Valid Forms for Sentential Looic Valid Argument Forms of Inference

| August 30, 2017

Question
Exercise 11 (p. 9) #1, 3, 7,
Exercise 13 (p. 11) #3, 7, 9, 11

Exercise 14 (p.12) #3, 5, 7

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

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).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

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