Question Details

(solution) Exercise 7 (p. 7) # 1, 3, 5, 7, 11, 13 Make sure you include the


Exercise 7 (p. 7) # 1, 3, 5, 7, 11, 13

Make sure you include the premises in your proof


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:

 

[email protected])l::[(pvqJvr]

 

,r'1o'r))::[email protected] 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. [email protected])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.) [email protected]

 

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).C1.'.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

 


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