SOLVED LOGIC PROBLEMS 1 |(Ex)Ix P 2 |(x)(Ix > Jx) / (Ex)Jx +--------------------- 3 |a|Ia A | +--------------------- 4 | |(x)(Ix > Jx) 2,R 5 | |Ia > Ja 4,UE 6 | |Ja 3,5,>E 7 | |(Ex)Jx 6,EI 8 |(Ex)Jx 1,3-7,EE SET D PROB A 1 |(Ex)(A > Px) / A > (Ex)Px +--------------------- 2 |b|A > Pb A | +--------------------- 3 | | |A A | | +--------------------- 4 | | |A > Pb 2,R 5 | | |Pb 3,4,>E 6 | | |(Ex)Px 5,EI 7 | |A > (Ex)Px 3-6,>I 8 |A > (Ex)Px 1,2-7,EE SET D PROB B 1 |(Ex)Hmx P 2 |(x)(~Hmx v Gxn) / (Ex)Gxn +--------------------- 3 |a|Hma A | +--------------------- 4 | |(x)(~Hmx v Gxn) 2,R 5 | |~Hma v Gan 4,UE 6 | |Gan 3,5,vE 7 | |(Ex)Gxn 6,EI 8 |(Ex)Gxn 1,3-7,EE SET D PROB C 1 |(Ex)(Cfx & Cxf) / (Ex)Cfx & (Ex)Cxf +--------------------- 2 |a|Cfa & Caf A | +--------------------- 3 | |Cfa 2,&E 4 | |Caf 2,&E 5 | |(Ex)Cfx 3,EI 6 | |(Ex)Cxf 4,EI 7 | |(Ex)Cfx & (Ex)Cxf 5,6,&I 8 |(Ex)Cfx & (Ex)Cxf 1,2-7,EE SET D PROB D 1 |(Ex)(Px v Qx) / (Ex)Px v (Ex)Qx +--------------------- 2 |a|Pa v Qa A | +--------------------- 3 | | |Pa A | | +--------------------- 4 | | |(Ex)Px 3,EI 5 | | |(Ex)Px v (Ex)Qx 4,vI 6 | | 7 | | |Qa A | | +--------------------- 8 | | |(Ex)Qx 7,EI 9 | | |(Ex)Px v (Ex)Qx 8,vI 10 | | 11 | |(Ex)Px v (Ex)Qx 2,3-5,7-9,AC 12 |(Ex)Px v (Ex)Qx 1,2-11,EE SET D PROB E 1 |(Ex)Px v (Ex)Qx / (Ex)(Px v Qx) +--------------------- 2 | |(Ex)Px A | +--------------------- 3 | |a|Pa A | | +--------------------- 4 | | |Pa v Qa 3,vI 5 | | |(Ex)(Px v Qx) 4,EI 6 | |(Ex)(Px v Qx) 2,3-5,EE 7 | 8 | |(Ex)Qx A | +--------------------- 9 | |a|Qa A | | +--------------------- 10 | | |Pa v Qa 9,vI 11 | | |(Ex)(Px v Qx) 10,EI 12 | |(Ex)(Px v Qx) 8,9-11,EE 13 | 14 |(Ex)(Px v Qx) 1,2-6,8-12,AC SET D PROB F 1 |(Ex)(Px > A) / (x)Px > A +--------------------- 2 |b|Pb > A A | +--------------------- 3 | | |(x)Px A | | +--------------------- 4 | | |Pb 3,UE 5 | | |Pb > A 2,R 6 | | |A 4,5,>E 7 | |(x)Px > A 3-6,>I 8 |(x)Px > A 1,2-7,EE SET D PROB G 1 |(x)(Px > A) / (Ex)Px > A +--------------------- 2 | |(Ex)Px A | +--------------------- 3 | |b|Pb A | | +--------------------- 4 | | |(x)(Px > A) 1,R 5 | | |Pb > A 4,UE 6 | | |A 3,5,>E 7 | |A 2,3-6,EE 8 |(Ex)Px > A 2-7,>I SET D PROB H 1 |(Ex)(Lxa ­ Lex) P 2 |(x)Lxa / (Ex)Lex +--------------------- 3 |b|Lba > Leb A | +--------------------- 4 | |(x)Lxa 2,R 5 | |Lba 4,UE 6 | |Leb 3,5,>E 7 | |(Ex)Lex 6,EI 8 |(Ex)Lex 1,3-7,EE SET D PROB I 1 |(x)(Gsx > ~Gxs) / (Ex)Gxs > (Ex)~Gsx +--------------------- 2 | |(Ex)Gxs A | +--------------------- 3 | |a|Gas A | | +--------------------- 4 | | |(x)(Gsx > ~Gxs) 1,R 5 | | |Gsa > ~Gas 4,UE 6 | | |~Gsa 3,5,DC 7 | | |(Ex)~Gsx 6,EI 8 | |(Ex)~Gsx 2,3-7,EE 9 |(Ex)Gxs > (Ex)~Gsx 2-8,>I SET D PROB J 1 |(Ex)(Px v Qx) P 2 |(x)(Px > Kx) P 3 |(x)(Qx > Kx) / (Ex)Kx +--------------------- 4 |a|Pa v Qa A | +--------------------- 5 | |(x)(Px > Kx) 2,R 6 | |Pa > Ka 5,UE 7 | |(x)(Qx > Kx) 3,R 8 | |Qa > Ka 7,UE 9 | |Ka 4,AC 10 | |(Ex)Kx 9,EI 11 |(Ex)Kx 1,4-10,EE SET D PROB K 1 |(Ex)(~Mxt v Mtx) P 2 |(x)(Mtx > Axx) / (Ex)(~Mxt v Axx) +--------------------- 3 |a|~Mat v Mta A | +--------------------- 4 | |(x)(Mtx > Axx) 2,R 5 | |Mta > Aaa 4,UE 6 | | |~Mat A | | +--------------------- 7 | | |~Mat v Aaa 6,vI 8 | | 9 | | |Mta A | | +--------------------- 10 | | |Mta > Aaa 5,R 11 | | |Aaa 9,10,>E 12 | | |~Mat v Aaa 11,vI 13 | | 14 | |~Mat v Aaa 3,6-7,9-12,AC 15 | |(Ex)(~Mxt v Axx) 14,EI 16 |(Ex)(~Mxt v Axx) 1,3-15,EE SET D PROB L 1 |(Ex)Hxg v (Ex)Nxf P 17 | | |(Ex)Cx 16,EI 2 |(x)(Hxg > Cx) P 18 | |(Ex)Cx 12,13-17,EE 3 |(x)(Nxf > Cx) / (Ex)Cx 19 | +--------------------- 20 |(Ex)Cx 1,4-10,12-18,AC 4 | |(Ex)Hxg A | +--------------------- 5 | |a|Hag A | | +--------------------- 6 | | |(x)(Hxg > Cx) 2,R 7 | | |Hag > Ca 6,UE 8 | | |Ca 5,7,>E 9 | | |(Ex)Cx 8,EI 10 | |(Ex)Cx 4,5-9,EE 11 | 12 | |(Ex)Nxf A | +--------------------- 13 | |a|Naf A | | +--------------------- 14 | | |(x)(Nxf > Cx) 3,R 15 | | |Naf > Ca 14,UE 16 | | |Ca 13,15,>E SET D PROB M 1 |(x)((Fx v Gx) > Lxx) P 2 |(Ex)~Lxx / (Ex)~Fx & (Ex)~Gx +--------------------- 3 |a|~Laa A | +--------------------- 4 | |(x)((Fx v Gx) > Lxx) 1,R 5 | |(Fa v Ga) > Laa 4,UE 6 | |~(Fa v Ga) 3,5,DC 7 | |~Fa & ~Ga 6,DM 8 | |~Fa 7,&E 9 | |~Ga 7,&E 10 | |(Ex)~Fx 8,EI 11 | |(Ex)~Gx 9,EI 12 | |(Ex)~Fx & (Ex)~Gx 10,11,&I 13 |(Ex)~Fx & (Ex)~Gx 2,3-12,EE SET D PROB N 1 |(x)(Fx > (Rxa v Rax)) P 2 |(Ex)~Rxa / (x)~Rax > (Ex)~Fx +--------------------- 3 |b|~Rba A | +--------------------- 4 | | |(x)~Rax A | | +--------------------- 5 | | |~Rab 4,UE 6 | | |(x)(Fx > (Rxa v Rax)) 1,R 7 | | |Fb > (Rba v Rab) 6,UE 8 | | |~Rba 3,R 9 | | |~Rba & ~Rab 5,8,&I 10 | | |~(Rba v Rab) 9,DM 11 | | |~Fb 10,7,DC 12 | | |(Ex)~Fx 11,EI 13 | |(x)~Rax > (Ex)~Fx 4-12,>I 14 |(x)~Rax > (Ex)~Fx 2,3-13,EE SET D PROB O 1 |(Ex)Qxj P 2 |(Ex)(QxjvDgx)>(x)Dgx / (x)(Dgx v Qjx) +--------------------- 3 |a|Qaj A | +--------------------- 4 | |Qaj v Dga 3,vI 5 | |(Ex)(Qxj v Dgx) 4,EI 6 | |(Ex)(QxjvDgx)>(x)Dgx 2,R 7 | |(x)Dgx 5,6,>E 8 | |Dgb 7,UE 9 | |Dgb v Qjb 8,vI 10 | |(x)(Dgx v Qjx) 9,UI 11 |(x)(Dgx v Qjx) 1,3-10,EE SET D PROB P 1 |(x)~Fx / ~(Ex)Fx +--------------------- 2 | |(Ex)Fx A | +--------------------- 3 | |a|Fa A | | +--------------------- 4 | | |(x)~Fx 1,R 5 | | |~Fa 4,UE 6 | | |~Fa v (P & ~P) 5,vI 7 | | |P & ~P 3,6,vE 8 | |P & ~P 2,3-7,EE 9 |~(Ex)Fx 8,(2-8),~I SET D PROB Q 1 |(x)~Fx / ~(Ex)Fx +--------------------- 2 |~(Ex)Fx 1,U~ (short cut) SET D PROB Q 1 |(Ex)~Fx / ~(x)Fx +--------------------- 2 |a|~Fa A | +--------------------- 3 | | |(x)Fx A | | +--------------------- 4 | | |Fa 3,UE 5 | | |~Fa 2,R 6 | |~(x)Fx 4,5,(3-5),~I 7 |~(x)Fx 1,2-6,EE SET D PROB R 1 |(Ex)~Fx / ~(x)Fx +--------------------- 2 |~(x)Fx 1,X~ (short cut) SET D PROB R 1 |(x)(Jxx > ~Jxf) / ~(Ex)(Jxx & Jxf) +--------------------- 2 | |(Ex)(Jxx & Jxf) A | +--------------------- 3 | |a|Jaa & Jaf A | | +--------------------- 4 | | |Jaa 3,&E 5 | | |Jaf 3,&E 6 | | |(x)(Jxx > ~Jxf) 1,R 7 | | |Jaa > ~Jaf 6,UE 8 | | |~Jaf 4,7,>E 9 | | |Jaf v (P & ~P) 5,vI 10 | | |P & ~P 8,9,vE 11 | |P & ~P 2,3-10,EE 12 |~(Ex)(Jxx & Jxf) 11,(2-11),~I SET D PROB S 1 |(Ex)Px v Qa P 2 |(x)~Px / (Ex)Qx +--------------------- 3 | |(Ex)Px A | +--------------------- 4 | |b|Pb A | | +--------------------- 5 | | |(x)~Px 2,R 6 | | |~Pb 5,UE 7 | | | |~(Ex)Qx A | | | +--------------------- 8 | | | |Pb 4,R 9 | | | |~Pb 6,R 10 | | |(Ex)Qx 8,9,(7-9),RD 11 | |(Ex)Qx 3,4-10,EE 12 | 13 | |Qa A | +--------------------- 14 | |(Ex)Qx 13,EI 15 | 16 |(Ex)Qx 1,3-11,13-14,AC SET D PROB T 1 |A > (Ex)Px / (Ex)(A > Px) 16 | |(Ex)(A > Px) 12,13-15,EE +--------------------- 17 | 2 |~A v (Ex)Px 1,C 18 |(Ex)(A > Px) 2,3-10,12-16,AC 3 | |~A A | +--------------------- 4 | | |A A | | +--------------------- 5 | | | |~Pb A | | | +--------------------- 6 | | | |~A 3,R 7 | | | |A 4,R 8 | | |Pb 6,7,(5-7),RD 9 | |A > Pb 4-8,>I 10 | |(Ex)(A > Px) 9,EI 11 | 12 | |(Ex)Px A | +--------------------- 13 | |a|Pa A | | +--------------------- 14 | | |A > Pa 13,W 15 | | |(Ex)(A > Px) 14,EI SET D PROB U