10
PART TWO: PREDICATE LOGIC
EXERCISE 7-1:
EXERCISE 7-2:
1. (1) x is bound.
EXERCISE 7-3:
EXERCISE 7-4:
10
EXERCISE 6-2:
(1) 1. p q
(3) 1. p q
(5) 1. p q
(7) 1. p q
10
(9) 1. ~ (p q)
10
3. ~ (x)(Px Ix)
EXERCISE 7-5:
EXERCISE 7-6:
1. (x)( Mx Px) (Mx = x is a marsupial , Px =xhas a pouch)
EXERCISE 7-7:
1. Some TV newscasters have pleasant personalities.
11
EXERCISE 7-8:
1. (Fa Ga) (Fb Gb)
EXERCISE 7-9:
(using obvious abbreviations)
EXERCISE 7-10:
(using obvious abbreviations)
1. ~ ( x)(Fx Lx)
13. ( x)[(Sx Bx) ~ Cx]
EXERCISE 8-1:
1. True:
3. True:
11
5. True:
Domain: People
7 True:
Domain: Whales
EXERCISE 8-2:
(For our answers assume the domain of discourse to be positive integers.)
1. Ax = x is an even number
5. Px = x > 2
EXERCISE 8-3:
(1) 1.(Aa Ba) (Ab Bb)
11
(5) 1. (Pa ~Qa) (Pb ~Qb)
(7) 1. (Pa Qa) (Pb Qb)
(9) 1. (Aa Ba) (Ab Bb)
EXERCISE 8-4:
(For our answers assume the domain of discourse to be positive integers.)
1. Ax = x is divisible by 4
EXERCISE 9-1:
(1) 4. (x) ~ Gx 1, 2 DS OR 4. Dx Gx 3UI
11
(3) 4. Ca AP/ ~Ca
(5) 3. ~ Bc 2UI
Exercise 9-2:
(1) 1. (x)[(Hx Kx)Mx]p
11
(5) 1. ( x)[(Px Qx)Rx]p
EXERCISE 9-3:
(1) 3. ~ Bx 2EI
(5) 3. Mx Sx 1UI
(7) 3. Ax Bx 1EI
4. ~ ~Fx ~ ~Rx 3DeM
5. Fx Rx 4DN (2x)
EXERCISE 9-4:
1. Incorrect. Correct use would
3. Incorrect. Correct use would
11
EXERCISE 9-5:
(1) 3. ~ ( x)Fx 2QN
(7) 3. ( x) ~ (Ax Gx) 2 QN
(9 ) 3. Fx AP/Lx
(11) 3. (y) ~ (Cy Dy) 2 QN
5. ~ (Ab Bb ) 3,4 MT
1
16
(19) 3. (x) (Sx Px)AP/ (x) (Sx Rx)
4. Sx AP/Rx
EXERCISE 10-1:
1. ~ Mcb
EXERCISE 10-2:
1. (x) (y)Sxy
117
EXERCISE 10-3:
1. ( x) (Sx (y) ~ Kxy)
EXERCISE 10-4:
1. ( x) ( y) [(Px Cy) Ayx] (Px =“x is a place”; Cx =“x is a cheater”; Axy =“x is
EXERCISE 10-5:
1. (x) {Dx Px · (y) (Ty ~ Lxy)]} (Dx =“x is a drama critic”; Px = “x is a
118
EXERCISE 10-6:
1. (x) [(Cx Bx)Dx] (Cx =“x is a company”; Bx =“x goes bankrupt”; Dx =“x
119
EXERCISE 10-7:
1. Everyone believes in God.
EXERCISE 10-8:
(1) 1. (Faa Fab) (Fba Fbb)
(3) 1. (Faa Fab ) (Fba Fbb)
120
(5) 1. (Faa Fab) (Fba Fbb)
EXERCISE 10-9:
(5) 1. Inference to line 4 violates the requirement that quantifiers be dropped only if
EXERCISE 10-10:
(1) 3. Ma AP/Ob
4. (y) [(Ma Py) (Oa Ob)] 2 UI
(5) 2. ~ (y) (Mya Oay)AP/ (y) (Mya Oay)
(7) 2. (y) (z)Azyw 1EI
(9) 2. ( y) ~ ( x) ~ (Bxy ·Byx) 1 QN
122
(13) 3. (y)Awy 1EI
EXERCISE 10-11:
(1) 3. Az (y) (Qy Lzy) 1 EI
(3) 2. (x)Fx AP/(y)Gy
(5) 3. Hx (y) (Ky Lxy) 2 EI
123
(7) 3. ~ ( y) (Gy Lx)AP/ ( z) (Ez Nzx)
4. (y) ~ (Gy Lx) 3 QN
(9) 4. (x) ~ (Axa ~Bxb) 1 QN
5. (x) ~ (Cxc Cbx) 2 QN
124
(11) 2. ~ ( z) (Bz Cxz)AP/ ~ ( y) (Ay Cxy)
3. (z) ~ (Bz Cxz) 2 QN
(13) 3. ~ (x) [(Fx Px) ( y)Gy]AP/ (x) [(Fx Px) ( y)Gy]
4. ( x) ~ [(Fx Px) ( y)Gy] 3 QN
125
EXERCISE 10-12:
(1) 1. (x) (y)Fxy AP/(y) (x)Fxy
2. (y)Fxy 1UI
(3) 1. ( x)Gx AP/ ( y)Gy
2. Gx 1EI