->B)<->(-A<->B) P 26 | |A A +--------------------- | +--------------------- 2 | |A&B A 27 | |-Av-B 24,R | +--------------------- 28 | |-Av--B 25,R 3 | |A 2,&E 29 | |-B 26,27,vE 4 | |B 2,&E 30 | |--B 26,28,vE 5 | |B>A 3,W 31 |-A 26-30,-I 6 | |A>B 4,W 32 | |-B A 7 | |A<->B 5,6,<>I | +--------------------- 8 | |(A<->B)<->(-A<->B) 1,R 33 | |-A 31,R 9 | |-A<->B 7,8,<>E 34 | |-B>-A 33,W 10 | |-A 4,9,<>E 35 | |A>B 34,CP 11 |-(A&B) 2-10,-I 36 | |-A>-B 32,W 12 | |A&-B A 37 | |B>A 36,CP | +--------------------- 38 | |A<->B 35,37,<>I 13 | |A 12,&E 39 | |(A<->B)<->(-A<->B) 1,R 14 | |-B 12,&E 40 | |-A<->B 38,39,<>E 15 | |A>-B 14,W 41 | |B 33,40,<>E 16 | |B>-A 15,CP 42 |B 32-41,RD 17 | |-B>A 13,W 43 |-A>B 42,W 18 | |-A>B 17,CP 44 |B>-A 31,W 19 | |-A<->B 16,18,<>I 45 |-A<->B 43,44,<>I 20 | |(A<->B)<->(-A<->B) 1,R 46 |A<->B 1,45,<>E 21 | |A<->B 19,20,<>E 47 |A 42,46,<>E 22 | |B 13,21,<>E 23 |-(A&-B) 12-22,-I 24 |-Av-B 11,DM 25 |-Av--B 23,DM 7-8. a) A sentence is a contradiction if and only if there is not an assignment of truth values to sentence letters which makes it true. Hence a set of sentences is inconsistent if and only if its conjunction is a contradiction. This characterization works only for finite sets of sentences, since there cannot be a conjunction of an infinite set of sentences. b) A finite set of sentences is shown to be inconsistent if there is a derivation which has the sentences in the set as premises and contradicting conclusions. c1) 1 |C<->G P 2 |G<->-C P +--------------------- 3 | |C A | +--------------------- 4 | |C<->G 1,R 5 | |G 3,4,<>E 6 | |G<->-C 2,R 7 | |-C 5,6,<>E 8 |-C 3-7,-I 9 | |-C A | +--------------------- 10 | |G<->-C 2,R 11 | |G 9,10,<>E 12 | |C<->G 1,R 13 | |C 11,12,<>E 14 |C 9-13,RD c2) 1 |FvT P 2 |(FvT)>(-F&-T) P +--------------------- 3 |-F&-T 1,2,>E 4 |-F 3,&E 5 |-T 3,&E 6 |T 1,4,vE c3) 1 |JvK P 2 |-Jv-K P 3 |J<->K P +--------------------- 4 | |J A | +--------------------- 5 | |J<->K 3,R 6 | |-Jv-K 2,R 7 | |K 4,5,<>E 8 | |-J 6,7,vE 9 | |J&-J 4,8,&I | 10 | |K A | +--------------------- 11 | |-Jv-K 2,R 12 | |-J 10,11,vE 13 | |J<->K 3,R 14 | |J 10,13,<>E 15 | |J&-J 12,14,&I 16 |J&-J 1,4-9,10-15,AC 17 |J 16,&E 18 |-J 16,&E c4) 1 |(GvK)>A P 2 |(AvH)>G P 3 |G&-A P +--------------------- 4 |G 3,&E 5 |-A 3,&E 6 |GvK 4,vI 7 |A 1,6,>E c5) 1 |D<->(-P&-M) P 2 |P<->(J&-F) P 3 |-Fv-D P 4 |D&J P +--------------------- 5 |D 4,&E 6 |J 4,&E 7 |-F 3,5,vE 8 |J&-F 6,7,&I 9 |P 2,8,<>E 10 |-P&-M 1,5,<>E 11 |-P 10,&E