Valid Forms for Sentential Looic Valid Argument Forms of Inference 1 . Modus Ponens(Mpl: p) 5 . Conlunction{Conj}: q p p l.'. q q l.'.p.q Modus Tollens(MT): b. HypotheticalSyllogism (flSl: p) q p) q - Q /:' - P q ) r l . ' .p ) r 3 . DisiunctiveSyllogism (DSl: pv q P l:.Pvq - p l..q ConstructiveDilemma ICD) : pv q -q 7 . Addition {Add): pv q l.'.p Simplification(Simp): g ) s l . ' .r v s P ' a / . ' .P P q /.'.q Valid Equivalence Forms (Rule of Replacementl Double Negation (DN): p:: -- p ( pl q ) : :( - q ) - p l DeMorgant Theorem (DeMl: -\\P q).:(-pv-q) -lpvqJ::Gp.-e) 't1. Commutation (Comm): (Pvql.:@vp) t 5 . lmplication(lmpl): ( p l q ) : :( - p v q ) 1 6 . Exportation (Expl: I(p dtrl::lp)(qtr)l 1 7 . Tautology (Taut): @ o ) : :l q ' P 1 p.:\\p.pl 12. Assocation(Assocl: Ipr@ur)l::[(pvqJvr] ,r'1o'r))::I@ q).r) 1 3 . Distribution (Dist): Ip lq v r)l:: llp. ql v \\p. rl) l p v ( q r ) l: : { ( p v q ) . ( p v r ) l Conditional and lndirect Proof 1 4 . Contraposition (Gontral: p:: \\pv p) (Equivl: 1 8 . Eguivalence (p-q)::t(plq).(q)pll ( p = q ): : l ( p q l " (- p. - q)l ConditionalProof A Pt . . q rr I I l^ p) q cP AP 1...p Bules for PredicatELogic Rule Ul: {uX Rule El: . t t . " 1 . . .w (3u)( RuleUG: \\ t:. \\ .) w. ..1 I l.'. lvil . . w Provided: 1. (. . . w. . .) results from replacing eachoccurr e n c eo f u f r e e i n { . . . u . . . ) w i t h a w t h a t i s fl e i t h e r a c o n s t a n t o r a v a r i a br e ei n ( . . . w . . . \\ (making otherchanges). no Provided: 1. w is not a constant. 2. w does not occur free previously the proof. in 3. l. . . w. . .) results irom replacing eachoccurrenceof ufree in (. . . u. . .) with a wthat rs free i n ( . . . w . . . ) ( m a k i n g o o t h e rc h a n g e s ) . n Provided: 1. u is not a constant. in 2. u does not occurfree previously a line obtarned El. by in u does not occurfree previously an assumed premisethat has not yet been discharged. (. . . w .. .) results from replacrng eachoccurr e n c eo f u f r e e i n ( . . . u . . . ) w i t ha w t h a t r s f r e e in (. . . w. . .) {making otherchanges} no and free occurrences h/ of there are no additional n a l r e a d v c o n t a i n ie d( . . . w . . . 1 . Rule EG: 1 . . u . . ) / . ' .l 3 v t | l RuleON: { u X . . . u . . . ) : :- ( 3 u-) ( . . . u . . .) (lu)(. . . u . . . ) : : - ( u ) - ( . . . u . . .) ( d - ( . . . u . . . ) : :- ( l u ) ( . . . u . . .) E u ) - ( . . . u . . . ) : : - l u l l . . . u . . .) Rule lD: (...u...1 u=w l.',( RulefR: l.'. \\xllx= x) w...) Provided: 1- (. . . w. . .) results from replacing least one at occurrence u, where u is a constantor a variof ablefree in (. . . u. . .) with a wthat is free in (. . . w. . .) (making otherchanges) there no and free occurrences w already ol are no additional in contained (. . . w. . .) {...u...) w=u l;. 1 w. ..1 e f y encrs -l^v-i/^'1,(ig5 Cwsl>".f { ' o n t / 4 6 i/u*,V t,\\ (A ) B)= (e u -A) rrulpurys I o, t .(* (a v : B))>(s 'a) c)(*B )'-4= $r,<) ,i) ..\\- = ((u -, A),(,(o A ) B)) _ 4)) (a r(.-) " A ='*A fj 'J) (A"-A)> B A) A ) (e ,n) Z Fxenr,j( R r ^ e , , c h 5 e n / t r t f D t .f l - , l r ( t fAv,tu5h o lt ,t( t f l ' l + 9 e a { * n t f{ ; , * t cvt fU bt'shl q^ul n crc *k (hy ;f ls (r,i s'[-fufi",'' tnrf>ntp q fr"rn. ov- it ,ot "( iA"i sah/et4ce # l.A 2. A)B 3. (AvB))C 4. Av@)q 5.(-AvB))C 6.-(AvB))C 7. -Av(Bf C) 8 . ( Av B ) ) - C 9.-IAv(BlC)l 10. -(-AvB))C 11. -[(AvB))CJ 12.-(AvB))-C 13. -[-(A.rB)rC] 14. -t-?evB)lCl 15.-t(-AvB))Q f t i a E A,r'- QUtL tuAih a.p b. -p c.pvq d.p)q e.-pvq f.-p)q E.-p)-q h.-(pvq) i. -(p)q) J.-(-p)q) k. (pvq))r l. pv(q)r) m. (- pv q)) r n. -(pvq))r o. (PVo)>-, p.-[pv(q)r)] q.-t(pvq))rl !^ o( d'tr? ftr d iluA,tn1 ^(eryPns,l 0 f t t|w { , l S r^ 'x or t vPi l- { v'tn* -{at ou los.7 ( fif / 1n o ,,/ TA l) z-) (<'7"erre.) A@B C ) ".'A ) B ,l) '.--fA) B) r) (a) 4"k= D).6vFrG) a (*r t {': *--. Exercise L i' a; Use MP, MT, DS, and HS to prove that the following argumentsare valid. la :. (l) 3 l. -R 2 . s I R / . . .- S (2) A.S (A.S))R/."R (3) - (H.K) Rv(I/.K)/.'.R (4) (PvQ)l(R'w) L)(PvQ)/.'. tr(R.w) (5) Rfs rlR -s/.'.-T (6) -M NfG NvM/.'.G (7) - D) E D)F -F/:.E (8) Gv H -Hvl -r/.'.G (e) -G)(AvB) -B A) D - G/.'.D (10) . (A) B)) C r 2. -DvA 3. -D f (Af B) 4. - A/.'. C (1r)1. Ar(Bf C) 2. -C 3. -D)A 4. Cv-D/.'.-B ( 1 2 )1 . - ( D . F ) 2. (LvM)vR 3. -T)-(LvM) 4.(D. F)v-T/.'.R ( 1 3 )l . 2. 3. 4. (AvB)r(BvO (B)C)vA (BlC)l(AvB) -Al.'.BvC ( 1 4 )l . ( P . O l l R v ( r . D l 2. (Tv R)r(P.O) 3. - (r.s) 4 , T v R / . ' .R Exercise 4y' Use the eight implicationalargument forms to provethat the following arguments valid. are (l) {B.M))R L)(B.M)/."L)R (2) RvS (A)L)'[(Rvs)rr] /.'.TvL (3) (s) (8) (FlG)vH -G -Ht...-F (10) A)(A.B) C)A A I (- B.C) C)D EvB A/.,.D.8 (e) A.B B)CI.,.C C)A A)(B.D) C 1,,, B L Zv-R (ZvR))-T / . ' .- R v B R.S T/.'.(TvL).(R.S) (4) (7) /.'.rc)(A B)1.(CtA) (6) A) B C.A/.'.BvD (11) l. Rv - W ( 1 7 )t . A 2. (BvC))D 3. (AvE))(B.C)/...D 2. -w)L 3. RIT/...TvL ( 1 2 )1 . ( R . A ) v E 2.(R.A))D 3. -D/...E.-D ( 1 3 )l . ( A . D ) I - C 2. (Rvs)r(A D) 3. -C)-(A.D) /.'.(RvS)r-6.D) ( 1 4 )r . A 2. (4v-D)r(R S) /.'.(R.S) v B A vB C)A (8.- c) I (D.- C) -A/...D ( 1 9 )l . ( - A . - B ) r ( c l B ) 2. B)A 3. -A/...-C ( 2 0 )1 . [ - A . - ( D . D ] ) @ ) _ D ) 2.-(D.E).-n 3. E) F 4. -Av(D.E) s.-(D.E)t(BvE) / . ' .- D v F (rs)l. 2. (CvA))L 3. Av D 4. (DvU))C/...L (16) l. R 2. -Rr(-i4.-N) -3. -(-Pv-M) -A A -- ZvR/.'.(-U ( 1 8 )l . 2. 3. 4. -N).2 Exercise ; 5 Using the eighteenvalid argumentforms, prove that the following argumentsare vaiid' (Theseproofs are very basic.None requiresmore than six additionallines to complete). ( 1 ) r . ( A. B ) r C 2. A/."8)C (2) l. -RvS 2. A) (R'S)i.'. -A (3) r. -MvN 2. -Rl-Nl:.M)R (4) r. A)B 2. -(8.-qt.'.A)c (5) r. -Ar(B.C) 2. -Cl:.A (6) r. F)G 2. - (H.G) 3. Hl."-F (7) l.-(F/v-K) 2. L)H/...L)M (t2) r. A.(B)C) 2 . - ( c . A ) / . . .- B (8) l. M=N/...-NvM ( 1 3 )1 . ( A . B ) v ( C . D ) 2, -A/...C (9) t. A)-A 2.eev-B))Ct.'.-A.C ( 1 0 )l . R l . t 2. R)T/...Rt(.t.r) ( 1 1 )l . H ) K 2. C=D 3. -c)-K/...H)D ( 1 4 )l . D v - A 2.-(A.-B)t-c 3. -D/."-c (r5)l. (A.B)) C 2 . A . - C / . . .- B Exercise C 6 Prove that the following argumentsare valid. These proofs especiallyemphasizeDist, Comm, and Assoc. This exerciseis fairly challenging.Rememberthat Dist, like all our equivalence rules, works in both directions. (l) (1) l . ( A B ) v ( C . D ) 1. Av(B.C) 2. -C/:.A l:.(C.D)vA (2) l. (Av B)v C 2.-(BvC)/:.A (8) 1 . ( A v B ) . C (3)'1 (AvB).c 2.-(B.C)t...C.A (9) 1. t(A B) ' D) v (C . A) t.'. A 2. -Av-C/.'.C.8 ( 1 0 )l . ( - R (4) l.(A'B)v(C.D) 2. -C/...A A)v-(QvR)/.'. -R (11) l. [(A v B) . (D . F)) v [(AvB).C\\1.'.CvF (s) l.(A'B)v(C.D) /.'.(A.B)vD ( 1 2 ) r . t ( A. B ) v ( D . 4 1 v ( 8 . C ) 2.-(D.nt.'.8 ( 6 ) l . ( A . B ) v ( C . D ) l . ' .D v A Exercise fI 7 Prove valid using the eighteen valid argument forms. (These proofs are moderately difFrcult. They will require betweensix and fifteen additionallines to complete.) (7) r. -H ( 1 ) l . ( A. B ) I R 2.A 3. C)-Rt...-(C B) (2) l. -A 2. (AvB):C -Bt.'.-(c J. (3) l. D) @.m)(M.n) /.'. (A .1{) r N (4) 1. S v ( - R . ] n ) 2. R l - s / . . . - R (5) l. H)K 2. (K.L))Mt...L)(H)M) (6) l. A)B 2. C ) D (BvD))E J. 4. - E l : . - ( A v C ) (12) P]R -P)(-RlS)/.'.RvS (l 3 ) -(DvC) -c)(Al-B) A=Bl.'.-A 2. HvK 3. L)H 4 . - ( K ' - L ) v ( - L . M ) t . ' .M (8) 1 . ( A ' B ) = C 2.-(Cv-A)l:.-B (e) r . (HvK))(A)B) 2, (HvM))(C)D\\ 3. (HvN)l(AvC) 4. L.Hl.'.BvD ( 1 0 )1 . W = Y 2. -Wv-Y 3. X)(Y.Z)/.'.-X (ll) 1. AvB 2.C 3.(A.C))D 4. -(-F.B)1.'.DvF (14) l. -(CvA) 2. B r ( - A ) c ) 1 . " - B -(A.B):-c @veI)C/"'E)A Exercise J g For each-of the following expressionsindicate (l) which variablesare free and which bound; (2) which letters serve as individual constants and which as property consranrs; (3) which free variablesare within the scopeor ro*" quuntifieror other and which individual constants not within the scopeof are any quantiirer. (x)(Fx ) Ga) 2. (3x) (Fa. Gx\\ I. 3. (_r)[Fx) (Gy v Hx)) (.r)Fx f (1y)(Gy v Dx) 4. 5. Fa v (x)[(Ga v Dx) ) (- Ky . Hb)] 6. (x)(Fa)Dx))(])tryt ?GxvFx)l Exercise , 9 Construct expansions a two-individual in universe ofdiscourse the followinssentences: for . Gx) 1. (.t)(Fr 8. -(3x)(FrvGx) 2. (3x)(Fx v Gr) 3. (-r)[F,r] (Gx v Hx)) 4. (fu)trr . (Gxv Hx)l 5. (x) - (Fx ) Gx) 6. (lx) - (Fx v Gx) 1. - (xXFx f Gx) Exercise U 9. 10. I l. 12. 13. t4. (rXFr I (Gx ) Hx)l (xXFr f - (Gx Hx)l (lxX(F.r . Gx) v (Hx Kx)l (xX(Fx.Gx) I (Hx. Kx)l (lx) - [(Fx ) Gx) v (Fx) Hx)] - (x) - l(Fx Gx). - (Hx. Kx)J lO Provethat the following arguments invalid. are ( l ) 1 . ()x)(Ax. Bx) 2 . (3x)(B.r'Cx) / .'. (3x)(Ax. Cx) (2) l. (x)(Ax ) Bx) ) (lx) - Ax l.'. (3x) - Bx ( 3 ) l . (lx)(A-r. - Bx) 2 . (3;r)(4.r.- Cx) 3 . (3rX- Bx. Dx) /.'. (3x)[Ax (- Bx . Dx)l (4) L (xXFx ) Gr) 2 . (x)(- Fx ) Ex) I .'. (x)(- Gx ) - Ex) ( 5 ) l . (].r)(Px.- Qx) 2. (x)(RxI Px) /.'. (l,rXRx - Q*) ( 6 ) I . (x)IQx. Qx) ) Rx) ') (lx)(Qx' - Rx) l:. (x)(- Px . - Qx) ( 7 ) l . (x)(Px ) Qx) ) (x)(Qx ) Rx) /.'. (x)(P.r.R.r) (8) l. (x)[Mx)(Nx)Px) 2. (x)(Qx ) Px) /:. (x)[Qx ) (Mx . Nx)) (e) 1. (lxXAx.B;) 2. (x)(- Bx v - Cx) /.'. (x)(- Ax v - Cx) (10) l. (3x)(Axv-B.r) 2. (x)l(Ax.-Bx))Cxl /.'. (1x)Cx Exercise * il Completethe following proofs using the rules for adding and removing quantifierswhere (l) l. (x)F; v (r) - Gx 2. - (x)Fx 3 (x)(Dx ) Gx) p P p /.'. (1xX- Dx v Gx) (2) l. ( . x ) [ A r v ( B x . - C . r ) ] 2 . (x)Cx p p | :. (3x)(Dx ) At) (3) L (x)[- Ax v (Bx . Cx)] 2. ( x X ( / x ) C x ) ) D x l (x)(Dx I - Cx) J. p p p l:. (fx) - Ax (4) p p p /.'. Bc L Ab) Bc z. (x)(Ax) Bx) J, (xX(tu)Bx))Axl (5) l. A b v B c 2 . (x) - Bx p p /.'. (ix)Ax ( 6 ) 1 . $)(Ry r - Gy) 2 . ()(87 v Gz) 3 . (y)Ry p p p t:. (v)Bv ( 1 ) 1 . (dlAz ) (- Bz ) Cz)l 2. - B a p p l:. Aa) Ca ( 8 ) l . (xX(&r .Ax) ) Tx) 2 . Ab 3 . (x)Rx p p pl:.Tb.Rb Exercise 1? iZ Which lines in the following are not valid? Explain why in eachcase. (l) l. 2. 3. 4. 5. .Kx)) Mxl (xX(FIx (]x)(Hx. Kx) Hx.Kx Mx (fx)Mx (2) r. (x)(Mx) Gx)) Fa 2. (x)(- Gx ) - Mx) 3.-Gy)-My a. (xX- Gx ) - Mx) ) Fa 5. (- Gx) - Mx)) Fa 6. Fa 7. (x)Fx (3) l. 2. 3. 4. (lx)(Fx.- Mx) (x)[(Gxv Hx) ) Mx] (Gy v Hy) ) My Fy.- My 5. - M1' 6. -(GyvHy) 7. (1x) - (Gx v Hx) (4) 1. (1x)(Px ' Qx) 2. Pv'Qv 3. Qy a.Qyv-R.y 5. (x)(Qxv-RJr) ( 5 ) l . ( 3 x ) [ ( P x. Q x ) v R x ] 2. 3. 4. 5. 6. 7. 8. (x) - Rx (3x)(Px v Rx) Pxv Rx -Px (x) - Px - Pv (z) - Pz ( 6 ) l. - (x)Fx 2. 3. 4. 5. 6. 7. 8. 9. (1x)Lx (x) - Fx -Fx I^a Lx Lx.- Fx (lxXl.x (x)Lx Fx) p p 2El I , 3M P 4EG p p 2Ul I Contra 4UI 3,5MP 6UG p p 2Ul IEI 4 Simp 3,5MT 6EG p IEI 2 Simp 3 Add 4UG p p I Simp 3EI 2,4DS 5UG 6UI 7UG p p p IUI 2El 2El a,6 C.qnj 7EG 5UG Exercise I f3 Prove valid. (l) l. (x)(Rx ) Bx) 2. (3x) - Bx p p /.'. (Lr) - Rr (2) 1. (x)(Fx ) Gx) 2 (y)(Gy ) Hy) (3) 1. Ka 2. (x)[Kx ) (y)Hy] p P /:. (z)(- Hz) - Fz) p p /.'. (x)Hx (4) l. (.r)(Fx I G.r) 2. (x)(Ax ) Fx) 3. (fx) - G; p p p /.'. (3x) - Ax (5) l. (x)(M-r I S.r) 2. (x)(- Bx v Mx) (6) l. (xXRr I Ox) p ) P /.'. (xX- ,Sx - Br) p p p /.'. (12)Pz 2' (3Y)- ov 3. (z)(- Rz) Pz) (7) t. ()x)(Ax. Bx) 2. (y)(A1, Cy) ) p p /.'. (3x)(Bx. Cx) ( 8 ) l . (1,Y),tt 2. ( x ) ( - G x f - R r ) 3 . (x)Mx p p p /:. (1x)Gx . (1x)Mx (e) l. p p /:. (1y) - Gy (x)[(FxvRr))-Gr] -Rx) 2. ( f u ) - ( - F x ( 1 0 ) l . (x)(Kx) - Lx) 2. (3x)(Mx. /x) (l l) l . (x)(Fx ) Gx) 2. (y)(Ey ) Fy) (z)-(Dz.-Ez) J. (12) l. (x)(l,x ) - Kx) 2. (12)(Rz. Kz) (y)t(- Ly Ry)) Byl J. P p l.'. (lx)(Mx' - Kx) p p p l:. (x)(Dx) Gx) p p p /.'. (1x)Bx Exercise 3 t,z Provevalid (note that theseproblemsare not necessarily order of diffrculty). in (l) (2) (3) l. (lr)fr v (fx)Gx 2. (x)-Fx l. l. (4) l. (s) l. 2. (6) L 2. p p l.'. (1x)Gx (x)(Hx) - Kx) p /:.-(ly)(Hy.Ky) - (x)A-r p /.'. (3x)(tu ) Bx) - (lx)F.r p /:. Fa ) Ga (1x)Fx ) (x) - Gx p (3x)Ex I - (x) - Fx p l.'. (1x)Ex ) - (3-r)Gx (3rXA.r. Bx) ) (y)Cy p -Ca p /.'. (x)(Ax) - Bx) (12) 1. (x)(Gx ) Hx) p ) (3.r)(1.r.-Hx) p 3 . (x)(- Fx v Gx) p /.'. (3x)(lx. -Fx) ( 1 3 ) L (x)l(Ax. Bx)) Cxl p 2. - c b p t . . .- ( A b . B b ) ( r 4 ) 1 . - (x)(Fx ) Gx) p 2. - (lxX- Gx. Hx) p /.'. (1x) - Hx ( l 5 ) l . (x)(Hx ) Kx) p 2. (3x)Hx v (Lr)Kx n r /.'. (lx)Kx ( 1 6 ) 1 . (3.r)Fx (lxXGx.Hx) I p 2. (lx)(Hx v Kx) ) (x)Lx p /... (x)(Fx) t-r) (7) l . (x)[(Fx v Hx) ) (Gx. Ax)] p p 2. - (x)(Ax. Gx) (8) 1 . - (x)(Hx v Kx) l:. (fx) - Hx p 2 . 0)t(- Kyv Ly)) Myl p /... (12)Mz (e) 1. (x)[(F.rv Gx) ) Hx) p (.rX(Hxv Kx) ) Lxl p 2. /:. (x)(Fx ) Lx)' (10) 1. (lx)tu)(x).9x p ) (.rXfx I R.r) p /.'. (1x)TxI (fr)Sr ( rl ) l . (x)[(A.rv Bx) ) Cx] p p 2. - (lyXCy v Dy) l.'. - (fx)Ax .Ax) ) Dx) (17) l. (x)[(Bx p 2. (3x)(Qx.Ax) p 3. (xX- Bx ) - Qx) p /.'. (lx)(Dx . Qx) (18) l. (rXPx) (Axv Bx)) p 2. (x)[(Bxv Cx)) Qx] p /:.(x)[(Px.-Ax))ex] (19) l. (rXPx f (Qr v Rr)l p . 2. (.rX(Sx Px) ) - Qxl p /.'. (xXSx) Px) ) (xXSxI rtr) (20) l. (x)l(A-r B.r)) (Cx.Dx)J p v /.'. (fx)(Ax v Cx) ) (]x)Cx Exercise n i >- Indicatewhich (if any) of the inferences the following proofs are invalid, and statewhy in they are invalid. (1) l. 2. 3. 4. 5. (lx)(y)Fry Q)Fxy Fxx (3y)Fyy (.rXiy)Fyr (2) L (fx)Fx 2. (]x)Gx 3. Fv 4.Gv 5. Fy.Gy 6. (lyXFy.Gr) '7. Gz)(1y)(Fy. Gz) (3) 1. 2. 3. 4. 5. (.r)(3yXFx Gy) I (?y)(Fx ) Gy) Fx :- Gy (x)(Fx ) Gy) (lyX-rXF.r f Gy) (4) l. (x)(lyXFx ) Gy) 2 . Fx 3 . (3y)(ry r Gy) Fy)Gy 5 . Gy 6 . (lw)(Fw I Gy) 1 Fw)Gy 8. (fw)(Fw ) Gw) 9 . (1w)[(Fw ) Gw p IEI 2Ul 3EG 4UG P p 1EI 2El 3,4 Conj 5EG 6EG p ltJI 2El 3UG 4EG p AP /.'. (lwX(rw ) Gw)'Gyl lUI 3EI 2,4WP 4EG 6EI 7EG 5,8 lu. 1 1 . (x)lFx I (3w)[(Fw Gw).Gyll ) (s) 1 . (x)br)[(z)Fzx' ' Hd)) (Gy 2. O ) I Q ) F z a ) ( G y . H A l 3. (z)Fza)(Ga.HA 4. Fba ) (Ga. Hd1 5 . (fy)lFby:_(Gy.Hd)l 6. Fbv 7. G y . H d 8 . Hd 9 . (lx)Hx 1 0 . Gy I l . (x 12. Fby ) (x)Gx 13. (v)FbyI (;r)G.r ((r,1 l. (xXtv)Fry Grl I 2, (t'\\Far ) Ca 3. l;ut') Go '' Gu l''4 lin' 5 | tr.,t..glirr I 7 ^ ' G u ) (x) - Fax 8 . ( l _ y )^ G y f ( x ) - F y x 10UG p lUI 2Ul 3UI 4EG AP /.'. (x)G.r 5,6MP 7 Simp 8EG 7 Simp IO UG 6-ll cP 12UG p ltJI zVr AP/.'. (x)- Fax 3,4MT 5UG MCP 7EG Exerciseill Z Ans.*er;, 2. a,d 10. a, d, f, n 12. a, d, l, g, n 14. a, b, i, i 4. a,c,l 6. a,d,f,n 8. a,d,k,o 4^s'wrts E x e r c i s e3 * (21 J. tI (4) ? I (6) 1 , 2M P 1 tD ttll 4.N 4.-H E 2 , 3D S -n 6. A) 1 , 4D S 2,6 DS 5. P.O 2,4MP 6. Bv (I. S) 7. R ( 14 ) 1 , 4D S (10) c.-, 6. - (Lv Ml 7. R 1 , 3D S 2 , 4M P (8) (12) 1 , 2H S 3,6 DS 3,5 MP 1 , 5M P 2,4DS B 3.5 MP -7/. 1 , 6M P Exercise ,lnlwers 4{ (21 3. (Rv$f f 2 Simp 4.7 5. fvL (4) 4 Add (10) 1 Simp t^ (8) 3. B +.v (6) 2,3MP 3. A 4.8 5. 8v D 5. -B.C 2 Simp 6.C 7. D 8. -8 9.E 1 0 .D . E 4. lvfr 5 .- r l12t (14) 1 . 3M P 6. -8 7. - Rv B 4. - (R. Al 1 , 3M P 4 Add (18) 1 , 4M P 5 Simp 2,6 MP 5 Simp 3,8 DS 7.9 Coni 1 Add 3,4 MP 2,5 DS 6 Add 2,3 MT 5.F 1 . 4D S 6. E.- D 3,5 Coni (20) 3. Av-D 4. F.S 5, (B'SlvB 5. -R 6.2 7. -M.-N 8. t- M.- Nt.Z 5.8 6. -C 7. B.-C 8. D.-C 9,D 6 . - ( D .E ) 7. - A 8. -A.-(D.E) 9. B) - D 1 0 .8 v E 1 1 .- D v F 1 Add 2,3MP 4 Add 1 . 3M T 4,5 DS 2,5 MP 6,7 Conj 1 , 4D S 2,4MT 5,6 Conj 3.7 MP 8 Simp 2 Simp 4,6 DS 6,7 Conj 1 , 8M P 5,6 MP 3,9,10CD Exercise{F Anc wrYS G (4) 3. Av(8vC) 'l Assoc 4.4 (21 2,3DS 3. t(A' 8)v Cl. tA' Bl v Dl 4. lA.BlvC 5. A.B 6.4 (6) 2. IlA. B) v Cl . l(A' 8) v Dl 3. A.Blv D 4. DvlA-Bl 5. (Dv Al.(Dv Bl 6. Dv A (8) 3. C'Av8l 4. (C.Al v(C.8) 5. - Cv - A 6. - (C.A) 7, C.B Exercise n l2l 7 1 Dist 2 Simp 3 Comm 4 Dist 5 Simp l Comm 3 Dist 2 Comm 5 DeM 4,6DS 4. [(AvB]f C1. lCl(AvB)l 5. Ct lAv 81 6. -4.-B 2 Equiv 4 Simp 1,3 Gonj 6 DeM 5.7 MT 9. -Cv-D 8 Add 1 0 .- ( c . o ) 9 DeM 3. (Sv-F)'(SvI) 4. Sv-8 5. - - Sv - B 1 Dist 3 Simp 5lmpl 7. R)-R 8. - 8v - R 9. -R 2.6 HS 5. - (8v D) 6. -8.-D 3,4 MT 7. - B 8. -A 6 Simp 9. -D 6 Simp 1 0 .- c 2,9 MT 11. - A.- C 12. - (AvCl (14) (10) 4DN 6. -Sl*F (6) t12l 8,10 Coni 7lmpl 8 Taut 5 DeM 1,7 MT 11DeM 3. - (A v C) 4. -(--AvC) 5. - (- A) 6. -8 3Dist 4Simp 5Comm 7.(-Rv-Rl .(-RvA) 8. -Bv-F 9. -B 1 Dist 3 Simp 2,4DS 5 Simp 2. 1-R.Atv - (Fv O) 3 . ( - B . A )v (-B'-O) 4. t(-F'A)v-Fl .l(-F.Alv-Ql 5. (-F.A)v-R 6. -Rv(-F.A) 6Dist TSimp STaut 3. l(D'FlvlA'8)l v (8. C) 4 . ( D . F l v [ { A' 8 ) v {8. C)l 5. A' A v {8. C) 6. (8'A) v (8. C) 7. B'lAvCl 8. B l Gomm 2DeM 1Comm 3 Assoc 2 , 4D S 5 Comm 6Dist 7 Simp tn*,*t'S 7. - (Av Bl 8. -C (4) (10) 1 Comm 3DN C) 4 tmpl 2.5MT t12l 3. -C.--A 4. -C 5 . t ( A .R t) C l . C) A . A l 6. G.A)C 7. - lA. St 8. -Av-B 9. --A 1 0 .- B 4. (W. Yl v l- W. - Yl 5. - (W. Yl 6. -W.-Y 7. -Y 8. -Yv-Z 9. - ly. Zl 1 0 .- x 3. -Bl-P 4. -nl(-F)Sl 5. (-F.-R)lS 6. -F)S 7. --FvS 8.8vS 2DeM 3Simp 1 Equiv SSimp 4,6MT TDeM 3Simp 8,9DS 1 Equiv 2 DeM 4,5DS 6Simp TAdd 8 DeM 3,9MT lGontra 2,3HS 4Exp STaut 6lmpl 7DN i Exercise f t2l J. -(R @ r(4>-(r'> S) 1 DeM 4. 3. - B v - - C 4. - B v C 2 DeM 3DN B) C A)C 4. - H v - G 5. - - H 6. - G (10) 2,3 MT 4lmpl b. -F 1 , 5H S 3DN 4,5 DS 1 Equiv 2 Simp 3lmpl llmpl 2lmpl (-Fvf) f) 3,4Conj SDist (14) r) 7. Rl(S 3. -Cv-A 6lmpl 2DeM 4. A 5, --A l12l 2 DeM 1 , 6M T 2. ( M ] M , ( N ] M ) 3. N ) M 4. - Nv M 3. -BvS 4. -RvT 5. (-BvS) 6. -Rv(S 1 Simp 4DN 6. -C 1. B)C B. -8 4. - A 5. --Av--'B 3,5DS -Bl SDeM 6.-lA 7. -C lSimp 6,7MT 1 , 3D S 4Add 2,6MP 8 An< u"-r 2. D. ( 1 )x i s b o u n d . c ; l { 2 ) a i s a n i n d i v i d u ac o n s t a n tF a n d G a r e p r o p e r t y o n s t a n t s ' (3) No free variables;a is within the scope of the (x) quantifier' are h ( 1 ) y a n d t h e { i r s t x v a r i a b l e ( n o t c o u n t i n g t h e x t h a t i s p a r tto fe q u a n t i f i e r ) i b o u n d ;t h e s e c o n dx v a r i a b l e s f r e e ' (2) No individualconstants;F, G, and D are propertyconstants' ( 3 1F r e ex v a r i a b l ei s w i t h i n t h e s c o p eo f t h e ( / q u a n t i f i e r ' free ( 1 )T h e f i r s t x v a r i a b l e a n d t h e y v a r i a b l ea r e b o u n d .T h e l a s tt w o x v a r i a b l e sa r e constant' a Ql F, G, and D are property constants; is an individual T ( 3 )T h e t w o f r e e x v a r i a b l e s r e w i t h i nt h e s c o p eo f t h e ( y ) q u a n t i f i e r .h e i n d i v i d u a l a a, is within the scopeof the (x) quantifier' constant, Exercise U 2. 4. 6. 8. g tlqsu/rrs lFa v Ga)v (Fb v Gb) lFa (Ga v Hall v tFb {Gb v Hb)l -lFavGa) v-(FbvGbl -[(Fav6a) v(FbvGb)l 10. [Fa] - (Ga' Hall. lFbl - (Gb. Hb)l 12. l(Fa. Ga) I lHa Kall t ( F b . G b )I ( H b . K b ) l 14. - {- llFa Gal - (Ha Kall - t(Fb Gbt.- (Hb Kb)l) Exercise:Il /f h"tsuers 4. Rx)-Gx lUl 2Ul 5. Bxv Gx 2Ul 5. Dx 6. Cxv-Bx 7.-8xvCx APl.'. Ax 4 Add 6 Comm 6. Bx 7. - Gx 3Ul 8. -8x \\21 3. Ax v (Bx. - Cxl 4. Cx 7DN 1 Ul (6) 4,6 MP 5,7 DS 9. lylBy 4. (Bb. Abl ) Tb 8UG 3,9 DS v--Cx 8. 8x 5. Fb 3Ul s-10cP 6. Rb.Ab 2,5 Conj 1 1E G 2Ut 3Ul 4.5MP 1 , 6M P 7. Tb 4,6 MP 8. Tb. Rb 5.7 Conj 8 DeM (4) 11, Dx) Ax 12. lSxl(Dx) Axl 4. Ab-r-Bb 5. (Ab I Bbt) Ab 6. Ab 7. Bc Exercise A t2l (4) (6) (8) lUl l? 4. Invalid. Quantifier must be removed first. 5. Invalid.The (x) quantifierdid not quantify the whole line. 6. Invalid.Antecedentof line 5 does not match line 3. 7. lnvalid. Can't universallygeneralize from a constant. 5. Invalid.can't use UG to bind a variablethat is free in a line that is justified by El. In this case y is free in line 2. 4. Invalid.The (x) quantifierdoes not quantify the whole line. 5. Invalid.Can't replacea variablewith a constantwhen using El. 9. Invalid.Can't universallygeneralize from a constant. Exercisef l2l 13 AntueyS 3. Fx = Gx 4. Gx -,:Hx (4) 5. Fx) Hx 6. -Hx)-Fx 7. (zll- Hz ) - Fzl 4. - Gx 5. Ax) Fx 6. Fx: Gx 7. Ax) Gx 8. -Ax (61 9. (3x) - Ax 4. - Ox 5. Rx= Ox 6. - Rx= Px 7. - Rx 8. Px 9. l3zlPz (8) 4, Rx 5.-Gxf-Bx 6. Mx 7. Rx) Gx 8. Gx 9. (3$Gx 10. lfxlMx 11. (SxlGx.ExlMx 1Ul 2Ul 3.4HS 5 Contra 6UG 3Et 2Ua 1Ul s,6 HS 4,7MT 8EG 2El 1Ul 3Ut 4.5MT 6.7MP 8EG lEl 2Ul 3Ul 5 Contra 4,7MP 8EG 6EG 9,10Conj {10} l12l 3. Mx.Lx 4. Kx)-Lx 5, lx 6. --l-x 7. - Kx L Mx 9. Mx' - Kx 10. (lxllMx. - Kxl 4. Rx. Kx 5. Lx)-Kx 6. (-tx.Rxl)Bx 7. Kx 8. --Kx 9. - Lx 10. Fx 11.- Lx.Rx 12. Bx 13. (3x)Bx 2El 1Ul 3 Simp 5DN 4,6MT 3 Simp 7,8Conj 9 EG 2El 1Ul 3Ul 4 Simp 7DN 5,8MT 4 Simp 9,10Coni 6,11 MP 12EG Exercise n (4) r/ 4nluei.s 2. Hx) - Kx 3.-Hxv-Kx 4. - lHx. Kxl 5. (y)- lHy. Kvl 6. - (]yl(Hv. Kyl 2. (xl - Fx 3. -Fa 1Ul (6) 2lmpl 3 DeM 4UG 5(}N 10N 2Ul 4. - Fav Ga 3 Add 5. Fa:. Ga 2EG 3()N 1 , 4M T 50N 6Ul 7 DeM 8lmpl 4lmpl llxl - (Hxv Kxl 10N 3El 4 DeM o- - K x 5 Simp (- Kxv Lxl) Mx 2Ul 8. - K x v L x 6 Add 9 . Mx 7,8MP I 0 . lfzlMz 9EG - (lx)Sx J. AP 4 . - (lxl,9x 1 , 3M T 5 . (x) - Rx 40N 6. - R x 5Ul 7 . Tx) Rx 2Ul 8. - T x 6,7 MT q lxl - Tx 8UG 1n 90N 1 1 . -(3x)Sx l- (fx)Ix 3-10 CP ' t 2 . (lx)Ix I (3x)Sx 11 Contra lx -Hx 2El 5 . - Fxv Gx 3Ul Gx) Hx 1Ul 7. rx)gx 5lmpl 8. Fx) Hx 6,7 HS q -Hx 4 Simp 10.-Fx 8,9 MT 1',t. lx 4 Simp tt. lx -Fx 10,11 onj C 1 3 . (lxXix. - Fx) 1 2E G J. 10N lSxl - (Fx) Gxl 4 . - (Fx ) Gxl 3El - (- Fxv 6x) 4lmpl o. - - F x . - G x 5 DeM 1. -Gx 6 Simp 8 . (xI - (- Gx Hxl 2(}N 9. - l- Gx. Hxl 8Ul 10. --Gxv-Hx 9 DeM 1 t . Gxv - Hx 10DN 12. - H x 7,11 S D t J . llxl - Hx 1 2E G 5. 9UG Fx (3xlFx 4 . - (Hx v Kxl 5. - H x - K x (1 0 ) 3. (]vl - Cy a. - (ylCy 5. - (lxXAx Bx) 6. (x) - (Ax. Bxl 7. -(Ax.Bxl 8. -Axv-Bx 9. Ax) - Bx 10. (xXAx) - Bxl 3EG 1 , 4M P 5El 6 Simp 7 Add 8EG 2,9 MP lxlLx Lx 10ul 2. Fx) Lx 3-11CP 3 . (xl(Fx) Lxl 1 2U G J. Px) (Axv Bx\\ 1 Ul 4. l B x v C x l = O x 2Ul Px. - Ax AP l.'. Qx o. Px 5 Simp 7. Axv Bx 3,6 MP R -Ax 5 Simp q Bx 7,8 DS 0. Bxv Cx 9 Add 1 . Ax 4 , 1 0M P 12. ( P x - A A ) A x 5 - 11 C P 1 3 . (xll(Px. - Axl ) Oxl 12 UG 2. - (]xlCx lixllGx. Hxl Gy. Hy Hy Hyv Ky (3x)(Hxv Kx) (18) 3 . ( x )- C x 2ON 4. - C x 3Ul (Axv Bxl ) (Cx. Dxl 1 Ul o- - l A x v B x l v (Cx Dxl Slmpl ( - A x . - B x lv (Cx. Dxl 6 DeM IF Ax. - Bxl v Cxl IF Ax.- 8x) v Dxl 7 Dist 9 . l- Ax. - Bxl v Cx 8 Simp 1 0 . - Ax. - Bx 4,9 DS 1 1 .- A x 1 0S i m p 12. - A x . - C x 4 , 1 1C o n j t J . - lAxv Cxl 12 DeM 1 4 . 8l - (Axv Cxl 13 UG 1 5 . - l3xl(Axv Cxl 14 ON t b . - taxtLx ) - (lxllAx v Cxl 2-15 CP 1 7 . Exl(Ax v Cxl > (3xlCx 16 Contra Valid Forms for Sentential Looic Valid Argument Forms of Inference 1 . Modus Ponens(Mpl: p) 5 . Conlunction{Conj}: q p p l.'. q q l.'.p.q Modus Tollens(MT): b. HypotheticalSyllogism (flSl: p) q p) q - Q /:' - P q ) r l . ' .p ) r 3 . DisiunctiveSyllogism (DSl: pv q P l:.Pvq - p l..q ConstructiveDilemma ICD) : pv q -q 7 . Addition {Add): pv q l.'.p Simplification(Simp): g ) s l . ' .r v s P ' a / . ' .P P q /.'.q Valid Equivalence Forms (Rule of Replacementl Double Negation (DN): p:: -- p ( pl q ) : :( - q ) - p l DeMorgant Theorem (DeMl: -\\P q).:(-pv-q) -lpvqJ::Gp.-e) 't1. Commutation (Comm): (Pvql.:@vp) t 5 . lmplication(lmpl): ( p l q ) : :( - p v q ) 1 6 . Exportation (Expl: I(p dtrl::lp)(qtr)l 1 7 . Tautology (Taut): @ o ) : :l q ' P 1 p.:\\p.pl 12. Assocation(Assocl: Ipr@ur)l::[(pvqJvr] ,r'1o'r))::I@ q).r) 1 3 . Distribution (Dist): Ip lq v r)l:: llp. ql v \\p. rl) l p v ( q r ) l: : { ( p v q ) . ( p v r ) l Conditional and lndirect Proof 1 4 . Contraposition (Gontral: p:: \\pv p) (Equivl: 1 8 . Eguivalence (p-q)::t(plq).(q)pll ( p = q ): : l ( p q l " (- p. - q)l ConditionalProof A Pt . . q rr I I l^ p) q cP AP 1...p Bules for PredicatELogic Rule Ul: {uX Rule El: . t t . " 1 . . .w (3u)( RuleUG: \\ t:. \\ .) w. ..1 I l.'. lvil . . w Provided: 1. (. . . w. . .) results from replacing eachoccurr e n c eo f u f r e e i n { . . . u . . . ) w i t h a w t h a t i s fl e i t h e r a c o n s t a n t o r a v a r i a br e ei n ( . . . w . . . \\ (making otherchanges). no Provided: 1. w is not a constant. 2. w does not occur free previously the proof. in 3. l. . . w. . .) results irom replacing eachoccurrenceof ufree in (. . . u. . .) with a wthat rs free i n ( . . . w . . . ) ( m a k i n g o o t h e rc h a n g e s ) . n Provided: 1. u is not a constant. in 2. u does not occurfree previously a line obtarned El. by in u does not occurfree previously an assumed premisethat has not yet been discharged. (. . . w .. .) results from replacrng eachoccurr e n c eo f u f r e e i n ( . . . u . . . ) w i t ha w t h a t r s f r e e in (. . . w. . .) {making otherchanges} no and free occurrences h/ of there are no additional n a l r e a d v c o n t a i n ie d( . . . w . . . 1 . Rule EG: 1 . . u . . ) / . ' .l 3 v t | l RuleON: { u X . . . u . . . ) : :- ( 3 u-) ( . . . u . . .) (lu)(. . . u . . . ) : : - ( u ) - ( . . . u . . .) ( d - ( . . . u . . . ) : :- ( l u ) ( . . . u . . .) E u ) - ( . . . u . . . ) : : - l u l l . . . u . . .) Rule lD: (...u...1 u=w l.',( RulefR: l.'. \\xllx= x) w...) Provided: 1- (. . . w. . .) results from replacing least one at occurrence u, where u is a constantor a variof ablefree in (. . . u. . .) with a wthat is free in (. . . w. . .) (making otherchanges) there no and free occurrences w already ol are no additional in contained (. . . w. . .) {...u...) w=u l;. 1 w. ..1 e f y encrs -l^v-i/^'1,(ig5 Cwsl>".f { ' o n t / 4 6 i/u*,V t,\\ (A ) B)= (e u -A) rrulpurys I o, t .(* (a v : B))>(s 'a) c)(*B )'-4= $r,<) ,i) ..\\- = ((u -, A),(,(o A ) B)) _ 4)) (a r(.-) " A ='*A fj 'J) (A"-A)> B A) A ) (e ,n) Z Fxenr,j( R r ^ e , , c h 5 e n / t r t f D t .f l - , l r ( t fAv,tu5h o lt ,t( t f l ' l + 9 e a { * n t f{ ; , * t cvt fU bt'shl q^ul n crc *k (hy ;f ls (r,i s'[-fufi",'' tnrf>ntp q fr"rn. ov- it ,ot "( iA"i sah/et4ce # l.A 2. A)B 3. (AvB))C 4. Av@)q 5.(-AvB))C 6.-(AvB))C 7. -Av(Bf C) 8 . ( Av B ) ) - C 9.-IAv(BlC)l 10. -(-AvB))C 11. -[(AvB))CJ 12.-(AvB))-C 13. -[-(A.rB)rC] 14. -t-?evB)lCl 15.-t(-AvB))Q f t i a E A,r'- QUtL tuAih a.p b. -p c.pvq d.p)q e.-pvq f.-p)q E.-p)-q h.-(pvq) i. -(p)q) J.-(-p)q) k. (pvq))r l. pv(q)r) m. (- pv q)) r n. -(pvq))r o. (PVo)>-, p.-[pv(q)r)] q.-t(pvq))rl !^ o( d'tr? ftr d iluA,tn1 ^(eryPns,l 0 f t t|w { , l S r^ 'x or t vPi l- { v'tn* -{at ou los.7 ( fif / 1n o ,,/ TA l) z-) (<'7"erre.) A@B C ) ".'A ) B ,l) '.--fA) B) r) (a) 4"k= D).6vFrG) a (*r t {': *--. Exercise L i' a; Use MP, MT, DS, and HS to prove that the following argumentsare valid. la :. (l) 3 l. -R 2 . s I R / . . .- S (2) A.S (A.S))R/."R (3) - (H.K) Rv(I/.K)/.'.R (4) (PvQ)l(R'w) L)(PvQ)/.'. tr(R.w) (5) Rfs rlR -s/.'.-T (6) -M NfG NvM/.'.G (7) - D) E D)F -F/:.E (8) Gv H -Hvl -r/.'.G (e) -G)(AvB) -B A) D - G/.'.D (10) . (A) B)) C r 2. -DvA 3. -D f (Af B) 4. - A/.'. C (1r)1. Ar(Bf C) 2. -C 3. -D)A 4. Cv-D/.'.-B ( 1 2 )1 . - ( D . F ) 2. (LvM)vR 3. -T)-(LvM) 4.(D. F)v-T/.'.R ( 1 3 )l . 2. 3. 4. (AvB)r(BvO (B)C)vA (BlC)l(AvB) -Al.'.BvC ( 1 4 )l . ( P . O l l R v ( r . D l 2. (Tv R)r(P.O) 3. - (r.s) 4 , T v R / . ' .R Exercise 4y' Use the eight implicationalargument forms to provethat the following arguments valid. are (l) {B.M))R L)(B.M)/."L)R (2) RvS (A)L)'[(Rvs)rr] /.'.TvL (3) (s) (8) (FlG)vH -G -Ht...-F (10) A)(A.B) C)A A I (- B.C) C)D EvB A/.,.D.8 (e) A.B B)CI.,.C C)A A)(B.D) C 1,,, B L Zv-R (ZvR))-T / . ' .- R v B R.S T/.'.(TvL).(R.S) (4) (7) /.'.rc)(A B)1.(CtA) (6) A) B C.A/.'.BvD (11) l. Rv - W ( 1 7 )t . A 2. (BvC))D 3. (AvE))(B.C)/...D 2. -w)L 3. RIT/...TvL ( 1 2 )1 . ( R . A ) v E 2.(R.A))D 3. -D/...E.-D ( 1 3 )l . ( A . D ) I - C 2. (Rvs)r(A D) 3. -C)-(A.D) /.'.(RvS)r-6.D) ( 1 4 )r . A 2. (4v-D)r(R S) /.'.(R.S) v B A vB C)A (8.- c) I (D.- C) -A/...D ( 1 9 )l . ( - A . - B ) r ( c l B ) 2. B)A 3. -A/...-C ( 2 0 )1 . [ - A . - ( D . D ] ) @ ) _ D ) 2.-(D.E).-n 3. E) F 4. -Av(D.E) s.-(D.E)t(BvE) / . ' .- D v F (rs)l. 2. (CvA))L 3. Av D 4. (DvU))C/...L (16) l. R 2. -Rr(-i4.-N) -3. -(-Pv-M) -A A -- ZvR/.'.(-U ( 1 8 )l . 2. 3. 4. -N).2 Exercise ; 5 Using the eighteenvalid argumentforms, prove that the following argumentsare vaiid' (Theseproofs are very basic.None requiresmore than six additionallines to complete). ( 1 ) r . ( A. B ) r C 2. A/."8)C (2) l. -RvS 2. A) (R'S)i.'. -A (3) r. -MvN 2. -Rl-Nl:.M)R (4) r. A)B 2. -(8.-qt.'.A)c (5) r. -Ar(B.C) 2. -Cl:.A (6) r. F)G 2. - (H.G) 3. Hl."-F (7) l.-(F/v-K) 2. L)H/...L)M (t2) r. A.(B)C) 2 . - ( c . A ) / . . .- B (8) l. M=N/...-NvM ( 1 3 )1 . ( A . B ) v ( C . D ) 2, -A/...C (9) t. A)-A 2.eev-B))Ct.'.-A.C ( 1 0 )l . R l . t 2. R)T/...Rt(.t.r) ( 1 1 )l . H ) K 2. C=D 3. -c)-K/...H)D ( 1 4 )l . D v - A 2.-(A.-B)t-c 3. -D/."-c (r5)l. (A.B)) C 2 . A . - C / . . .- B Exercise C 6 Prove that the following argumentsare valid. These proofs especiallyemphasizeDist, Comm, and Assoc. This exerciseis fairly challenging.Rememberthat Dist, like all our equivalence rules, works in both directions. (l) (1) l . ( A B ) v ( C . D ) 1. Av(B.C) 2. -C/:.A l:.(C.D)vA (2) l. (Av B)v C 2.-(BvC)/:.A (8) 1 . ( A v B ) . C (3)'1 (AvB).c 2.-(B.C)t...C.A (9) 1. t(A B) ' D) v (C . A) t.'. A 2. -Av-C/.'.C.8 ( 1 0 )l . ( - R (4) l.(A'B)v(C.D) 2. -C/...A A)v-(QvR)/.'. -R (11) l. [(A v B) . (D . F)) v [(AvB).C\\1.'.CvF (s) l.(A'B)v(C.D) /.'.(A.B)vD ( 1 2 ) r . t ( A. B ) v ( D . 4 1 v ( 8 . C ) 2.-(D.nt.'.8 ( 6 ) l . ( A . B ) v ( C . D ) l . ' .D v A Exercise fI 7 Prove valid using the eighteen valid argument forms. (These proofs are moderately difFrcult. They will require betweensix and fifteen additionallines to complete.) (7) r. -H ( 1 ) l . ( A. B ) I R 2.A 3. C)-Rt...-(C B) (2) l. -A 2. (AvB):C -Bt.'.-(c J. (3) l. D) @.m)(M.n) /.'. (A .1{) r N (4) 1. S v ( - R . ] n ) 2. R l - s / . . . - R (5) l. H)K 2. (K.L))Mt...L)(H)M) (6) l. A)B 2. C ) D (BvD))E J. 4. - E l : . - ( A v C ) (12) P]R -P)(-RlS)/.'.RvS (l 3 ) -(DvC) -c)(Al-B) A=Bl.'.-A 2. HvK 3. L)H 4 . - ( K ' - L ) v ( - L . M ) t . ' .M (8) 1 . ( A ' B ) = C 2.-(Cv-A)l:.-B (e) r . (HvK))(A)B) 2, (HvM))(C)D\\ 3. (HvN)l(AvC) 4. L.Hl.'.BvD ( 1 0 )1 . W = Y 2. -Wv-Y 3. X)(Y.Z)/.'.-X (ll) 1. AvB 2.C 3.(A.C))D 4. -(-F.B)1.'.DvF (14) l. -(CvA) 2. B r ( - A ) c ) 1 . " - B -(A.B):-c @veI)C/"'E)A Exercise J g For each-of the following expressionsindicate (l) which variablesare free and which bound; (2) which letters serve as individual constants and which as property consranrs; (3) which free variablesare within the scopeor ro*" quuntifieror other and which individual constants not within the scopeof are any quantiirer. (x)(Fx ) Ga) 2. (3x) (Fa. Gx\\ I. 3. (_r)[Fx) (Gy v Hx)) (.r)Fx f (1y)(Gy v Dx) 4. 5. Fa v (x)[(Ga v Dx) ) (- Ky . Hb)] 6. (x)(Fa)Dx))(])tryt ?GxvFx)l Exercise , 9 Construct expansions a two-individual in universe ofdiscourse the followinssentences: for . Gx) 1. (.t)(Fr 8. -(3x)(FrvGx) 2. (3x)(Fx v Gr) 3. (-r)[F,r] (Gx v Hx)) 4. (fu)trr . (Gxv Hx)l 5. (x) - (Fx ) Gx) 6. (lx) - (Fx v Gx) 1. - (xXFx f Gx) Exercise U 9. 10. I l. 12. 13. t4. (rXFr I (Gx ) Hx)l (xXFr f - (Gx Hx)l (lxX(F.r . Gx) v (Hx Kx)l (xX(Fx.Gx) I (Hx. Kx)l (lx) - [(Fx ) Gx) v (Fx) Hx)] - (x) - l(Fx Gx). - (Hx. Kx)J lO Provethat the following arguments invalid. are ( l ) 1 . ()x)(Ax. Bx) 2 . (3x)(B.r'Cx) / .'. (3x)(Ax. Cx) (2) l. (x)(Ax ) Bx) ) (lx) - Ax l.'. (3x) - Bx ( 3 ) l . (lx)(A-r. - Bx) 2 . (3;r)(4.r.- Cx) 3 . (3rX- Bx. Dx) /.'. (3x)[Ax (- Bx . Dx)l (4) L (xXFx ) Gr) 2 . (x)(- Fx ) Ex) I .'. (x)(- Gx ) - Ex) ( 5 ) l . (].r)(Px.- Qx) 2. (x)(RxI Px) /.'. (l,rXRx - Q*) ( 6 ) I . (x)IQx. Qx) ) Rx) ') (lx)(Qx' - Rx) l:. (x)(- Px . - Qx) ( 7 ) l . (x)(Px ) Qx) ) (x)(Qx ) Rx) /.'. (x)(P.r.R.r) (8) l. (x)[Mx)(Nx)Px) 2. (x)(Qx ) Px) /:. (x)[Qx ) (Mx . Nx)) (e) 1. (lxXAx.B;) 2. (x)(- Bx v - Cx) /.'. (x)(- Ax v - Cx) (10) l. (3x)(Axv-B.r) 2. (x)l(Ax.-Bx))Cxl /.'. (1x)Cx Exercise * il Completethe following proofs using the rules for adding and removing quantifierswhere (l) l. (x)F; v (r) - Gx 2. - (x)Fx 3 (x)(Dx ) Gx) p P p /.'. (1xX- Dx v Gx) (2) l. ( . x ) [ A r v ( B x . - C . r ) ] 2 . (x)Cx p p | :. (3x)(Dx ) At) (3) L (x)[- Ax v (Bx . Cx)] 2. ( x X ( / x ) C x ) ) D x l (x)(Dx I - Cx) J. p p p l:. (fx) - Ax (4) p p p /.'. Bc L Ab) Bc z. (x)(Ax) Bx) J, (xX(tu)Bx))Axl (5) l. A b v B c 2 . (x) - Bx p p /.'. (ix)Ax ( 6 ) 1 . $)(Ry r - Gy) 2 . ()(87 v Gz) 3 . (y)Ry p p p t:. (v)Bv ( 1 ) 1 . (dlAz ) (- Bz ) Cz)l 2. - B a p p l:. Aa) Ca ( 8 ) l . (xX(&r .Ax) ) Tx) 2 . Ab 3 . (x)Rx p p pl:.Tb.Rb Exercise 1? iZ Which lines in the following are not valid? Explain why in eachcase. (l) l. 2. 3. 4. 5. .Kx)) Mxl (xX(FIx (]x)(Hx. Kx) Hx.Kx Mx (fx)Mx (2) r. (x)(Mx) Gx)) Fa 2. (x)(- Gx ) - Mx) 3.-Gy)-My a. (xX- Gx ) - Mx) ) Fa 5. (- Gx) - Mx)) Fa 6. Fa 7. (x)Fx (3) l. 2. 3. 4. (lx)(Fx.- Mx) (x)[(Gxv Hx) ) Mx] (Gy v Hy) ) My Fy.- My 5. - M1' 6. -(GyvHy) 7. (1x) - (Gx v Hx) (4) 1. (1x)(Px ' Qx) 2. Pv'Qv 3. Qy a.Qyv-R.y 5. (x)(Qxv-RJr) ( 5 ) l . ( 3 x ) [ ( P x. Q x ) v R x ] 2. 3. 4. 5. 6. 7. 8. (x) - Rx (3x)(Px v Rx) Pxv Rx -Px (x) - Px - Pv (z) - Pz ( 6 ) l. - (x)Fx 2. 3. 4. 5. 6. 7. 8. 9. (1x)Lx (x) - Fx -Fx I^a Lx Lx.- Fx (lxXl.x (x)Lx Fx) p p 2El I , 3M P 4EG p p 2Ul I Contra 4UI 3,5MP 6UG p p 2Ul IEI 4 Simp 3,5MT 6EG p IEI 2 Simp 3 Add 4UG p p I Simp 3EI 2,4DS 5UG 6UI 7UG p p p IUI 2El 2El a,6 C.qnj 7EG 5UG Exercise I f3 Prove valid. (l) l. (x)(Rx ) Bx) 2. (3x) - Bx p p /.'. (Lr) - Rr (2) 1. (x)(Fx ) Gx) 2 (y)(Gy ) Hy) (3) 1. Ka 2. (x)[Kx ) (y)Hy] p P /:. (z)(- Hz) - Fz) p p /.'. (x)Hx (4) l. (.r)(Fx I G.r) 2. (x)(Ax ) Fx) 3. (fx) - G; p p p /.'. (3x) - Ax (5) l. (x)(M-r I S.r) 2. (x)(- Bx v Mx) (6) l. (xXRr I Ox) p ) P /.'. (xX- ,Sx - Br) p p p /.'. (12)Pz 2' (3Y)- ov 3. (z)(- Rz) Pz) (7) t. ()x)(Ax. Bx) 2. (y)(A1, Cy) ) p p /.'. (3x)(Bx. Cx) ( 8 ) l . (1,Y),tt 2. ( x ) ( - G x f - R r ) 3 . (x)Mx p p p /:. (1x)Gx . (1x)Mx (e) l. p p /:. (1y) - Gy (x)[(FxvRr))-Gr] -Rx) 2. ( f u ) - ( - F x ( 1 0 ) l . (x)(Kx) - Lx) 2. (3x)(Mx. /x) (l l) l . (x)(Fx ) Gx) 2. (y)(Ey ) Fy) (z)-(Dz.-Ez) J. (12) l. (x)(l,x ) - Kx) 2. (12)(Rz. Kz) (y)t(- Ly Ry)) Byl J. P p l.'. (lx)(Mx' - Kx) p p p l:. (x)(Dx) Gx) p p p /.'. (1x)Bx Exercise 3 t,z Provevalid (note that theseproblemsare not necessarily order of diffrculty). in (l) (2) (3) l. (lr)fr v (fx)Gx 2. (x)-Fx l. l. (4) l. (s) l. 2. (6) L 2. p p l.'. (1x)Gx (x)(Hx) - Kx) p /:.-(ly)(Hy.Ky) - (x)A-r p /.'. (3x)(tu ) Bx) - (lx)F.r p /:. Fa ) Ga (1x)Fx ) (x) - Gx p (3x)Ex I - (x) - Fx p l.'. (1x)Ex ) - (3-r)Gx (3rXA.r. Bx) ) (y)Cy p -Ca p /.'. (x)(Ax) - Bx) (12) 1. (x)(Gx ) Hx) p ) (3.r)(1.r.-Hx) p 3 . (x)(- Fx v Gx) p /.'. (3x)(lx. -Fx) ( 1 3 ) L (x)l(Ax. Bx)) Cxl p 2. - c b p t . . .- ( A b . B b ) ( r 4 ) 1 . - (x)(Fx ) Gx) p 2. - (lxX- Gx. Hx) p /.'. (1x) - Hx ( l 5 ) l . (x)(Hx ) Kx) p 2. (3x)Hx v (Lr)Kx n r /.'. (lx)Kx ( 1 6 ) 1 . (3.r)Fx (lxXGx.Hx) I p 2. (lx)(Hx v Kx) ) (x)Lx p /... (x)(Fx) t-r) (7) l . (x)[(Fx v Hx) ) (Gx. Ax)] p p 2. - (x)(Ax. Gx) (8) 1 . - (x)(Hx v Kx) l:. (fx) - Hx p 2 . 0)t(- Kyv Ly)) Myl p /... (12)Mz (e) 1. (x)[(F.rv Gx) ) Hx) p (.rX(Hxv Kx) ) Lxl p 2. /:. (x)(Fx ) Lx)' (10) 1. (lx)tu)(x).9x p ) (.rXfx I R.r) p /.'. (1x)TxI (fr)Sr ( rl ) l . (x)[(A.rv Bx) ) Cx] p p 2. - (lyXCy v Dy) l.'. - (fx)Ax .Ax) ) Dx) (17) l. (x)[(Bx p 2. (3x)(Qx.Ax) p 3. (xX- Bx ) - Qx) p /.'. (lx)(Dx . Qx) (18) l. (rXPx) (Axv Bx)) p 2. (x)[(Bxv Cx)) Qx] p /:.(x)[(Px.-Ax))ex] (19) l. (rXPx f (Qr v Rr)l p . 2. (.rX(Sx Px) ) - Qxl p /.'. (xXSx) Px) ) (xXSxI rtr) (20) l. (x)l(A-r B.r)) (Cx.Dx)J p v /.'. (fx)(Ax v Cx) ) (]x)Cx Exercise n i >- Indicatewhich (if any) of the inferences the following proofs are invalid, and statewhy in they are invalid. (1) l. 2. 3. 4. 5. (lx)(y)Fry Q)Fxy Fxx (3y)Fyy (.rXiy)Fyr (2) L (fx)Fx 2. (]x)Gx 3. Fv 4.Gv 5. Fy.Gy 6. (lyXFy.Gr) '7. Gz)(1y)(Fy. Gz) (3) 1. 2. 3. 4. 5. (.r)(3yXFx Gy) I (?y)(Fx ) Gy) Fx :- Gy (x)(Fx ) Gy) (lyX-rXF.r f Gy) (4) l. (x)(lyXFx ) Gy) 2 . Fx 3 . (3y)(ry r Gy) Fy)Gy 5 . Gy 6 . (lw)(Fw I Gy) 1 Fw)Gy 8. (fw)(Fw ) Gw) 9 . (1w)[(Fw ) Gw p IEI 2Ul 3EG 4UG P p 1EI 2El 3,4 Conj 5EG 6EG p ltJI 2El 3UG 4EG p AP /.'. (lwX(rw ) Gw)'Gyl lUI 3EI 2,4WP 4EG 6EI 7EG 5,8 lu. 1 1 . (x)lFx I (3w)[(Fw Gw).Gyll ) (s) 1 . (x)br)[(z)Fzx' ' Hd)) (Gy 2. O ) I Q ) F z a ) ( G y . H A l 3. (z)Fza)(Ga.HA 4. Fba ) (Ga. Hd1 5 . (fy)lFby:_(Gy.Hd)l 6. Fbv 7. G y . H d 8 . Hd 9 . (lx)Hx 1 0 . Gy I l . (x 12. Fby ) (x)Gx 13. (v)FbyI (;r)G.r ((r,1 l. (xXtv)Fry Grl I 2, (t'\\Far ) Ca 3. l;ut') Go '' Gu l''4 lin' 5 | tr.,t..glirr I 7 ^ ' G u ) (x) - Fax 8 . ( l _ y )^ G y f ( x ) - F y x 10UG p lUI 2Ul 3UI 4EG AP /.'. (x)G.r 5,6MP 7 Simp 8EG 7 Simp IO UG 6-ll cP 12UG p ltJI zVr AP/.'. (x)- Fax 3,4MT 5UG MCP 7EG Exerciseill Z Ans.*er;, 2. a,d 10. a, d, f, n 12. a, d, l, g, n 14. a, b, i, i 4. a,c,l 6. a,d,f,n 8. a,d,k,o 4^s'wrts E x e r c i s e3 * (21 J. tI (4) ? I (6) 1 , 2M P 1 tD ttll 4.N 4.-H E 2 , 3D S -n 6. A) 1 , 4D S 2,6 DS 5. P.O 2,4MP 6. Bv (I. S) 7. R ( 14 ) 1 , 4D S (10) c.-, 6. - (Lv Ml 7. R 1 , 3D S 2 , 4M P (8) (12) 1 , 2H S 3,6 DS 3,5 MP 1 , 5M P 2,4DS B 3.5 MP -7/. 1 , 6M P Exercise ,lnlwers 4{ (21 3. (Rv$f f 2 Simp 4.7 5. fvL (4) 4 Add (10) 1 Simp t^ (8) 3. B +.v (6) 2,3MP 3. A 4.8 5. 8v D 5. -B.C 2 Simp 6.C 7. D 8. -8 9.E 1 0 .D . E 4. lvfr 5 .- r l12t (14) 1 . 3M P 6. -8 7. - Rv B 4. - (R. Al 1 , 3M P 4 Add (18) 1 , 4M P 5 Simp 2,6 MP 5 Simp 3,8 DS 7.9 Coni 1 Add 3,4 MP 2,5 DS 6 Add 2,3 MT 5.F 1 . 4D S 6. E.- D 3,5 Coni (20) 3. Av-D 4. F.S 5, (B'SlvB 5. -R 6.2 7. -M.-N 8. t- M.- Nt.Z 5.8 6. -C 7. B.-C 8. D.-C 9,D 6 . - ( D .E ) 7. - A 8. -A.-(D.E) 9. B) - D 1 0 .8 v E 1 1 .- D v F 1 Add 2,3MP 4 Add 1 . 3M T 4,5 DS 2,5 MP 6,7 Conj 1 , 4D S 2,4MT 5,6 Conj 3.7 MP 8 Simp 2 Simp 4,6 DS 6,7 Conj 1 , 8M P 5,6 MP 3,9,10CD Exercise{F Anc wrYS G (4) 3. Av(8vC) 'l Assoc 4.4 (21 2,3DS 3. t(A' 8)v Cl. tA' Bl v Dl 4. lA.BlvC 5. A.B 6.4 (6) 2. IlA. B) v Cl . l(A' 8) v Dl 3. A.Blv D 4. DvlA-Bl 5. (Dv Al.(Dv Bl 6. Dv A (8) 3. C'Av8l 4. (C.Al v(C.8) 5. - Cv - A 6. - (C.A) 7, C.B Exercise n l2l 7 1 Dist 2 Simp 3 Comm 4 Dist 5 Simp l Comm 3 Dist 2 Comm 5 DeM 4,6DS 4. [(AvB]f C1. lCl(AvB)l 5. Ct lAv 81 6. -4.-B 2 Equiv 4 Simp 1,3 Gonj 6 DeM 5.7 MT 9. -Cv-D 8 Add 1 0 .- ( c . o ) 9 DeM 3. (Sv-F)'(SvI) 4. Sv-8 5. - - Sv - B 1 Dist 3 Simp 5lmpl 7. R)-R 8. - 8v - R 9. -R 2.6 HS 5. - (8v D) 6. -8.-D 3,4 MT 7. - B 8. -A 6 Simp 9. -D 6 Simp 1 0 .- c 2,9 MT 11. - A.- C 12. - (AvCl (14) (10) 4DN 6. -Sl*F (6) t12l 8,10 Coni 7lmpl 8 Taut 5 DeM 1,7 MT 11DeM 3. - (A v C) 4. -(--AvC) 5. - (- A) 6. -8 3Dist 4Simp 5Comm 7.(-Rv-Rl .(-RvA) 8. -Bv-F 9. -B 1 Dist 3 Simp 2,4DS 5 Simp 2. 1-R.Atv - (Fv O) 3 . ( - B . A )v (-B'-O) 4. t(-F'A)v-Fl .l(-F.Alv-Ql 5. (-F.A)v-R 6. -Rv(-F.A) 6Dist TSimp STaut 3. l(D'FlvlA'8)l v (8. C) 4 . ( D . F l v [ { A' 8 ) v {8. C)l 5. A' A v {8. C) 6. (8'A) v (8. C) 7. B'lAvCl 8. B l Gomm 2DeM 1Comm 3 Assoc 2 , 4D S 5 Comm 6Dist 7 Simp tn*,*t'S 7. - (Av Bl 8. -C (4) (10) 1 Comm 3DN C) 4 tmpl 2.5MT t12l 3. -C.--A 4. -C 5 . t ( A .R t) C l . C) A . A l 6. G.A)C 7. - lA. St 8. -Av-B 9. --A 1 0 .- B 4. (W. Yl v l- W. - Yl 5. - (W. Yl 6. -W.-Y 7. -Y 8. -Yv-Z 9. - ly. Zl 1 0 .- x 3. -Bl-P 4. -nl(-F)Sl 5. (-F.-R)lS 6. -F)S 7. --FvS 8.8vS 2DeM 3Simp 1 Equiv SSimp 4,6MT TDeM 3Simp 8,9DS 1 Equiv 2 DeM 4,5DS 6Simp TAdd 8 DeM 3,9MT lGontra 2,3HS 4Exp STaut 6lmpl 7DN i Exercise f t2l J. -(R @ r(4>-(r'> S) 1 DeM 4. 3. - B v - - C 4. - B v C 2 DeM 3DN B) C A)C 4. - H v - G 5. - - H 6. - G (10) 2,3 MT 4lmpl b. -F 1 , 5H S 3DN 4,5 DS 1 Equiv 2 Simp 3lmpl llmpl 2lmpl (-Fvf) f) 3,4Conj SDist (14) r) 7. Rl(S 3. -Cv-A 6lmpl 2DeM 4. A 5, --A l12l 2 DeM 1 , 6M T 2. ( M ] M , ( N ] M ) 3. N ) M 4. - Nv M 3. -BvS 4. -RvT 5. (-BvS) 6. -Rv(S 1 Simp 4DN 6. -C 1. B)C B. -8 4. - A 5. --Av--'B 3,5DS -Bl SDeM 6.-lA 7. -C lSimp 6,7MT 1 , 3D S 4Add 2,6MP 8 An< u"-r 2. D. ( 1 )x i s b o u n d . c ; l { 2 ) a i s a n i n d i v i d u ac o n s t a n tF a n d G a r e p r o p e r t y o n s t a n t s ' (3) No free variables;a is within the scope of the (x) quantifier' are h ( 1 ) y a n d t h e { i r s t x v a r i a b l e ( n o t c o u n t i n g t h e x t h a t i s p a r tto fe q u a n t i f i e r ) i b o u n d ;t h e s e c o n dx v a r i a b l e s f r e e ' (2) No individualconstants;F, G, and D are propertyconstants' ( 3 1F r e ex v a r i a b l ei s w i t h i n t h e s c o p eo f t h e ( / q u a n t i f i e r ' free ( 1 )T h e f i r s t x v a r i a b l e a n d t h e y v a r i a b l ea r e b o u n d .T h e l a s tt w o x v a r i a b l e sa r e constant' a Ql F, G, and D are property constants; is an individual T ( 3 )T h e t w o f r e e x v a r i a b l e s r e w i t h i nt h e s c o p eo f t h e ( y ) q u a n t i f i e r .h e i n d i v i d u a l a a, is within the scopeof the (x) quantifier' constant, Exercise U 2. 4. 6. 8. g tlqsu/rrs lFa v Ga)v (Fb v Gb) lFa (Ga v Hall v tFb {Gb v Hb)l -lFavGa) v-(FbvGbl -[(Fav6a) v(FbvGb)l 10. [Fa] - (Ga' Hall. lFbl - (Gb. Hb)l 12. l(Fa. Ga) I lHa Kall t ( F b . G b )I ( H b . K b ) l 14. - {- llFa Gal - (Ha Kall - t(Fb Gbt.- (Hb Kb)l) Exercise:Il /f h"tsuers 4. Rx)-Gx lUl 2Ul 5. Bxv Gx 2Ul 5. Dx 6. Cxv-Bx 7.-8xvCx APl.'. Ax 4 Add 6 Comm 6. Bx 7. - Gx 3Ul 8. -8x \\21 3. Ax v (Bx. - Cxl 4. Cx 7DN 1 Ul (6) 4,6 MP 5,7 DS 9. lylBy 4. (Bb. Abl ) Tb 8UG 3,9 DS v--Cx 8. 8x 5. Fb 3Ul s-10cP 6. Rb.Ab 2,5 Conj 1 1E G 2Ut 3Ul 4.5MP 1 , 6M P 7. Tb 4,6 MP 8. Tb. Rb 5.7 Conj 8 DeM (4) 11, Dx) Ax 12. lSxl(Dx) Axl 4. Ab-r-Bb 5. (Ab I Bbt) Ab 6. Ab 7. Bc Exercise A t2l (4) (6) (8) lUl l? 4. Invalid. Quantifier must be removed first. 5. Invalid.The (x) quantifierdid not quantify the whole line. 6. Invalid.Antecedentof line 5 does not match line 3. 7. lnvalid. Can't universallygeneralize from a constant. 5. Invalid.can't use UG to bind a variablethat is free in a line that is justified by El. In this case y is free in line 2. 4. Invalid.The (x) quantifierdoes not quantify the whole line. 5. Invalid.Can't replacea variablewith a constantwhen using El. 9. Invalid.Can't universallygeneralize from a constant. Exercisef l2l 13 AntueyS 3. Fx = Gx 4. Gx -,:Hx (4) 5. Fx) Hx 6. -Hx)-Fx 7. (zll- Hz ) - Fzl 4. - Gx 5. Ax) Fx 6. Fx: Gx 7. Ax) Gx 8. -Ax (61 9. (3x) - Ax 4. - Ox 5. Rx= Ox 6. - Rx= Px 7. - Rx 8. Px 9. l3zlPz (8) 4, Rx 5.-Gxf-Bx 6. Mx 7. Rx) Gx 8. Gx 9. (3$Gx 10. lfxlMx 11. (SxlGx.ExlMx 1Ul 2Ul 3.4HS 5 Contra 6UG 3Et 2Ua 1Ul s,6 HS 4,7MT 8EG 2El 1Ul 3Ut 4.5MT 6.7MP 8EG lEl 2Ul 3Ul 5 Contra 4,7MP 8EG 6EG 9,10Conj {10} l12l 3. Mx.Lx 4. Kx)-Lx 5, lx 6. --l-x 7. - Kx L Mx 9. Mx' - Kx 10. (lxllMx. - Kxl 4. Rx. Kx 5. Lx)-Kx 6. (-tx.Rxl)Bx 7. Kx 8. --Kx 9. - Lx 10. Fx 11.- Lx.Rx 12. Bx 13. (3x)Bx 2El 1Ul 3 Simp 5DN 4,6MT 3 Simp 7,8Conj 9 EG 2El 1Ul 3Ul 4 Simp 7DN 5,8MT 4 Simp 9,10Coni 6,11 MP 12EG Exercise n (4) r/ 4nluei.s 2. Hx) - Kx 3.-Hxv-Kx 4. - lHx. Kxl 5. (y)- lHy. Kvl 6. - (]yl(Hv. Kyl 2. (xl - Fx 3. -Fa 1Ul (6) 2lmpl 3 DeM 4UG 5(}N 10N 2Ul 4. - Fav Ga 3 Add 5. Fa:. Ga 2EG 3()N 1 , 4M T 50N 6Ul 7 DeM 8lmpl 4lmpl llxl - (Hxv Kxl 10N 3El 4 DeM o- - K x 5 Simp (- Kxv Lxl) Mx 2Ul 8. - K x v L x 6 Add 9 . Mx 7,8MP I 0 . lfzlMz 9EG - (lx)Sx J. AP 4 . - (lxl,9x 1 , 3M T 5 . (x) - Rx 40N 6. - R x 5Ul 7 . Tx) Rx 2Ul 8. - T x 6,7 MT q lxl - Tx 8UG 1n 90N 1 1 . -(3x)Sx l- (fx)Ix 3-10 CP ' t 2 . (lx)Ix I (3x)Sx 11 Contra lx -Hx 2El 5 . - Fxv Gx 3Ul Gx) Hx 1Ul 7. rx)gx 5lmpl 8. Fx) Hx 6,7 HS q -Hx 4 Simp 10.-Fx 8,9 MT 1',t. lx 4 Simp tt. lx -Fx 10,11 onj C 1 3 . (lxXix. - Fx) 1 2E G J. 10N lSxl - (Fx) Gxl 4 . - (Fx ) Gxl 3El - (- Fxv 6x) 4lmpl o. - - F x . - G x 5 DeM 1. -Gx 6 Simp 8 . (xI - (- Gx Hxl 2(}N 9. - l- Gx. Hxl 8Ul 10. --Gxv-Hx 9 DeM 1 t . Gxv - Hx 10DN 12. - H x 7,11 S D t J . llxl - Hx 1 2E G 5. 9UG Fx (3xlFx 4 . - (Hx v Kxl 5. - H x - K x (1 0 ) 3. (]vl - Cy a. - (ylCy 5. - (lxXAx Bx) 6. (x) - (Ax. Bxl 7. -(Ax.Bxl 8. -Axv-Bx 9. Ax) - Bx 10. (xXAx) - Bxl 3EG 1 , 4M P 5El 6 Simp 7 Add 8EG 2,9 MP lxlLx Lx 10ul 2. Fx) Lx 3-11CP 3 . (xl(Fx) Lxl 1 2U G J. Px) (Axv Bx\\ 1 Ul 4. l B x v C x l = O x 2Ul Px. - Ax AP l.'. Qx o. Px 5 Simp 7. Axv Bx 3,6 MP R -Ax 5 Simp q Bx 7,8 DS 0. Bxv Cx 9 Add 1 . Ax 4 , 1 0M P 12. ( P x - A A ) A x 5 - 11 C P 1 3 . (xll(Px. - Axl ) Oxl 12 UG 2. - (]xlCx lixllGx. Hxl Gy. Hy Hy Hyv Ky (3x)(Hxv Kx) (18) 3 . ( x )- C x 2ON 4. - C x 3Ul (Axv Bxl ) (Cx. Dxl 1 Ul o- - l A x v B x l v (Cx Dxl Slmpl ( - A x . - B x lv (Cx. Dxl 6 DeM IF Ax. - Bxl v Cxl IF Ax.- 8x) v Dxl 7 Dist 9 . l- Ax. - Bxl v Cx 8 Simp 1 0 . - Ax. - Bx 4,9 DS 1 1 .- A x 1 0S i m p 12. - A x . - C x 4 , 1 1C o n j t J . - lAxv Cxl 12 DeM 1 4 . 8l - (Axv Cxl 13 UG 1 5 . - l3xl(Axv Cxl 14 ON t b . - taxtLx ) - (lxllAx v Cxl 2-15 CP 1 7 . Exl(Ax v Cxl > (3xlCx 16 Contra