Practice Midterm 2(B) Solutions
1. An individual constant is a lowercase letter (chosen from the beginning of the alphabet) that abbreviates a name. An individual variable is a variable that can take as substitution instance any name or individual constant. Use letters from the end of the alphabet (x, y, z) as individual variables.
2. Universal generalization and existential instantiation have flagging restrictions. Universal instantiation and existential generalization don't.
3. A universal sentence asserts that some propositional function is true of everything. An existential sentence asserts that some propositional function is true of at least one thing.
4. Truth functional compound.
5. No. The premise is a negation, not an existential statement.
6. Let Px = x is a person.
Ax = x gets an A
Ix = x is intelligent.
Lx = x loves Cheezy Poofs.
Mx = x has big MTV hair.
We can symbolize the argument as follows:
1. (x)((Px&Ax) -> (Ix & Lx))
2. (Ex)((Px &Ax) & Mx)
Therefore, (Ex)((Px & Mx) & Lx)
7.
1. (x)(Ax -> Bx)
2. (x)(Bx -> Cx)
3. Aa -> Ba 1, UI
4. Ba -> Ca 2, UI
5. Aa -> Ca 3, 4 HS
6. (Ex)(Ax -> Cx) 5, EG
8.
1. (x)(Rx -> Sx)
2. ~Sb
3. Rb -> Sb 1, UI
4. ~Rb 2, 3 MT
5. (Ex)(~Rx) 4, EG
9.
01. (x)[Sx -> (Ax v Bx)]
02. (x)(Ax -> Px)
03. ~(x)(Sx -> Px)
04. (Ex)(Sx & ~Px) 3 CQN
05. Sa & ~Pa 4, EI, flag a
06. Sa 5, SIMP
07. ~Pa 5 SIMP
08. Aa -> Pa 2 UI
09. ~Aa 7, 8 MT
10. Sa -> (Aa v Ba) 1 UI
11. Aa v Ba 10, 6 MP
12. Ba 9, 11 DS
13. Sa & Ba 6, 12 CONJ
14. (Ex)(Sx &Bx) 13, EG
15. ~(x)(Sx -> ~Bx) 14, CQN
10.
1. (x)((Px&Ax) -> (Ix & Lx))
2. (Ex)((Px &Ax) & Mx)
3. (Pa & Aa) & Ma 2 EI flag a
4. (Pa & Aa) -> (Ia & La) 1, UI
5. Pa &Aa 3, Simp
6. Pa 5 simp
7. Ia & La 4, 5 MP
8. Ma 3 simp
9. Pa & Ma 6, 8 conj
10. La 7 simp
11. (Pa & Ma) & La 9, 10 conj
12. (Ex)((Px & Mx) & Lx)
11.
01. -> ~(Ex)(Fx&Gx) ACP
02. | ->(x)(Fx) ACP
03. | | ->(Ex)(Gx) AIP
04. | | | Ga 3, EI flag a
05. | | | Fa 2, UI
06. | | | (x) ~(Fx & Gx) 1, QN
07. | | | ~(Fa & Ga) 6, UI
08. | | | (Fa & Ga) 4, 5 CONJ
09. | | | (Fa & Ga) & ~(Fa & Ga) 7, 8 CONJ
10. | | |- ~(Ex)(Gx) 3-9 IP
11 | |- (x)(Fx) -> ~(Ex)(Gx) 2-10, CP
12. |- ~(Ex)(Fx&Gx) -> ((x)(Fx) -> ~(Ex)(Gx)) 1-11 CP
12.
1. -> (x)Fx & (x)Gx ACP
2. | -> flag a FSUG
3. | | (x)Fx 1 SIMP
4. | | (x)Gx 1 SIMP
5. | | Fa 3 UI
6. | | Ga 4 UI
7. | | Fa & Ga 5, 6 CONJ
8. | |- (x)(Fx & Gx) 2-7 UG
9. |- ((x)Fx & (x)Gx) -> ((x)(Fx & Gx)) 1-8 CP
13.
01. Fa -> (x) (Hx & Gx)
02. (x)Jx
03. (x)Fx
04. (Jb &Hb) -> (x)(Kx)
05. Fa 3 UI
06. Jb 2, UI
07. (x)(Hx & Gx) 5, 1 MP
08. Hb & Gb 7, UI
09. Hb 8, simp
10. Jb & Hb 6, 9 conj
11. (x)Kx 4, 10 MP
12. Ka 11, UI
13. Fa & Ka 5, 12 UI