Symbolize and provide
derivations.
1. All students read (some) long novels. Any novel is exciting. Therefore, all students read (some) things that are exciting.
(Ax)[Sx -> (Ey)(Ny & Ly & Rxy)], (x)(Nx -> Ex) |- (x)[Sx -> (Ey)(Ey & Rxy)]
1. (x)[Sx -> (Ey)(Ny & Ly & Rxy)]
2. (x)(Nx ->
Ex)
/ (x)[Sx -> (Ey)(Ey &
Rxy)]
3.
Sa
HCP
4. Sa -> (Ey)(Ny & Ly & Ray)
1, UI
5. (Ey)(Ny & Ly &
Ray)
3,4, MP
6. Nb & Lb &
Rab
5, EI
7. Nb ->
Eb
2, UI
8.
Nb
6, Simp
9.
Eb
7,8, MP
10.
Rab
6, Simp
11. Eb &
Rab
9,10, Conj
12. (Ey)(Ey &
Ray)
11, EG
13. Sa -> Ey(Ey &
Ray)
3-12, CP
14. (x)[Sx -> Ey(Ey & Rxy)]
13, UG
2. No student reads every novel. Al is a student. Therefore, there are novels that Al does not read.
(x)[Sx -> (Ey)(Ny & ~Rxy)], Sa |- (Ey)(Ny & ~Ray)
1. (x)[(Sx -> Ey(Ny & ~Rxy)]
2.
Sa
/ (Ey)(Ny & ~Ray)
3. Sa -> (Ey)(Ny &
~Ray)
1 UI
4. (Ey)(Ny &
~Ray)
2,3. MP
3. Bob and Carol are students. Don is a teacher. Every teacher likes all students. Therefore, Don likes Bob and Carol.
Sb & Sc, Td, (x)[Tx -> (y)(Sy -> Lxy] |- Ldb & Ldc
1. Sb &
Sc
2.
Td
3. (x)[Tx -> (y)(Sy -> Lxy)]
/ show
Ldb
& Ldc
4. Td -> (y)(Sy ->
Ldy)
3, UI
5. (y)(Sy ->
Ldy)
2,4, MP
6. Sb ->
Ldb
5, UI
7.
Sb
1, Simp
8.
Ldb
6,7, MP
9. Sc ->
Ldc
5, UI
10.
Sc
1, Simp
11.
Ldc
9,10, MP
12. Ldb &
Ldc
8,11, Conj
4. If every student is ambitious, then there are individuals who take some hard courses. No one takes any hard courses. Therefore, there are some students who are not ambitious.
(x)(Sx -> Ax) -> (Ex)(Ey)(Hy & Txy), ~(Ex)(Ey)(Hy &
Txy) |-
(Ex)(Sx & ~Ax)
Note that the first premise does not tell us that any student who is
ambitious takes some hard courses.
Instead, we are setting up a condition that states that if every
student is ambitious then you would have at least some individuals
taking hard courses.
1. (x)(Sx -> Ax) -> (Ex)(Ey)(Hy &
Txy)
2. ~(Ex)(Ey)(Hy &
Txy)
/ show (Ex)(Sx & ~Ax)
3. ~(Ex)(Ey)(Hy & Txy) -> ~(x)(Sx ->
Ax) 1,
Trans
4. ~(x)(Sx ->
Ax)
2,3, MP
5. (Ex)(Sx &
~Ax)
4, QN
5. Students and teachers like anything that is fun. Any logic problem is fun. Ted is a student. Therefore, Ted likes all logic problems. (Hint: symbolize "students and teachers" as a disjunction, not a conjunction.)
(x)[(Sx v Tx) -> (y)(Fy -> Lxy)], (x)(Lx -> Fx), St |- (x)(Lx -> Ltx)
1. (x)[(Sx v Tx) -> (y)(Fy -> Lxy)]
2. (x)(Lx ->
Fx)
3.
St
/
(x)(Lx
-> Ltx)
4. (St v Tt) -> (y)(Fy ->
Lty)
1, UI
5. St v
Tt
3, Add
6. (y)(Fy ->
Lty)
4,5, MP
7. La ->
Fa
2, UI
8. Fa ->
Lta
6, UI
9.
La
HCP
10.
Fa
7,9, MP
11.
Lta
8,10, MP
12. La ->
Lta
9-11, CP
13. (x)(Lx ->
Ltx)
12, UG
